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[
[
"Collective excitations of a quantized vortex in $^3P_2$ superfluids in\n neutron stars"
],
[
"Abstract We discuss collective excitations (both fundamental and solitonic excitations) of quantized superfluid vortices in neutron $^3P_2$ superfluids, which likely exist in high density neutron matter such as neutron stars.",
"Besides the well-known Kelvin modes (translational zero modes), we find a gapfull mode whose low-energy description takes the simple form of a double sine-Gordon model.",
"The associated kink solution and its effects on spontaneous magnetization inside the vortex core are analyzed in detail."
],
[
"Introduction",
"The properties of high density nuclear matter are still not sufficiently understood.",
"However, in recent years, neutron star (NS) observations began to place stronger restrictions , , on the equation of state (EoS) of dense matter and to probe structure and composition of NS cores.",
"Most valuable information into the state of their interiors came from the detections of pulsar glitches via pulsar timing measurements, optical and X-ray observations of cooling and accreting NSs and from neutrino emission measurements from proto-neutron stars.",
"These observations provide evidence for superfluidity in the interiors of NSs .",
"The first direct evidence , that the NS core should exist in a superfluid state has been reported in the study of the NS in the supernova remnant known as Cassiopeia A.",
"It has been measured that Cassiopeia A's surface temperature has rapidly decreased from $2.12\\times 10^6$ K to $2.04\\times 10^6$ K. This pronounced drop in surface temperature can be naturally explained , , if one assumes that neutrons have recently become superfluid in the NS core.",
"As the neutrons combine to form Cooper pairs, a splash of neutrinos is emitted accelerating the NS cooling process.",
"Another evidence for a superfluid NS core comes from the observation of pulsar glitches, which are sudden jumps in the NS rotation frequency.",
"Glitches like the ones observed in the Crab and Velar pulsar can be explained by two main physical mechanism – either they are due to starquakes from the NS core , or crust , , or they are caused by the sudden unpinning and displacement of a large number of vortices , , in the NS superfluid.",
"In the starquake glitch model , , , the glitches are caused by a sudden reduction in the moment of inertia of some solid component of the NS.",
"For example, the differential rotation between the crust and the superfluid NS core can produce stresses in the NS crust leading to crustquakes.",
"The resulting crustquakes distort the star's shape and hence generate sudden jumps in the NS rotational frequency, seen as glitches.",
"In contrast, in the pinned superfluid model angular momentum is suddenly transferred from the superfluid NS core to the non-superfluid crust via vortex unpinning, spinning it up.",
"Neutron superfluid vortices can become pinned to nuclear clusters in the inner crust or to magnetic flux tubes in the core .",
"This prevents angular momentum from being transferred to the NS crust.",
"Hence, a differential rotation builds up between the NS core and crust.",
"When this differential becomes large enough angular momentum is suddenly transferred from the core to the crust through the catastrophic unpinning of vortices, resulting in a sudden spin-up in the NS.",
"Furthermore, the observed long time relaxation after glitches can be explained only by assuming the coexistence of normal and superfluid components , .",
"Therefore, understanding the properties of neutron superfluidity and the formation and dynamics of superfluid vortices can give us further insights into the evolution of NSs and their composition.",
"At the high density central part of NSs the neutron superfluidity is attributed to the $^3 P_2$ pairing interaction , , , , , rather than to $^1 S_0$ pairing familiar from the conventional BCS theory of superconductors.",
"Analysis of nucleon-nucleon scattering data shows that the transition from an isotropic $^1 S_0$ superfluid to an anisotropic $^3P_2$ superfluid occurs at densities above $1.6\\times 10^{14}\\, g/\\text{cm}^3$ .",
"The Ginzburg-Landau (GL) energy functional generalized for $^3P_2$ superfluids was developed in Refs.",
", in the weak coupling limit.",
"Note that the general GL free energy formally agrees with the angular momentum $l=2$ GL functional solved by Mermin ; depending on the GL parameters, the ground state is in either nematic, ferromagnetic or cyclic phase.",
"The ground states of $^3P_2$ superfluids in the weak coupling limit were found in Ref.",
"to be in the nematic phase in the absence of a magnetic field.",
"They are continuously degenerated when we ignore the sixth order term and can be decomposed by unbroken $O(2)$ , $D_2$ and $D_4$ symmetry groups into the uniaxial, $D_2$ - and $D_4$ -biaxial nematic phases, respectively.",
"(This is also known from spin-2 spinor Bose-Einstein condensates, see Refs.",
", .)",
"This degeneracy is lifted and the uniaxial nematic phase becomes the unique ground state once the sixth order term is taken into account.",
"The ground states in the presence of a magnetic field were recently determined ; the ground state is in the uniaxial nematic phase for smaller magnetic fields relevant for ordinary NSs and in the $D_2$ - or $D_4$ -biaxial nematic phase for large magnetic field relevant for magnetars.",
"Beyond the GL theory, the ground states in the $(T,H)$ phase diagram have been obtained recently in the Bogoliubov-de Gennes (BdG) formalism .",
"The $D_2$ - and $D_4$ -biaxial nematic phases appear in lower $T$ and $H$ region and in higher $T$ and $H$ region, respectively.",
"The phase boundary is of second order at higher temperature while it is of first order at lower temperature, and a tricritical point connects these boundaries.",
"One of the most important results of the BdG formalism is that $^3P_2$ superfluids were found to be topological superfluids predicting gapless Majorana fermions on the surface .",
"The strong-coupling corrections to the $^3 P_2$ NS matter GL free energy including spin-orbit and central forces were calculated in Ref. .",
"As phenomenological aspects relevant for NSs physics are concerned, $^3P_2$ superfluids provide new mechanisms for neutrino emission of dense neutron matter , , , , , , , , , , explain the entrainment of superconducting protons by rotating superfluid neutrons in NSs and offer possible explanations of the anomalously rapid cooling of NSs , , .",
"Since superfluids are rotating inside NSs, a large number ($\\sim 10^{19}$ ) of superfluid vortices exist in their interior.",
"The rich structure and magnetic properties of vortices emerging in the GL equations for $^3 P_2$ superfluids were explored in Refs.",
", , , , .",
"$^3P_2$ vortices in NS matter turn out to be structurally different from their counterparts in $^1 S_0$ superfluids.",
"For example, different to $^1 S_0$ vortices, $^3 P_2$ vortices exhibit spontaneous magnetization in the vortex core region , .",
"For magnetic field strengths of orders of magnitude as they appear in magnetars, the ground state is in the $D_4$ -biaxial nematic phase, in which the first homotopy group $\\pi _1$ is non-Abelian, thereby admitting non-Abelian (non-commutative) vortices which carry half-quantized circulation .",
"In this article, we study low-energy collective modes [or (pseudo) moduli] and solitonic excitations of an integer vortex in a $^3P_2$ superfluid.",
"By solving numerically the GL equation, we reconstruct the axially symmetric integer vortex solution in the absence of magnetic fields and sixth order terms.",
"Because of the off-diagonal elements of the tensor order parameter, there exists spontaneous magnetization at the vortex core as described above.",
"Here, we calculate the net magnetic moment per femtometer along vortex line to be one order less than the neutron magnetic moment.",
"We then study collective modes in the presence of a single vortex.",
"As usual, there are gapless (massless) Kelvin modes (translational moduli) due to the spontaneously broken translational symmetry in the presence of the vortex.",
"In addition, the phase $\\delta $ of the off-diagonal elements of the tensor order parameter gives rise to a gapfull (massive) mode.",
"We construct the low-energy (long distance) effective free energy of this gapfull mode and find it is a double sine-Gordon model, consisting of the potential terms cos $\\delta $ and cos $2\\delta $ .",
"For GL parameter values describing typical NSs, it allows only $2\\pi $ kink solutions Sine-Gordon kinks on vortices were studied in the Skyrme model , .",
"In this case, the kink represents a Skyrmion, carrying a non-zero baryon number.. We find that the core magnetization of the vortex flips its direction at the kink.",
"The article is structured as follows.",
"After introducing the GL free energy relevant for $^3P_2$ superfluids in the weak coupling limit, $^3P_2$ vortex solutions are constructed in Section and their spontaneous magnetization in the vortex core region is evaluated.",
"Then, in Section , we discuss the effect of the parameter $\\delta $ on the free energy density by writing down the associated effective free energy functional which takes the simple form of a double sine-Gordon model.",
"The associated kink soliton solution is derived in Section .",
"Finally, our conclusions and possible future lines of investigation are presented in Section .",
"To facilitate the reader to reproduce our numerical results, we add two appendices to this article.",
"In Appendix , we briefly review the GL theory for $^3P_2$ superfluid states and state all the parameter values used in our numerical simulations.",
"We list explicitly all the vortex equations together with the imposed boundary conditions in Appendix .",
"Note that a detailed investigation of vortex structure and dynamics in the presence of nonzero external magnetic fields and higher order terms, in particular the inclusion of the sixth order term, will be published in a forthcoming paper."
],
[
"Ginzburg-Landau Description of Vortices in $^3P_2$ Neutron Superfluids",
"In this section we discuss the GL free energy and determine the vortex configurations numerically.",
"A discussion of spontaneous magnetization is included at the end of the section."
],
[
"Ginzburg-Landau Free Energy",
"The GL free energy for the $^3P_2$ superfluidity was originally derived in Refs.",
", , , by generalizing Gor'kov's procedure.",
"In this case the original $^1S_0$ contact interaction was generalised to a $^3P_2$ contact interaction by introducing a derivative coupling.",
"A short explanation of the derivation is given in Appendix .",
"The tensorial order parameter $A_{\\alpha i}$ transforms under the symmetry group as $A \\rightarrow e^{i \\theta } \\mathbf {g}A \\mathbf {g}^T, \\quad \\text{ with }\\quad e^{i \\theta } \\in U(1),\\quad \\text{ and }\\quad \\mathbf {g}\\in SO(3)_{L+S}.$ Here, $SO(3)_{L+S}$ is the diagonal subgroup of the full group $SO(3)_{L} \\times SO(3)_{S}$ and is generated by the total angular momentum $J = L + S$ .",
"The GL free energy density $F$ can be written as a function of the tensor $A_{\\mu i}$ $F= \\frac{1}{G}\\int d^3 \\rho \\ \\left(F_{\\rm grad} + F_{2+4}+F_6+F_H\\right), $ where $F_{\\rm grad}$ is the gradient term, $F_{2+4}$ and $F_6$ are the free energy densities up to fourth order and up to sixth order, respectively and $F_H$ is the magnetic term.",
"Here, we rescaled for later convenience the free energy functional by $G=3\\pi ^2/\\left(m_n k_F^3\\right)\\approx 115.39\\,\\text{MeV}\\,\\text{fm}^5$ where $k_F$ is the Fermi momentum and $m_n$ represents the nucleon mass, see Appendix for the numerical values of all model parameters.",
"The free energy density contributions are explicitly given by $F_{\\rm grad} &= K_1\\, \\partial _iA_{\\alpha j}\\partial _iA^{\\dagger }_{\\alpha j}+ K_2\\, \\left(\\partial _iA_{\\alpha i}\\partial _jA^{\\dagger }_{\\alpha j}+\\partial _iA_{\\alpha j}\\partial _jA^{\\dagger }_{\\alpha i}\\right) , \\\\F_{2+4} &= \\alpha \\,\\, {\\rm Tr}AA^{\\dagger }+\\beta \\,\\,\\left[\\left({\\rm Tr}AA^{\\dagger }\\right)^2-{\\rm Tr}A^2A^{\\dagger 2}\\right], \\\\F_6 =&\\gamma _6 \\,\\left[-3\\left({\\rm Tr}AA^{\\dagger }\\right)|{\\rm Tr}AA|^2+4({\\rm Tr}AA^{\\dagger })^3\\right.",
"\\nonumber \\\\&\\left.+12({\\rm Tr}AA^{\\dagger }){\\rm Tr}(AA^{\\dagger })^2+ 6({\\rm Tr}AA^{\\dagger }){\\rm Tr}(A^2A^{\\dagger 2})\\right.\\qquad \\nonumber \\\\&\\left.+8{\\rm Tr}(AA^{\\dagger })^3 +12{\\rm Tr}[(AA^{\\dagger })^2A^{\\dagger }A]\\right.",
"\\nonumber \\\\&\\left.-12{\\rm Tr}[AA^{\\dagger }A^{\\dagger }A^{\\dagger }AA]-12{\\rm Tr}AA({\\rm Tr}AA^{\\dagger }AA)^{\\ast }\\right],\\\\F_H &= g^{\\prime }_H\\, H^2 {\\rm Tr}\\left(A A^{\\dagger }\\right)+g_H \\,H_{\\alpha }\\left(AA^{\\dagger }\\right)_{\\alpha \\beta }H_{\\beta },$ where $H$ is the strength of the external magnetic field and we implicitly sum over repeated indices.",
"Note that for simplicity, we set $g^{\\prime }_H=0$ in Eq. ().",
"In this paper we also neglect the sixth order term because of small values of $\\gamma _6$ .",
"Existence of nonzero $\\gamma _6$ would make our system metastable, however we assume that small value of $\\gamma _6$ would make the relaxation time much longer than the time scale of the system.",
"All the following calculations will be performed in the weak coupling limit by considering only the excitations around the Fermi surface , , .",
"In this limit, $K_1$ and $K_2$ in Eq.",
"(REF ) take the same value which is set to be $K$ in the following.",
"To simplify comparison with the works , , , , we fix the numerical values of the model parameters appearing in Eq.",
"(REF ) as discussed in Appendix and Table REF .",
"As mentioned in the introduction, the ground state of the GL free energy is continuously degenerate in the absence of $\\gamma _6$ and of external magnetic fields.",
"Namely, the ground state order parameter in the cartesian basis can always be diagonalised using $SO(3)$ and $U(1)$ rotations to the form $A_{gs} = \\sqrt{\\frac{|\\alpha |}{\\beta (1 + \\eta ^2 + (1 + \\eta )^2)}}\\left(\\begin{array}{ccc}\\eta & 0 & 0 \\\\0 &-(1+\\eta ) & 0 \\\\0 & 0 & 1\\end{array}\\right)\\,,$ where $\\eta $ is a dimensionless numerical parameter , , taking values within the interval $-1 \\le \\eta \\le -\\frac{1}{2}$ ."
],
[
"Vortex Configuration",
"Vortex configurations in the absence of a magnetic field and sixth order terms were first studied in Refs.",
", .",
"Since the order parameter is tensorial, we can choose a basis which diagonalizes the tensor at large distances from the vortex core.",
"In Refs.",
", , the configuration in which the tensor is diagonalized in the cylindrical basis was considered.",
"Later, this configuration was compared with the configuration in which the tensor is diagonalized in the Cartesian basis, and it was confirmed that the former gives the lowest energy configuration .",
"One interesting feature is that the vortex profile functions eventually choose asymptotically only one point among the whole degenerate ground state values.",
"This means that the continuos degeneracy of the ground state, as mentioned in Eq.",
"(REF ), is lifted , , in the presence of a vortex at large distances.",
"In this subsection, we shall derive the equations of motion for the vortex profile functions using the following ansatz for the order parameter of a vortex state $&& A^{(x,y,z)}= \\sqrt{\\frac{|\\alpha |}{6\\beta }}R(\\theta )A^{(\\rho , \\theta , z)}R^T(\\theta )e^{i \\theta }, \\nonumber \\\\&& A^{(\\rho , \\theta , z)} =\\left(\\begin{array}{ccc}f_1 & ige^{i\\delta } & 0 \\\\ige^{ i\\delta } & f_2 & 0 \\\\0 & 0 & -f_1-f_2\\end{array}\\right), $ where $(\\rho ,\\theta ,z)$ denote cylindrical coordinates and $\\delta $ is a constant parameter.",
"$A^{(x,y,z)}$ is the order parameters in the Cartesian basis and $A^{(\\rho ,\\theta ,z)}$ is the order parameters in the cylindrical basis which are related by a rotation matrix $R$ , given by $R(\\theta )=\\left(\\begin{array}{ccc}{\\rm {cos}}\\theta & -{\\rm {sin}}\\theta & 0 \\\\{\\rm {sin}}\\theta & {\\rm {cos}}\\theta & 0 \\\\0 & 0 & 1\\end{array}\\right) .$ For a brief motivation of the ansatz (REF ), we refer the reader to Appendix.",
"and Refs.",
", , , .",
"In Eq.",
"(REF ), $f_1$ , $f_2$ and $g$ are profile functions depending only on the radial coordinate $\\rho $ , and satisfying the boundary conditions $& f_1\\rightarrow -\\sqrt{\\frac{6}{2\\eta ^2+2\\eta +2}}\\eta =1.1116\\,,\\quad \\mbox{as }\\rho \\rightarrow \\infty \\, \\nonumber \\\\& f_2 \\rightarrow \\sqrt{\\frac{3}{2}}\\frac{2\\eta +2}{\\sqrt{2\\eta ^2+2\\eta +2}}= 0.8841\\quad \\mbox{as }\\rho \\rightarrow \\infty \\, ,\\nonumber \\\\& g \\rightarrow 0 \\quad \\mbox{as }\\rho \\rightarrow \\infty \\, \\\\& f_1,f_2 \\rightarrow 0,\\quad g \\rightarrow 0 \\,\\,\\,\\,\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\mbox{as } \\rho \\rightarrow 0\\,,$ where the dimensionless parameter $\\eta $ has a boundary value at spatial infinity determined by energy minimisation to be given by $\\eta = 6 -\\sqrt{43} \\sim -0.557$ , , ; the continuous degeneracy of the ground state is lifted in the vortex background as described above.",
"As the profile function $g$ vanishes at the origin and at spatial infinity, it creates ring-shaped soliton solutions and in general one may replace $g$ by $g e^{im\\theta }$ , where $m$ denotes how many times the phase of $g$ is twisted along the ring.",
"Note that in Ref.",
"winding numbers that are locally defined in the core region were used to classify vortex-core structures in spin-1 Bose-Einstein condensates.",
"In Refs.",
", , vortex solutions in $^{3}P_2$ superfluids have been constructed and analysed using the ansatz (REF ) for the order parameter of a vortex state.",
"However, the effects of a non-zero phase parameter $\\delta $ on the vortex structure have not been considered so far in the existing literature.",
"Here, we introduce the constant phase $\\delta $ in (REF ) to gain further insight into the collective excitations localised around a vortex.",
"Around the vicinity of the vortex, one off-diagonal component was considered so far in the literature , , .",
"However, the authors only discussed the implications of a purely imaginary off-diagonal element.",
"As a further generalisation of the vortex ansatz, we allow for real off-diagonal element contributions by introducing the non-zero phase $\\delta $ in (REF ) as a minimal extension.",
"This generalised ansatz is partly motivated by the idea that the phase might be a Nambu-Goldstone mode as it is often the case for solitons that a phase of a localised profile can be identified with a Nambu-Goldstone mode localised around the soliton.",
"However, as we will learn in the following there exists a mass gap and therefore the phase $\\delta $ cannot be considered as a Nambu-Goldstone mode.",
"Using the ansatz (REF ) we may rewrite the free energy (REF ) as $F&=\\int \\text{d}^2\\rho \\,\\frac{|\\alpha |}{6\\beta }\\frac{K}{G}\\left\\lbrace \\left(t_1+ t_2\\right)+ \\frac{\\alpha }{K} t_3+\\frac{|\\alpha |}{6 K}t_4\\right\\rbrace \\,,$ where the individual energy density contributions $t_1$ , $t_2$ , $t_3$ and $t_4$ take the following form $t_1&=2\\left(f_1^{\\prime 2}+f_2^{\\prime 2}+f_1^\\prime f_2^\\prime +g^{\\prime 2}\\right)+\\frac{1}{\\rho ^2}\\left(4f_1^2+4f_2^2-2f_1 f_2+10g^2\\right.\\nonumber \\\\ &\\left.-8\\left(f_1-f_2\\right)g\\cos \\delta \\right)\\,,\\\\t_2&=2\\left(f_1^{\\prime 2}+g^{\\prime 2}\\right)+\\frac{2}{\\rho ^2}\\left\\lbrace f_2^2+5g^2+\\left(f_1-f_2\\right)^2\\right.\\nonumber \\\\&\\left.-2g\\left(f_1-f_2\\right)\\cos \\delta +4f_2g\\cos \\delta \\right\\rbrace +\\frac{1}{\\rho }\\left\\lbrace -2\\left(f_1^\\prime +f_2^\\prime \\right)g\\cos \\delta \\right.\\nonumber \\\\&\\left.+2\\left(f_1+f_2\\right)g^\\prime \\cos \\delta +2\\left(f_1^\\prime +f_2^\\prime \\right)\\left(f_1-f_2\\right)\\right\\rbrace \\nonumber ,\\\\t_3&=2\\left(f_1^2+f_2^2+f_1f_2+g^2\\right)\\,,\\\\t_4&=2f_1^4+4f_1^3f_2+6f_1^2f_2^2+4f_1f_2^3+2f_2^4+\\Big \\lbrace \\left(6+2\\cos 2\\delta \\right)f_1^2 \\nonumber \\\\&+4f_1f_2+\\left(6+2\\cos 2\\delta \\right) f_2^2+4 f_1f_2\\Big \\rbrace g^2+2g^4\\,.$ Here the prime denotes the derivative with respect to $\\rho $ .",
"For numerical calculations it is useful to rescale the radial coordinate by $\\rho \\rightarrow \\tilde{\\rho } = \\rho /\\xi $ where $1/\\xi ^2 = 12 |\\alpha |/K$ with $\\xi $ being the coherence length ($\\sim 30$ fm) and to change variables to $u = f_1 +f_2$ and $v=f_1 - f_2$ .",
"After rescaling and change of variables, the free energy (REF ) can be written as $F &= \\int \\,\\text{d}^2\\tilde{\\rho }\\,\\frac{2|\\alpha |^2 \\xi ^2}{G \\beta }\\left\\lbrace \\left(\\tilde{t}_1+ \\tilde{t}_2\\right)+ \\frac{\\alpha }{12|\\alpha |} t_3+\\frac{1}{72}t_4\\right\\rbrace $ where the energy contributions read $\\tilde{t}_1&=\\frac{3}{2}u^{\\prime 2}+\\frac{1}{2}v^{\\prime 2}+2g^{\\prime 2}+\\frac{1}{\\tilde{\\rho }^2}\\left(\\frac{3}{2}u^2+\\frac{5}{2}v^2+10g^2-8vg\\cos \\delta \\right)\\,,\\\\\\tilde{t}_2&=\\frac{1}{2}\\left(u^\\prime +v^\\prime \\right)^2+2g^{\\prime 2}+\\frac{1}{\\tilde{\\rho }^2}\\left(\\frac{1}{2}u^2+\\frac{5}{2}v^2-uv+10g^2 \\right.\\nonumber \\\\&\\left.+ 4g(u - 2v)\\cos \\delta \\right)+\\frac{1}{\\tilde{\\rho }}\\left( 2(u g^{\\prime } - gu^{\\prime })\\cos \\delta +2u^\\prime v\\right)\\,,\\\\t_3&=\\frac{3}{2}u^2+\\frac{1}{2}v^2+2g^2\\,,\\\\t_4&=\\frac{1}{8}\\left(3u^2+v^2\\right)^2+\\left[(4u^2 + 2 v^2) + (u^2 + v^2)\\cos 2\\delta \\right]g^2+2g^4\\,.$ Here, the prime denotes the derivative with respect to $\\tilde{\\rho }$ .",
"Note that $ \\tilde{t}_1 = \\xi ^2 t_1(\\rho ) = t_1\\left(\\rho /\\xi \\right) $ .",
"In the following, we calculate the static vortex solution for $\\delta =0$ .",
"In the next section, the resulting vortex profile functions will be used to study perturbatively the effects of nonzero $\\delta $ .",
"We summarise the equations of motion derived from the free energy functional (REF ) for vanishing $\\delta $ together with the imposed boundary conditions in Appendix .",
"Figure: We display the profile functions f 1 f_1, f 2 f_2 and gg for a vortex configuration of winding number one as a function of the distance ρ ˜\\tilde{\\rho } from the vortex center.",
"Here, the order parameter is expressed in the cylindrical basis (n=1n=1) and the vortex solution of lowest energy is found for the boundary value η=-0.557\\eta =-0.557.",
"Numerical results are shown for the case m=0m=0.The obtained profile functions $f_1$ , $f_2$ and $g$ are displayed in Fig.",
"REF as a function of the rescaled radial distance $\\tilde{\\rho }$ from the vortex center.",
"To obtain solutions such as those shown in Fig.",
"REF , we solve the equations of motion () with the boundary conditions stated in Eqs.",
"() using a numerical relaxation method.",
"The minimal energy solutions in the model (REF ) are constructed by solving the gradient flow equations with a crude initial guess (an approximation in terms of the hyperbolic tangents for the functions $u$ and $v$ and a gaussian curve approximation for the profile function $g$ ).",
"The initial configurations are relaxed on a spatial grid with 2001 points and spacing $\\Delta \\tilde{\\rho }= 0.1$ .",
"$^3 P_2$ vortices are known to differ from their counterparts in $^{1} S_0$ superfluids by their nonzero magnetic moments , .",
"From the vortex profile functions displayed in Fig.",
"REF , we can compute the resulting small spontaneous magnetization in the $^3 P_2$ vortex core region.",
"The vortex magnetization is given in Refs.",
", by $\\mathbf {M}&=&\\frac{\\gamma _n \\hbar }{2} \\mathbf { \\sigma }\\, ,$ where $\\mathbf {\\sigma }$ is computed as $\\mathbf {\\sigma }&=&C_m\\frac{|\\alpha |}{6\\beta }g(\\rho )\\left[f_1(\\rho )-f_2(\\rho )\\right]{\\rm cos}\\delta \\,\\, \\hat{\\mathbf {z}}\\,.$ In the following, we take $\\gamma _n = -1.832 471 72 \\times 10^4 s^{-1}G^{-1}$ as experimental value for the neutron gyromagnetic ratio.",
"Here, $C_m = \\frac{4}{9}N^{\\prime }(0)k_F^2$ and $N^{\\prime }(0) = m_n^2 /\\left( 2\\pi ^2 k_F\\right)$ is the number density of states $N = m_n k /\\left(2\\pi ^2\\right)$ differentiated with respect to the energy $E=k^2/2m_n$ and evaluated at the Fermi surface $k=k_F$ .",
"For the numerical parameter values stated in Appendix , $N^{\\prime }(0)$ takes the value $1.341 \\times 10^{-5} \\text{MeV}^{-2} \\text{fm}^{-2}$ and $C_m$ is computed to be $ 2.882 \\times 10^{-5} \\text{MeV}^{-2} \\text{fm}^{-5}$ .",
"We display in Fig.",
"REF the resulting magnetization $\\sigma $ (REF ) for vanishing $\\delta $ as a function of the rescaled distance $\\tilde{\\rho }$ from the vortex centre.",
"The magnetization densities are evaluated using the vortex profile functions $f_1$ , $f_2$ and $g$ plotted in Fig.",
"REF .",
"Figure: Magnetization σ(ρ ˜)\\sigma (\\tilde{\\rho }) for δ=0\\delta =0 plotted as a function of the rescaled distance ρ ˜\\tilde{\\rho } from the vortex centre.Substituting the vortex profile functions computed for vanishing $\\delta $ in (REF ), we can also find an estimate for the magnetization of the system as a function of $\\delta $ .",
"Integrating the magnetization (REF ) over the $xy$ -plane gives the magnetic moment per unit length $\\mathbf {m}(z)& =& \\frac{\\gamma _n \\hbar }{2} 2\\pi \\xi ^2\\int \\tilde{\\rho } d\\tilde{\\rho } \\mathbf { \\sigma }(\\tilde{\\rho })\\nonumber \\\\&=&C_m\\frac{|\\alpha |}{6\\beta } \\frac{\\gamma _n \\hbar }{2} 2\\pi \\xi ^2\\int \\tilde{\\rho } d\\tilde{\\rho } g(\\tilde{\\rho })(f_1(\\tilde{\\rho })-f_2(\\tilde{\\rho })){\\rm cos}\\delta \\,\\, \\hat{\\mathbf {z}}\\,.\\nonumber \\\\$ The integration in (REF ) can be performed numerically using the vortex profile function depicted in Fig.",
"REF .",
"The integral is found to take the following value $2\\pi \\int \\tilde{\\rho } d\\tilde{\\rho } g(\\tilde{\\rho })(f_1(\\tilde{\\rho })-f_2(\\tilde{\\rho })) = -404.83$ .",
"Hence, for vanishing $\\delta $ the magnetic moment per unit length is given by $m(z)\\approx 4.196\\times 10^{-19}\\text{MeV}\\text{fm}^{-1}\\text{G}^{-1}\\approx 0.6 \\times 10^{-27} \\text{J} \\,\\text{T}^{-1} \\text{fm}^{-1}$ .",
"Therefore, the magnetic moment of a 10 fm string is found to be comparable to the neutron magnetic moment."
],
[
"The long distance Effective description of free energy of a vortex",
"In this section we discuss the effective description of collective excitations along a vortex line by using the so-called moduli approximation, originally used for monopoles and applied to various topological solitons, e. g. .",
"At large distances, superfluid vortices can be characterized by Kelvin modes.",
"When orientating the vortex axis along the $z$ -axis, these two moduli correspond to translations in the $x$ - and $y$ - direction.",
"Physically these modes can be interpreted as the massless Nambu-Goldstone excitations generated from the spontaneous breaking of the translational symmetry , .",
"These modes are well studied in the literature of superfluids and are not an issue of this paper.",
"In this article, we want to explore collective excitations localized around the vortex.",
"We introduce a massive mode $\\delta $ which may be important for understanding the spontaneous magnetization of the $^3 P_2$ vortex.",
"Interestingly, when we switch on the mode $\\delta $ in Eq.",
"(REF ), the spontaneous magnetization changes inside the vortex core.",
"In this article we solved the equations of motion by imposing cylindrical symmetry, which makes the profile function independent of the $z$ -coordinate.",
"The parameter $\\delta $ has been introduced as a constant phase in the off-diagonal element for a vortex solution in the $xy$ -plane.",
"As we know, the vortex profile functions eventually reach their ground state values (REF ) at large distances from the vortex core and $g(\\rho )$ forms a concentric ring around the vortex.",
"So $g(\\rho )$ becomes practically zero outside the vortex, see Fig.",
"REF .",
"Therefore, we may treat $\\delta $ as a collective coordinate (or a modulus) of a vortex and may take it as a function of $z$ in the effective free energy functional.",
"In principle, $\\delta $ also depends on time but here we are not considering time dependence.",
"In our case, $\\delta $ is a slowly varying function of the $z$ -coordinate.",
"The parameter $\\delta $ depends on $z$ and the $z$ derivative of $\\delta $ also contributes to the total effective energy.",
"We insert the ansatz $&& A^{(\\rho , \\theta , z)}_{\\delta } =e^{i\\theta } \\left(\\begin{array}{ccc}f_1^0(\\rho ) & ig_0(\\rho )e^{i\\delta (z)} & 0 \\\\ig_0(\\rho )e^{ i\\delta (z)} & f^0_2(\\rho ) & 0 \\\\0 & 0 & -f_1^0-f_2^0\\end{array}\\right) $ into the free energy functional () with the sixth order and magnetic term set to be zero and with the gradient and up to fourth order term given in Eqs.",
"(REF ) and (), respectively.",
"With the ansatz (REF ) substituted into (), the effective free energy takes the following form $F= &\\frac{4\\alpha ^2 \\xi ^4}{\\beta G} \\int \\text{d}z\\,\\text{d}^2\\tilde{\\rho }\\, \\left[\\frac{1}{\\xi ^2}\\left\\lbrace \\left(\\frac{2g_0}{\\tilde{\\rho }^2}\\left(u_0 - 4v_0 \\right) + \\frac{1}{\\tilde{\\rho }} (u_0 {g_0}^{\\prime } - {g_0}{u_0}^{\\prime })\\right)(\\cos \\delta -1)+\\frac{g^2}{144} (u_0^2 + v_0^2)(\\cos 2\\delta - 1)\\right\\rbrace + (\\partial _z\\delta )^2 g_0^2\\right]\\,,$ where $f_1^0$ , $f_2^0$ and $g_0$ are the vortex profile functions for zero $\\delta $ , see Fig.",
"REF .",
"$u_0(\\tilde{\\rho })$ and $v_0(\\tilde{\\rho })$ are defined as $u_0 = f_1^0 + f_2^0$ and $v_0 = f_1^0 - f_2^0$ , respectively.",
"The rescaled radial distance $\\tilde{\\rho }$ is given by $\\tilde{\\rho }= \\rho /\\xi $ , where $\\xi $ is the coherence length of the system.",
"Eq.",
"(REF ) can be expressed more neatly as, $&F_{\\text{eff}} = \\int \\text{d}z\\,\\,\\frac{4\\alpha ^2 \\xi ^4}{\\beta G} \\left[\\frac{1}{\\xi ^2}\\left\\lbrace \\lambda _1(\\cos \\delta -1) +\\lambda _2(\\cos 2\\delta - 1)\\right\\rbrace \\right.\\nonumber \\\\&\\left.\\qquad \\qquad \\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}+ (\\partial _z\\delta )^2\\lambda _3\\large \\right]\\,,$ where the coefficients are defined as $\\lambda _1& = \\int d^2 \\tilde{\\rho }\\left(\\frac{2g_0}{\\tilde{\\rho }^2}\\left(u_0 - 4v_0 \\right) + \\frac{1}{\\tilde{\\rho }} (u_0 g_0^{\\prime } - g_0u_0^{\\prime })\\right)\\,, \\,\\,\\\\\\lambda _2 &= \\int d^2 \\tilde{\\rho }\\,\\,\\frac{g^2}{144} (u_0^2 + v_0^2)\\,,\\\\\\lambda _3 &= \\int d^2 \\tilde{\\rho }g_0^2\\,.$ The coefficients $\\lambda _1, \\lambda _2$ and $\\lambda _3$ have been evaluated numerically to be given by $\\lambda _1 = -4.889\\,, \\quad \\lambda _2 = 0.411\\,,\\quad \\lambda _3 = 17.748\\,.$ Figure: We display the effective free energy potential F eff F_{\\text{eff}} () as a function of the δ\\delta parameter.",
"We evaluate the free energy for the λ\\lambda coefficients listed in Eq. ().",
"The GL parameters are given by α=-0.1\\alpha =-0.1 and β=2.57664(MeVfm) -2 \\beta = 2.57664\\, \\text{(MeV\\, fm)}^{-2}.",
"The coherence length of the system is set to 30.3634fm30.3634\\,\\text{fm}.",
"Note that the energy values are given in units of MeV fm -1 \\text{fm}^{-1}.Note that the signs and numerical values of the $\\lambda $ coefficients are very important in this work.",
"The graph of the effective potential (REF ) should exhibit a bump at the bottom because of the negative sign of the $\\lambda _2$ contribution.",
"However, as can be seen from Fig.",
"REF , the minima occur at $\\delta = 2\\pi n$ with $n$ being an integer.",
"To get any local extrema other than $n\\pi $ , $\\delta $ has to satisfy the equation $\\cos \\delta = \\frac{|\\lambda _1|}{4\\lambda _2}\\,.$ It can be checked that the numerically evaluated $\\lambda $ values (REF ) yield $\\frac{|\\lambda _1|}{4\\lambda _2} > 1$ .",
"Hence, the obtained $\\lambda $ coefficients do not support any other local minima or maxima.",
"This finding supports our expansion of the system around $\\delta =0$ ."
],
[
"Double sine-Gordon Kink solution on a vortex",
"In the last section, we have derived the effective free energy functional (REF ) that describes the low-energy dynamics of the system along with other dynamical terms.",
"Since in this article we are solely interested in time independent configurations, we discuss in the following the static solutions of the system.",
"Note that the system (REF ) is known as the double sine-Gordon model.",
"However, the potential graph shown in Fig.",
"REF almost agrees with the sine-Gordon case because of $|\\lambda _2|<<|\\lambda _1|$ , see Eq.",
"(REF ).",
"Here, we shall derive the kink solution and describe the physics behind the existence of the kink.",
"We use the technique similar to Bogomol'nyi-Prasad-Sommerfield (BPS) monopoles , .",
"Let us rewrite the free energy (REF ) as, $&F_{\\text{eff}} = \\frac{4\\alpha ^2 |\\lambda _1| \\xi ^2}{\\beta G} \\int \\text{d}z\\,\\, \\left[\\left\\lbrace (1 - \\cos \\delta ) + \\frac{\\lambda }{2} (\\cos 2\\delta - 1)\\right\\rbrace \\right.\\nonumber \\\\&\\left.+ \\frac{\\xi ^2\\lambda _3}{|\\lambda _1|} (\\partial _z\\delta )^2\\right]\\,,$ where $ \\lambda = \\frac{2\\lambda _2}{|\\lambda _1|}\\,.$ The numerical value of $\\lambda $ is evaluated to be $0.168735$ .",
"After rescaling $z$ , the effective free energy in Eq.",
"(REF ) takes the form $&F_{\\text{eff}} =& \\frac{M_{\\rm E}}{2} \\int \\text{d}\\zeta \\,\\, \\left[\\left\\lbrace 2(1 - \\cos \\delta ) + \\lambda (\\cos 2\\delta - 1)\\right\\rbrace + (\\partial _{\\zeta }\\delta )^2\\right]\\, ,$ where $\\zeta = \\sqrt{ \\frac{|\\lambda _1|}{2 \\lambda _3}} \\,\\frac{z}{\\xi }\\, $ and $M_E = \\frac{4\\alpha ^2\\xi ^3\\sqrt{2|\\lambda _1| \\lambda _3}}{\\beta G}\\,.$ Although the second term in Eq.",
"(REF ) is not positive, the potential part of Eq.",
"(REF ) is positive definite.",
"So, we can use the BPS technique to find the kink solution.",
"To do that we write the $F_{\\text{eff}}$ as, $F_{\\text{eff}} =& \\frac{M_E}{2} \\int \\text{d}\\zeta \\left[ \\left\\lbrace \\partial _{\\zeta }\\delta \\pm \\sqrt{2(1 - \\cos \\delta ) + \\lambda (\\cos 2\\delta - 1)}\\right\\rbrace ^2 \\right.",
"\\nonumber \\\\& \\left.+ M_E\\, \\left| \\partial _\\zeta \\delta \\sqrt{ 2(1 - \\cos \\delta ) + \\lambda (\\cos 2\\delta - 1)}\\right|.",
"\\right]\\,$ Then $F_{\\text{eff}}$ satisfy the inequality $F_{\\text{eff}} \\ge E_{\\rm {BPS}} =& M_E \\left| \\int d\\zeta \\partial _\\zeta \\delta \\sqrt{ 2(1 - \\cos \\delta ) + \\lambda (\\cos 2\\delta - 1)}\\right|.$ The minimum energy kink configurations would saturate the bound and satisfy the so-called BPS equation $\\partial _{\\zeta }\\delta = \\pm \\sqrt{ 2(1 - \\cos \\delta ) + \\lambda (\\cos 2\\delta - 1)}.$ It is easy to show that the solutions of above equation also satisfy the original equations of motion.",
"The (anti)soliton solution of the BPS equation can be written as, $\\delta (\\zeta )_{}\\pm &= \\pi \\pm 2 \\tan ^{-1} \\left(\\Lambda \\,\\sinh \\Lambda \\zeta \\right)\\, ,$ where $\\Lambda = \\sqrt{1 - 2\\lambda }\\,$ is a real positive constant.",
"Figure: The double sine-Gordon kink solution δ(ζ)\\delta (\\zeta ) () shown for Λ=0.813959\\Lambda = 0.813959.It can be easily seen that for $\\lambda = 0$ , the above solution reduces to the usual sine-Gordon kink $\\delta _{sG}(\\zeta ) = 4\\arctan e^{\\pm \\zeta }$ .",
"The BPS energy bound for the (anti)soliton can be expressed as $E_{\\rm {BPS}} =& M_E \\int ^{\\delta (\\infty )=2\\pi }_{\\delta (-\\infty )=0} d\\delta \\sqrt{ 2(1 - \\cos \\delta ) + \\lambda (\\cos 2\\delta - 1)}\\nonumber \\\\=& \\frac{M_E}{\\sqrt{2\\lambda }} \\int ^{\\sqrt{2\\lambda }}_{-\\sqrt{2\\lambda }} dt \\sqrt{ 1 - t^2} \\nonumber \\\\=& 1.88 \\, M_E\\, .$ For a nuclear matter density of $\\rho _n=6\\times 10^{17}\\, \\text{kg}\\,\\text{m}^{-3}$ and a temperature of $T=0.16$ MeV (as assumed in Appendix and Table REF ), the total kink energy can be evaluated as $E_{\\rm {BPS}} = 93.3\\,\\, \\text{MeV}\\,.$ We display in Fig.",
"REF the double sine-Gordon kink solution $\\delta (\\zeta )$ obtained for the $\\lambda $ values given in Eq.",
"(REF ).",
"Figure: The magnetization 2m(ζ) |γ n |ℏ=-σ(ζ)\\left(\\frac{2m(\\zeta )}{|\\gamma _n|\\hbar }= -\\sigma (\\zeta )\\right) plotted as a function of the rescaled zz-coordinate ζ\\zeta .In Fig.",
"REF , we plot the change in the vortex core magnetization in the presence of the kink solution along the $z$ -axis.",
"When substituting (REF ) in Eq.",
"(REF ), it is observed that the magnetization changes its value continuously and reaches its opposite value at the center of the kink.",
"Note that in this article we observed a sign flip of the trapped magnetic moment inside the core of a $^3P_2$ neutron superfluid vortex.",
"For this case it is natural to think of the system as a chain of aligned magnetic moments with a flipped moment at the position of the kink.",
"Hence, $\\nabla \\cdot \\mathbf {M} \\ne 0$ on the junctions, where $\\mathbf {M}$ is defined in Eq.",
"(REF ).",
"Since the total magnetic field $\\mathbf {B} $ is divergenceless, $\\nabla \\cdot \\mathbf {B} = 0$ , there would be a leakage of a magnetic field, say $\\mathbf {H}$ , from the junctions.",
"In this case $\\nabla \\cdot \\mathbf {H} \\ne 0$ at the junctions and the total magnetic field is defined as $\\mathbf {B} = \\left(\\mathbf {H} + 4 \\pi \\mathbf {M}\\right)$ .",
"We schematically show in Fig.",
"REF an example of a magnetic field configuration with a flipped magnetic moment together with the resulting leakage of the magnetic field.",
"Figure: A schematic figure of a magnetic field configuration to illustrate the flipped magnetic moment and the leakage.",
"The dotted red arrows and the solid blue arrows denote the magnetic field 𝐇\\mathbf {H} and magnetic moment 𝐌\\mathbf {M}, respectively."
],
[
"Conclusions",
"In this article, we have derived an effective theory of $^3P_2$ neutron superfluid vortices at large distances and expressed solely in terms of the phase $\\delta $ of the vortex profile function $g$ .",
"$g$ lives in one of the off-diagonal components of the tensorial $^3P_2$ order parameter $A_{\\alpha \\,i}$ .",
"Here, $\\delta $ plays a crucial role as it gives a nonzero spontaneous magnetisation in the core region of $^3P_2$ neutron superfluid vortices for the angular independent profile function $g$ .",
"Variations in $\\delta $ change the induced magnetic moment accordingly.",
"In our analysis we have found that at low energies the $^3P_2$ superfluid vortices are described by a double sine-Gordon model in addition to the well-known Kelvin (translational zero) modes part.",
"We have derived a kink solution of this particular double sine-Gordon model and have determined its BPS energy bound.",
"We have found that the vortex core magnetisation flips its direction at the kink on the vortex.",
"Since the number of vortices in a rotating NS is of the order of $10^{19}$ and the NS radius is about 10 km, there should exist a large number of magnetic strings in the NS interior like the ones discussed in this paper.",
"A pair of kink and anti-kink may be created spontaneously during the creation of the vortex through phase transition.",
"Possible implications of these objects for NS physics should be one of the most important future problems.",
"In this article, we have neglected higher order terms (esp.",
"sixth order terms) in the GL free energy.",
"We also did not take into account the effect of an external magnetic field.",
"However, in the core of neutron stars where it is most likely to find neutron superfluid, the inclusion of high external magnetic fields and of the sixth order term would be important.",
"So it would be interesting to study the effective free energy in the presence of high external magnetic fields and with nonzero sixth order term included.",
"We have also considered only static configuration in this article.",
"In order to discuss dynamics, we have to include time dependence.",
"To this end, we need a time dependent GL equation, which has not been obtained yet.",
"For the time-dependent problem, the phase $\\delta $ has to be promoted to a time-dependent function and can be treated as a field which lies on the two-dimensional string world sheet.",
"Fluctuations of $\\delta $ may indicate changes in the magnetization on the world sheet.",
"In the case of conventional superfluids, Kelvin modes will propagate with quadratic dispersion relation for small system sizes, see e. g. Ref. .",
"However, it is well known that for large system sizes the dispersion relation is given by $\\epsilon = k \\log k$ with wave vector $k$ .",
"Recently, a formula for arbitrary system sizes has been obtained in Ref. .",
"On the other hand, the gapfull mode $\\delta $ was previously unknown.",
"We expect $\\delta $ to have relativistic dynamics.",
"In this article, we have discussed only the integer vortex.",
"For strong magnetic fields, the ground state is in the $D_4$ -biaxial nematic phase admitting half-quantized non-Abelian vortices .",
"The vortex core magnetization is ten times larger than that of the integer vortex.",
"For this case we should also derive the low-energy collective coordinate approximation and construct a magnetic lump."
],
[
"Acknowledgements",
"This work is supported by the Ministry of Education, Culture, Sports, Science (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No.",
"S1511006).",
"C. C. acknowledges support as an International Research Fellow of the Japan Society for the Promotion of Science (JSPS).",
"Some of the work of M. H. was undertaken at the Department of Mathematics and Statistics, University of Massachusetts, financially supported by FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-606096).",
"The work of M. N. is supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No.",
"16H03984), and by a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant No.",
"15H05855) and “Nuclear Matter in Neutron Stars Investigated by Experiments and Astronomical Observations” (KAKENHI Grant No.",
"15H00841) from the the Ministry of Education, Culture, Sports, Science (MEXT) of Japan."
],
[
"Ginzburg-Landau Description for $^3P_2$ Superfluids",
"In this appendix, we briefly discuss the Ginzburg-Landau (GL) construction of $^3P_2$ superfluids and the associated GL parameter values.",
"The Hamiltonian with $^3P_2$ contact interaction can be written as $H &= \\int d^3 \\rho \\ \\psi ^{\\dagger }\\left(-\\frac{\\mathbf {\\nabla }^2}{2M}-\\mu \\right)\\psi -\\frac{1}{2}gT^{\\dagger }_{\\alpha \\beta }(\\mathbf {\\rho })T_{\\alpha \\beta }(\\mathbf {\\rho }), \\\\T^{\\dagger }_{\\alpha \\beta } (\\mathbf {\\rho })&= \\psi ^{\\dagger }_{\\sigma }(\\mathbf {\\rho })(t^{\\ast }_{\\alpha \\beta })_{\\sigma \\sigma ^{\\prime }}(\\mathbf {\\nabla })\\psi ^{\\dagger }_{\\sigma ^{\\prime }}(\\mathbf {\\rho }) ,$ where $\\mathbf {\\rho }$ denotes the space coordinates, $\\psi $ is a neutron field, $\\mu $ is a baryon chemical potential, $M$ is the neutron mass, and $g(>0)$ is the coupling constant.",
"Here, $\\alpha ,\\beta $ are the space indices, and the tensor $T_{\\alpha \\beta }$ represents the $^3P_2$ pair creation and annihilation operator.",
"The differential operator $t$ is defined as $(t_{\\alpha \\beta })_{\\sigma \\sigma ^{\\prime }}(\\mathbf {\\nabla })&=&\\frac{1}{2}((S_{\\alpha })_{\\sigma \\sigma ^{\\prime }}\\nabla _{\\beta }+\\nabla _{\\alpha }(S_{\\beta })_{\\sigma \\sigma ^{\\prime }}) \\nonumber \\\\&&-\\frac{1}{3}\\delta _{\\alpha \\beta }(\\mathbf {S})_{\\sigma \\sigma ^{\\prime }}\\cdot \\mathbf {\\nabla },$ and $\\mathbf {S}$ is given by $(S_{\\alpha })_{\\sigma \\sigma ^{\\prime }} =i(\\sigma _y \\sigma _{\\alpha })_{\\sigma \\sigma ^{\\prime }}$ with $\\alpha =x,y,z$ and pauli matrices $\\sigma $ .",
"The $^3P_2$ order parameter can be represented in general as a $3 \\times 3$ traceless symmetric complex tensor $A_{\\mu i}$ , which is defined in Refs.",
", as $\\Delta =\\sum _{\\alpha i} i\\sigma _{\\alpha }\\sigma _y A_{\\alpha i} k_i,$ where $\\Delta $ is the gap parameter.",
"As above, Greek subscripts stand for spin indices while Roman indices denote the spatial coordinates.",
"The tensor $A_{\\alpha i}$ is related to $T_{\\alpha \\beta }$ as given in Ref.",
", $A_{\\alpha \\beta } = g \\langle T_{\\alpha \\beta }(\\mathbf {\\rho })\\rangle _T,$ where the angular brackets are defined as an ensemble average in the grand canonical ensemble.",
"Table: Summary of the GL parameters computed in the weak coupling limit.",
"We refer the reader to the last paragraph of section for the numerical values of all model parameters used in our numerical simulations."
],
[
"The Large Distance Bahaviour of the Order Parameter for a Vortex",
"An ansatz for the order parameter $A$ of a vortex state has been developed first in Refs.",
", .",
"Here, we briefly sketch its derivation.",
"For a detailed derivation we refer the interested reader to Ref. .",
"For an isolated vortex solution with cylindrical symmetry and phase $ m \\theta $ , where $m$ measures the circulation about the vortex line, we first choose the order parameter at large distances by minimising the potential in Eq.",
"() as $A &\\sim & C(\\eta )e^{im\\theta }R(\\theta ) \\left(\\begin{array}{ccc}\\eta & 0 & 0 \\\\0 & - (1 + \\eta ) & 0 \\\\0 & 0 & 1\\end{array}\\right)R^T(\\theta )\\,, \\\\C(\\eta ) &=& \\sqrt{\\frac{|\\alpha |}{\\beta (1 + \\eta ^2 + (1 + \\eta )^2)}} \\,.",
"\\nonumber $ Here $R(\\theta )$ is the rotation matrix which rotates the axis from $(\\rho , \\theta , z)$ into $(x, y, z)$ .",
"Unlike in the case of the ground state, the order parameter can also have extra contributions from the kinetic terms in the free energy functional.",
"Both the $K_1$ and $K_2$ terms in Eq.",
"(REF ) give logarithmic contributions to the total energy.",
"Minimizing the coefficient of the logarithmic term breaks the ground state degeneracy for all $-1 \\le \\eta \\le -\\frac{1}{2}$ and the lowest energy is achieved for $\\eta = 6 -\\sqrt{43} \\sim -0.557$ .",
"Hence, for a vortex configuration of minimal energy we may choose the matrix order parameter of a vortex state as given in Refs.",
", , , $A\\sim \\sqrt{\\frac{|\\alpha |}{6\\beta }}e^{i\\theta }R(\\theta ) \\left(\\begin{array}{ccc}f_1 & 0 & 0 \\\\0 & f_2 & 0 \\\\0 & 0 & -f_1-f_2\\end{array}\\right)R^T(\\theta )\\, \\,, $ where the boundary conditions for the profile functions are chosen according to the large distance behaviour in Eq.",
"(REF ).",
"In Table REF , we summarise the coefficients $\\alpha $ , $\\beta $ , $K_1$ and $K_2$ (the GL parameters) calculated in the weak coupling limit by considering only the excitations around the Fermi surface , , .",
"In this limit, $K_1$ and $K_2$ in Eq.",
"(REF ) take the same value.",
"To simplify comparison with the works , , , the GL parameters are calculated for a nuclear mass density of $\\rho _n=6\\times 10^{17}\\, \\text{kg}\\,\\text{m}^{-3}$ , Fermi momentum $k_F = 2.1987 \\,\\text{fm}^{-1}$ , nucleon mass $\\text{m}_n = 0.02413\\, \\text{MeV}^{-1} \\text{fm}^{-2}$ , a critical temperature for ${}^3P_2$ pairing of $T_C=0.2\\,\\text{MeV}$ and a temperature of $T=0.8T_C$ ."
],
[
"Equations of Motion",
"In this appendix, we list explicitly all the vortex equations together with the imposed boundary conditions.",
"Note that all vortex equations are given in the cylindrical basis ($n=1$ ).",
"The equations of motion derived from (REF ) are given by $&\\tilde{\\rho }^2 \\left(4 u^{\\prime \\prime }+v^{\\prime \\prime }\\right)+\\tilde{\\rho }(4 u^\\prime +3v ^\\prime -4g^\\prime )-\\left(4u-v+4g\\right)\\nonumber \\\\&-\\frac{\\tilde{\\rho }^2}{144}\\left(\\frac{36 \\alpha }{|\\alpha |}+9u^2+3 v^2+20g^2\\right) u=0\\,,\\\\&\\tilde{\\rho }^2 \\left(u^{\\prime \\prime }+2v^{\\prime \\prime }\\right)+\\tilde{\\rho }\\left(2v^\\prime -u^\\prime \\right)- \\left(10v-u-16g\\right)\\nonumber \\\\&-\\frac{\\tilde{\\rho }^2}{144}\\left(\\frac{12\\alpha }{|\\alpha |}+ 3u^2+v^2+12 g^2\\right)v=0\\,,\\\\&4\\tilde{\\rho }^2 g^{\\prime \\prime }+2\\tilde{\\rho }(2g^\\prime + u^\\prime )+2\\left(4v-u-10 g\\right) \\nonumber \\\\&-\\frac{\\tilde{\\rho }^2}{72}\\left(\\frac{12\\alpha }{|\\alpha |} + 5u^2+3v^2+4g^2\\right)g=0\\,.$ All profile function $u$ , $v$ and $g$ are required to vanish at the origin.",
"At spatial infinity we impose the following boundary conditions which are compatible to the above equations of motion $&&u^2 = (f_1+f_2)^2 = \\frac{6}{(\\eta ^2 + (1+\\eta )^2+1)},\\\\&&v^2=(f_1-f_2)^2 = \\frac{6(2\\eta +1)^2}{(\\eta ^2 + (1+\\eta )^2+1)},\\\\&&u^{\\prime }=v^{\\prime }=g^{\\prime }=g=0.$ Hence, with the vortex boundary state selected to be given by the lowest energy state $\\eta =-0.557$ (due to the fourth order term), the linear combinations $u=f_1 +f_2$ and $v=f_1 - f_2$ have to fullfill the boundary conditions $u(\\infty )=1.9956$ and $v(\\infty )=0.2275$ ."
]
] | 1612.05588 |
[
[
"The curious case of high-energy deuterons in Galactic cosmic rays"
],
[
"Abstract A new analysis of cosmic ray (CR) data collected by the SOKOL experiment in space found that the deuteron-to-helium ratio at energies between 500 and 2000 GeV/nucleon takes the value d/He ~ 1.5.",
"As we will show, this result cannot be explained by standard models of secondary CR production in the interstellar medium and points to the existence of a high-energy source of CR deuterons.",
"To account for the deuteron excess in CRs, we argue that the only viable solution is hadronic interaction processes of accelerated particles inside old supernova remnants (SNRs).",
"From this mechanism, however, the B/C ratio is also expected to increase at energy above ~50 of GeV/nucleon, in conflict with new precision data just released by the AMS-02 experiment.",
"Hence, if this phenomenon is a real physical effect, hadronic production of CR deuterons must occur in SNRs characterized by low metal abundance.",
"In such a scenario, the sources accelerating C-N-O nuclei are not the same as those accelerating helium or protons, so that the connection between d/He ratio and B/C ratio is broken, and the latter cannot be used to place constraints on the production of light isotopes or antiparticles."
],
[
"Introduction",
"Deuteron isotopes $^{2}$H are rare particles in Galactic cosmic rays (CRs).",
"They are destroyed rather than formed in thermonuclear reactions in stellar interior so that, from supernova remnants (SNRs) as sources of CRs, no significant amount of deuteron is expected to be released by diffusive shock acceleration (DSA).",
"An important process by which high-energy deuterons can be created in the Galaxy is nuclear fragmentation of CR nuclei with the gas of the interstellar medium (ISM).",
"The main source of deuteron production is fragmentation of $^{4}$He and $^{3}$He isotopes, along with N-O nuclei, in collisions with interstellar hydrogen and helium.",
"An important contribution comes from the reaction p+p$\\rightarrow $d+$\\pi $ .",
"From nuclear fragmentation, interactions of CRs with the ISM are also known to generate Li-Be-B nuclei and antiparticles, that are otherwise rare from stellar nucleosynthesis processes.",
"The measured abundances of these elements in the cosmic radiation enables us to pose tight constraints on the astrophysical models of CR propagation in the Galaxy and, in the context of indirect searches of dark matter, to asses the astrophysical background of CRs antiparticles [13].",
"Very recently, the Alpha Magnetic Spectrometer (AMS-02) experiment has reported a new precise measurement of the B/C ratio in CRs at energy between $\\sim $ 0.5 and 1000 GeV/n [2].",
"At kinetic energies above $\\sim $ 10 GeV/n, the ratio is found to decrease steadily with increasing energy.",
"Since boron nuclei are mainly produced by N-O fragmentation in the ISM, the observed trend of the B/C ratio reflects the conception that CRs diffusively propagate in the Galactic magnetic fields with an average diffusion coefficient that increases with energy.",
"According to this picture, the ratio between $^{2}$H and its main progenitor $^{4}$He must follow the same behavior in the GeV-TeV energy range.",
"Recent calculations have shown, indeed, that the observed boron and deuteron abundances at GeV/n energies can both be self-consistently described by diffusion models of CR propagation [8], [26].",
"In the GeV-TeV energy region, the d/He ratio is expected to decrease rapidly, as fast as the B/C ratio does, but CR deuterons have never been detected at these energies.",
"Quite unexpectedly, a new analysis of the data collected by the satellite mission SOKOL has determined the d/He ratio at 0.5-2 TeV/n energy [32].",
"In this measurement, the discrimination between $^{1,2}$H and $^{3,4}$He isotopes was performed by means of neural network analyses of the topology of hadronic showers developed by these particles.",
"By means of two different Monte Carlo simulations, consistent results have been obtained: 0.114$\\pm $ 0.023 and 0.099$\\pm $ 0.021 for the $^{2}$H/($^{1}$H$+$$^{2}$H) ratio, 1.64$\\pm $ 0.30 and 1.43$\\pm $ 0.27 for the $^{2}$H/$^{4}$He ratio.",
"In this Letter, we show that the above results represent a striking anomaly in CR physics that cannot be explained by standard models of CR propagation and secondary production in the ISM.",
"Then, we argue that a deuteron excess can be explained in terms of hadronic production occurring inside SNRs, which was proposed in [5] to explain the positron excess in CRs [15], [25].",
"Using new evaluations of fragmentation cross-sections, we calculate for the first time the high-energy production of CR deuterons in SNRs and in the ISM, demonstrating that this mechanism can account for the new d/He data.",
"Along with the d/He ratio, we also compute the B/C ratio under the same framework, showing that there are conflicting results in the model predictions for the two observables.",
"We therefore conclude that this tension can be resolved if the deuterons progenitors are not accelerated in the same sources of boron progenitors.",
"We discuss our results and their implications for the interpretation of antiproton data."
],
[
"Calculations",
"We compute the spectrum of CR nuclei accelerated in SNRs within the linear DSA theory and including the production of secondary fragments.",
"Similar calculations are done in earlier works [5], [6], [18], [19], [31], [14].",
"We follow closely the derivation of [30].",
"In the shock rest-frame ($x = 0$ ), the upstream plasma flows in from $x < 0$ with speed $u_{1}$ (density $n_{1}$ ) and the downstream plasma flows out to $x > 0$ with speed $u_{2}$ (density $n_{2}$ ).",
"The compression ratio is $r=u_{1}/u_{2} = n_{2}/n_{1}$ .",
"For a nucleus with charge $Z$ and mass number $A$ , the equation describing diffusion and convection at the shock reads $ u \\frac{\\partial f}{\\partial x} = D \\frac{\\partial ^{2}f}{\\partial x^{2}} +\\frac{1}{3}\\frac{du}{dx}p\\frac{\\partial f}{\\partial p}-\\Gamma ^{\\rm tot}{f} + Q \\,,$ where $f$ is the phase space density, $D(p)$ is the diffusion coefficient near the shock, $u$ is the fluid speed, and $\\Gamma ^{\\rm tot} = c n \\sigma ^{\\rm tot}$ is the total fragmentation rate for cross-sections $\\sigma ^{\\rm tot}$ and background density $n$ , which is assumed to be composed of H and He like the average ISM.",
"The source term includes particle injection at the shock, $Q = Y \\delta (x) \\delta (p-p^{\\rm inj})$ , with $p^{\\rm inj}\\equiv Z R^{\\rm inj}$ and $R^{\\rm inj}\\cong 0.5\\,$ GV for all nuclei.",
"The $Y$ -constants set the normalization of each species.",
"We assume strong shocks ($r\\approx 4$ ) and a diffusion coefficient $D = \\kappa _{B}\\frac{p/Z}{3B}$ , where $\\kappa _{B}$ parameterizes the deviation of $D(p)$ from the Bohm value due to magnetic damping.",
"The resulting acceleration rate at momentum $p$ is $\\Gamma ^{\\rm acc} \\sim u_{1}^{2}/20\\,D$ .",
"For an SNR of age $\\tau _{\\rm snr}$ , the condition $\\Gamma ^{\\rm acc}=\\tau ^{-1}_{\\rm snr}$ defines the maximum momentum scale attainable by DSA.",
"In the presence of hadronic interactions, the additional requirement $\\Gamma ^{\\rm tot} \\ll \\Gamma ^{\\rm acc}$ must be fulfilled.",
"The downstream solution reads $f_{2}(x,p) = f_{0}(p) \\left( 1 - \\frac{\\Gamma ^{\\rm tot}_{2}}{u_{2}}x \\right) + \\frac{q_{2}}{u_{2}}x \\,,$ where $f_{0}(p)$ is the distribution function at the shock.",
"As found in [18], $f_{0}(p)$ is given by $\\begin{aligned}f_{0}(p) = \\alpha \\int _{0}^{p} \\left( \\frac{p^{\\prime }}{p} \\right)^{\\alpha } Y\\delta (p^{\\prime }-p^{\\rm inj}) e^{-\\chi (p,p^{\\prime })} \\frac{dp^{\\prime }}{p^{\\prime }}\\\\+ \\alpha \\int _{0}^{p} \\left( \\frac{p^{\\prime }}{p} \\right)^{\\alpha } \\frac{q_{1} D}{u^{2}_{1}}\\left( 1 + r^{2} \\right) e^{-\\chi (p,p^{\\prime })} \\frac{dp^{\\prime }}{p^{\\prime }} \\,,\\end{aligned}$ with $\\alpha = 3r/(r-1)$ , $\\chi \\approx \\alpha (\\Gamma ^{\\rm tot}_{1}/\\Gamma ^{\\rm acc}) [ D(p) - D(p^{\\prime }) ]$ , and the subscript $i=1$ ($i=2$ ) indicates the upstream (downstream) region.",
"The first term of Eq.",
"REF describes primary particles injected at the shock and it is of the form $\\sim p^{-\\alpha } e^{-\\chi }$ .",
"The second term of Eq.",
"REF describes the production and acceleration of CR fragments from heavier progenitors.",
"For each $k\\rightarrow j$ process, the $q$ –term of Eq.",
"REF is given by $q_{kj}^{\\rm sec}(p) = \\xi _{kj}^{-3} f_{j}(x,p/\\xi _{kj}) \\Gamma ^{\\rm fr}_{kj}$ , where $\\Gamma ^{\\rm fr}_{kj}= \\sigma _{kj}n \\beta c$ is the secondary production rate, and $\\sigma _{kj}$ is the corresponding cross-section.",
"The momentum inelasticity factor $\\xi _{kj} = A_{j}/A_{k}$ expresses the conservation of kinetic energy per nucleon between progenitor and fragment.",
"Equation REF is solved for all relevant species.",
"We considered primary nuclei (with $Y>0$ ) p, $^{4}$He, N, O tuned to recent data as in [19], and secondary B production from N-O collisions with hydrogen and helium gas.",
"The adopted fragmentation cross-sections are those re-evaluated in [29].",
"We account for deuteron production from collisions of p, $^{3}$He, and $^{4}$He off hydrogen and helium.",
"Measurements and calculations for these reactions are available only below 10 GeV/n of energy.",
"Thus, we have performed new calculations at $E=$ 10-1000 GeV/n using the hadronic Monte Carlo generator QGSJET-II-04 [23].",
"The cross-sections for the dominant channels are shown in Fig.",
"REF .",
"The solid lines are our parameterizations adapted from [26] and extended to higher energy.",
"Cross-sections for collisions off He target are larger by nearly a factor of two and contribute to $\\sim $ 20 %.",
"The contribution from fragmentation of heavier nuclei is negligible.",
"In all reactions, kinetic energy per nucleon is approximately conserved except for the p$+$p reaction, which peaks at a proton energy of $\\sim $ 600 MeV/n and produces deuterons within a narrow energy range around 150 MeV/n [20].",
"Hence, these deuterons experience DSA acceleration similarly to primary components with non-zero $Y$ -factor, i.e., to a power-law spectrum $\\sim p^{-\\alpha }$ .",
"Figure: Cross-section data for deuteron production from 4 ^{4}He-p, 3 ^{3}He-p, and p-pcollisions along with new calculations obtained from QGSJET-II-04 and our parameterizations.Data are from the compilations in and .In contrast, deuterons ejected by helium fragmentation have an initial spectrum $p^{-\\alpha }$ so that, for those produced within a distance $d\\lesssim \\,D/u_{1}$ from the shock and undergo DSA, their final spectrum reaches the form $f^{\\rm sec}_{0}\\sim f^{\\rm pri}_{0}D(p) \\propto p^{-\\alpha +1}$ (see Eq.",
"REF ).",
"This means that, for Bohm-type diffusion, the secondary deuteron spectrum is one power harder than that of its progenitors.",
"The total CR flux produced by SNRs is computed as $\\mathcal {S}^{\\rm snr}(p) = 4\\pi p^{2}\\mathcal {R_{\\rm SN}} \\int _{0}^{\\tau _{\\rm snr}u_{2}} 4 \\pi x^{2} f_{2}(x,p) dx$ where $\\mathcal {R}_{\\rm SN}\\cong $ 25 Myr$^{-1}$ kpc$^{-2}$ is the explosion rate per unit volume and $\\tau _{\\rm snr}\\cong $ 40 kyr is the age of the SNR.",
"To model the subsequent propagation of CRs in the ISM, we adopt a two-halo model of CR diffusion and nuclear interactions [27], [28].",
"The Galaxy is modeled as a disk of half-thickness $h\\cong $ 100 pc containing SNRs and gas with number density $\\tilde{n}\\cong $ 1 cm$^{-3}$ .",
"The disk is surrounded by a diffusive halo of half-thickness $L$ and zero matter density.",
"We give a one-dimensional description in the thin disk limit ($h \\ll L$ ).",
"For each CR nucleus, the transport equation reads $\\frac{\\partial \\mathcal {N}}{\\partial t} = \\frac{ \\partial }{\\partial z} \\left( K(z) \\frac{\\partial \\mathcal {N}}{\\partial z} \\right)-2h \\hat{\\delta }(z) \\tilde{\\Gamma }^{\\rm tot} \\mathcal {N} + 2h\\hat{\\delta }(z) \\mathcal {S}^{\\rm tot} \\,,$ where $\\mathcal {N}$ is its density, $\\hat{\\delta }$ is a Dirac function of the $z$ -coordinate, $K(z)$ is the diffusion coefficient of CRs in the Galaxy, and $\\tilde{\\Gamma }^{\\rm tot} = \\beta c \\tilde{n} \\sigma ^{\\rm tot}$ is the destruction rate in the ISM at velocity $\\beta c$ and cross-section $\\sigma ^{\\rm tot}$ .",
"The source term $\\mathcal {S}^{\\rm tot}$ is split into a primary term $\\mathcal {S}^{\\rm snr}$ , obtained from Eq.",
"REF as solution of the DSA equation, and a secondary production term $\\mathcal {S}^{\\rm sec}= \\sum _{k} \\tilde{\\Gamma }_{k}^{\\rm fr} \\mathcal {N}_{k}$ , from fragmentation of $k$ -type nuclei in the ISM with rate $\\tilde{\\Gamma }_{k}^{\\rm fr}$ .",
"To compute the interaction rates in the ISM $\\tilde{\\Gamma }^{\\rm in/fr}$ we adopt the same fragmentation network (and same cross-sections) as occurring inside SNRs.",
"Equation REF is solved in steady-state conditions $\\partial \\mathcal {N}/\\partial t =0$ .",
"The derivation of the full solution is in [27].",
"The diffusion coefficient is taken of the type $K(p,z) = \\beta K_{0}((pc/Ze)/GV)^{\\delta (z)}$ where $K_{0}$ expresses its normalization.",
"For the scaling index, $\\delta (z)$ , we adopt $\\delta =\\delta _{0}$ in the region of $|z|<\\xi \\,L$ (inner halo) and $\\delta =\\delta _{0}+\\Delta $ for $|z|>\\xi \\,L$ (outer halo).",
"Our default parameters are set as $\\xi \\,L\\cong \\,0.1$ $\\delta _{0}\\cong $ 1/3, $\\Delta \\cong $ 0.55, and $K_{0}/L\\cong $ 0.01 kpc Myr$^{-1}$ .",
"The differential energy fluxes of each species are given by $J(E) = \\frac{\\beta c}{4 \\pi }\\mathcal {N}$ .",
"Solar modulation is described in force-field approximation using the parameter $\\phi =500$ MV for a medium-level modulation strength.",
"The d/He and B/C ratios as function of kinetic energy per nucleon are eventually calculated as $J_{^{2}{\\rm H}}/J_{^{4}{\\rm He}}$ and $(J_{^{10}{\\rm B}}+J_{^{11}{\\rm B}})/(J_{^{12}{\\rm C}}+J_{^{13}{\\rm C}})$ , respectively.",
"In the following, we consider two model implementations representing two alternative scenarios: Scenario # 1 (B/C-driven, conservative) — standard model without interactions in sources, which is the case of a CR flux released by young SNRs with amplified magnetic fields ($B\\gtrsim $ 100 $\\mu \\,G$ ) and/or low background density ($n_{1}\\approx \\,10^{-3}$ cm$^{-3}$ ).",
"In this model, secondary production of CRs deuterons or Li-Be-B occurs only in the ISM.",
"This model is tuned to match the new B/C data from AMS-02 at GeV/n-TeV/n energies.",
"Scenario # 2 (d/He-driven, speculative) — model with copious production and acceleration of secondary particles in SNR shockwaves, which is the case for a GeV-TeV flux provided by old SNRs with damped magnetic fields, slow shock speed, or dense ambient medium, i.e., with the combination $n_{1}\\kappa _{B}B^{-1}u_{8}^{-2}\\sim $ 400.",
"This model is tuned against the d/He data including the new SOKOL measurement at TeV/n energies.",
"Figure: Model calculations from Scenario # 1 (green solid lines) andScenario # 2 (red dashed lines) for the B/C ratio (left) and d/He ratio (right).Data are from AMS-02 , BESS , , CAPRICE ,IMAX , AMS-01 , and SOKOL ."
],
[
"Results and Discussion",
"Model calculations are shown in Fig.",
"REF for the B/C ratio the d/He ratio at energies between $\\sim $ 0.5 GeV and 2 TeV per nucleon.",
"Scenario # 1 is plotted as green solid lines.",
"In this model, secondary nuclei such as d or B are entirely generated in the ISM, i.e., without SNR components, and thus secondary/primary ratios decrease steadily as $J_{s}/J_{p}\\propto (L/K_{0})\\left[ \\xi + (1-\\xi )\\rho ^{-\\Delta } \\right]\\rho ^{-\\delta _{0}}$ where $\\rho =(pc/Ze)/GV$ .",
"In the high-energy limit one has $J_{s}/J_{p} \\propto E^{-1/3}$ .",
"It can be seen that this model fits remarkably well the new AMS-02 data on the B/C ratio and dictates a similar trend for the d/He ratio, which is predicted to reach the level of $\\sim \\,10^{-2}$ in the TeV/n energy scale.",
"We therefore conclude that the SOKOL measurement of the d/He ratio is at least two orders of magnitude higher than that expected from standard models where CR deuterons are produced by fragmentation in the ISM.",
"In Scenario # 2, shown as red dashed lines, hadronic interaction processes inside SNRs generate a source component of secondary nuclei, which is harder than that arising from CR collisions with the ISM and, as discussed, even harder than that of primary p-He spectra.",
"It is then possible, with fragmentation inside SNRs, having secondary/primary ratios that increase with energy.",
"Figure REF shows that Scenario # 2 matches fairly well the d/He ratio measurements at GeV/n and at TeV/n energies, therefore providing an explanation for the new SOKOL data.",
"Under this model, however, the B/C ratio is also predicted to increase, at energies above $\\sim $ 50 GeV/n, in remarkable contrast with the new AMS-02 data.",
"While interactions inside SNRs seem to be the only mechanism capable of explaining a rise in the d/He ratio, it is apparent that the observed decrease of the B/C ratio conflicts with this mechanism.",
"We also note that, at the $\\sim $ 1 GeV/n energy region where secondary CR production in the ISM dominates, the two ratios are consistently described by both models # 1 and # 2, at least within the precision of the current data.",
"As we see it, the only solution to this tension is a situation where the connection between d/He ratio and B/C ratio is broken.",
"This situation is realized if the sources accelerating helium and protons (and producing deuterons) are not the same as those accelerating heavier N-O nuclei and, in particular, deuterons must be accelerated by a low-metallicity source, which may be the case for SNRs expanding over H-dominated molecular clouds.",
"Such a possibility was also discussed in [7] (see Sect.",
"VI), and proposed in other works [10], [17], all focused on antiparticle excesses in CRs.",
"In such a scenario, the B/C ratio can no longer be used to place constraints on antimatter spectra.",
"In this respect, it is important to note that the connection between d/He ratio and antimatter/matter ratios would still be preserved because, in contrast to Li-Be-B nuclei, secondary deuterons share their progenitors with positrons and antiprotons.",
"Figure: Antiproton/proton ratio from our calculations in comparison with new data from AMS-02 and PAMELA , .It is then interesting to calculate the antiproton/proton ratio in light of new data released by AMS-02 [3].",
"Calculations were performed in [6] and subsequent works [19].",
"In this work, the antiproton distribution at the shock $f_{0}(p)$ is calculated numerically, as done in [14], in order to drop the “inelasticity approximation” that links the antiproton momentum $p$ to the primary proton momentum $p_{p}$ through an assumedly constant factor $\\xi \\equiv \\langle p/p_{p}\\rangle $ .",
"It was noted that such an approximation leads to an overestimate of the high-energy antiproton production in SNRs [15].",
"Along with antiproton production from p-He collisions with hydrogen and helium gas, we also account for tertiary reactions (such as $\\bar{p}+p\\rightarrow \\bar{p}^{\\prime }+X$ ) and for destruction processes.",
"All these processes are implemented in both acceleration and propagation.",
"The corresponding cross-sections are taken from [11].",
"The resulting $\\bar{p}$ /$p$ ratio is shown in Fig.",
"REF .",
"It can be seen that Scenario #2 is preferred by the AMS-02 data.",
"Reducing nuclear uncertainties in antiproton production is clearly essential for a complete discrimination between the different models [11]."
],
[
"Experimental Challenges",
"Given the implications of these new data on the phenomenology of CR propagation, we believe that the situation deserves more clarification on the experimental side.",
"The SOKOL analysis relies on unconventional techniques of deuteron/proton mass separation, which is always a very challenging task.",
"For instance the d/He measurement might be overestimated due to undetected background arising, e.g., from the mass distribution tails of CR protons or from $^{4}$He nuclei fragmenting in the top of the instrument.",
"On the other hand, it is very unlikely for such a background to affect the results by two orders of magnitude.",
"The deuteron spectrum and the d/He ratio are being precisely measured by the AMS-02 experiment at $E\\sim $ 0.1-10 GeV/n with standard spectrometric techniques.",
"In addition, AMS-02 is also equipped with a Transition Radiation Detector (TRD), designed for lepton/hadron mass separation, which can provide direct measurements of the Lorentz factor $\\gamma $ at TeV/n energies [22].",
"In standard magnetic spectrometers, the CR mass is derived from velocity and momentum measurements, $M\\propto \\,p/(\\gamma \\beta c) = \\frac{p}{\\beta c}\\sqrt{1-\\beta ^{2}}$ , so that its corresponding resolution $\\delta M/M$ is given by $\\left(\\frac{ \\delta M}{M}\\right)^{2} = \\left( \\frac{\\delta p}{p} \\right)^{2} + \\gamma ^{4}\\left( \\frac{\\delta \\beta }{\\beta } \\right)^{2},$ showing that the mass resolution degradates rapidly, at relativistic energies, due to the $\\gamma ^{4}$ -factor.",
"In contrast, with the opportunity of performing direct TRD-based $\\gamma $ -measurements, AMS-02 may have the capability to detect CR deuterons at the $\\mathcal {O}({\\rm TeV})$ energy scale."
],
[
"Conclusions",
"This work is aimed at interpreting new data, registered by the SOKOL experiment in space, that revealed a surprisingly high abundance of CR deuterons in the TeV/n region.",
"In contrast to antiparticle excesses that can be explained, e.g., by pulsar models or dark matter annihilation [25], [12], the SOKOL data demand an enhanced high-energy production of CR deuterons from hadronic interactions.",
"We found that no explanation for this measurement can be provided in terms of standard collisions of CRs with the gas of the ISM.",
"As we have shown, a viable solution for this puzzle is the occurrence of nuclear fragmentation inside SNRs, but this mechanism conflicts with the new AMS-02 data on the B/C ratio.",
"Thus, if the SOKOL measurement is taken as face value, we conclude that the sources accelerating helium and protons (thereby producing deuteron) may not be the same as those accelerating N-O nuclei (otherwise producing Li-Be-B nuclei), and that the former are more efficient in the production and acceleration of secondary particles.",
"Under such a scenario, the connection between B/C ratio and antiparticle/particle ratios would also be broken, and thus the B/C ratio should not be used to place constraints on the astrophysical antimatter background.",
"On the other hand, the d/He ratio would still represent a direct diagnostic tool for assessing this background.",
"We are grateful to Tanguy Pierog, Colin Baus, and Ralf Ulrich for the CRMC interface to MC generators.",
"J.F.",
"acknowledges support from China Scholarship Council and the Taiwanese Ministry of Science and Technology under grants No.",
"104-2112-M-001-024 and No.",
"105-2112-M-001-003.",
"NT acknowledges support from MAtISSE - Multichannel Investigation of Solar Modulation Effects in Galactic Cosmic Rays.",
"This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 707543."
]
] | 1612.05651 |
[
[
"Using Collisions of AGN Outflows with ICM Shocks as Dynamical Probes"
],
[
"Abstract In this paper we lay out a simple set of relationships connecting the dynamics of fast plasma jets to the dynamical state of their ambient media.",
"The objective is to provide a tool kit that can be used to connect the morphologies of radio AGNs in galaxy clusters to the dynamical state of the local ICM.",
"The formalism is intended to apply to jets whether they are relativistic or non-relativistic.",
"Special attention is paid to interactions involving ICM shocks, although the results can be applied more broadly.",
"Our formalism emphasizes the importance of the relative Mach number of the impacting ICM flow and the internal Mach number of the AGN jet in determining how the AGN outflows evolve."
],
[
"Introduction",
"Clusters of galaxies are the last and most massive class of bound structures to form out of the Big Bang ($z $ <$$$\\sim $ 1, R Mpc, M1014-1015M$).",
"Cluster assembly combines irregular accretionfrom surrounding diffuse matter, especially from filaments, and violent mergers with other clusters.",
"As in the universeat large, most cluster matter is non-baryonic, ``dark matter^{\\prime \\prime }.",
"The baryonicmatter within clusters is predominantly hot, diffuse plasma ($ kT $>$$\\sim $ keV, ne $<$$\\sim $ 10-2cm-3$); the intracluster medium (ICM).",
"The ICMs are shocked andstirred throughout their formation \\cite {krav12,vazz16,zing16}.",
"The resulting ICM flow structures provide vitalinformation about the large scale dynamics of cluster formation, as well as about thephysics of the ICM plasma\\cite {schek06}.",
"Cluster-scale shocks, in particular, are tell-taleconsequences ofmerger events \\cite {mark07}.",
"Tracing and deciphering these ICM flow structures are essentialto our understanding of cluster formation.$ While thermal X-rays have successfully revealed some ICM shocks and other flow features [5], [6], [7], distinct X-ray signatures are often subtle and hard to isolate and measure cleanly.",
"Observations of the Sunyaev-Zeldovich effect add some statistical information about the ICM pressure and velocity distributions[8].",
"But, a full, clear picture requires additional, complementary tools.",
"Thankfully, non-thermal synchrotron emissions from $>$ GeV electrons (CRe) in the ICM and in embedded objects may help reach this goal.",
"So far, the principal non-X-ray tool in efforts to characterize ICM dynamics has been cluster-scale, diffuse radio emissions (radio “halos”, “mini-halos” and “relics”) by electrons within the ICM that are energized or re-energized by cluster shocks and/or turbulence [9].",
"There are multiple plausible origins for those diffuse CRe, which in various models may be recent or ancient [10], [11].",
"Their links to local, current ICM dynamics result from the fact that they have rather short lifetimes against synchrotron losses and inverse Compton scattering of CMB photons ($<$ 100 Myr).",
"Observed CRe must be “energetically younger” than this.",
"Since ICM flow velocities are characteristically $\\sim 1$ kpc/Myr, such embedded CRe must have been energized relatively close by on cluster scales.",
"Although these emissions place essential constraints on ICM physics, their translation into distinct and firm ICM dynamical properties has so far proven to be quite challenging [11], [12].",
"At the same time, radio-bright, bipolar plasma outflows from active galactic nuclei (radio AGN, henceforth, simply AGN) are common in clusters [13], [14].",
"Interactions between those outflows and ICM dynamical structures on scales of 10s to 100s of kpc offer an additional and potentially very useful diagnostic probe of the ICM dynamics and physics.",
"That is quite separate from the increasing evidence that AGN-ICM feedback loops on those scales in the cores of relatively undisturbed clusters are central to the properties of those ICMs [15].",
"The focus of our discussion here, instead, is the consequences of AGN outflow encounters with cluster scale ICM flow structures (especially shocks), and, in particular, impact on the consequent dynamical form of the AGN structures that develop as the AGN plasma penetrates and generate cavities in the ICM.",
"Specifically, we explore major distortions to the AGN flows and the potential for using such distortions to identify and characterize ICM flow structures, including shocks.",
"In subsequent publications we will present analyses of high resolution 3D MHD simulations of AGN outflows in related dynamical ICM situations.",
"There we will examine details such as coherence and disruption of the AGN flows, development of turbulence, magnetic field evolution, as well as relativistic electron propagation, acceleration and emissions.",
"Our purpose now is more restricted; namely it is to outline a simple, but coherent formalism that we have found to be useful in predicting behaviors of the above simulation experiments and that hopefully captures essential physics needed to interpret observed AGN interactions with dynamical ICMs.",
"We note, in the context of the HEDLA workshop that inspired this work, that many of the dynamical components laid out below should be testable in laboratory experiments.",
"The best known and most widely discussed AGN distortion produced by asymmetric ICM interactions are those that have formed tails as a result of relative motions between the AGN host galaxy and the surrounding ICM.",
"So-called “Narrow Angle Tailed” (NAT) AGNs, in which the originally collimated AGN outflows appear deflected into a pair of quasi-parallel trails in the wake of the galaxy are especially common in disturbed clusters, where relatively high velocity ICM motions are likely [16].",
"If there are apparent, but less dramatic deflections, the AGNs are typically called “Wide Angle Tailed” (WAT) AGNs [17].",
"Both are obvious candidates as probes of ICM dynamics [20], [18], [19], [21].",
"The idea that ICM shocks encountering plasma bubbles or cavities left behind by expired AGN might rejuvenate their CRe populations has also been widely discussed [22], [23].",
"To the best of our knowledge, however, expected behaviors of active AGN outflows colliding with ICM shocks has not been previously outlined, although there have been several recent suggestions that observed NAT AGN may, in fact, be associated with cluster radio relics [24].",
"Here we do not address that issue directly, but do outline the essentials of dynamical encounters of that kind, in order to facilitate future discussions.",
"Basics of the formalism we outline have been in the literature for some time.",
"Our objective is to set them down as a package, extending them as needed for application to the issues at hand, while looking for parameterizations that may be useful for diagnostic purposes.",
"We specifically target AGN outflow/shock encounters, although the necessary formalism has broader applications.",
"We also include along the way some preliminary results from simulations in order to illustrate the kinds of behaviors that result.",
"We ignore here a number of complications that will ultimately be important to detailed models, but that do not seem essential to a “first order” picture.",
"Although our goal in this paper is to set out a simple set of analytic relationships, we do show illustrations of behaviors from high resolution numerical simulations that will be discussed in detail in subsequent publications.",
"Those simulations were carried out using the WOMBAT MHD code employing a second order MHDTVD Riemann solver [25].",
"An adiabatic equation of state with $\\gamma = 5/3$ was employed.",
"Steady bi-polar jets were formed within a small cylindrical volume, where the jet pressure, $P_j$ , was kept approximately equal to the ambient pressure, and the jet plasma was accelerated to a velocity, $v_j$ .",
"All the simulations shown in this paper kept the density within the jet-source cylinder, $\\rho _j$ , at 1% of the undisturbed ambient density.",
"We have, however, varied these parameters to confirm relations described below.",
"We also allowed jet power to cycle off in some cases.",
"The simulated jets were weakly magnetized with a toroidal geometry and nominal $\\beta = P_g/P_m = 75$ within the jet-source cylinder.",
"Magnetic stresses are too small to influence dynamics at the levels relevant to this paper.",
"The simulations did include CRe; in follow-up discussions we will discuss in detail synchrotron emission properties.",
"That is beyond the scope of the present paper, however.",
"The paper outline is this.",
"Section reviews the interactions between a plane shock and a low density cavity.",
"In section we outline in simple terms the propagation of a jet into a head or tail wind (section REF ) and into a cross wind (section REF ) that would, for instance, represent post-shock ambient conditions.",
"For completeness section sets out a simple model to estimate the form of the plasma cocoon inflated by jets nominally prior to the above interactions."
],
[
"An ICM Shock Colliding with a Cavity",
"Typically the initial encounter between an ICM shock and an AGN will involve the shock penetrating the AGN cocoon/cavity inflated by the AGN outflow.",
"The basic dynamics of such an encounter are qualitatively simple: the core of the cavity is crushed relatively quickly, while the perimeters of the cavity roll into a vortex ring structure [22].",
"Here we simply outline of the physics.",
"Consider then a planar ICM shock of Mach number, $\\mathcal {M}_i = v_{si}/a_i$ , where $v_{si}$ is the speed of the shock in the ICM and $a_i = \\sqrt{\\gamma _i P_i/\\rho _i}$ is the ICM sound speed.",
"$P_i$ and $\\rho _i$ are the pre-shock ICM pressure and mass density, while $\\gamma _i$ ($=5/3$ ) is the ICM adiabatic index.",
"Typical ICM shocks will have modest Mach numbers, $\\mathcal {M}_i $ <$$$\\sim $ 4$, \\cite {rkhj03} but toallow a wider application of this discussion we do not restrict $ Mi$.We then assume the shock encounters a low density cavity, which we takeinitially to be in pressureequilibrium with the ICM ($ Pc = Pi$).",
"Let the undisturbed density inside the cavity be $ c = i$, with $ 1$.The initial cavity sound speed is, $ ac = c/i ai/ ai/ ai$, where, acknowledging thatthe cavity includes relativistic plasma, we allow the cavitypotentially to have an adiabatic index, $ c i$, distinct from the ICM.$ As the shock encounters the cavity, it penetrates at a speed $v_{sc} = \\mathcal {M}_c a_c \\ge a_c$ pulling ICM gas with it.",
"Pressure balance between the original cavity and un-shocked ICM leads to the condition, $v_{sc} \\ge v_{si}/(\\delta ^{1/2} \\mathcal {M}_i)$ .",
"The original ICM-cavity contact discontinuity (CD) follows the shock at a speed, $v_{cD}$ , intermediate between the ICM shock and the cavity shock.",
"At the same time a rarefaction propagates back into the post-shock ICM at the post-shock ICM sound speed.",
"Pfrommer and Jones [23] presented the full, nonlinear, 1D Riemann solution to this problem.",
"Here we need only a few, approximate behaviors in order to understand what happens during the encounter.",
"Generally, even though the shock speed is greater inside the cavity, the shock strength is less than the incident shock; that is, $\\mathcal {M}_c < \\mathcal {M}_i$ .",
"There is no general, analytic formula for $\\mathcal {M}_c$ , although it can be found numerically from the Riemann solution.",
"In the strong shock limit, when $\\mathcal {M}_c \\gg 1$ , Pfrommer and Jones found $v_{sc} \\sim 2~v_{si}$ ; that is $\\mathcal {M}_c \\sim 2~ \\delta ^{1/2} \\mathcal {M}_i$ .",
"In the weak shock limit (as $\\mathcal {M}_c \\rightarrow 1$ ), of course, $v_{sc} \\sim v_{si} \\delta ^{-1/2}$ .",
"In either regime the internal shock speed considerably exceeds the incident, ICM shock speed whenever $\\delta \\ll 1$ .",
"Consequently, the shock passes through the cavity much more quickly than it propagates around it.",
"The velocity of the CD within the cavity, $v_{CD}$ , obtained in the Riemann solution is, $v_{CD} = \\frac{2}{\\gamma _c+1}\\frac{(\\mathcal {M}_c^2 - 1)}{\\mathcal {M}_c^2}v_{sc}.$ This measures the cavity collapse rate.",
"It will generally be less than $v_{sc}$ .",
"However, Pfrommer and Jones found, so long as $\\mathcal {M}_i $ >$$$\\sim $ 2$ and $ 1$, that$ vCD > vsi$.",
"Under those circumstances, the cavity is crushed (the initial CD pushes all the way through the cavity) {\\it before} theICM shock propagates around it.$ We note that if the cavity has previously developed a thick boundary layer within which $\\delta $ is not very small (e.g., through turbulent mixing; see section ), the internal shock speed within this layer will be slower than above, but $\\mathcal {M}_c$ can remain comparable to $\\mathcal {M}_i$ through that boundary layer.",
"This can, for instance, significantly influence such things as particle acceleration during shock passage and shock amplification of turbulence.",
"If the initial cavity is “round” (e.g., a sphere), the shock intrusion into the cavity begins sooner and is usually stronger at the normal, “point of first contact.” On the other hand, the oblique penetration towards the extremes of the cavity boundary also generates vorticity.",
"An initially spherical cavity will, thus, evolve into a torus, or vortex ring that mixes ICM and cavity plasma [22], [23].",
"Analogously, even if the cavity boundary is planar, but the shock normal is oblique to the cavity boundary, refraction of the shock and penetrating CD will produce vorticity and mixing [28].",
"Examples of shocks impacting ICM cavities have been presented previously [22], [23].",
"Outcomes are also evident in Figures REF and REF ."
],
[
"Jet Propagation in a (Post-shock) Wind",
"If an AGN jet remains active following shock impact, so that it drives into the post-shock ICM, its propagation will be modified because the post-shock ICM is denser than the pre-shock ICM and also because the post-shock ICM plasma is put in motion by the shock.",
"In this section we outline a toy model to address these effects.",
"The basic geometry is illustrated in Figure REF .",
"Figure: Cartoon illustrating the basic geometry of bipolar AGN jetspropagating into an ambient wind.We approximate the dynamics as that of a jet within an ambient wind, decomposing the wind velocity into components parallel (head wind or tail wind) and orthogonal (cross wind) to the jet velocity vector, $\\vec{v_j} = v_j\\hat{v_j}$ , near its terminus.",
"That is, we set $\\vec{v}_w = v_{w,\\parallel }\\hat{v}_j + v_{w,\\perp } \\hat{v}_\\perp ,$ where $\\hat{v}_j\\cdot \\hat{v}_\\perp = 0$ .",
"When $v_{w,\\parallel } \\ne 0$ ; that is, whenever there is a tail wind, $v_{w,\\parallel }> 0$ , or a head wind, $v_{w,\\parallel } < 0$ , the rate of advance of the jet terminus will be modified; when $v_{w,\\perp } \\ne 0$ the jet propagation will be deflected laterally.",
"Although we present the model in the context of a post-shock flow, the formalism applies to any relative motion between the AGN and its ambient medium.",
"Indeed the model is essentially a merger and modest extension of classical cartoons of AGN jet propagation [26], [20].",
"Simply stated, jet advance and deflection are influenced by parallel and transverse ram pressures created by the wind."
],
[
"Advance of a Jet Terminus in an Ambient Head or Tail Wind",
"We begin by considering the influence of an ambient wind motion aligned with the jet velocity; that is, consequences of $v_{w,\\parallel } \\ne 0$ .",
"To keep the discussion simple we consider here only steady, collimated jets (zero opening angle) and homogeneous ambient media.",
"We adopt the standard picture that the advance of the jet terminus into the ICM can be expressed in terms of the propagation of a contact discontinuity that forms at the head of the jet [26].",
"Relative to the AGN, that boundary (the jet “head”) propagates at velocity, $v_h\\hat{v}_j = d \\ell _j/ dt~\\hat{v}_j$ , where $\\ell _j$ represents the length of the jet from the AGN.",
"There will be a bow shock propagating into the wind, ahead of the head, as well as a reverse, “terminal” shock in the jet.",
"The simple, 1D cartoon model ignores these details, drawing a box around all this and assumes the jet thrust; that is, the total jet momentum flux coming in through an area, $A_j = \\pi r_j^2$ , is balanced by the total wind momentum flux coming in on the opposite side of the box, again, through an area, $A_j$ .",
"In reality, because both jet and wind plasma will, as “back flow”, exit the face of the box through which the jet enters (and box sides), the effective sizes of the box faces need to be larger than $A_j$ .",
"We can crudely account for this asymmetry by setting the effective area allocated to the wind, $A_h$ , to be larger than the nominal cross section of the jet.",
"We comment, as well, that even for a collimated jet the radius, $r_j$ , will not generally be a constant along the length of the jet, especially when it becomes over or under pressured with respect to its surroundings.",
"In that case, for instance, the propagating jet plasma will execute a sequence of expansions and rarefactions around an equilibrium radius.",
"Consequently, the jet velocity, $v_j$ , density, $\\rho _j$ , and pressure, $P_j$ , will all vary along a real jet.",
"Our analysis here does not try to account for dynamics at that level of detail, so we will assume below that the jet has some suitably chosen characteristic radius, $r_j$ , density, $\\rho _j$ and pressure, $P_j$ .",
"We allow both the jet velocity and equation of state (EoS) to be either non-relativistic or relativistic, although we will assume the propagation of the wind and the head through the wind in the AGN reference frame are non-relativistic.",
"Then, the jet 4-velocity is $U_j = \\Gamma _j v_j$ , with $\\Gamma _j$ the Lorentz factor of the jet velocity, $v_j$ , while the enthalpy density of the jet plasma in the jet plasma frame is $w_j c^2 = e_j + P_j = \\rho _j c^2 + e_j^{\\prime } + P_j = \\rho _j c^2 \\tilde{w}$ , with $e_j$ the internal energy density of the jet plasma, including rest mass energy, $\\rho _j c^2$ .",
"The jet momentum flux density in the AGN frame can be written [29] $T_{mj} = w_j U_j^2 + P_j = \\rho _j \\tilde{w} U_j^2 + P_j.$ The jet thrust is $N_j = T_{mj} A_j = (w_j U_j^2 + P_j) A_j = \\rho _j U_j^2 ( \\tilde{w}_j+ \\frac{P_j}{\\rho _j U_j^2} ) A_j.$ For a non-relativistic EoS, where $e \\approx \\rho c^2$ , $\\tilde{w} = (e + P)/(\\rho c^2) \\approx 1 + (e^{\\prime }+P)/(\\rho c^2) = 1 + a^2/((\\gamma - 1) c^2) \\approx 1$ , while for a relativistic EoS, where $P \\approx (1/3) e$ , $\\tilde{w} \\sim 4 P/(\\rho c^2) \\gg 1$ .",
"It will be useful later on also to have expressions for the jet energy flux density in the AGN frame.",
"Subtracting off the rest mass energy this is $T_{ej} = U_j \\Gamma _j (w c^2 - \\rho _j c^2/\\Gamma _j)\\nonumber \\\\= U_j \\Gamma _j \\rho _j c^2 (\\tilde{w} - 1/\\Gamma _j)\\nonumber \\\\= \\Gamma _j^2 \\rho _j c^2 v_j \\left[(1 - 1/\\Gamma _j) + (e_j^{\\prime } + P_j)/(\\rho _j c^2)\\right],$ so that the “luminosity” of the jet is $L_j = T_{ej} A_j = v_j \\Gamma _j^2 \\rho _j c^2 (\\tilde{w} - 1/\\Gamma _j) A_j.$ We define the internal jet Mach number as, $\\mathcal {M}_j = \\Gamma _j v_j /(\\Gamma _{s,j} a_j)$ ,[30] where $a_j = c \\sqrt{\\partial P_j/\\partial e_j}$ is the jet sound speed, with $\\Gamma _{s,j}^2 = 1/(1 - (a_j/c)^2)$ .",
"For a relativistic EoS $a^2 = (1/3) c^2$ , while $\\Gamma _s^2 = 3/2$ , so $\\Gamma _s^2 a^2 = (1/2) c^2$ .",
"Note in this limit that the jet Mach number, $\\mathcal {M}_j \\rightarrow \\sqrt{2}~\\Gamma _j (v_j/c)$ , which, if $v_j \\rightarrow c$ gives $\\mathcal {M}_j = \\sqrt{2}~\\Gamma _j$ .",
"This makes clear that the fact that a jet velocity is close to the speed of light does not, by itself, imply a high Mach number for the jet dynamics.",
"A non-relativistic EoS with $P = (\\gamma - 1) e^{\\prime }$ recovers the familiar $a^2 = \\gamma P/\\rho $ .",
"With these definitions the momentum flux density is $T_{mj} = \\rho _j \\tilde{w} \\Gamma _j^2 v_j^2 + P_j = \\mathcal {M}_j^2 P_j \\left[\\frac{a_j^2}{c^2 - a_j^2} \\tilde{w}\\frac{\\rho _j c^2}{P_j} + \\frac{1}{\\mathcal {M}_j^2}\\right].$ Setting $\\gamma ^{\\prime } = (\\Gamma _{s,j}^2 a_j^2 w_j)/P_j$ , this becomes simply $T_{mj} = \\gamma ^{\\prime } \\mathcal {M}_j^2 P_j \\left[1 + \\frac{1}{\\gamma ^{\\prime } \\mathcal {M}_j^2}\\right].$ For a relativistic EoS $\\gamma ^{\\prime } = 2$ , whereas for a non-relativistic EoS, $\\gamma ^{\\prime } = \\gamma $ , the usual gas adiabatic index.",
"Written in this form the expression for the jet momentum flux is virtually the same for relativistic or non-relativistic jets.",
"If, as is often the case, $\\mathcal {M}_j^2 \\gg 1$ we can neglect the second term in square brackets.",
"Then, the jet thrust for either relativistic or non-relativistic flow is $N_j = T_{mj} A_j \\approx \\gamma ^{\\prime } \\mathcal {M}_j^2 P_j A_j.$ In either regime the thrust depends simply on the internal jet pressure and the internal jet Mach number.",
"The jet luminosity is not quite so tidy, but still simple to express.",
"For a non-relativistic EoS the luminosity would be $L_j \\approx \\mathcal {M}_j \\Gamma _j \\rho _j c^2~(\\Gamma _{s,j}a_j)~\\left[1 + \\frac{a_j^2}{(\\gamma _j - 1) c^2} - \\frac{1}{\\Gamma _j}\\right] A_j,$ which, with $v_j \\ll c$ , but $v_j/a_j = \\mathcal {M}_j \\gg 1$ and $\\gamma _j = 5/3$ takes the simple familiar form, $L_j \\approx \\frac{1}{2} \\rho _j v_j^3 A_j = \\frac{5}{6} \\mathcal {M}_j^2 v_j P_j A_j.$ At the other extreme, with a relativistic EoS, $\\tilde{w}_j = 4 P_j/(\\rho _j c^2)$ , and $\\Gamma _j \\gg 1$ , the luminosity is also simple to express; namely, $L_j \\approx 2 \\mathcal {M}_j^2 c P_j A_j,$ since, now $2 \\Gamma _j^2 \\approx \\mathcal {M}_j^2$ .",
"Again, in this form the relativistic and non-relativistic expressions are almost the same.",
"The jet luminosity can, like the thrust, be described in terms of the jet pressure and Mach number, but now also in terms of a jet speed, $v_j$ , (which may $\\rightarrow c$ ).",
"We return now to estimating the rate at which the jet terminus propagates through its surrounding plasma.",
"The momentum flux balance condition determining the propagation velocity, $\\vec{v}_h$ , is actually measured in the frame of the head, so we should transform the momentum flux relations to that frame.",
"However, provided $v_h/v_j \\ll (c/v_j)^2/\\Gamma _j^2$ , it is easy to show that the fractional change in $T_{mj}$ is small.",
"We will neglect that correction below.",
"The momentum balance condition becomes, $N_j = A_j T_{mj} = A_h [\\rho _w (v_h - v_{w,\\parallel })^2 + P_w ].$ Using equation REF this leads to $\\frac{(v_h - v_{w,\\parallel })^2}{a_w^2}\\left[ 1 + \\frac{a_w^2}{\\gamma _w (v_h \\mp v_w)^2}\\right] \\approx \\mathcal {M}_j^2 \\frac{\\gamma ^{\\prime }}{\\gamma _w}\\frac{A_j P_j}{A_h P_w} $ or, $(\\mathcal {M}_h \\mp \\mathcal {M}_{w,\\parallel })^2\\left[ 1 + \\frac{1}{\\gamma _w (\\mathcal {M}_h \\mp \\mathcal {M}_w)^2}\\right] \\approx \\mathcal {M}_j^2 \\frac{\\gamma ^{\\prime }}{\\gamma _w}\\frac{A_j}{A_h}\\frac{P_j}{P_w}$ where $\\mathcal {M}_h = v_h/a_w$ and $\\mathcal {M}_{w,\\parallel } = |v_{w,\\parallel }|$ are the Mach numbers of the jet head advance and the head/tail wind with respect to the AGN.",
"Then, of course, $a_w = \\sqrt{\\gamma _w P_w/\\rho _w}$ is the wind sound speed.",
"In equations REF and REF the upper (lower) sign corresponds to a tail (head) wind.",
"If $(\\mathcal {M}_h \\mp \\mathcal {M}_{w,\\parallel })^2 \\gg 1$ we have the simple result $|\\mathcal {M}_h \\mp \\mathcal {M}_{w,\\parallel }| \\approx \\mathcal {M}_j \\sqrt{\\frac{A_j}{A_h}}\\sqrt{\\frac{P_j}{P_w}}\\sqrt{\\frac{\\gamma ^{\\prime }}{\\gamma _w}}.$ That is, the Mach number of the advance of the head relative to its ambient medium is similar to the internal Mach number of the jet modified by a factor that depends mostly on the ratio of the integrated jet pressure, $ P_j A_j$ , to the integrated wind pressure (isotropic pressure, not ram pressure) “across the head”, $P_w A_h$ .",
"In our simulation experiments with steady, fixed axis, non-relativistic jets $A_j/A_h \\sim 1/2$ provides reasonable matches between simulated jet propagation and equations REF , REF and REF .",
"Some insights into appropriate $P_j/P_w$ are useful going forward.",
"Our discussions relate to jets that are pressure confined as they propagate.",
"So, we expect $P_j \\sim P_{a}$ , where $P_{a}$ represents the pressure of whatever medium is immediately surrounding the jet.",
"If a jet enters a region where it is out of pressure balance, it will generally expand or converge to compensate.",
"As noted above, the pressures within simulated jets are actually quite non-uniform as a result.",
"On average, however, we find, independent of the pressure assignd to a simulated jet at its origin, the average pressure along the propagating jet becomes roughly comparable to the ambient pressure.",
"We also find in simulations and argue in section that the pressure inside jet cocoons are commonly roughly similar to those in the undisturbed surroundings, even though the cocoon formation drives a shock into its surroundings.",
"Consequently, we expect within a factor of a few that $P_j/P_w \\sim 1$ .",
"Nonetheless, we leave such ratios as undetermined in our analyses, in order to reveal their roles.",
"We note also for non-relativistic jets with non-relativistic EoS that, since for both the jet and the ambient medium, $a^2 = \\gamma P/\\rho $ , equation REF can be written $|v_h \\mp |v_{w,\\parallel }|| \\approx v_j \\frac{\\sqrt{A_j/A_h}}{\\sqrt{\\rho _w/\\rho _j}},$ independent of $P_j/P_w$ .",
"This recovers the commonly applied assumption that, modulo “an efficiency factor” ($\\sqrt{A_j/A_h}$ ) the advance speed of the jet head is roughly the jet speed reduced by a factor of the square root of the density ratio between the ambient medium and the jet when the jet and its advance are both highly supersonic [26].",
"It is further evident from these relations, as we would expect intuitively, that the advance speed of the jet with respect to the AGN, $v_h$ , is greater when it propagates downstream with a wind (a “tail wind”) than if it propagates upstream into a wind (a “head wind”).",
"In fact a jet propagating into a sufficiently strong head wind can be stopped, or even reversed.",
"Using equation REF the approximate condition for the head wind to stop forward progress of the head is $|\\mathcal {M}_{w,\\parallel }| >$ $\\sim $ Mj Aj Ah' Pjw Pw.",
"We have verified this condition in our simulations (see Figure REF ).",
"Although the above results would be applicable for any AGN relative motion through its ambient medium when the AGN jets are aligned with the relative motion, we introduced the issue in the context of post-shock ICM flows.",
"In that case it is useful to express these relations in terms of wind properties resulting from ICM shocks.",
"To keep it simple, we assume that the pre-shock ICM plasma was at rest with respect to the AGN, although that is easily modified.",
"Again expressing the ICM shock Mach number as $\\mathcal {M}_i$ we have (assuming $\\gamma _w = \\gamma _i = 5/3$ ), $\\rho _w = \\frac{4\\mathcal {M}_i^2}{\\mathcal {M}_i^2+3}\\rho _i,\\\\P_w = \\frac{5\\mathcal {M}_i^2-1}{4}P_i,\\\\v_w = \\frac{3}{4}\\frac{\\mathcal {M}_i^2-1}{\\mathcal {M}_i}a_{i},\\\\a_w^2 = \\frac{(\\mathcal {M}_i^2 + 3)(5 \\mathcal {M}_i^2 - 1)}{16 \\mathcal {M}_i^2} a_i^2.$ where, as above, the subscript index `i' identifies the unshocked ICM, and `w' indicates post-shock ICM (the “wind”).",
"Applying equations REF - to equation REF we obtain an approximate relation for the strength, $\\mathcal {M}_{is}$ , of an ICM shock that can stop the advance of an approaching jet of internal Mach number, $\\mathcal {M}_j$ in a “head on collision” when the AGN is at rest in the undisturbed ICM; namely $\\mathcal {M}_i =\\mathcal {M}_{i,s} ~>$ $\\sim $ 23 3'5 Mj PjPiAjAi1G(Mi), where $G(\\mathcal {M}_i) = (1 - 1/\\mathcal {M}_i^2)/\\sqrt{1 + 3/\\mathcal {M}_i^2}) \\le 1$ .",
"As noted above, we expect for steady jets with fixed axes, $A_j/A_i \\sim 1/2$ .",
"The function $G(\\mathcal {M}_i) \\sim 0.6 - 0.9$ , for 2<$$$\\sim $ Mi $<$$\\sim $ 4$,corresponding to expected ``merger-related^{\\prime \\prime } ICM shock strengths.As a rough rule of thumb, then,an AGN jet running head on into a shock will be stopped or reversed if theMach number of the shock is comparable to or even a bit less than the Mach number of the jet.$ The above behaviors are illustrated in Figure REF for two, simulated non-relativistic shock-jet interactions with different relative shock strengths.",
"The views are 3D volume renderings of jet mass fraction; that is, only plasma that originated from the AGN is shown.",
"In both cases, $\\mathcal {M}_j = 3.5$ , and the shock normal was aligned with the (bi-polar) jet axis.",
"ICM shock propagation was left-to-right, although in this view the shock plane was rotated 40 degrees from the line of sight, in order to reveal the structures more clearly.",
"The location of the AGN is marked by an X in each case.",
"The color map of the mass fraction tracer runs from “white” (100%) through yellow, red, green and blue ($\\sim $ 30%).",
"The figure upper panel represents the outcome for an ICM shock with $\\mathcal {M}_i = 2 = 0.57 \\mathcal {M}_j$ , whereas the lower panel involves a $\\mathcal {M}_i = 4 = 1.1 \\mathcal {M}_j$ shock.",
"Both are shown at approximately the same time interval since initial contact between the shock and the left-facing (upwind) jet.",
"In the bottom panel the stronger shock has left the computational domain to the right after reversing the left-facing jet and crushing the two original jet plasma cocoons and stripping them from the jets.",
"The “smoke ring” to the right is the resulting vortex ring structure, whose formation out of the pre-shock cavities was outlined in the previous section.",
"The flaring seen at the right end of the remaining (downwind) jet represents the head of that jet.",
"The shocked jet plasma is unable to propagate back to the AGN through the strong right-facing wind behind this shock.",
"It is not able to refill a cocoon (see section ).",
"The jet remains “naked”.",
"Several AGNs in disturbed clusters have been seen that are candidates for this interaction.",
"Probably the best known example is the radio source `C' in A2256, [31], [32] which has a roughly 1 Mpc, very thin (unresolved) “tail” extending to the west of an AGN, but nothing evident to the east.",
"Indeed, that AGN is seen projected near a strong “radio relic“ in the cluster, suggesting that it could have passed through a moderately strong cluster merger shock.",
"The weaker and slower ICM shock involved in the upper Figure REF panel dynamics is still in the volume illustrated, although not directly visible in this rendering.",
"In the undisturbed ICM its position along the jet axis is about 2/3 of the distance from the AGN X to the right end of the right-facing jet head.",
"Inside the right-side cocoon, the shock has just reached the head at this time.",
"We found experimentally in this case that an incident shock with $\\mathcal {M}_i \\approx (3/4) \\mathcal {M}_j$ would just stop the approaching (left facing in the figure) jet.",
"Figure: Illustration from simulations of ICM shocks colliding head on with ℳ j =3.5\\mathcal {M}_j = 3.5AGN jet pairs.",
"Both panels are volume renderings of a passive jet mass-fraction tracer.",
"Shockpropagation was left to right.",
"'X' marks AGN location.",
"The jet axes are rotated 40 degrees out of the plane of the sky with the left jets approaching the observer.",
"Top: ℳi=2\\mathcal {M}i = 2.",
"Bottom: ℳ i =4\\mathcal {M}_i = 4.",
"See text for details.Thus, evidence for relative AGN foreshortening to one side has the potential to find and even measure the strengths of ICM shocks or winds."
],
[
"Deflection of a Jet by a Cross Wind",
"In the presence of a cross wind, $v_{w,\\perp } \\ne 0$ (see Figure REF ), the jet is subjected to an unbalanced transverse ram pressure force, $\\rho _w v_{w,\\perp }^2$ .",
"Particularly, if that cross wind results from a crossing shock, the jet cocoon (cavity) will be crushed and stripped away from the propagating jet (see Figure REF ).",
"From that point on the jet interacts directly with the wind, and we can estimate the induced transverse pressure gradient within the jet as $\\partial P_j/\\partial x_{\\perp } \\approx \\rho _w v_{w,\\perp }^2/(2 r_j).$ Then the transverse acceleration of a steady jet is determined by the relation $\\frac{w \\Gamma _j^2 v_j}{c^2}\\frac{\\partial v_{j,\\perp }}{\\partial \\ell } \\approx \\frac{\\rho _w v_{w,\\perp }^2}{2 r_j}(1 - \\Gamma _j^2 v_{j,\\perp }^2/c^2),$ We can use $\\partial v_{j,\\perp }/\\partial \\ell \\sim v_j/\\ell _b$ as a way to estimate the length $\\ell _b$ over which the transverse ram pressure from the wind will deflect the jet by 90 degrees; that is, the “ jet bending length”.",
"Neglecting the (initially small) second term inside the parentheses in equation REF this gives, $\\ell _b \\approx 2 r_j \\frac{w \\Gamma _j^2 v_j^2}{\\rho _w v_{w,\\perp }^2} \\approx 2 r_j \\frac{T_{mj}}{\\rho _w v_{w,\\perp }^2},$ where the final expression has used equation REF .",
"Analogous to the results of the previous subsection we can also write this final form in terms of the jet Mach number, $\\mathcal {M}_j$ and the cross-wind Mach number, $\\mathcal {M}_{w,\\perp }$ , $\\ell _b \\approx 2 r_j \\frac{\\gamma ^{\\prime }}{\\gamma _w}\\frac{\\mathcal {M}_j^2}{\\mathcal {M}_{w,\\perp }^2}\\frac{P_j}{P_w}.$ If the jet and wind pressures are comparable, the ratio of the jet bending length, $\\ell _b$ , to the jet diameter, $2 r_j$ , is roughly $(\\mathcal {M}_j/\\mathcal {M}_w)^2$ .",
"A little bit of algebra shows that equation REF matches the bending radius of curvature derived in [20] for an AGN moving supersonically through an ambient medium at right angles to the jet axis.",
"Of course, only relative motion matters, and here we emphasize that it is not necessary that the cross wind is supersonic for the bending to develop.",
"The bending radius will, however, scale inversely with the square of the Mach number of relative motion.",
"So, once again, obvious distortion in the AGN points to relatively large Mach number of the relative motion between the AGN and its immediate surroundings.",
"Assuming the cross wind under discussion comes entirely from an ICM shock propagating transverse to the jet axis, we can use equations REF - to give a relation between the jet bending length, the jet radius, the jet Mach number and the ICM shock Mach number, $\\mathcal {M}_i$ , $\\ell _b \\approx \\frac{8}{9}\\frac{\\gamma ^{\\prime }}{\\gamma _i}\\frac{\\mathcal {M}_j^2 (\\mathcal {M}_i^2 + 3)}{(\\mathcal {M}_i^2 - 1)^2}\\frac{P_j}{P_i}~ r_j.$ By convention, when the jet length, $\\ell _j$ satisfies $\\ell _j~$ >$$$\\sim $ b$, but $ brj$ the resultingAGN morphology would be describe as a ``wide angle tail^{\\prime \\prime } radio galaxy, or a ``WAT\".",
"Asthe ratio $ b/rj$ becomes smaller, the morphological label would shift to ``narrow angle tail^{\\prime \\prime } or ``NAT^{\\prime \\prime }.$ Figure REF illustrates results from a simulation of a $\\mathcal {M}_i = 4$ shock that crossed from the left and collided at right angles with a pair of $\\mathcal {M}_j = 10$ jets oriented vertically in the image.",
"Equation REF predicts $\\ell _b/r_j = 6.7$ , which is actually very close to the empirical result of the simulation.",
"As in Figure REF this image shows a volume rendering of the passive jet mass fraction tracer.",
"The shock plane has again been rotated 40 degrees from the line of sight to make 3D structures more distinct.",
"We comment on the close resemblance between the structures visible in Figure REF and the “classic” NAT source, NGC1265 [33], whose morphology has long been modeled in terms of a strong cross wind in the rest frame of the host galaxy.",
"Here the wind is a feature of the post-shock environment.",
"That has also been suggested for NGC1265 [23].",
"Figure: Simulation of an ℳ i =4\\mathcal {M}_i = 4 shock that crossed fromthe left and collided at rightangles with an ℳ j =10\\mathcal {M}_j = 10AGN jet pair (vertical jet axis in the image).",
"Shown is a volume rendering of a passive jet mass-fraction tracerafter the shock has passed through the volume shown.",
"'X' marks AGN location.",
"View orientation is the same as Figure .",
"See text for details.Note that the two jets in Figure REF remain stable long after they are deflected by $~\\sim ~90$ degrees into “tails”, even though turbulent mixing regions develop around them and enlarge downstream.",
"The tails merge at the far right into a pair of merging vortex rings that formed along the lines outlined in section as the ICM shock crushed the two jet cocoons produced before shock impact.",
"The jets can be traced as coherent structures almost to the vortex rings.",
"The shock itself has left the observed volume to the right at the time shown.",
"Finally, we comment on the more complex case where an incident shock collides at an arbitrary angle with respect to the AGN jet axis, or more generally when the AGN jets encounter an oblique wind.",
"Since $v_{w,\\perp }$ is initially the same for both jets, the rate of jet deflection indicated by equation REF would, before deformation, be the same for both jets (see Figure REF ).",
"The aligned wind velocity, $v_{w,\\parallel }$ , has the same magnitude, but opposite sign on the two jets.",
"The downwind jet, where $v_{w,\\parallel } > 0$ , thus advances more rapidly.",
"As the jets begin to be deflected, however, the upwind jet, where $v_{w,\\parallel } < 0$ , is deflected to be more nearly transverse to the wind ($v_{w,\\perp }$ increases along the jet on that side), while the downwind jet is deflected to be more nearly aligned with the wind ($v_{w,\\parallel }$ increases along this jet).",
"Consequently, the upwind jet becomes more sharply bent than the downwind jet.",
"We have confirmed in simulations a regular transitioun along these lines between the aligned wind interactions described in section REF .",
"and cross wind interactions outlined at the beginning of this section.",
"Evidently, such asymmetries can provide information about both the relative Mach numbers of the jets and a wind and the relative orientations of the wind and the jets.",
"The shape evolution of the two jets depends distinctively on the ratio of two Mach numbers as well as the orientation between the AGN jet axis and the wind velocity vector, $\\vec{v}w$ .",
"Thus, the resulting shape provides a means to determine both $\\mathcal {M}_j/\\mathcal {M}_w$ and $\\hat{v}_w\\cdot \\hat{v}_j$ .",
"Also, it is obvious that if the “weather conditions” encountered by jets vary as they extend through the ICM, additional morphological features are likely to develop that can be used to identify and characterize these ICM structures."
],
[
"Formation of a Backflow Cocoon",
"We began this discussion with an outline of shock propagation through a pre-existing cavity formed by AGN outflows; that is a jet cocoon.",
"Then we outlined propagation of the jets themselves in the post-shock flow.",
"The shocks both crush the cocoon and may, if the wind “stripping” is faster than replacement from the jet terminus, remove the cocoon (see Figures REF and REF ).",
"We did not, however, address what might be learned from the cocoons themselves, In order to tie the pieces together, here we outline briefly some of the related physics connecting the jet propagation to the formation of the jet cocoon before shock impact.",
"Since our purpose is only to lay out a rough picture of cocoon inflation, we ignore in this section complications such as relative motions between the AGN and its ambient ICM, the presence of large scale pressure or density gradients, or ICM turbulence.",
"All of these will modify cocoon morphology and quantitative measures, although they should not fundamentally change the basic picture presented here.",
"In simple terms the cocoon represents the reservoir of plasma that has previously passed through the jet.",
"The cocoon plasma will generally be of substantially lower density than the surrounding ICM, with the two media nominally separated by a contact discontinuity.",
"Because it is also a “slip surface”, that boundary is likely to be unstable to Kelvin-Helmholtz instabilities (KHI), so that some degree of mixing will take place (unless, for instance, magnetic fields in or around the cocoon are strong enough and coherent enough to stabilize the contact discontinuity [34], [35]).",
"Figure shows volume renderings of cocoons formed from two simulated steady AGN jet pairs.",
"In each case the AGN (marked by an `X') expelled identical, but oppositely directed jets.",
"The rendered quantity is the same passive mass fraction tracer shown in Figures REF and REF .",
"In this case the jet axis is viewed in the plane of the sky.",
"The upper image corresponds to a $\\mathcal {M}_j = 3.5$ jet pair, while the lower image represents the cocoons of a $\\mathcal {M}_j = 10$ jet pair.",
"The structures are viewed when $\\ell _j \\approx 140-150 r_j$ .",
"The jets, themselves, are faintly recognizable inside the cocoons and along the cocoon axes.",
"The cocoon boundaries are clearly influenced by KHI; some mixing has occurred.",
"Indeed the small differences between the left and right cocoons come from detailed differences in the KHI on the two sides that are seeded by the mismatch between a mathematically circular jet and a Cartesian numerical grid.",
"KHI details depend on the exact placement of the AGN on the numerical grid.",
"The jet mass fraction dominating the images $$ >$$$\\sim $ 80%$.",
"Despite suchcomplications, there isa relatively clear cocoon boundary, so we assume below that the jet cocoon and the surrounding ICM are cleanly separated.\\begin{figure}\\includegraphics [width=0.48height=0.41]{figure4}\\caption { Jet ``backflow^{\\prime \\prime } cocoons formed by \\mathcal {M}_j = 3.5 ({\\bf {Top}}) and \\mathcal {M}_j = 10 ({\\bf {Bottom}}) steady jets.",
"A passive, jet mass-fractiontracer is volumed rendered in the images.",
"The jets are ``in the plane of the sky^{\\prime \\prime }and shown when they have approximately the same lengths, \\ell _j.The location of the AGN is marked by an `X^{\\prime }.",
"See text for details.",
"}\\end{figure}$ The energy deposition from the jet into the cocoon will generally drive a shock laterally outward into the ICM.",
"We will call this the “inflation shock”.",
"Provided the jet terminus propagates supersonically into the ICM ($\\mathcal {M}_h \\mp \\mathcal {M}_{w,\\parallel } > 1$ , equation REF ) there will be a bow shock attached to the jet head, which will generally merge with the inflation shock.",
"In Figure the tips of the two cocoons are confined by bow shocks.",
"The tighter “Mach cone” of the higher Mach number jet leads to “sharper points” in the cocoons.",
"Although the bow shock may be quite strong, both observations and simulations suggest that the inflation shock is typically rather weak.",
"For example the $\\mathcal {M}_j = 10$ jets shown in the bottom of Figure and in Figure REF produce bow shocks with $\\mathcal {M}_h \\approx 7$ ($v_{w,\\parallel } = 0$ ) on the nose of the jet in agreement with expectations from equation REF .",
"On the other hand, over much of their surfaces the accompanying both inflation shocks have Mach numbers, $\\mathcal {M} $ <$$$\\sim $ 1.5$ for the duration of the simulation (see Figure \\ref {fig:noshock-mach}).",
"There is nofixed value, however.",
"The relatively small Mach numbers of most of the inflation shock surfacesalso point to the fact that the pressure inside the shocks and inside the cocoons isnot much greater than the ambient pressure, $ Pi$ (see equation \\ref {eq:jump-p}).$ Because these properties do not inherently lead to scale-free structures, we do not try to model cocoon formation in terms of self-similar behavior [36], [37].",
"Figure: Volume rendering of the shocks associated with theℳ j =10\\mathcal {M}_j = 10 simulation shown in the bottom image ofFigure .",
"The Top image matches the orientation of the bottom image in Figure , while the Bottom image has the jet axis rotated 40 degrees, with the left jet approaching (so similar to Figures and .",
"Shocks are color coded, with the strongest shocks, ℳ s ≈7\\mathcal {M}_s \\approx 7 shown in white, with ℳ s ≈1.5\\mathcal {M}_s \\approx 1.5 represented by purple.",
"'X' marks AGN location.",
"See text for details.Still, it makes sense to estimate the volume of the cocoon simply by comparing the work required to inflate the cocoon to the energy that has passed down the jet into the cocoon.",
"Very simply, we set $P_i V_c(t) = K_w \\int L_j dt = K_w \\langle L_j \\rangle t$ , where $K_w < 1$ is a numerical factor that accounts for energy lost by the cocoon plasma as it inflated.",
"Accounting for adiabatic work done on the ICM would lead to $K_w = (\\gamma _j - 1)/\\gamma _j$ .",
"For a non-relativistic jet with $\\gamma _j = 5/3$ , this would give $K_w = 2/5$ , while for a jet with a relativistic EoS and $\\gamma _j = 4/3$ , the equivalent consequences would give $K_w = 1/4$ .",
"Empirical estimates from non-relativistic simulations do, indeed, suggest $K_w \\sim 1/2$ .",
"[38] The exact value for $K_w$ is not essential to our purposes here.",
"As a primitive model for cocoon geometry we assume the cocoon length is set by $\\int v_h(t) dt = \\langle v_h\\rangle t$ , giving $V_c = \\langle v_h\\rangle A_c \\approx \\frac{K_w \\langle L_j\\rangle }{P_i},$ where $A_c$ represents a characteristic cocoon cross sectional area.",
"We do not imply in this that the cocoon cross section is constant, and, indeed, it generally will not be (see Figure ).",
"In practice we treat it as a geometric average cross section.",
"To simplify our treatment further, we express the jet luminosity in the form applicable to both relativistic and non-relativistic jets suggested by equations REF and REF ; that is, $L_j \\approx \\alpha _j \\mathcal {M}_j^2 v_j P_j A_j,$ where $1 \\le \\alpha \\le 2$ .",
"For a relativistic jet $v_j \\rightarrow c$ in this expression.",
"From equation REF we write $v_h \\approx a_i \\mathcal {M}_j \\sqrt{\\frac{A_j}{A_h}}\\sqrt{\\frac{P_j}{P_i}}\\sqrt{\\frac{\\gamma ^{\\prime }}{\\gamma _i}}$ Combining equations REF , REF and REF gives a simple estimate for $A_c$ , $A_c \\approx \\sqrt{A_j A_h}~\\mathcal {M}_j \\frac{v_j}{a_j} \\sqrt{\\frac{a_j^2\\gamma _i}{a_i^2\\gamma ^{\\prime }}}\\sqrt{\\frac{P_j}{P_i}}~(\\alpha K_w).$ We expect $\\alpha K_w $ <$$$\\sim $ 1$, and, indeed using equation \\ref {eq:ac-nonrel}, below, and the simulation parameters for the cocoons shown inFigure \\ref {fig:noshock-color} we get consistent estimates for $ Ac$ using$ Kw 1/2$.",
"Note that, if these various jet parameters are timeindependent, the effective cocoon cross section, $ Ac$ is steady in time.",
"That is, the ratio $ Ac/j 1/t$.We briefly address alternative possibilities below.$ For a relativistic jet with a relativistic EoS, $v_j/a_j \\rightarrow \\sqrt{3}$ , while $\\tilde{w}_j \\rightarrow 4P_j/(\\rho _j c^2)$ , and equation REF becomes, $A_c \\approx \\sqrt{A_j A_h}~\\mathcal {M}_j\\left(\\sqrt{\\frac{\\rho _i}{\\rho _j}}~\\frac{P_j}{P_i}\\right)\\sqrt{\\frac{\\rho _j c^2}{2 P_j}}~(\\alpha K_w)\\\\= \\sqrt{A_j A_h}~\\mathcal {M}_j\\sqrt{\\frac{P_j}{P_i}}\\sqrt{\\gamma _i}\\frac{c}{a_i}~(\\alpha K_w).\\nonumber $ while for a non-relativistic jet, $A_c \\approx \\sqrt{A_j A_h}~\\mathcal {M}_j^2\\left( \\sqrt{\\frac{\\rho _i}{\\rho _j}}~ \\frac{P_j}{P_i}\\right)~(\\alpha K_w).$ The only difference between equations REF and REF is a relative factor $(1/\\mathcal {M}_j)\\sqrt{\\rho _j c^2/(2 P_j)}$ in equation REF .",
"We can see from these relations that the cocoon cross section scales with the geometric mean of the jet cross section, $A_j$ and the jet head cross section, $A_h$ .",
"It is strongly boosted if the jet Mach number is large (especially for non-relativistic jets), and, with less sensitivity, to a large density contrast between the ICM and the jet, $\\rho _i/\\rho _j$ .",
"That is, cocoons will be somewhat fatter for jets with larger density contrast, $\\rho _i / \\rho _j$ , but especially fatter for larger Mach numbers.",
"We are assuming in these comments that, as argued previously, we should expect $P_j/P_i \\sim 1$ .",
"The Mach number sensitivity of $A_c$ is evident in a comparison of the non-relativistic jet cocoons shown in Figure , where the only difference in the two simulations was jet Mach number; above, $\\mathcal {M}_j = 3.5$ , while below, $\\mathcal {M}_j = 10$ .",
"The $\\mathcal {M}_j = 10$ cocoon, in particular, is, as noted above, strongly tapered, representing the fact that the Mach cone at its nose has strongly confined it.",
"We associate $A_c$ with the mean cross sectional area, which here would be roughly midway between the AGN and the jet terminus.",
"The reader may have noticed at the same time that the cocoons from the jets shown in Figure are relatively skinny compared to typical, observed radio galaxy cocoons.",
"That is characteristic of the properties of moderate Mach number, non-relativistic AGN jet simulations that are steady and maintain a precise axis for the jets.",
"On the other hand, if either the jet power cycles substantially or the axis of the jet wanders or precesses, the relative rate at which the jet advances is reduced [38].",
"In order to maintain the same cocoon volume indicated in equation REF the cross section must increase.",
"This effect can, be roughly accounted for in equations REF - REF by adjusting the effective head area, $A_h$ .",
"For instance, a precessing jet will balance its thrust during a precession period over an area, $A_h \\sim 4 \\pi r_j \\ell _j \\gg \\pi r_j^2$ , where $\\ell _j = \\ell _j(t)$ is the instantaneous length of the jet.",
"Applied into equation REF the head advance rate, $v_h = d\\ell _j/dt \\propto 1/t^{1/2}$ .",
"If we maintain constant jet luminosity, $V_c \\propto t$ , so $A_c \\propto t^{1/2}$ .",
"Now, $A_c /\\ell _j =$ const, rather than $A_c/\\ell _j \\propto 1/t$ .",
"Simultaneously, the volume swept out by the precessing jet itself will be $\\sim \\ell _j^3 \\sin {\\theta }\\sin {2\\theta }$ , where $\\theta $ is the opening angle of the precession cone.",
"This sets a scale for the effective $A_c \\sim 2~\\ell _j^2 \\sin {\\theta }^2$ .",
"These effects both fatten the cocoons relative to what we derived above.",
"Finally, we mention that equation REF predicts rather fat cocoons for high Mach number relativistic jets propagating in an environment where $c\\gg a_i$ .",
"That is consistent with results of numerical simulations of relativistic hydrodynamical jets [39], and also simulations of jets dominated by their Poynting flux (so, internally highly relativistic) [40]."
],
[
"Summary",
"The plasma media in galaxy clusters, ICMs, are dynamical.",
"They are stirred by many processes associated with cluster assembly.",
"The dynamical state of an ICM provides unique information about how this takes place, so it is important to find and evaluate ICM dynamical conditions, especially those far away from equilibrium.",
"X-ray observations can reveal critical information about ICM thermodynamical properties, some statistical characteristics of ICM dynamical states and find relatively strong shocks in higher density ICM regions.",
"But, much of the story is not visible in the X-ray window.",
"On the other hand, AGNs in so-called “radio mode” are common in clusters, and especially in disturbed clusters.",
"Those AGNs expel fast plasma jets that plow through the ICM.",
"Those interactions will influence the ICM and its evolution.",
"More to the point of this paper, however, is the considerable impact of ICM dynamics on the trajectories and of the AGN jets and the distributions of their debris.",
"In particular, impacts between ICM shocks and winds will distort and even disrupt the AGN structures that form.",
"Through an understanding of how those distortions develop and how they depend on the AGN and ICM properties we hope to open a clearer window to revealing the dynamical structures of ICMs.",
"By relating those structures to other information about the dynamical state of the cluster, and, for instance, evidence for merging, strong accretion events or gravitational disturbance by dark matter halos, these insights can provide vital probes of cluster formation processes and their relative roles.",
"We have laid out in this paper a simple summary of some of the essential dynamical relationships involved in AGN/ICM interactions, with a special focus on interactions involving ICM shocks.",
"The nominal target application is ICM shocks colliding with AGN outflows.",
"We pointed out, however, that the relations we developed have application to any relative motion between the AGN and its immediate environment.",
"The formalism allows AGN jets that are either non-relativistic or relativistic.",
"We set down basic relations to evaluate shock interactions with the low density cavities created by AGN jets, as well as to follow the propagation of the jets within post-shock flows.",
"For completeness, we used the same formalism to provide a basic context for the formation of the cavities themselves.",
"One notable aspect of the relationships we derive is the essential roles of the internal Mach number of the jet flow and the Mach number of the ICM shock.",
"More directly, the ratio of these two Mach numbers seems central to evaluating the interactions between the AGN and a dynamical ambient environment.",
"This provides a potentially useful link that can help develop quantitative understandings of ICM dynamical states that otherwise are likely to remain obscure for some time.",
"TWJ and BO were supported in this work at the University of Minnesota by NSF grant AST1211595.",
"CN was supported by an NSF Graduate Fellowship under Grant 000039202.",
"TWJ, CN and BO gratefully acknowledge support and hospitality of the Minnesota Supercomputing Insitute.",
"We thank an anonymous referee for constructive comments that improved the manuscript."
]
] | 1612.05700 |
[
[
"Optimal Target Assignment and Path Finding for Teams of Agents"
],
[
"Abstract We study the TAPF (combined target-assignment and path-finding) problem for teams of agents in known terrain, which generalizes both the anonymous and non-anonymous multi-agent path-finding problems.",
"Each of the teams is given the same number of targets as there are agents in the team.",
"Each agent has to move to exactly one target given to its team such that all targets are visited.",
"The TAPF problem is to first assign agents to targets and then plan collision-free paths for the agents to their targets in a way such that the makespan is minimized.",
"We present the CBM (Conflict-Based Min-Cost-Flow) algorithm, a hierarchical algorithm that solves TAPF instances optimally by combining ideas from anonymous and non-anonymous multi-agent path-finding algorithms.",
"On the low level, CBM uses a min-cost max-flow algorithm on a time-expanded network to assign all agents in a single team to targets and plan their paths.",
"On the high level, CBM uses conflict-based search to resolve collisions among agents in different teams.",
"Theoretically, we prove that CBM is correct, complete and optimal.",
"Experimentally, we show the scalability of CBM to TAPF instances with dozens of teams and hundreds of agents and adapt it to a simulated warehouse system."
],
[
"Introduction",
"Teams of agents often have to assign targets among themselves and then plan collision-free paths to their targets.",
"Examples include autonomous aircraft towing vehicles [12], automated warehouse systems [22], office robots [19] and game characters in video games [15].",
"For example, in the near future, autonomous aircraft towing vehicles might tow aircraft all the way from the runways to their gates (and vice versa), reducing pollution, energy consumption, congestion and human workload.",
"Today, autonomous warehouse robots already move inventory pods all the way from their storage locations to the inventory stations that need the products they store (and vice versa), see Figure REF .",
"Figure: A typical Kiva warehouse system .We therefore study the TAPF (combined target-assignment and path-finding) problem for teams of agents in known terrain.",
"The agents are partitioned into teams.",
"Each team is given the same number of unique targets (goal locations) as there are agents in the team.",
"The TAPF problem is to assign agents to targets and plan collision-free paths for the agents from their current locations to their targets in a way such that each agent moves to exactly one target given to its team, all targets are visited and the makespan (the earliest time step when all agents have reached their targets and stop moving) is minimized.",
"Any agent in a team can be assigned to a target of the team, and the agents in the same team are thus exchangeable.",
"However, agents in different teams are not exchangeable.",
"The TAPF problem generalizes the anonymous and non-anonymous MAPF (multi-agent path-finding) problems: The anonymous MAPF problem (sometimes called goal-invariant MAPF problem) results from the TAPF problem if only one team exists (that consists of all agents).",
"It is called “anonymous” because any agent can be assigned to a target, and the agents are thus exchangeable.",
"The anonymous MAPF problem can be solved optimally in polynomial time [23].",
"Anonymous MAPF solvers use, for example, the polynomial-time max-flow algorithm on a time-expanded network [23] (an idea that originated in the operations research literature [1]) or graph-theoretic algorithms [11].",
"The non-anonymous MAPF problem (often just called MAPF problem) results from the TAPF problem if every team consists of exactly one agent and the number of teams thus equals the number of agents.",
"It is called “non-anonymous” because only one agent can be assigned to a target (meaning that the assignments of agents to targets are pre-determined), and the agents are thus non-exchangeable.",
"The non-anonymous MAPF problem is NP-hard to solve optimally and even NP-hard to approximate within any constant factor less than 4/3 [10].",
"Non-anonymous MAPF solvers use, for example, reductions to problems from satisfiability, integer linear programming or answer set programming [24], [6], [17] or optimal, bounded suboptimal or suboptimal search algorithms [16], [8], [20], [14], [3], [9], [5], [2], [4], such as the optimal CBS (conflict-based search) algorithm [13].",
"Research so far has concentrated on these two extreme cases.",
"Yet, many real-world applications fall between the extreme cases because the number of teams is larger than one but smaller than the number of agents, which is why we study the TAPF problem in this paper.",
"The TAPF problem is NP-hard to solve optimally and even NP-hard to approximate within any constant factor less than 4/3 if more than one team exists [10].",
"It is unclear how to generalize anonymous MAPF algorithms to solving the TAPF problem.",
"Straightforward ways of generalizing non-anonymous MAPF algorithms to solving the TAPF problem have difficulties with either scalability (due to the resulting large state spaces), such as searching over all assignments of agents to targets to find optimal solutions, or solution quality, such as assigning agents to targets with algorithms such as [18], [26] and then planning collision-free paths for the agents with non-anonymous MAPF algorithms (perhaps followed by improving the assignment and iterating [21]) to find sub-optimal solutions."
],
[
"Contribution",
"We present the CBM (Conflict-Based Min-Cost-Flow) algorithm to bridge the gap between the extreme cases of anonymous and non-anonymous MAPF problems.",
"CBM solves the TAPF problem optimally by simultaneously assigning agents to targets and planning collision-free paths for them, while utilizing the polynomial-time complexity of solving the anonymous MAPF problem for all agents in a team to scale to a large number of agents.",
"CBM is a hierarchical algorithm that combines ideas from anonymous and non-anonymous MAPF algorithms.",
"It uses CBS on the high level and a min-cost max-flow algorithm [7] on a time-expanded network on the low level.",
"Theoretically, we prove that CBM is correct, complete and optimal.",
"Experimentally, we show the scalability of CBM to TAPF instances with dozens of teams and hundreds of agents and adapt it to a simulated warehouse system.",
"In this section, we formalize the TAPF problem and show how it can be solved via a reduction to the integer multi-commodity flow problem on a time-expanded network."
],
[
"Definition and Properties",
"For a TAPF instance, we are given an undirected connected graph $G = (V,E)$ (whose vertices $V$ correspond to locations and whose edges $E$ correspond to ways of moving between locations) and $K$ teams $team_1 \\ldots team_K$ .",
"Each team $team_i$ consists of $K_i$ agents $a^i_1 \\ldots a^i_{K_i}$ .",
"Each agent $a^i_j$ has a unique start vertex $s^i_j$ .",
"Each team $team_i$ is given unique targets (goal vertices) $g^i_1 \\ldots g^i_{K_i}$ .",
"Each agent $a^i_j$ must move to a unique target $g^i_{j^{\\prime }}$ .",
"An assignment of agents in team $team_i$ to targets is thus a one-to-one mapping $\\varphi ^i$ , determined by a permutation on $1\\ldots K_i$ , that maps each agent $a^i_j$ in $team_i$ to a unique target $g^i_{j^{\\prime }} = \\varphi ^i(a^i_j)$ of the same team.",
"A path for agent $a^i_j$ is given by a function $l^i_j$ that maps each integer time step $t = 0\\ldots \\infty $ to the vertex $l^i_j(t) \\in V$ of the agent in time step $t$ .",
"A solution consists of paths for all agents that obey the following conditions: $\\forall i, j:~l^i_j(0) = s^i_j$ (each agent starts at its start vertex); $\\forall i, j~\\exists \\mbox{a minimal}~T^i_j~\\forall t \\ge T^i_j:~l^i_j(t) = \\varphi ^i(a^i_j)$ (each agent ends at its target); $\\forall i, j, t:~(l^i_j(t) = l^i_j(t+1)$ or $(l^i_j(t), l^i_j(t+1))\\in E)$ (each agent always stays at its current vertex or moves to an adjacent vertex); $\\forall a^i_j, a^{i^{\\prime }}_{j^{\\prime }}, t~\\mbox{with}~a^i_j \\ne a^{i^{\\prime }}_{j^{\\prime }}:l^i_j(t)\\ne l^{i^{\\prime }}_{j^{\\prime }}(t)$ (there are no vertex collisions since different agents never occupy the same vertex at the same time); $\\forall a^i_j, a^{i^{\\prime }}_{j^{\\prime }}, t~\\mbox{with}~a^i_j \\ne a^{i^{\\prime }}_{j^{\\prime }}:(l^i_j(t) \\ne l^{i^{\\prime }}_{j^{\\prime }}(t+1)$ or $l^{i^{\\prime }}_{j^{\\prime }}(t) \\ne l^i_j(t+1))$ (there are no edge collisions since different agents never move along the same edge in different directions at the same time).",
"Given paths for all agents in team $team_i$ , the team cost of team $team_i$ is $\\max _{j}T^i_j$ (the earliest time step when all agents in the team have reached their targets and stop moving).",
"Given paths for all agents, the makespan is $\\max _{i,j} T^i_j$ (the earliest time step when all agents have reached their targets and stop moving).",
"The task is to find an optimal solution, namely one with minimal makespan.",
"Note that a (non-anonymous) MAPF instance can be obtained from a TAPF instance by fixing the assignments of agents to targets.",
"Any solution of a TAPF instance is thus also a solution of a (non-anonymous) MAPF instance on the same graph for a suitable assignment of agents to targets.",
"Since the makespan of any optimal (non-anonymous) MAPF solution is bounded by $O(|V|^3)$ [25], the makespan of any optimal TAPF solution is also bounded by $O(|V|^3)$ .",
"We define a collision between an agent agent $a^i_j$ in team $team_i$ and a different agent $a^{i^{\\prime }}_{j^{\\prime }}$ in team $team_{i^{\\prime }}$ to be either a vertex collision ($team_i$ , $team_{i^{\\prime }}$ , $l$ , $t$ ) [if $l = l^i_j(t) =l^{i^{\\prime }}_{j^{\\prime }}(t)$ and thus both agents occupy the same vertex at the same time] or an edge collision ($team_i$ , $team_{i^{\\prime }}$ , $l_1$ , $l_2$ , $t$ ) [if $l_1= l^i_j(t) = l^{i^{\\prime }}_{j^{\\prime }}(t+1)$ and $l_2 = l^{i^{\\prime }}_{j^{\\prime }}(t) = l^i_j(t+1)$ and thus both agents move along the same edge in different directions at the same time].",
"Likewise, we define a constraint to be either a vertex constraint ($team_i$ , $l$ , $t$ ) [that prohibits any agent in $team_i$ from occupying vertex $l$ in time step $t$ ] or an edge constraint ($team_i$ , $l_1$ , $l_2$ , $t$ ) [that prohibits any agent in team $team_i$ from moving from vertex $l_1$ to vertex $l_2$ between time steps $t$ and $t+1$ ]."
],
[
"Solution via Reduction to Flow Problem",
"Given a TAPF instance on undirected graph $G=(V,E)$ and a limit $T$ on the number of time steps, we construct a $T$ -step time-extended network using a reduction that is similar to that from the (non-anonymous) MAPF problem to the integer multi-commodity flow problem [24] (the idea of which is an extension of [23]).",
"A $T$ -step time-extended network is a directed network $\\mathcal {N} = (\\mathcal {V},\\mathcal {E})$ with vertices $\\mathcal {V}$ and directed edges $\\mathcal {E}$ that have unit capacity.",
"Each vertex $v \\in V$ is translated to a vertex $v_t^{out} \\in \\mathcal {V}$ for all $t=0\\ldots T$ (which represents vertex $v$ at the end of time step $t$ ) and a vertex $v_t^{in} \\in \\mathcal {V}$ for all $t=1\\ldots T$ (which represents vertex $v$ in the beginning of time step $t$ ).",
"There is a supply of one unit of commodity type $i$ at vertex ${(s^i_j)}_0^{out}$ and a demand of one unit of commodity type $i$ at vertex ${(g^i_j)}^{out}_T$ for all $i = 1\\ldots K$ and $j = 1\\ldots K_i$ .",
"Each vertex $v \\in V$ is also translated to an edge $(v_t^{out},v_{t+1}^{in}) \\in \\mathcal {E}$ for all $t=0\\ldots T-1$ (which represents an agent staying at vertex $v$ between time steps $t$ and $t+1$ ).",
"Each vertex $v \\in V$ is also translated to an edge $(v_t^{in},v_t^{out}) \\in \\mathcal {E}$ for all $t=1\\ldots T$ (which prevents vertex collisions of the form $(*, *, v, t)$ among all agents since only one agent can occupy vertex $v$ between time steps $t$ and $t+1$ ).",
"Each edge $(u,v) \\in E$ is translated to a gadget of vertices in $\\mathcal {V}$ and edges in $\\mathcal {E}$ for all $t=0\\ldots {T-1}$ , which consists of two auxiliary vertices $w, w^{\\prime } \\in \\mathcal {V}$ that are unique to the gadget (but have no subscripts here for ease of readability) and the edges $(u_t^{out},w),(v_t^{out},w), (w,w^{\\prime }), (w^{\\prime },u_{t+1}^{in}), (w,v_{t+1}^{in}) \\in \\mathcal {E}$ .",
"This gadget prevents edge collisions of the forms $(*, *, u, v,t)$ and $(*, *, v, u, t)$ among all agents since only one agent can move along the edge $(u,v)$ in any direction between time steps $t$ and $t+1$ .",
"Figure REF shows a simple example.",
"The following theorem holds by construction and can be proved in a way similar to the one for the reduction of the (non-anonymous) MAPF problem to the integer multi-commodity flow problem [24]: Theorem 1 There is a correspondence between all feasible integer multi-commodity flows on the $T$ -step time-extended network of a number of unit that equals the number of agents and all solutions of the TAPF instance with makespans of at most $T$ .",
"An optimal solution can therefore be found by starting with $T=0$ and iteratively checking for increasing values of $T$ whether a feasible integer multi-commodity flow of a number of units that equals the number of agents exists for the corresponding $T$ -step time-expanded network (which is an NP-hard problem), until an upper bound on $T$ is reached (such as the one provided in [25]).",
"Each $T$ -step time-expanded network is translated in the standard way into an ILP (integer linear program), which is then solved with an ILP algorithm.",
"We evaluate this ILP-based TAPF solver experimentally in Section REF .",
"The anonymous MAPF problem results from the TAPF problem if only one team exists (that consists of all agents).",
"The following corollary thus follows from [23]: Corollary 2 The TAPF problem can be solved optimally in polynomial time if only one team exists."
],
[
"Conflict-Based Min-Cost Flow",
"In this section, we present the CBM (Conflict-Based Min-Cost-Flow) algorithm, a hierarchical algorithm that solves TAPF instances optimally.",
"On the high level, CBM considers each team to be a meta-agent.",
"It uses CBS to resolve collisions among meta-agents, that is, agents in different teams.",
"CBS is a form of best-first search on a tree, where each node contains a set of constraints and paths for all agents that obey these constraints, move all agents to unique targets of their teams and result in no collisions among agents in the same team.",
"On the low level, CBM uses a polynomial-time min-cost max-flow algorithm [7] on a time-expanded network to assign all agents in a single team to unique targets of the same team and plan paths for them that obey the constraints imposed by the currently considered high-level node and result in no collisions among the agents in the team.",
"Since the running time of CBS on the high level can be exponential in the number of collisions that need to be resolved [13], CBM uses edge weights on the low level to bias the search so as to reduce the possibility of creating collisions with agents in different teams.",
"The idea of biasing the search on the low level has been used before for solving the (non-anonymous) MAPF problem with CBS [2].",
"Similarly, the idea of grouping some agents into a meta-agent on the high level and planning paths for each group on the low level has been used before for solving the (non-anonymous) MAPF problem with CBS [13] but faces the difficulty of having to identify good groups of agents.",
"The best way to group agents can often be determined only experimentally and varies significantly among MAPF instances.",
"On the other hand, grouping all agents in a team into a meta-agent for solving the TAPF problem is a natural way of grouping agents since the assignments of agents in the same team to targets and their paths strongly depend on each other and should therefore be planned together on the low level.",
"For example, if an agent is assigned to a different target, then many of the agents in the same team typically need to be assigned to different targets as well and have their paths re-planned.",
"Also, the lower level can then use a polynomial-time max-flow algorithm on a time-expanded network to assign all agents in a single team to targets and find paths for them due to the polynomial-time complexity of the corresponding anonymous MAPF problem."
],
[
"High-Level Search of CBM",
"[t] High-Level Search of CBM $Root.constraints \\leftarrow \\emptyset $ $Root.paths \\leftarrow \\emptyset $ each $team_i$ Lowlevel($team_i$ , $Root$ ) returns no paths No solution exists Add the returned paths to $Root.paths$ $Root.key \\leftarrow $ Makespan($Root.paths$ ) $Priorityqueue \\leftarrow \\lbrace Root\\rbrace $ $Priorityqueue \\ne \\emptyset $ $N \\leftarrow Priorityqueue.pop()$ Findcollisions($N.paths$ ) returns no collisions Solution is $N.paths$ $Collision \\leftarrow $ earliest collision found each $team_i$ involved in $Collision$ $N^{\\prime }\\leftarrow $ Newnode() /* with parent $N$ */ $N^{\\prime }.constraints \\leftarrow N.constraints$ $N^{\\prime }.paths \\leftarrow N.paths$ Add one new constraint for $team_i$ to $N^{\\prime }.constraints$ Lowlevel($team_i$ , $N^{\\prime }$ ) returns paths Update $N^{\\prime }.paths$ with the returned paths $N^{\\prime }.key \\leftarrow $ $max$ (Makespan($N^{\\prime }.paths$ ), $N.key$ ) $Priorityqueue.insert(N^{\\prime })$ No solution exists On the high level, CBM performs a best-first search on a binary tree, see Algorithm REF .",
"Each node $N$ contains constraints $N.constraints$ and paths for all agents $N.paths$ that obey these constraints, move all agents to unique targets of their teams and result in no collisions among agents in the same team.",
"All nodes are stored in a priority queue.",
"The priority queue initially consists of only the root node $Root$ with no constraints and paths for all agents that move all agents to unique targets of their teams, result in no collisions among agents in the same team and minimize the team cost of each team [Lines 1-8].",
"If the priority queue is empty, then CBM terminates unsuccessfully [Line 23].",
"Otherwise, CBM always chooses a node $N$ in the priority queue with the smallest key [Line 10].",
"The key of a node is the makespan of its paths.",
"(Ties are broken in favor of the node whose paths have the smallest number of colliding teams.)",
"If the paths of node $N$ have no colliding agents, then they are a solution and CBM terminates successfully with these paths [Lines 11-12].",
"Otherwise, CBM determines all collisions between two agents (which have to be in different teams) and then resolves a collision $Collision$ whose time step $t$ is smallest [Line 13].",
"(We have evaluated different ways of prioritizing the collisions, including the one suggested in [3], but have not observed significant differences in the resulting running times of CBM.)",
"Let the two colliding agents be in $team_i$ and $team_{i^{\\prime }}$ .",
"CBM then generates two child nodes $N_1$ and $N_2$ of node $N$ , both of which inherit the constraints and paths from their parent node [Lines 15-17].",
"If the collision is a vertex collision $(team_i, team_{i^{\\prime }}, l, t)$ or, equivalently, $(team_{i^{\\prime }}, team_i, l, t)$ , then CBM adds the vertex constraint $(team_i, l, t)$ to the constraints of node $N_1$ and the vertex constraint $(team_{i^{\\prime }}, l, t)$ to the constraints of node $N_2$ [Line 18].",
"If the collision is an edge collision $(team_i,team_{i^{\\prime }}, l_1, l_2, t)$ or, equivalently, $(team_{i^{\\prime }}, team_i, l_2, l_1, t)$ , then CBM adds the edge constraint $(team_i, l_1, l_2, t)$ to the constraints of node $N_1$ and the edge constraint $(team_{i^{\\prime }}, l_2, l_1, t)$ to the constraints of node $N_2$ [Line 18].",
"For each of the two new nodes, say node $N_1$ , the low-level search is called to assign all agents in team $team_i$ to unique targets of the same team and find paths for them that obey the constraints of node $N_1$ and result in no collisions among the agents in the team.",
"If the low-level search successfully returns such paths, then CBM updates the paths of node $N_1$ by replacing the paths of all agents in team $team_i$ with the returned ones, updates the key of node $N_1$ and inserts it into the priority queue [Lines 19-22].",
"Otherwise, it discards the node."
],
[
"Low-Level Search of CBM",
"On the low level, Lowlevel($team_i$ ,$N$ ) assigns all agents in team $team_i$ to unique targets of the same team and finds paths for them that obey all constraints of node $N$ (namely all vertex constraints of the form ($team_i$ , *, *) and all edge constraints of the form ($team_i$ , *, *, *)) and result in no collisions among the agents in the team.",
"Given a limit $T$ on the number of time steps, CBM constructs the $T$ -step time-expanded network from Section REF with the following changes: a) There is only a single commodity type $i$ since CBM considers only the single team $team_i$ .",
"There is a supply of one unit of this commodity type at vertex ${(s^i_j)}_0^{out}$ and a demand of one unit of this commodity type at vertex ${(g^i_j)}^{out}_T$ for all $j = 1\\ldots K_i$ .",
"b) To obey the vertex constraints, CBM removes the edge $(l^{in}_t, l^{out}_t)$ from $\\mathcal {E}$ for each vertex constraint of the form $(team_i, l, t)$ .",
"c) To obey the edge constraints, CBM removes the edges $((l_1)^{out}_t, w)$ and $(w^{\\prime }, (l_2)^{in}_{t+1})$ from $\\mathcal {E}$ for all gadgets that correspond to edge $(l_1, l_2) \\in E$ for each edge constraint of the form $(team_i, l_1,l_2, t)$ .",
"Let $\\mathcal {V} = \\mathcal {V^{\\prime }}$ be the set of (remaining) vertices and $\\mathcal {E^{\\prime }}$ be the set of remaining edges.",
"Similar to the procedure from Section REF , CBM iteratively checks for increasing values of $T$ whether a feasible integer single-commodity flow of $K_i$ units exists for the corresponding $T$ -step time-expanded network, which can be done with the polynomial-time max-flow algorithm that finds a feasible maximum flow.",
"CBM can start with $T$ being the key of the parent node of node $N$ since it is a lower bound on the new key of node $N$ due to Line 21.",
"(For $N = Root$ , CBM starts with $T=0$ .)",
"During the earliest iteration when the max-flow algorithm finds a feasible flow of $K_i$ units, the call returns successfully with the paths for the agents in the team that correspond to the flow.",
"If $T$ reaches an upper bound on the makespan of an optimal solution (such as the one provided in [25]) and no feasible flow of $K_i$ units was found, then the call returns unsuccessfully with no paths.",
"CBM actually implements Lowlevel($team_i$ ,$N$ ) in a more sophisticated way to avoid creating collisions between agents in team $team_i$ and agents in other teams by adding edge weights to the $T$ -step time-expanded network.",
"CBM sets the weights of all edges in $\\mathcal {E^{\\prime }}$ to zero initially and then modifies them as follows: a) To reduce vertex collisions, CBM increases the weight of edge $(v^{in}_t, v^{out}_t) \\in \\mathcal {E^{\\prime }}$ by one for each vertex $v =l^{i^{\\prime }}_{j^{\\prime }}(t) \\in V$ in the paths of node $N$ with $i^{\\prime } \\ne i$ to reduce the possibility of an agent of team $team_i$ occupying the same vertex at the same time step as an agent from a different team.",
"b) To reduce edge collisions, CBM increases the weight of edge $(v^{in}_t, w) \\in \\mathcal {E^{\\prime }}$ by one for each edge $(u = l^{i^{\\prime }}_{j^{\\prime }}(t), v = l^{i^{\\prime }}_{j^{\\prime }}(t+1)) \\in E$ in the paths of node $N$ with $i^{\\prime } \\ne i$ (where $w$ is the auxiliary vertex of the gadget that corresponds to edge $(u,v)$ and time step $t$ ) to reduce the possibility of an agent of team $team_i$ moving along the same edge in a different direction but at the same time step as an agent from a different team.",
"CBM uses the procedure described above, except that it now uses a min-cost max-flow algorithm (instead of a max-flow algorithm) that finds a flow of minimal weight among all feasible maximal flows.",
"In particular, it uses the successive shortest path algorithm [7], a generalization of the Ford-Fulkerson algorithm that uses Dijkstra's algorithm to find a path of minimal weight for one unit of flow.",
"The complexity of the successive shortest path algorithm is $O(U(|\\mathcal {E^{\\prime }}| + |\\mathcal {V^{\\prime }}|\\log |\\mathcal {V^{\\prime }}|))$ , where $O(|\\mathcal {E^{\\prime }}| + |\\mathcal {V^{\\prime }}|\\log |\\mathcal {V^{\\prime }}|)$ is the complexity of Dijkstra's algorithm and $U$ is the value of the feasible maximal flow, which is bounded from above by $K_i$ .",
"The number of times that the successive shortest path algorithm is executed is bounded from above by the chosen upper bound on the makespan of an optimal solution, which in turn is bounded from above by $O(|V|^3)$ .",
"Thus, each low-level search runs in polynomial time."
],
[
"Analysis of Properties",
"We use the following properties to prove that CBM is correct, complete and optimal.",
"Property 1 There is a correspondence between all feasible integer flows of $K_i$ units on the $T$ -step time-extended network constructed for team $team_i$ and node $N$ and all paths for agents in team $team_i$ that a) obey the constraints of node $N$ , b) move all agents from their start vertices to unique targets of their team, c) result in no collisions among agents in team $team_i$ and d) result in a team cost of team $team_i$ of at most $T$ .",
"Reason.",
"The property holds by construction and can be proved in a way similar to the one for the reduction of the (non-anonymous) MAPF problem to the integer multi-commodity flow problem [24]: Left to right: Assume that a flow is given that has the stated properties.",
"Each unit flow from a source to a sink corresponds to a path through the time-extended network from a unique source to a unique sink.",
"Thus, it can be converted to a path for an agent such that all such paths together have the stated properties: Properties a and d hold by construction of the time-extended network; Property b holds because a flow of $K_i$ units uses all supplies and sinks; and Property c holds since the flows neither share vertices nor edges.",
"Right to left: Assume that paths are given that have the stated properties.",
"If necessary, we extend the paths by letting the agents stay at their targets.",
"Each path now corresponds to a path through the time-extended network (due to Properties a and d) from a unique source to a unique sink (due to Property b) that does not share directed edges with the other such paths (due to Property c).",
"Thus, it can be converted to a unit flow such that all such unit flows together respect the unit capacity constraints and form a flow of $K_i$ units.",
"Property 2 CBM generates only finitely many nodes.",
"Reason.",
"The constraint added on Line 18 to a child node is different from the constraints of its parent node since the paths of its parent node do not obey it.",
"Overall, CBM creates a binary tree of finite depth since only finitely many different vertex and edge constraints exist and thus generates only finitely many nodes.",
"Property 3 Whenever CBM inserts a node into the priority queue, its key is finite.",
"Reason (by induction).",
"The property holds for the root node.",
"Assume that it holds for the parent node of some child node.",
"The key of the child node is the maximum of the key of the parent node and the team costs of all teams for the paths of the child node.",
"The key of the parent node is finite due to the induction assumption.",
"The low level returned the paths for each team successfully at some point in time and all team costs are thus finite as well.",
"Property 4 Whenever CBM chooses a node on Line 10 and the paths of the node have no colliding agents, then CBM correctly terminates with a solution with finite makespan of at most the value of its key.",
"Reason.",
"The key of the node is finite according to Property REF , and the makespan of its paths is at most the value of its key due to Line 21.",
"Property 5 CBM chooses nodes on Line 10 in non-decreasing order of their keys.",
"Reason.",
"CBM performs a best-first search, and the key of a parent node is most the key of any of its child nodes due to Line 21.",
"Property 6 The smallest makespan of any solution that obeys the constraints of a parent node is at most the smallest makespan of any solution that obeys the constraints of any of its child nodes.",
"Reason.",
"The solutions that obey the constraints of a parent node are a superset of the solutions that obey the constraints of any of its child nodes since the constraints of the parent node are a subset of the constraints of any of its child nodes.",
"Property 7 The key of a node is at most the makespan of any solution that obeys its constraints.",
"Reason (by induction).",
"The property holds for the root node.",
"Assume that it holds for the parent node $N$ of any child node $N^{\\prime }$ and that the paths for team $team_i$ were updated in the child node.",
"Let $x$ be the smallest makespan of any solution that obeys the constraints of the parent node and $y$ be the smallest makespan of any solution that obeys the constraints of the child node.",
"We show in the following that the key of the parent node and the team costs of all teams for the paths of the child node are all at most $y$ .",
"Then, the key of the child node is also at most $y$ since it is the maximum of all these quantities, and the property holds.",
"First, consider the key of the parent node.",
"The key of the parent node is at most $x$ due to the induction assumption, which in turn is at most $y$ due to Property REF .",
"Second, consider any team different from team $team_i$ .",
"Then, the team cost of the team for the paths of the child node is equal to the team cost of the team for the paths of the parent node (since the paths were not updated in the child node and are thus identical), which in turn is at most the key of the parent node (since the key of the parent node is the maximum of several quantities that include the team cost of the team for the paths of the parent node), which in turn is at most $y$ (as shown directly above).",
"Finally, consider team $team_i$ .",
"When the low level finds new paths for team $team_i$ , it starts with $T$ being the key of the parent node, which is at most $y$ (as shown directly above).",
"Thus, the max-cost min-flow algorithm on a $T$ -step time-expanded network constructed for team $team_i$ and the child node finds a feasible integer flow of $K_i$ units for $T \\le y$ since there exists a solution with makespan $y$ that obeys the constraints of the child node.",
"The team cost of the corresponding paths for team $team_i$ is at most $T$ due to Property REF .",
"Theorem 3 CBM is correct, complete and optimal.",
"Assume that no solution to a TAPF instance exists and CBM does not terminate unsuccessfully on Line 5.",
"Then, whenever CBM chooses a node on Line 10, the paths of the node have colliding agents (because otherwise a solution would exist due to Property REF ).",
"Thus, the priority queue eventually becomes empty and CBM terminates unsuccessfully on Line 23 since it generates only finitely many nodes due to Property REF .",
"Now assume that a solution exists and the makespan of an optimal solution is $x$ .",
"Assume, for a proof by contradiction, that CBM does not terminate with a solution with makespan $x$ .",
"Thus, whenever CBM chooses a node on Line 10 with a key of at most $x$ , the paths of the node have colliding agents (because otherwise CBM would correctly terminate with a solution with makespan at most $x$ due to Property REF ).",
"A node whose constraints the optimal solution obeys has a key of at most $x$ due to Property 7.",
"The root note is such a node since the optimal solution trivially obeys the (empty) constraints of the root node.",
"Whenever CBM chooses such a node on Line 10, the paths of the node have colliding agents (as shown directly above since its key is at most $x$ ).",
"CBM thus generates the child nodes of this parent node, the constraints of at least one of which the optimal solution obeys and which CBM thus inserts into the priority queue with a key of at most $x$ .",
"Since CBM chooses nodes on Line 10 in non-decreasing order of their keys due to Property REF .",
"it chooses infinitely many nodes on Line 10 with keys of at most $x$ , which is a contradiction with Property REF ."
],
[
"Experiments",
"In this section, we describe the results of four experiments on a 2.50 GHz Intel Core i5-2450M PC with 6 GB RAM.",
"First, we compare CBM to four other TAPF or MAPF solvers.",
"Second, we study how CBM scales with the number of agents in each team.",
"Third, we study how CBM scales with the number of agents.",
"Fourth, we apply CBM to a simulated warehouse system."
],
[
"Experiment 1: Alternative Solvers",
"We compare our optimal TAPF solver CBM to two optimal (non-anonymous) MAPF solvers, namely a) the CBS solver provided by the authors of [13] and b) the ILP-based MAPF solver provided by the authors of [24], and two optimal TAPF solvers, namely a) an unweighted version of CBM that runs the polynomial-time max-flow algorithm on a time-expanded network without edge weights (instead of the min-cost max-flow algorithm on a time-expanded network with edge weights) on the low level and b) an ILP-based TAPF solver (based on the ILP-based MAPF solver) that casts a TAPF instance as a series of integer multi-commodity flow problems as described in Section REF , each of which it models as an ILP and solves with the ILP solver Gurobi 6.0 (www.gurobi.com).",
"For Experiment 1, each team consists of 5 agents but the number of agents varies from 10 to 50, resulting in $2\\ldots 10$ teams.",
"For each number of agents, we generate 50 TAPF instances from the same 50 $30 \\times 30$ 4-neighbor grids with $10\\%$ randomly blocked cells by randomly assigning unique start cells to agents and unique targets to teams.",
"For the MAPF solvers, we convert each TAPF instance to a (non-anonymous) MAPF instance by randomly assigning the agents in each team to unique targets of the same team.",
"Table REF shows the success rates as well as the means of the makespans and running times (in seconds) over the instances that are solved within a time limit of 5 minutes each.",
"Red entries indicate that some instances are not solved within the time limit, while dashed entries indicate that all instances are not solved within the time limit.",
"CBM solves all TAPF instances within the time limit.",
"Both MAPF solvers solve most of the MAPF instances within the time limit.",
"The running times of CBM and CBS are similar because, on the low level, both the min-cost max-flow algorithm of CBM (for a single team) and the A* algorithm of CBS (for a single agent) are fast.",
"Optimal solutions of the TAPF instances have smaller makespans than optimal solutions of the MAPF instances due to the freedom of assigning agents to targets for the TAPF instances rather than assigning them randomly for the MAPF instances."
],
[
"Unweighted CBM",
"Unweighted CBM solves less than half of all TAPF instances within the time limit if the number of agents is larger than 10 due to the large number of collisions among agents in different teams produced by the max-flow algorithm on the low level in tight spaces with many agents, which results in a large number of node expansions by CBS on the high level.",
"We conclude that biasing the search on the low level is important for CBM to solve all TAPF instances within the time limit."
],
[
"ILP-Based TAPF Solver",
"The ILP-based TAPF solver solves less than half of all TAPF instances within the time limit if the number of agents is larger than 30, and its running time is much larger than that of CBM.",
"The success rates and running times of the the ILP-based TAPF solver tend to be larger than those of the ILP-based MAPF solver even though the ILP formulation of a TAPF instance has fewer variables than that of the corresponding MAPF instance (since the number of commodity types equals the number of teams for the TAPF instance but the number of agents for the MAPF instance).",
"However, the variables in the ILP formulation of the MAPF instance are Boolean variables while those in the ILP formulation of the TAPF instance are integer variables.",
"Furthermore, the ILP-based MAPF solver uses the maximum over all agents of the length of a shortest path of each agent as the starting value of $T$ for the time-expanded network while the ILP-based TAPF solver solves the LP formulation of the max-flow problem that finds paths for each team (ignoring other teams) and then uses the maximum over all teams of the team costs of the paths as the starting value $T$ for the time-expanded network."
],
[
"Experiment 2: Team Size",
"For Experiment 2, there are 100 agents but the number of agents in a team (team size) varies from 50 to 2, resulting in $K=2\\ldots 50$ teams.",
"For each team size, we generate 50 TAPF instances as described before.",
"Table REF shows the means of the makespans and running times (in seconds) over the instances that are solved within a time limit of 5 minutes each.",
"CBM solves all TAPF instances within the time limit.",
"For large team sizes and thus small numbers of teams, the makespans are small because CBM has more freedom to assign agents to targets.",
"The running times are also small because the min-cost max-flow algorithm on the low level is fast even for large numbers of agents while CBS on the high level is fast because it needs to resolve collisions among agents in different teams but there are only a small number of teams.",
"Thus, it is advantageous for teams to consist of as many agents as possible."
],
[
"Experiment 3: Number of Agents",
"For Experiment 3, each team consists of 5 agents but the number of agents varies from 100 to 450, resulting in $20\\ldots 90$ teams.",
"For each number of agents, we generate 50 TAPF instances as described before.",
"Table REF shows the success rates as well as the means of the makespans and running times (in seconds) over the instances that are solved within a time limit of 5 minutes each.",
"For 250 agents or fewer, the success rate is larger than 85%.",
"Current (non-anonymous) MAPF algorithm are not able to handle instances of this scale.",
"As the number of agents increases, the success rates decrease and the makespans and running times increase due to the increasing number of collisions among agents in different teams produced by the min-cost max-flow algorithm on the low level.",
"For 450 agents, for example, more than half of the unblocked cells are occupied by agents and thus many start cells of agents are also targets for other agents."
],
[
"Experiment 4: Warehouse System",
"We now apply CBM to a simulated Kiva (now: Amazon Robotics) warehouse system [22].",
"Figure REF shows a typical grid layout with inventory stations on the left side and storage locations in the storage area to the right of the inventory stations.",
"Each inventory station has an entrance (purple cells) and an exit (pink cells).",
"Each storage location (green cell) can store one inventory pod.",
"Each inventory pod consists of a stack of trays, each of which holds bins with products.",
"The autonomous warehouse robots are called drive units.",
"Each drive unit is capable of picking up, carrying and putting down one inventory pod at a time.",
"As a team, the drive units need to move inventory pods all the way from their storage locations to the inventory stations that need the products they store (to ship them to customers) and then back to the same or different empty storage locations.",
"After a drive unit enters an inventory station, the requested product is removed from its inventory pod by a worker.",
"Once drive units have delivered all requested products for one shipment to the same inventory station, the worker prepares the shipment to the customer.",
"Figure REF shows a randomly generated Kiva instance.",
"The light grey cells are free space.",
"The dark grey cells are storage locations occupied by inventory pods and thus blocked.",
"There are 7 inventory stations on the left side.",
"The red cells are their exits, and the other 7 cells with graduated blue-green colors are their entrances.",
"Drive units can enter and leave the inventory stations one at a time through their entrances and exits, respectively.",
"The cells with graduated blue-green colors in the storage area are occupied by drive units.",
"Each drive unit needs to carry the inventory pod in its current cell to the inventory station of the same color.",
"For Experiment 4, we generate 50 TAPF instances.",
"Each instance has 420 drive units.",
"210 “incoming” drive units start at randomly determined storage locations: 30 drive units each need to move their inventory pods to the 7 inventory stations.",
"In order to create difficult Kiva instances, we generate the start cells of these drive units randomly among all storage locations rather than cluster them according to their target inventory stations.",
"210 “outgoing” drive units start at the inventory stations: 30 drive units each need to move their inventory pods from the 7 inventory stations to the storage locations vacated by the incoming drive units.",
"The task is to assign the 210 outgoing drive units to the vacated storage locations and plan collision-free paths for all 420 drive units in a way such that the makespan is minimized.",
"The incoming drive units that have the same inventory station as target are a team (since they can arrive at the inventory station in any order), and all outgoing drive units are a team.",
"So far, we have assumed that, for any TAPF instance, all start vertices are unique, all targets are unique and each of the teams is given the same number of targets as there are agents in the team but these assumptions are not necessarily satisfied here.",
"1) The outgoing drive units that start at the same inventory station all start at its exit.",
"In this case, we change the construction of the $T$ -step time-extended network for the team of outgoing drive units so that there is a supply of one unit at vertex $v_t^{out} \\in \\mathcal {V^{\\prime }}$ for all $t = 0 \\ldots 29$ and all vertices $v \\in V$ that correspond to exits of inventory stations.",
"This construction forces the outgoing drive units that start at the same inventory station to leave it one after the other during the first 30 time steps.",
"No further changes are necessary.",
"2) The incoming drive units that have the same inventory station as target all end at its entrance.",
"In this case, we change the construction of the $T$ -step time-extended network for each team of incoming drive units so that there is an auxiliary vertex with a demand of 30 units and vertex $v_t^{out} \\in \\mathcal {V^{\\prime }}$ for all $t = 0 \\ldots T$ is connected to the auxiliary vertex with an edge with unit capacity and zero edge weight, where $v \\in V$ corresponds to the entrance of the inventory station.",
"This construction forces the incoming drive units to enter the inventory station at different time steps.",
"No further changes are necessary.",
"3) There could be more empty storage locations than outgoing drive units.",
"In this case, no changes are necessary.",
"CBM finds solutions for 40 of the 50 Kiva instances within a time limit of 5 minutes each, yielding a success rate of 80%.",
"The mean of the makespan over the solved Kiva instances is 63.73, and the mean of the running time is 91.61 seconds.",
"Since early Kiva warehouse systems typically had about 200 drive units in more spacious (and thus less challenging) warehouses and even bounded-suboptimal (non-anonymous) MAPF algorithms that were specifically designed for simulated Kiva warehouse systems do not scale well to hundreds of agents [4], we conclude that CBM is a promising TAPF algorithm for applications of real-world scale."
],
[
"Conclusions",
"In this paper, we studied the TAPF (combined target-assignment and path-finding) problem for teams of agents in known terrain to bridge the gap between the extreme cases of anonymous and non-anonymous MAPF problems, as required by many applications.",
"We presented CBM, a hierarchical algorithm that is correct, complete and optimal for solving the TAPF problem.",
"CBM outperforms (non-anonymous) MAPF algorithms in terms of both scalability and solution quality in our experiments.",
"It also generalizes to applications with dozens of teams and hundreds of agents, which demonstrates its promise."
],
[
"Acknowledgments",
"We thank Jingjin Yu for making the code of their ILP-based MAPF solver and Guni Sharon for making the code of their CBS solver available to us.",
"Our research was supported by NASA via Stinger Ghaffarian Technologies as well as NSF under grant numbers 1409987 and 1319966 and a MURI under grant number N00014-09-1-1031.",
"The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the sponsoring organizations, agencies or the U.S. government."
]
] | 1612.05693 |
[
[
"Error Estimates for the Kernel Gain Function Approximation in the\n Feedback Particle Filter"
],
[
"Abstract This paper is concerned with the analysis of the kernel-based algorithm for gain function approximation in the feedback particle filter.",
"The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian.",
"The kernel-based method -- introduced in our prior work -- allows one to approximate this solution using {\\em only} particles sampled from the probability distribution.",
"This paper describes new representations and algorithms based on the kernel-based method.",
"Theory surrounding the approximation is improved and a novel formula for the gain function approximation is derived.",
"A procedure for carrying out error analysis of the approximation is introduced.",
"Certain asymptotic estimates for bias and variance are derived for the general nonlinear non-Gaussian case.",
"Comparison with the constant gain function approximation is provided.",
"The results are illustrated with the aid of some numerical experiments."
],
[
"Introduction",
"This paper is concerned with the analysis of the kernel-based algorithm for numerical approximation of the gain function in the feedback particle filter algorithm; cf. [11].",
"The filter represents a numerical solution of the following continuous-time nonlinear filtering problem: $&\\text{Signal:} \\quad &\\,\\mathrm {d}X_t &= a(X_t)\\,\\mathrm {d}t + \\,\\mathrm {d}B_t,\\quad X_0\\sim p_0^*,&\\\\&\\text{Observation:} \\quad &\\,\\mathrm {d}Z_t &= h(X_t)\\,\\mathrm {d}t + \\,\\mathrm {d}W_t,&$ where $X_t\\in \\mathbb {R}^d$ is the (hidden) state at time $t$ , the initial condition $X_0$ has the prior density $p_0^*$ , $Z_t \\in \\mathbb {R}$ is the observation, and $\\lbrace B_t\\rbrace $ , $\\lbrace W_t\\rbrace $ are mutually independent standard Wiener processes taking values in $\\mathbb {R}^d$ and $\\mathbb {R}$ , respectively.",
"The mappings $a(\\cdot ): \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ and $h(\\cdot ): \\mathbb {R}^d \\rightarrow \\mathbb {R}$ are given $C^1$ functions.",
"The goal of the filtering problem is to approximate the posterior distribution of the state $X_t$ given the time history of observations (filtration) ${\\cal Z}_t :=\\sigma (Z_s: 0\\le s \\le t)$ .",
"The feedback particle filter (FPF) is a controlled stochastic differential equation (sde), $\\begin{aligned}&\\text{FPF:} \\nonumber \\\\&\\,\\mathrm {d}X^i_t = a(X^i_t) \\,\\mathrm {d}t + \\,\\mathrm {d}B^i_t + {\\sf K}_t(X^i_t) \\circ (\\,\\mathrm {d}Z_t -\\frac{h(X^i_t) + \\hat{h}_t}{2}\\,\\mathrm {d}t),\\;\\; X_0^i \\sim p_0^*, \\end{aligned}$ for $i=1,\\ldots ,N$ , where $X_t^i\\in \\mathbb {R}^d$ is the state of the $i^\\text{th}$ particle at time $t$ , the initial condition $X^i_0\\sim p_0^*$ , $B^i_t$ is a standard Wiener process, and $\\hat{h}_t := {\\sf E}[h(X_t^i)|\\mathcal {Z}_t]$ .",
"Both $B^i_t$ and $X^i_0$ are mutually independent and also independent of $X_t,Z_t$ .",
"The $\\circ $ indicates that the sde is expressed in its Stratonovich form.",
"The gain function ${\\sf K}_t$ is obtained by solving a weighted Poisson equation: For each fixed time $t$ , the function $\\phi $ is the solution to a Poisson equation, $\\text{PDE:} && &\\begin{aligned}\\nabla \\cdot (p(x,t) \\nabla \\phi (x,t) ) & = - (h(x)-\\hat{h}) p(x,t),\\\\\\int \\phi (x,t) p(x,t) \\,\\mathrm {d}x & = 0 \\qquad \\text{(zero-mean)},\\end{aligned} &$ where $\\nabla $ and $\\nabla \\cdot $ denote the gradient and the divergence operators, respectively, and $p$ denotes the conditional density of $X_t^i$ given $\\mathcal {Z}_t$ .",
"In terms of the solution $\\phi $ , the gain function is given by, $&\\text{Gain Function:}& \\quad {\\sf K}_t(x)&= \\nabla \\phi (x,t)\\, .",
"&$ The gain function ${\\sf K}_t$ is vector-valued (with dimension $d\\times 1$ ) and it needs to be obtained for each fixed time $t$ .",
"For the linear Gaussian case, the gain function is the Kalman gain.",
"FPF is an exact algorithm: If the initial condition $X^i_0$ is sampled from the prior $p_0^*$ then ${\\sf P}[X_t \\in A\\mid {\\cal Z}_t ] = {\\sf P}[X_t^i \\in A\\mid {\\cal Z}_t ], \\quad \\forall \\;A\\subset \\mathbb {R}^d,\\;\\;t>0.$ In a numerical implementation, a finite number, $N$ , of particles is simulated and ${\\sf P}[X_t^i \\in A\\mid {\\cal Z}_t ] \\approx \\frac{1}{N}\\sum _{i=1}^N \\text{\\rm \\large 1}[ X^i_t\\in A]$ by the Law of Large Numbers (LLN).",
"The challenging part in the numerical implementation of the FPF algorithm is the solution of the PDE (REF ).",
"This has been the subject of a number of recent studies: In our original FPF papers, a Galerkin numerical method was proposed; cf., [13], [14].",
"A special case of the Galerkin solution is the constant gain approximation formula which is often a popular choice in practice [13], [10], [12], [2].",
"The main issue with the Galerkin approximation is to choose the basis functions.",
"A proper orthogonal decomposition (POD)-based procedure to select basis functions is introduced in [3] and certain continuation schemes appear in [8].",
"Apart from the Galerkin procedure, probabilistic approaches based on dynamic programming appear in [9].",
"In a recent work, we introduced a basis-free kernel-based algorithm for approximating the solution of the gain function [11].",
"The key step is to construct a Markov matrix on the $N$ -node graph defined by the $N$ particles $\\lbrace X^i_t\\rbrace _{i=1}^N$ .",
"The value of the function $\\phi $ for the particles, $\\phi (X^i_t)$ , is then approximated by solving a fixed-point problem involving the Markov matrix.",
"The fixed-point problem is shown to be a contraction and the method of successive approximation applies to numerically obtain the solution.",
"The present paper presents a continuation and refinement of the analysis for the kernel-based method.",
"The contributions are as follows: A novel formula for the gain function is derived for the kernel-based approximation.",
"A procedure for carrying out error analysis of the approximation is introduced.",
"Certain asymptotic estimates for bias and variance are derived for the general nonlinear non-Gaussian case.",
"Comparison with the constant gain approximation formula are provided.",
"These results are illustrated with the aid of some numerical experiments.",
"The outline of the remainder of this paper is as follows: The mathematical problem of the gain function approximation together with a summary of known results on this topic appears in Sec. .",
"The kernel-based algorithm including the novel formula for gain function, referred to as (G2), appears in Sec. .",
"The main theoretical results of this paper including the bias and variance estimates appear in Sec. .",
"Some numerical experiments for the same appear in Sec. .",
"Notation.",
"$\\mathbb {Z}_+$ denotes the set of positive integers and $\\mathbb {Z}_+^d$ is the set of $d$ -tuples.",
"For vectors $x,y\\in \\mathbb {R}^d$ , the dot product is denoted as $x\\cdot y$ and $|x|:=\\sqrt{x\\cdot x}$ .",
"Throughout the paper, it is assumed that the probability measures admit a smooth Lebesgue density.",
"A density for a Gaussian random variable with mean $\\mu $ and variance $\\Sigma $ is denoted as ${\\cal N}(\\mu ,\\Sigma )$ .",
"$C^k$ is used to denote the space of $k$ -times continuously differentiable functions.",
"For a function $f$ , $\\nabla f = \\frac{\\partial f}{\\partial x_i}$ is used to denote the gradient.",
"$L^2(\\mathbb {R}^d,\\rho )$ is the Hilbert space of square integrable functions on $\\mathbb {R}^d$ equipped with the inner-product, $\\big <\\phi ,\\psi \\big >_{L^2}:=\\int \\phi (x)\\psi (x) \\rho (x)\\,\\mathrm {d}x$ .",
"The associated norm is denoted as $\\Vert \\phi \\Vert ^2_2:=\\big <\\phi ,\\phi \\big >$ .",
"The space $H^1(\\mathbb {R}^d,\\rho )$ is the space of square integrable functions $\\phi $ whose derivative (defined in the weak sense) is in $L^2(\\mathbb {R}^d,\\rho )$ .",
"For the remainder of this paper, $L^2$ and $H^1$ is used to denote $L^2(\\mathbb {R}^d,\\rho )$ and $H^1(\\mathbb {R}^d,\\rho )$ , respectively.",
"The Poisson equation (REF ) is expressed as, $\\text{PDE} && &\\begin{aligned}-\\Delta _\\rho \\phi & = h-\\hat{h},\\\\\\int \\phi \\rho \\,\\mathrm {d}x & = 0 \\qquad \\text{(zero-mean)},\\end{aligned} &$ where $\\rho $ is a probability density on $\\mathbb {R}^d$ , $\\Delta _\\rho \\phi :=\\frac{1}{\\rho }\\nabla \\cdot (\\rho \\nabla \\phi )$ .",
"The gain function ${\\sf K}(x):=\\nabla \\phi (x)$ .",
"Problem statement: Given $N$ independent samples $\\lbrace X^1,\\hdots ,X^i,\\hdots ,X^N\\rbrace $ drawn from $\\rho $ , approximate the gain function $\\lbrace {\\sf K}(X^1),\\hdots ,{\\sf K}(X^i),\\hdots ,{\\sf K}(X^N)\\rbrace $ .",
"The density $\\rho $ is not explicitly known.",
"The appropriate function space for the solutions of (REF ) is the co-dimension 1 subspace $L^2_0:=\\lbrace \\phi \\in L^2; \\int \\phi \\rho \\,\\mathrm {d}x =0\\rbrace $ and $H^1_0 :=\\lbrace \\phi \\in H^1; \\int \\phi \\rho \\,\\mathrm {d}x =0\\rbrace $ ; cf.",
"[7], [13]."
],
[
"Existence-Uniqueness",
"On multiplying both side of (REF ) by test function $\\psi $ , one obtains the weak-form of the PDE: $\\int \\nabla \\phi \\cdot \\nabla \\psi \\;\\rho \\,\\mathrm {d}x = \\int (h-\\hat{h}) \\psi \\; \\rho \\,\\mathrm {d}x,\\quad \\forall \\psi \\in H^1.$ The following is assumed throughout the paper: (i) Assumption A1: The probability density is of the form $\\rho (x)= e^{-V(x)}$ where $V \\in C^2$ with $\\liminf _{x\\rightarrow \\infty }\\;\\; [-\\Delta V(x) + \\frac{1}{2}|\\nabla V(x)|^2]=\\infty .$ (ii) Assumption A2: The function $h,\\nabla h \\in L^2$ .",
"Under the Assumption A1, the density $\\rho $ admits a spectral gap (or Poincaré inequality) ([1] Thm 4.6.3), i.e., $\\exists \\lambda _1 >0$ such that, $\\int f^2 \\, \\rho \\,\\mathrm {d}x \\le \\frac{1}{\\lambda _1} \\int |\\nabla f|^2 \\, \\rho \\,\\mathrm {d}x ,\\quad \\forall f \\in H^1_0.$ The Poincaré inequality implies the existence and uniqueness of a weak solution to the weighted Poisson equation.",
"Theorem 1 [Theorem 2.2 in [7]].",
"Assume (A1)-(A2).",
"Then there exists a unique weak solution $\\phi \\in H_0^1(\\mathbb {R}^d;\\rho )$ satisfying (REF ).",
"Moreover, the gain function ${\\sf K}=\\nabla \\phi $ is controlled by the size of the data: $\\int |{\\sf K}|^2 \\;\\rho \\,\\mathrm {d}x \\le \\frac{1}{\\lambda _1} \\int |h-\\hat{h}|^2 \\;\\rho \\,\\mathrm {d}x.$ There are two special cases where the exact solution can be found: (i) Scalar case where the state dimension $d=1$ ; (ii) Gaussian case where the density $\\rho $ is a Gaussian.",
"The results for these two special cases appear in the following two subsections."
],
[
"Exact Solution in the Scalar Case",
"In the scalar case (where $d=1$ ), the Poisson equation is: $-\\frac{1}{\\rho (x)}\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}x}(\\rho (x) \\frac{\\,\\mathrm {d}\\phi }{\\,\\mathrm {d}x}(x)) = h-\\hat{h}.$ Integrating twice yields the solution explicitly, $\\begin{aligned}{\\sf K}(x) = \\frac{\\,\\mathrm {d}\\phi }{\\,\\mathrm {d}x}(x) &= -\\frac{1}{\\rho (x)}\\int _{-\\infty }^x\\rho (z)(h(z)-\\hat{h})\\,\\mathrm {d}z.\\end{aligned}$ For the particular choice of $\\rho $ as the sum of two Gaussians ${\\cal N}(-1,\\sigma ^2)$ and ${\\cal N}(+1,\\sigma ^2)$ with $\\sigma ^2=0.2$ and $h(x)=x$ , the solution obtained using (REF ) is depicted in Fig.",
"REF .",
"Figure: The exact solution to the Poisson equation using theformula ().",
"The density ρ\\rho is the sum of two GaussiansN(-1,σ 2 )N(-1,\\sigma ^2) and N(+1,σ 2 )N(+1,\\sigma ^2), and h(x)=xh(x)=x.",
"The density is depicted as the shaded curve in the background."
],
[
"Exact Spectral Solution for the Gaussian Density",
"Under Assumption (A1), the spectrum is known to be discrete with an ordered sequence of eigenvalues $0=\\lambda _0<\\lambda _1\\le \\lambda _2\\le \\hdots $ and associated eigenfunctions $\\lbrace e_n\\rbrace $ that form a complete orthonormal basis of $L^2$ [Corollary 4.10.9 in [1]].",
"The trivial eigenvalue $\\lambda _0=0$ with associated eigenfunction $e_0=1$ .",
"On the subspace of zero-mean functions, the spectral decomposition yields: For $\\phi \\in L^2_0$ , $-\\Delta _\\rho \\phi = \\sum _{m=1}^\\infty \\lambda _m <e_m,\\phi >e_m.$ The spectral gap condition (REF ) implies that $\\lambda _1>0$ .",
"The spectral representation (REF ) yields the following closed-form solution of the PDE (REF ): $\\phi = \\sum _{m=1}^N \\frac{1}{\\lambda _m}<e_m,h-\\hat{h}>e_m.$ The spectral representation formula (REF ) is used to obtain the exact solution for the Gaussian case where the eigenvalues and the eigenfunctions are explicitly known in terms of Hermite polynomials.",
"Definition 1 The Hermite polynomials are recursively defined as $\\hbar _{n+1}(x) = 2 \\, \\hbar _{n}(x) - \\hbar ^{\\prime }_n(x),\\quad \\hbar _0(x) = 1,$ where the prime $^{\\prime }$ denotes the derivative.",
"Proposition 1 Suppose the density $\\rho $ is Gaussian ${\\cal }(\\mu ,\\Sigma )$ where the mean $\\mu \\in \\mathbb {R}^d$ and the covariance $\\Sigma $ is assumed to be a strictly positive definite symmetric matrix.",
"Express $\\Sigma = VDV^T$ where $D=\\text{diag}(\\sigma _1^2,\\ldots ,\\sigma _d^2)$ and $V = [V_1|\\hdots |V_j|\\hdots |V_d]$ is an orthonormal matrix with the $j^{\\text{th}}$ column denoted as $V_j \\in \\mathbb {R}^d$ .",
"For $n=(n_1,\\ldots ,n_d) \\in \\mathbb {Z}_+^d$ , (i) The eigenvalues are, $\\lambda _n = \\sum _{j=1}^d\\frac{n_j}{\\sigma _j^2}.$ (ii) The corresponding eigenfunctions are, $e_n (x) = \\prod _{j=1}^d \\hbar _{n_j}(\\frac{V_j \\cdot (x-\\mu )}{\\sigma _j}),$ where $\\hbar _{n_j}$ is the Hermite polynomial.",
"An immediate corollary is that the first non-zero eigenvalue is $\\frac{1}{\\sigma _{\\text{max}}^2}$ and the corresponding eigenfunction is $\\psi (x)=V_{\\text{max}} \\cdot (x-\\mu )$ , where $\\sigma _{\\text{max}}^2$ is the largest eigenvalue of the covariance matrix and $V_{\\text{max}}\\in \\mathbb {R}^d$ is the corresponding eigenvector.",
"Example 1 Suppose the density $\\rho $ is a Gaussian ${\\cal }(\\mu ,\\Sigma )$ .",
"(i) The observation function $h(x) =H \\cdot x$ , where $H \\in \\mathbb {R}^d$ .",
"Then, $\\phi = \\Sigma H\\cdot x$ and the gain function ${\\sf K}= \\Sigma \\, H$ is the Kalman gain.",
"(ii) Suppose $d=2$ , $\\mu = [0,0]$ , $\\Sigma =\\begin{bmatrix} \\sigma _1^2& 0 \\\\0&\\sigma _2^2\\end{bmatrix}$ , and the observation function $h(x_1,x_2)=x_1\\,x_2$ .",
"Then, $\\phi (x) =\\frac{1}{\\frac{1}{\\sigma _1^2}+\\frac{1}{\\sigma _2^2}}x_1x_2\\quad \\text{and}\\quad {\\sf K}(x_1,x_2) =\\frac{1}{\\frac{1}{\\sigma _1^2}+\\frac{1}{\\sigma _2^2}} \\begin{bmatrix}x_2 \\\\ x_1\\end{bmatrix}.$ In the general non-Gaussian case, the solution is not known in an explicit form and must be numerically approximated.",
"Note that even in the two exact cases, one may need to numerically approximate the solution because the density is not given in an explicit form.",
"A popular choice is the constant gain approximation briefly described next.",
"Figure: Constant gain approximation: Approximating nonlinear 𝖪{\\sf K} byits expected value 𝖤[𝖪]{\\sf E}[{\\sf K}]."
],
[
"Constant gain approximation",
"The constant gain approximation is the best – in the least-square sense – constant approximation of the gain function (see Fig.",
"REF ).",
"Precisely, consider the following least-square optimization problem: $\\kappa ^\\ast = \\arg \\min _{\\kappa \\in \\mathbb {R}^d} {\\sf E}_\\rho [|{\\sf K}- \\kappa |^2].$ By using a standard sum of square argument, $\\kappa ^\\ast = {\\sf E}_\\rho [{\\sf K}].$ The expected value admits an explicit formula: In the weak-form (REF ), choose the test functions to be the coordinate functions: $\\psi _k(x) = x_k$ for $k=1,2,\\hdots ,d$ .",
"Writing $\\psi (x) = (\\psi _1,\\psi _2,\\hdots ,\\psi _d)^T = x$ , $\\kappa ^* = {\\sf E}_\\rho [{\\sf K}] &= {\\sf E}_\\rho [ (h-\\hat{h}) \\psi ] = \\int _{\\mathbb {R}^d} (h(x)-\\hat{h})\\;x \\; \\rho (x) \\,\\mathrm {d}x.$ On computing the integral using only the particles, one obtains the formula for the gain function approximation: ${\\sf K}\\approx \\frac{1}{N}\\sum _{i=1}^N\\; (h(X^i)-\\hat{h}^{(N)}) \\; X^i,$ where $\\hat{h}^{(N)} = N^{-1} \\sum _{i=1}^N h(X^i)$ .",
"This formula is referred to as the constant gain approximation of the gain function; cf., [13].",
"It is a popular choice in applications [13], [10], [12], [2].."
],
[
"Kernel-based Approximation",
"Semigroup: The spectral gap condition (REF ) implies that $\\lambda _1>0$ .",
"Consequently, the semigroup $e^{t\\Delta _\\rho }\\phi := \\sum _{m=1}^{\\infty } e^{-t\\lambda _m}<e_m,\\phi >e_m$ is a strict contraction on the subspace $L^2_0$ .",
"It is also easy to see that $\\mu $ is an invariant measure and $\\int e^{t\\Delta _\\rho }\\phi (x)\\,\\mathrm {d}\\mu (x) = \\int \\phi (x) \\,\\mathrm {d}\\mu (x) = 0$ for all $\\phi \\in L^2_0$ .",
"The semigroup formula (REF ) is used to obtain the solution of the Poisson equation (REF ) by solving the following fixed-point equation for any fixed positive value of $t$ : $\\phi = e^{t\\Delta _\\rho } \\phi + \\int _0^t e^{s\\Delta _\\rho } (h-\\hat{h}) \\,\\mathrm {d}s.$ A unique solution exists because $e^{t\\Delta _\\rho }$ is a contraction on $L^2_0$ .",
"Kernel-based method: In the kernel-based algorithm, one approximates the solution of the fixed point problem (REF ) by approximating the semigroup by an integral operator for $t=\\epsilon $ .",
"The approximation, introduced in [11], has three main steps: $\\text{Exact}:&~&\\phi &= e^{\\epsilon \\Delta _\\rho } \\phi + \\int _0^\\epsilon e^{s\\Delta _\\rho } (h-\\hat{h}) \\,\\mathrm {d}s \\\\\\text{Kernel approx:}&~ &{\\phi }_{\\epsilon }&= {T}_{\\epsilon }{\\phi }_{\\epsilon }+ \\int _0^\\epsilon T_s(h-\\hat{h}) \\,\\mathrm {d}s \\\\\\text{Empirical approx:}&~ &{\\phi }^{(N)}_\\epsilon &= {T}^{(N)}_\\epsilon {\\phi }^{(N)}_\\epsilon +\\int _0^{\\epsilon }T_s^{(N)} (h-\\hat{h}) \\,\\mathrm {d}s$ The justification for these steps is as follows: (i) A solution of the Poisson equation (REF ) is also a solution of the fixed-point problem (REF ) where $\\epsilon >0$ is arbitrary.",
"A unique solution exists because $e^{\\epsilon \\Delta _\\rho } $ is contraction on $L^2_0$ .",
"(ii) The Kernel approximation () involves approximating the semigroup $e^{\\epsilon \\Delta _\\rho }$ by an integral operator ${T}_{\\epsilon }$ , ${T}_{\\epsilon }f := \\int _{\\mathbb {R}^d}k_\\epsilon (x,y)f(y)\\rho (y)\\,\\mathrm {d}y$ where the exact form of ${k}_{\\epsilon }$ appears in the Appendix, where it is also shown that $e^{\\epsilon \\Delta _\\rho } \\approx {T}_{\\epsilon }$ as $\\epsilon \\downarrow 0$ .",
"The approximation of the semigroup by the integral operator appears in [4], [5].",
"(iii) The empirical approximation () involves approximating the integral operator empirically in terms of the particles, ${T}^{(N)}_\\epsilon f(x) := \\frac{1}{N}\\sum _{i=1}^N {k}_{\\epsilon }^{(N)}(x,X^i)f(X^i),$ justified by the LLN.",
"The gain ${{\\sf K}}_\\epsilon ^{(N)}$ is computed by taking the gradient of the fixed-point equation ().",
"For this purpose, denote, $\\begin{aligned}\\nabla {T}^{(N)}_\\epsilon f (x) &:= \\frac{1}{N}\\sum _{j=1}^N \\nabla {k}_{\\epsilon }^{(N)}(x,X^j)f(X^j) \\\\&= \\frac{1}{2\\epsilon } \\bigg [\\frac{1}{N}\\sum _{i=1}^N {k}_{\\epsilon }^{(N)}(x,X^i)X^if(X^i) \\\\- &\\left(\\frac{1}{N}\\sum _{i=1}^N {k}_{\\epsilon }^{(N)}(x,X^i)X^i\\right)\\left(\\frac{1}{N}\\sum _{i=1}^N {k}_{\\epsilon }^{(N)}(x,X^i)f(X^i)\\right)\\bigg ].\\end{aligned}$ Next, two approximate formulae for ${{\\sf K}}_\\epsilon ^{(N)}$ are presented based on two different approximations of the integral $\\int _0^\\epsilon T_s^{(N)}(h-\\hat{h}) \\,\\mathrm {d}s$ : Approximation 1: The integral term is approximated by $\\epsilon (h-\\hat{h})$ and the resulting formula for the gain is, $&\\text{(G1)}&\\quad {K}_\\epsilon ^{(N)}(x)&:= \\nabla {T}^{(N)}_\\epsilon {\\phi }^{(N)}_\\epsilon (x) + \\epsilon \\nabla h(x).&$ By approximating the integral term differently, one can avoid the need to take a derivative of $h$ .",
"Approximation 2: The integral term is approximated by ${T}^{(N)}_\\epsilon (h-\\hat{h})$ .",
"The resulting formula for the gain is, $&(\\text{G2})&\\quad {K}_\\epsilon ^{(N)}(x)&:=\\nabla {T}^{(N)}_\\epsilon {\\phi }^{(N)}_\\epsilon (x) + \\epsilon \\nabla {T}^{(N)}_\\epsilon (h-\\hat{h})(x).&$ Remark 1 Although $\\nabla {T}^{(N)}_\\epsilon $ and ${K}_\\epsilon ^{(N)}$ are ultimately important in the numerical algorithm (described next), it is useful to introduce the limiting (as $N\\rightarrow \\infty $ ) variables $\\nabla {T}_{\\epsilon }$ and ${K}_\\epsilon $ .",
"The operator $\\nabla {T}_{\\epsilon }$ is defined as follows: $\\begin{aligned}\\nabla {T}_{\\epsilon }f(x) &= \\int _{\\mathbb {R}^d} \\nabla _x{k}_{\\epsilon }(x,y) f(y)\\rho (y)\\,\\mathrm {d}y \\\\&= \\frac{1}{2\\epsilon }\\bigg [{T}_{\\epsilon }(ef)-{T}_{\\epsilon }(e){T}_{\\epsilon }(f)\\bigg ](x)\\end{aligned}$ where $e$ is the identity function $e(x)=x$ .",
"In terms of $\\nabla {T}_{\\epsilon }$ , the gain function ${K}_\\epsilon $ is defined by taking of the gradient of the fixed-point equation ().",
"This leads to the limiting counterpart of the approximation (G1) and (G2).",
"In particular, analogous to (G2), ${K}_\\epsilon (x):=\\nabla {T}_{\\epsilon }{\\phi }_{\\epsilon }(x) + \\epsilon \\nabla {T}_{\\epsilon }(h-\\hat{h})(x),$ where ${\\phi }_{\\epsilon }$ is the solution of the fixed-point equation ().",
"Numerical Algorithm: A numerical implementation involves the following steps: (i) Assemble a $N\\times N$ Markov matrix to approximate the finite rank operator ${T}^{(N)}_\\epsilon $ in (REF ).",
"The $(i,j)$ -entry of the matrix is given by, ${\\bf T}_{ij}= \\frac{1}{N}\\sum _{j=1}^N {k}_{\\epsilon }^{(N)}(X^i,X^j).$ (ii) Use the method of successive approximation to solve the discrete counterpart of the fixed-point equation (), $\\Phi = {\\bf T}\\Phi + \\epsilon ({\\bf h}-\\hat{h}^{(N)})$ where $\\Phi :=[{\\phi }^{(N)}_\\epsilon (X^1),\\ldots ,{\\phi }^{(N)}_\\epsilon (X^N)] \\in \\mathbb {R}^N$ is the (unknown) solution, ${\\bf h}:=[h(X^1),\\ldots ,h(X^N)]\\in \\mathbb {R}^N$ is given, and $\\hat{h}^{(N)} = \\frac{1}{N}\\sum _{i=1}^Nh(X^i)$ .",
"In filtering applications, the solution from the previous time-step is typically used to initialize the algorithm.",
"(iii) Once $\\Phi $ has been computed, the gain function $\\lbrace {\\sf K}(X^1),\\hdots ,{\\sf K}(X^i),\\hdots ,{\\sf K}(X^N)\\rbrace $ is obtained by using either (G1) or (G2).",
"Note that the discrete counterpart of $\\nabla {T}^{(N)}_\\epsilon $ is obtained using the Markov matrix ${\\bf T}$ .",
"The overall algorithm is tabulated as Algorithm 1 where (G2) is used for the gain function approximation.",
"Kernel-based gain function approximation $\\lbrace X^i\\rbrace _{i=1}^N$ , $H:=\\lbrace h(X^i)\\rbrace _{i=1}^N$ ,$\\Phi _0:=\\lbrace \\phi _0(X^i)\\rbrace _{i=1}^N$ $\\Phi :=\\lbrace \\phi (X^i)\\rbrace _{i=1}^N$ , $\\lbrace \\nabla \\phi (X^i)\\rbrace _{i=1}^N$ Calculate $g_{ij}:=\\exp (-|X^i-X^j|^2/4\\epsilon )$ for $i,j=1$ to $N$ .",
"Calculate $k_{ij}:=\\frac{g_{ij}}{\\sqrt{\\sum _l g_{il}}\\sqrt{\\sum _l g_{jl}}}$ for $i,j=1$ to $N$ .",
"Calculate $T_{ij}:=\\frac{k_{ij}}{\\sum _l k_{il}}$ for $i,j=1$ to $N$ .",
"Calculate $\\hat{h}^{(N)}=\\frac{1}{N}\\sum _{i=1}^N H_i$ .",
"$t=1$ to $T$ Solve $\\Phi _{t}= T \\Phi _{t-1} + \\epsilon (H-\\hat{h})$ .",
"$\\Phi _{t} = \\Phi _t - \\frac{1}{N}\\sum _{i=1}^N \\Phi _{t,i}$ Calculate ${\\sf K}(X^i)= \\frac{1}{2\\epsilon }\\sum _{j=1}^N \\left[T_{ij}(\\Phi _j+\\epsilon (H_j - \\hat{h}))\\left(X^j- \\sum _{k=1}^N T_{ik}X^k\\right)\\right]$"
],
[
"Error Analysis",
"The objective is to characterize the approximation error $ {\\sf E} [\\Vert {\\sf K}_\\epsilon ^{(N)} - {\\sf K}\\Vert _2]$ .",
"Using the triangle inequality, $ {\\sf E} [\\Vert {\\sf K}_\\epsilon ^{(N)} - {\\sf K}\\Vert _2] \\;\\; \\le \\;\\;\\underbrace{ {\\sf E} [\\Vert {\\sf K}_\\epsilon ^{(N)} -{\\sf K}_\\epsilon \\Vert _2]}_{\\text{Variance}} \\; +\\; \\underbrace{\\Vert {\\sf K}_\\epsilon - {\\sf K}\\Vert _2}_{\\text{Bias}},$ where ${\\sf K}=\\nabla \\phi $ denotes the exact gain function, and ${\\sf K}_\\epsilon (x)=\\nabla \\phi _{\\epsilon }(x)$ and ${\\sf K}_\\epsilon ^{(N)}(x)=\\nabla \\phi _{\\epsilon }^{(N)}(x)$ are defined by taking the gradient of the fixed-point equation () and (), respectively.",
"The following Theorem provides error estimates for the gain function in the asymptotic limit as $\\epsilon \\downarrow 0$ and $N \\rightarrow \\infty $ .",
"These estimates apply to either of the two approximations, (G1) or (G2), used to obtain the gain function.",
"A sketch of the proof appears in the Appendix.",
"Theorem 2 Suppose the assumptions (A1)-(A2) hold for the density $\\rho $ and the function $h$ , with spectral gap constant $\\lambda _1$ .",
"Then (Bias) In the asymptotic limit as $\\epsilon \\downarrow 0$ , $\\Vert {\\sf K}_\\epsilon - {\\sf K}\\Vert _2 \\; \\le \\; C\\epsilon + \\text{h.o.t.",
"},$ (Variance) In the asymptotic limit as $\\epsilon \\downarrow 0$ and $N \\rightarrow \\infty $ , $\\begin{aligned} {\\sf E} [\\Vert {\\sf K}_\\epsilon - {\\sf K}_\\epsilon ^{(N)}\\Vert _2]&\\; \\le \\;\\frac{C}{N^{1/2}\\epsilon ^{1+d/4}} + \\text{h.o.t.",
"},\\end{aligned}$ where the constant $C$ depends upon the function $h$ .",
"Figure: The figure shows explicit error ∥𝖪 ϵ -𝖪∥ 2 \\Vert {\\sf K}_\\epsilon - {\\sf K}\\Vert _2"
],
[
"Difference between (G1) and (G2)",
"In the asymptotic limit as $\\epsilon \\downarrow 0$ , the two approximations (G1) and (G2) yield identical error estimates.",
"The difference arises as $\\epsilon $ becomes larger.",
"The following Proposition provides explicit error estimates for the bias in the special linear Gaussian case.",
"Proposition 2 Suppose the density $\\rho $ is a Gaussian $N(0,\\sigma ^2I)$ and $h(x)=H\\cdot x$ .",
"Then the bias for the two approximations is given by the following closed-form formula: $\\text{Bias for (G1):} \\quad \\Vert {\\sf K}_\\epsilon - {\\sf K}\\Vert _2 &= \\epsilon \\frac{\\sigma ^2 - 4\\epsilon }{\\sigma ^2 + 4\\epsilon } |H|, \\\\\\text{Bias for (G2):} \\quad \\Vert {\\sf K}_\\epsilon - {\\sf K}\\Vert _2 &= \\epsilon \\frac{\\sigma ^6}{(\\sigma ^2+4\\epsilon )(\\sigma ^4+3\\epsilon \\sigma ^2 + 4\\epsilon ^2)} |H|.",
"$ Figure: Numerical gain function approximation (G2) for a range ofϵ\\epsilon .",
"The dimension d=1d=1 and the number of particlesN=200N=200.Note that the bias has the same scaling, $\\sim \\epsilon |H|$ , as $\\epsilon \\downarrow 0$ .",
"However as $\\epsilon $ gets larger, the two approximations behave very differently.",
"For (G1), the bias grows unbounded as $\\epsilon \\rightarrow \\infty $ .",
"Remarkably, for (G2), the bias goes to zero as $\\epsilon \\rightarrow \\infty $ .",
"Figure REF depicts the bias error for a scalar example where $\\sigma ^2=1$ and $H=1$ .",
"The following Proposition shows that the limit $\\epsilon \\rightarrow \\infty $ is well-behaved for the (G2) approximation more generally.",
"In fact, one recovers the constant gain approximation in that limit.",
"Proposition 3 Consider the gain approximation (G2) given by (REF ).",
"Then, $\\lim _{\\epsilon \\rightarrow \\infty } \\;\\;{\\sf K}_{\\epsilon } &= {\\sf E}[K],\\\\\\lim _{\\epsilon \\rightarrow \\infty } \\;\\;{\\sf K}_{\\epsilon }^{(N)} &= \\frac{1}{N}\\sum _{i=1}^N(h(X^i)-\\hat{h}^{(N)})X^i.$"
],
[
"Numerics",
"Suppose the density $\\rho $ is a mixture of two Gaussians, $\\frac{1}{2}{\\cal N}(-\\mu ,\\sigma ^2I) + \\frac{1}{2}{\\cal N}(+\\mu ,\\sigma ^2I)$ , where $\\mu =[1,0,\\ldots ,0]\\in \\mathbb {R}^d$ , and $\\sigma ^2=0.2$ .",
"The observation function $h(x)=x_1$ .",
"In this case, the exact gain function ${\\sf K}(x) = [{\\sf K}_1(x),0,\\ldots ,0]$ where ${\\sf K}_1(\\cdot )$ is obtained using the explicit formula (REF ) as in the scalar case.",
"Figure REF depicts a comparison between the exact solution and the approximate solution obtained using the kernel approximation formula (G2).",
"The dimension $d=1$ and the number of particles $N=200$ .",
"The part (a) of the figure depicts the gain function for a range of (relatively large) $\\epsilon $ values $\\lbrace 0.1,0.2,0.4,0.8\\rbrace $ where the error is dominated by the bias.",
"The constant gain approximation is also depicted and, consistent with Proposition REF , the (G2) approximation converges to the constant as $\\epsilon $ gets larger.",
"The part (b) of the figure depicts a comparison for a range of (very small) $\\epsilon $ values $\\lbrace 0.01,0.001\\rbrace $ .",
"At $N=200$ particles, the error in this range is dominated by the variance.",
"This is manifested in a somewhat irregular spread of the particles for these $\\epsilon $ values.",
"In the next study, we experimentally evaluated the error for a range of $\\epsilon $ and $d$ , again with a fixed $N=200$ .",
"For a single simulation, the error is defined as $\\text{Error}:=\\sqrt{\\frac{1}{N}\\sum _{i=1}^N |{\\sf K}_\\epsilon ^{(N)}(X^i)-{\\sf K}(X^i)|^2}.$ Figure REF and REF depict the averaged error obtained from averaging over $M=100$ simulations.",
"In each simulation, the parameters $\\epsilon $ and $d$ are fixed but a different realization of $N=200$ particles is sampled from the density $\\rho $ .",
"Figure REF depicts the averaged error as $\\epsilon $ and $d$ are varied.",
"As $\\epsilon $ becomes large, the kernel gain converges the constant gain formula.",
"For relatively large values of $\\epsilon $ , the error is dominated by bias which is insensitive to the size of dimension $d$ .",
"Figure REF depicts the averaged error for small values of $\\epsilon $ .",
"The logarithmic scale is used to better assess the asymptotic characteristics of the error as a function of $\\epsilon $ and $d$ .",
"Recall that the estimates in Theorem REF predict that the error scales as $\\epsilon ^{-1-d/4}$ for the small $\\epsilon $ large $N$ limit.",
"To verify the prediction, an empirical exponent was computed by fitting a linear curve to the error data on the logarithmic scale.",
"The empirical exponents together with the error estimates predicted by Theorem REF are tabulated in Table REF .",
"It is observed that the empirical exponents are smaller than the predictions.",
"The gap suggests that the error bound may not be tight.",
"A more thorough comparison is a subject of continuing investigation.",
"Table: Comparison of empirically obtained exponents (α\\alpha ) with thetheoretical exponents 1+d/41+d/4.",
"Empirical exponents are obtained bycurve fitting the data in Fig.",
".Figure: Averaged error over M=100M=100 simulations with N=200N=200 particles: (a) Linear scaleover a range of ϵ\\epsilon and (b) Logarithmic scale for smallϵ\\epsilon ."
],
[
"Kernel-based algorithm",
"This section provides additional details for the kernel-based algorithm presented in Sec. .",
"We begin with some definitions and then provide justification for the three main steps, fixed-point equations (REF )-().",
"Definitions: The Gaussian kernel is denoted as $g_\\epsilon (x,y):=\\exp (-\\frac{\\Vert x-y\\Vert ^2}{4\\epsilon })$ .",
"The approximating family of operators $\\lbrace {T}_{\\epsilon },\\epsilon \\ge 0\\rbrace $ are defined as follows: For $f:\\mathbb {R}^d \\rightarrow \\mathbb {R}$ , ${T}_{\\epsilon }f(x) := \\int _{\\mathbb {R}^d}k_\\epsilon (x,y)f(y)\\rho (y)\\,\\mathrm {d}y,$ where ${k}_{\\epsilon }:(x,y)= \\frac{1}{n_\\epsilon (x)}\\frac{g_\\epsilon (x,y)}{\\sqrt{\\int g_\\epsilon (y,z) \\rho (z) \\,\\mathrm {d}z}}, $ and $n_\\epsilon $ is the normalization factor, chosen such that $\\int {k}_{\\epsilon }(x,y)\\rho (y) \\,\\mathrm {d}y=1$ .",
"The finite-$N$ approximation of these operators, denoted as $\\lbrace {T}^{(N)}_\\epsilon \\rbrace _{\\epsilon \\ge 0, N \\in \\mathbb {N}}$ , is defined as, ${T}^{(N)}_\\epsilon f(x) := \\frac{1}{N}\\sum _{i=1}^N {k}_{\\epsilon }^{(N)}(x,X^i)f(X^i),$ where ${k}_{\\epsilon }^{(N)}(x,y) := \\frac{1}{n_\\epsilon ^{(N)}(x)}\\frac{{g}_{\\epsilon }(x,y) }{\\sqrt{\\sum _{j=1}^N g_\\epsilon (y,X^j)}},$ and $n_\\epsilon ^{(N)}$ is chosen such that $\\frac{1}{N}\\sum _{i=1}^N{k}_{\\epsilon }(x,X^i)=1$ .",
"Justification of the fixed-point equations (REF )-(): (i) Definition of the semigroup $e^{\\epsilon \\Delta _\\rho }$ implies: $e^{\\epsilon \\Delta _\\rho } f = f + \\int _0^\\epsilon e^{s\\Delta _\\rho }\\Delta _\\rho f \\,\\mathrm {d}s$ On choosing $f= \\phi $ where $\\Delta _\\rho \\phi = - (h-\\hat{h})$ yields the exact fixed point equation (REF ).",
"(ii) The justification for approximation ${T}_{\\epsilon }\\approx e^{\\epsilon \\Delta _\\rho }$ is the following Lemma.",
"Lemma 1 Consider the family of Markov operators $\\lbrace {T}_{\\epsilon }\\rbrace _{\\epsilon \\ge 0}$ .",
"Fix a smooth function $f$ .",
"Then ${T}_{\\epsilon }f(x) = f(x) + \\epsilon \\Delta _\\rho f(x) +O(\\epsilon ^2).$ Introduce the heat semigroup ${G}_\\epsilon $ as, $G_\\epsilon f (x) := \\int _{\\mathbb {R}^d} g_\\epsilon (x-y) f(y) \\rho (y)\\,\\mathrm {d}y.$ The following are the two properties of the heat semigroup: $G_0 f(x) &= \\rho (x)f(x), \\\\\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}\\epsilon }G_\\epsilon f(x) \\big |_{\\epsilon =0}&= \\Delta (\\rho f)(x).$ In terms of ${G}_\\epsilon $ , the operator ${T}_{\\epsilon }$ is expressed as, ${T}_{\\epsilon }f(x) = \\frac{1}{n_\\epsilon (x)}{G}_\\epsilon (\\frac{f}{\\sqrt{{G}_\\epsilon 1}})(x),$ where $n_\\epsilon (x) = {G}_\\epsilon (\\frac{1}{\\sqrt{{G}_\\epsilon 1}})(x)$ .",
"The Taylor expansion of ${T}_{\\epsilon }f(x)$ yields, $T_\\epsilon f(x) &= T_0 f(x) + \\epsilon \\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}\\epsilon } {T}_{\\epsilon }f (x)\\big |_{\\epsilon =0} + O(\\epsilon ^2).$ Now, the properties (REF ) and () can be used to show that $T_0 f(x)=f(x)$ and $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}\\epsilon } {T}_{\\epsilon }f (x)\\big |_{\\epsilon =0} = \\Delta _\\rho f(x)$ .",
"(iii) The justification for the third step is Law of Large numbers.",
"Moreover the following lemma provides a bound for the $L^2$ error.",
"Lemma 2 Consider the Markov operators ${T}_{\\epsilon }$ and ${T}^{(N)}_\\epsilon $ defined in (REF ) and (REF ).",
"Then $\\forall f \\in L^2(\\rho )$ , $ {\\sf E} \\left[\\Vert {T}_{\\epsilon }f -{T}^{(N)}_\\epsilon f\\Vert _{L^2(\\rho )}^2\\right] \\le \\frac{C}{N\\epsilon ^{d/2}}.$ The bound is proved by explicitly evaluating the error $ {\\sf E} \\left[\\Vert {G}_\\epsilon f -G_\\epsilon ^{(N)}f\\Vert _{L^2(\\rho )}^2\\right]$ where ${G}_\\epsilon $ is defined in (REF ) and $G_\\epsilon ^{(N)}f(x):=\\frac{1}{N}\\sum _{i=1}^N g_\\epsilon (x,X^i)f(X^i)$ .",
"Since $\\lbrace X^i\\rbrace $ are i.i.d., ${\\sf E}\\left[\\Vert {G}_\\epsilon f -G_\\epsilon ^{(N)}f\\Vert _2\\right] \\le \\frac{1}{N}\\int \\int g_\\epsilon ^2(x,y)f(y)^2 \\rho (y)\\rho (x)\\,\\mathrm {d}y \\,\\mathrm {d}x.$ Subsequently, using the fact that $g_\\epsilon ^2(x,y) =\\frac{C}{\\epsilon ^{d/2}}g_{2\\epsilon }(x,y)$ and $\\int g_{2\\epsilon }(x,y)\\rho (x)\\,\\mathrm {d}x \\le C$ , one obtains, $ {\\sf E} \\left[\\Vert {G}_\\epsilon f -{G}_\\epsilon ^{(N)}f\\Vert _{L^2(\\rho )}^2\\right] \\le \\frac{C\\Vert f\\Vert ^2_{L^2(\\rho )}}{N\\epsilon ^{d/2}},$ and the estimate follows because of (REF )."
],
[
"Sketch of the Proof of Theorem ",
"Estimate for Bias: The crucial property is that ${T}_{\\epsilon }$ is a bounded strictly contractive operator on $H_0^1$ with $\\Vert (I-{T}_{\\epsilon })^{-1}\\Vert _{H^1_0(\\rho )} = \\frac{1}{\\epsilon \\lambda _1} +O(1),$ where $\\lambda _1$ is the spectral bound for $\\Delta _{\\rho }$ .",
"Since ${\\phi }_{\\epsilon }$ solves the fixed-point equation (), ${\\phi }_{\\epsilon }= {T}_{\\epsilon }{\\phi }_{\\epsilon }+ \\epsilon (h-\\hat{h}) + O(\\epsilon ^2).$ Therefore, $\\phi - {\\phi }_{\\epsilon }& = \\phi - {T}_{\\epsilon }\\phi + {T}_{\\epsilon }(\\phi - {\\phi }_{\\epsilon }) - \\epsilon (h-\\hat{h}) + O(\\epsilon ^2) \\\\& = -\\epsilon \\Delta _{\\rho }\\phi + {T}_{\\epsilon }(\\phi - {\\phi }_{\\epsilon }) - \\epsilon (h-\\hat{h}) + O(\\epsilon ^2),$ where we have used Lemma REF .",
"Noting $-\\Delta _{\\rho }\\phi =(h-\\hat{h})$ , $\\phi - {\\phi }_{\\epsilon }= {T}_{\\epsilon }(\\phi - {\\phi }_{\\epsilon }) + O(\\epsilon ^2).$ The bias estimate now follows from using the norm estimate (REF ).",
"Estimate for variance: The variance estimate follows from using Lemma REF .",
"The key steps are to show that $\\Vert {T}^{(N)}_\\epsilon f - {T}_{\\epsilon }f\\Vert _{H^1_0(\\rho )} \\rightarrow 0\\quad \\text{a.s},$ which follows from the LLN, and that ${T}^{(N)}_\\epsilon $ are bounded and compact on $H^1_0(\\rho )$ .",
"This allows one to conclude that $\\Vert (I-{T}^{(N)}_\\epsilon )^{-1}\\Vert _{H^1_0(\\rho )}$ is bounded.",
"These forms of approximation error bounds in a somewhat more general context of compact operators appears in [Chapter 7 of [6]]."
],
[
"Proof of proposition ",
"For the Gaussian density ${\\cal N}(0,\\sigma ^2 I)$ , the completion of square is used to obtain an explicit form for the operator ${T}_{\\epsilon }$ : ${T}_{\\epsilon }f(x) = \\int \\frac{1}{\\sqrt{4\\pi \\epsilon (1-\\delta _\\epsilon )}}\\exp \\left[-\\frac{(y-(1-\\delta _\\epsilon )x)^2}{4\\epsilon (1-\\delta _\\epsilon )}\\right]f(y)\\,\\mathrm {d}y,$ where $\\delta _\\epsilon :=\\epsilon \\frac{\\sigma ^2+4\\epsilon }{\\sigma ^4+3\\epsilon \\sigma ^2+4\\epsilon ^2}$ .",
"For the linear function $h(x)=H\\cdot x$ , the fixed-point equation () admits an explicit solution, $\\phi _\\epsilon = \\frac{\\epsilon }{\\delta _\\epsilon }H\\cdot x$ where we used the fact that ${T}_{\\epsilon }x = 1-\\delta _\\epsilon x$ .",
"Since the solution $\\phi _\\epsilon $ is known in an explicit form, one can easily compute the gain function solution in an explicit form: $\\begin{aligned}\\text{(G1)}\\quad {\\sf K}_\\epsilon &=\\frac{\\epsilon }{\\delta _\\epsilon }H=\\sigma ^2 H-\\epsilon \\frac{\\sigma ^2-4\\epsilon }{\\sigma ^2 + 4\\epsilon }H,\\\\\\text{(G2)}\\quad {\\sf K}_\\epsilon &= \\sigma ^2 H+ \\frac{\\epsilon \\sigma ^6}{(\\sigma ^2 + 4\\epsilon )(\\sigma ^4 + 3\\epsilon \\sigma ^2 + 4\\epsilon ^2)}H.\\end{aligned}$ The error estimates follow based on the exact Kalman gain solution ${\\sf K}= \\sigma ^2H$ ."
],
[
"Proof of Proposition ",
"The proof relies on the fact that $\\epsilon ^{d/2}{g}_{\\epsilon }(x,y)$ converges to a constant as $\\epsilon \\rightarrow \\infty $ .",
"This would imply that ${k}_{\\epsilon }(x,y)\\rightarrow 1$ as $\\epsilon \\rightarrow \\infty $ .",
"Therefore for a fixed function $f$ , $\\lim _{\\epsilon \\rightarrow \\infty }{T}_{\\epsilon }f (x) = \\int f(x)\\rho (x)\\,\\mathrm {d}x =: \\hat{f}$ Define the limit $T_\\infty :=\\lim _{\\epsilon \\rightarrow \\infty }{T}_{\\epsilon }$ and observe: $\\lim _{\\epsilon \\rightarrow \\infty } \\frac{{\\phi }_{\\epsilon }}{\\epsilon } = \\lim _{\\epsilon \\rightarrow \\infty }(I-{T}_{\\epsilon })^{-1}(h-\\hat{h}) =(I-T_\\infty )^{-1}(h-\\hat{h})=h-\\hat{h}$ where the last step uses the fact that $\\hat{h}=T_\\infty h$ and we assumed $(I-{T}_{\\epsilon })^{-1}h\\rightarrow (I-T_\\infty )h$ .",
"Then the gain approximation formula (REF ) implies: $\\begin{aligned}\\lim _{\\epsilon \\rightarrow \\infty } {\\sf K}_\\epsilon (x)&=\\lim _{\\epsilon \\rightarrow \\infty } \\frac{1}{2}\\left[{T}_{\\epsilon }(e\\frac{{\\phi }_{\\epsilon }}{\\epsilon }) - {T}_{\\epsilon }(e){T}_{\\epsilon }(\\frac{{\\phi }_{\\epsilon }}{\\epsilon })\\right] \\\\&+\\lim _{\\epsilon \\rightarrow \\infty }\\frac{1}{2}\\left[{T}_{\\epsilon }(eh) - {T}_{\\epsilon }(e){T}_{\\epsilon }(h)\\right] \\\\&=\\int x(h(x)-\\hat{h})\\rho (x)\\,\\mathrm {d}x\\end{aligned}$ The argument for the finite-$N$ case is identical and omitted on account of space."
]
] | 1612.05606 |
[
[
"On the Weber integral equation and solution to the Weber-Titchmarsh\n problem"
],
[
"Abstract We derive sufficient conditions for the existence of the Weber formal solution of the corresponding integral equation, related to the familiar Weber-Orr integral transforms.",
"This gives a solution to the old Weber-Titchmarsh problem (posed in Proc.",
"Lond.",
"Math.",
"Soc.",
"22(2) (1924), pp.15, 16.)",
"Our method involves properties of the inverse Mellin transform of integrable functions.",
"The Mellin-Parseval equality and some integrals with the associated Legendre functions are used."
],
[
"Introduction and preliminary results",
"In [4] E.C.",
"Titchmarsh formally showed that an arbitrary complex-valued function $g(x),\\ x \\in \\mathbb {R}_+$ can be expanded in terms of the following repeated integral $g(x)= {x\\over J_\\nu ^2(ax)+ Y_\\nu ^2(ax)} \\int _a^\\infty C_\\nu (xt, xa) t \\int _0^\\infty C_\\nu (t\\xi , a\\xi ) g(\\xi ) d\\xi dt,\\qquad \\mathrm {(1)}$ where $a >0$ , $\\nu \\in \\mathbb {C}$ , $J_\\nu (z), Y_\\nu (z)$ are Bessel functions of the first and second kind [1], Vol.",
"II and $C_\\nu (\\alpha , \\beta )= J_\\nu (\\alpha ) Y_\\nu (\\beta ) - Y_\\nu (\\alpha ) J_\\nu (\\beta ).\\qquad \\mathrm {(2)}$ Expansion (1) is related to the familiar Weber-Orr integral expansions of an arbitrary function $f(x)$ as repeated integrals $f(x)= \\int _0^\\infty {t \\ C_\\nu (xt, at)\\over J_\\nu ^2(at)+ Y_\\nu ^2(at)}\\int _a^\\infty C_\\nu (\\xi t, at) \\xi f(\\xi ) d\\xi dt,\\qquad \\mathrm {(3)}$ $f(x)= \\int _a^\\infty C_\\nu (xt, xa) t \\int _0^\\infty { C_\\nu (t \\xi , a\\xi ) \\over J_\\nu ^2(a\\xi )+ Y_\\nu ^2(a\\xi )} \\xi f (\\xi ) d\\xi dt,\\qquad \\mathrm {(4)}$ which are different from (1).",
"Combining with (3), (4), Titchmarsh proved formally (1) for $a=1$ (see [4], p. 15).",
"He posed the problem to find sufficient conditions for the validity of expansion (1) in order to solve the following Weber integral equation with respect to $g$ $f(x)= \\int _0^\\infty C_\\nu (x \\xi , a\\xi ) g(\\xi ) d\\xi ,\\qquad \\mathrm {(5)}$ where $f(x),\\ x \\in \\mathbb {R}_+$ is a given function.",
"As far as the author is aware this question is still open.",
"Our method will be based on the use of the Mellin transform [5].",
"Precisely, the Mellin transform is defined in $L_{\\mu , p}(\\mathbb {R}_+),\\ 1 < p \\le 2$ by the integral $f^*(s)= \\int _0^\\infty f(x) x^{s-1} dx,\\qquad \\mathrm {(6)}$ being convergent in mean with respect to the norm in $L_q(\\mu - i\\infty , \\mu + i\\infty ),\\ q=p/(p-1)$ .",
"Moreover, the Parseval equality holds for $f \\in L_{\\mu , p}(\\mathbb {R}_+),\\ g \\in L_{1-\\mu , q}(\\mathbb {R}_+)$ $\\int _0^\\infty f(x) g(x) dx= {1\\over 2\\pi i} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } f^*(s) g^*(1-s) ds.\\qquad \\mathrm {(7)}$ The inverse Mellin transform is given accordingly $f(x)= {1\\over 2\\pi i} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } f^*(s) x^{-s} ds,\\qquad \\mathrm {(8)}$ where the integral converges in mean with respect to the norm in $L_{\\mu , p}(\\mathbb {R}_+)$ $||f||_{\\mu ,p} = \\left( \\int _0^\\infty |f(x)|^p x^{\\mu p-1} dx\\right)^{1/p}.\\qquad \\mathrm {(9)}$ In particular, letting $\\mu = 1/p$ we get the usual space $L_1(\\mathbb {R}_+)$ .",
"A special class of functions related to the Mellin transform (6) and its inversion (8), was introduced in [7].",
"Indeed, we have Definition 1 ([7]).",
"Denote by ${\\mathcal {M}}^{-1}(L_c)$ the space of functions $f(x), \\ x \\in \\mathbb {R}_+$ , representable by inverse Mellin transform (8) of integrable functions $f^{*}(s) \\in L_{1}(c)$ on the vertical line $c =\\lbrace s \\in \\mathbb {C}: \\mu ={\\rm Re s} = c_0\\rbrace $ .",
"The space ${\\mathcal {M}}^{-1}(L_c)$ with the usual operations of addition and multiplication by scalar is a linear vector space.",
"If the norm in ${\\mathcal {M}}^{-1}(L_c)$ is introduced by the formula $ \\big \\vert \\big \\vert f \\big \\vert \\big \\vert _{{\\mathcal {M}}^{-1}(L_c)}= {1\\over 2\\pi }\\int ^{+\\infty }_{-\\infty } |f^{*}\\left(c_0 +it\\right)| dt,\\qquad \\mathrm {(10)}$ then it becomes a Banach space.",
"Definition 2 ([7], [8]).",
"Let $\\mu \\ne 0,\\ c_1, c_2 \\in \\mathbb {R}$ be such that $2 \\hbox{sign}\\ c_1 + \\hbox{sign}\\ c_2 \\ge 0$ .",
"By ${\\mathcal {M}}_{c_1,c_2}^{-1}(L_c)$ we denote the space of functions $f(x), x \\in \\mathbb {R}_+$ , representable in the form (8), where $s^{c_2}e^{\\pi c_1|s|} f^*(s) \\in L_1(c)$ .",
"It is a Banach space with the norm $ \\big \\vert \\big \\vert f \\big \\vert \\big \\vert _{{\\mathcal {M}}_{c_1,c_2}^{-1}(L_c)}= {1\\over 2\\pi }\\int _{c} e^{\\pi c_1|s|}|s^{c_2} f^{*}(s) ds|.$ In particular, letting $c_1=c_2=0$ we get the space ${\\mathcal {M}}^{-1}(L_c)$ .",
"Moreover, it is easily seen the inclusion ${\\mathcal {M}}_{d_1,d_2}^{-1}(L_c) \\subseteq {\\mathcal {M}}_{c_1,c_2}^{-1}(L_c)$ when $2 \\hbox{sign}(d_1- c_1) + \\hbox{sign}(d_2-c_2) \\ge 0$ ."
],
[
"Solution to the Weber-Titchmarsh problem",
"The goal of this Note is to prove the following Theorem.",
"Let $a > 0, \\ \\nu \\in \\mathbb {C}, \\ 0< {\\rm Re } \\nu < 1/2, g(x) \\in {\\mathcal {M}}_{0,1} ^{-1}(L_c)$ with $c =\\lbrace s \\in \\mathbb {C}: -1< {\\rm Re} s < 0\\rbrace $ .",
"Then for almost all $x >0$ expansion $(1)$ holds, where the inner and outer integrals are understood in the improper sense.",
"We begin, writing $g(\\xi )$ in terms of the inverse Mellin transform (8) of the reciprocal function $g^*(s) \\in L_1(c)$ .",
"Then substituting this expression into (5), we change the order of integration when $\\xi \\in [0, N]$ to find $f(x)= {1\\over 2\\pi i} \\lim _{N\\rightarrow \\infty } \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\int _0^N C_\\nu (x \\xi , a\\xi ) \\xi ^{-s} d\\xi ds,\\qquad \\mathrm {(11)}$ where the interchange is ensured by the Fubini theorem via absolute convergence of the repeated integral.",
"Indeed, it is due to conditions of the theorem and asymptotic behavior of Bessel functions at infinity and near the origin [1] via the assumption $-1< \\mu < 0.$ Then kernel (2) behaves as follows $ C_\\nu (x \\xi , a\\xi ) = O( 1),\\ \\xi \\rightarrow 0+,\\ \\nu \\ne 0,$ $ C_\\nu (x \\xi , a\\xi ) = O( \\log \\xi ),\\ \\xi \\rightarrow 0+,\\ \\nu = 0,$ $ C_\\nu (x \\xi , a\\xi ) = - {2\\over \\pi \\xi \\sqrt{a x}} \\left[ \\sin (\\xi (x-a)) + O\\left( {1\\over \\xi } \\right)\\right],\\ \\xi \\rightarrow \\infty .$ Hence, for fixed $x, a, N$ $\\int _{\\mu - i\\infty }^{\\mu +i\\infty } \\left| g^*(s) \\right|\\int _0^N \\left| C_\\nu (x \\xi , a\\xi )\\right| \\xi ^{-\\mu } d\\xi |ds| < \\infty .$ In the meantime, the integral with respect to $\\xi $ over $\\mathbb {R}_+$ can be calculated with the use of relation (2.13.15.4) in [3], Vol.",
"2, Boltz's and self-transformation formulae for the Gauss hypergeometric function and basic relations between the associated Legendre functions (see details in [1], Vol.",
"I).",
"Thus we obtain $\\int _0^\\infty C_\\nu (x \\xi , a\\xi ) \\xi ^{-s} d\\xi = {2^{1 -s}\\over \\pi } \\ e^{i\\nu \\pi } \\left(x^2- a^2\\right)^{(s-1)/2}\\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} Q^{-\\nu }_{(s-1)/2} \\left( {x^2+a^2 \\over x^2-a^2}\\right),\\qquad \\mathrm {(12)}$ where $x > a > 0, \\Gamma (z)$ is Euler's gamma-function and $Q^\\nu _\\mu (z)$ is the associated Legendre function of the second kind [1], Vol.",
"I, [6].",
"Hence, we write (11) in the form $ f(x)= { e^{i\\nu \\pi } \\over \\pi ^2 i} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} Q^{-\\nu }_{(s-1)/2} \\left( {x^2+a^2 \\over x^2-a^2}\\right) 2^{-s} \\left(x^2- a^2\\right)^{(s-1)/2} ds$ $- {1\\over 2\\pi i} \\lim _{N\\rightarrow \\infty } \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\int _N^\\infty C_\\nu (x \\xi , a\\xi ) \\xi ^{-s} d\\xi ds\\qquad \\mathrm {(13)}$ and will prove that $\\lim _{N\\rightarrow \\infty } \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\int _N^\\infty C_\\nu (x \\xi , a\\xi ) \\xi ^{-s} d\\xi ds= 0,\\ -1< \\mu < 0.$ To do this, we will need the exact asymptotic behavior at infinity of the kernel (2) (see above).",
"Then, substituting the main asymptotic term and integrating by parts in the integral by $\\xi $ , we deduce for fixed $x > a$ $ - {2\\over \\pi \\sqrt{a x} } \\int _N^\\infty \\sin (\\xi (x-a)) \\xi ^{-s-1} d\\xi = - {2 \\cos (N (x-a)) \\over \\pi (x-a) \\sqrt{a x} } N^{-s-1} + O( (s+1) N^{-s-1} ), \\ N \\rightarrow \\infty .$ Hence, under conditions of the theorem $ \\left| \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\int _N^\\infty C_\\nu (x \\xi , a\\xi ) \\xi ^{-s} d\\xi ds\\right| \\le C\\ N^{-\\mu -1} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } \\left| g^*(s) \\right| (|s| +2)|ds| \\rightarrow 0,\\ N \\rightarrow \\infty , $ where $C >0$ is an absolute constant.",
"We note that will use the same notation below for different positive constants.",
"Now, taking into account (13), let us consider the following sequence of functions (see (1)) $G_N(x)= { e^{i\\nu \\pi } \\over \\pi ^2 i} \\int _a^N C_\\nu (xt, xa) t \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s)\\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt.\\qquad \\mathrm {(14)}$ Choosing a complex variable $w$ from the vertical strip $-1 < {\\rm Re} w < - 1/2$ , we multiply both sides of (14) by $x^w$ and integrate with respect to $x$ over $\\mathbb {R}_+$ .",
"Hence, changing the order of integration in its right-hand side by Fubini's theorem owing to the same motivation as above when $N$ is fixed.",
"Hence, it becomes $\\int _0^\\infty G_N(x) x^w dx = { e^{i\\nu \\pi } \\over \\pi ^2 i} \\int _a^N t \\int _0^\\infty C_\\nu (xt, xa) x^w dx \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt.$ Making use (12), we find the equality $\\int _0^\\infty G_N(x) x^w dx = { 2^{1+w} e^{2 i\\nu \\pi } \\over \\pi ^3 i} \\frac{\\Gamma ((1+w)/2)}{ \\Gamma ( (1- 2\\nu -w)/2)}\\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} 2^{-s} $ $\\times \\int _a^N t \\left(t^2- a^2\\right)^{(s- w-2)/2} Q^{-\\nu }_{- (w+1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) dt ds.\\qquad \\mathrm {(16)}$ Then due to elementary substitutions the inner integral with respect to $t$ in (16) can be written as $\\int _a^N t \\left(t^2- a^2\\right)^{(s- w-2)/2} Q^{-\\nu }_{- (w+1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) dt $ $= {1\\over 2} a^{s-w} \\int _1^\\infty \\left(t- 1\\right)^{(s- w-2)/2} Q^{-\\nu }_{- (w+1)/2} \\left( {t+1 \\over t-1}\\right) Q^{-\\nu }_{(s-1)/2} \\left( {t+1 \\over t- 1}\\right) dt - I_N(s,w), $ where $ I_N(s,w) = {1\\over 2} a^{s-w} \\int _{(N/a)^2}^\\infty \\left(t- 1\\right)^{(s- w-2)/2} Q^{-\\nu }_{- (w+1)/2} \\left( {t+1 \\over t-1}\\right) Q^{-\\nu }_{(s-1)/2} \\left( {t+1 \\over t-1}\\right) dt.\\qquad \\mathrm {(17)}$ Meanwhile, Parseval equality (7) and relation (8.4.42.5) in [3], Vol.",
"3 permit to get the equality ${1\\over 2} a^{s-w} \\int _1^\\infty \\left(t- 1\\right)^{(s- w-2)/2} Q^{-\\nu }_{- (w+1)/2} \\left( {t+1 \\over t-1}\\right) Q^{-\\nu }_{(s-1)/2} \\left( {t+1 \\over t- 1}\\right) dt $ $= {e^{-2i\\nu \\pi } \\over 16\\pi i}\\ a^{s-w} \\Gamma ( (1+ s)/2) \\Gamma ( (1- w)/2) \\Gamma ( (1- 2\\nu + s)/2) \\Gamma ( (1- 2\\nu - w)/2)$ $\\times \\int _{\\gamma - i\\infty }^{\\gamma +i\\infty } \\frac{\\Gamma ( (\\nu +w +1)/2 - \\tau ) \\Gamma ( (w+1-\\nu )/2 -\\tau ) \\Gamma ( (\\nu -s -1)/2 + \\tau ) \\Gamma (\\tau - (1+ s + \\nu )/2 )}{\\Gamma ( 1+ \\nu /2 -\\tau ) \\Gamma ( 1- \\nu /2 -\\tau )\\Gamma ( \\tau +\\nu /2 ) \\Gamma ( \\tau - \\nu /2 )} d\\tau ,$ where $-1 < {\\rm Re} s < 0, \\ -1 < {\\rm Re} w < -1/2$ and $ {1\\over 2} \\left( 1+ {\\rm Re} (s + \\nu ) \\right) < \\gamma < {1\\over 2} \\left( 1+ {\\rm Re} (w - \\nu ) \\right).$ However, the latter integral can be calculated as the sum of residues at the left-hand poles of gamma-functions with the use of Slater's theorem [2], and values of the hypergeometric function ${}_2F_1$ at the unity [1], Vol.",
"I, [3], Vol.",
"3.",
"Thus we obtain the result ${1\\over 2\\pi i} \\int _{\\gamma - i\\infty }^{\\gamma +i\\infty } \\frac{\\Gamma ( (\\nu +w +1)/2 - \\tau ) \\Gamma ( (w+1-\\nu )/2 -\\tau ) \\Gamma ( (\\nu -s -1)/2 + \\tau ) \\Gamma (\\tau - (1+ s + \\nu )/2 )}{\\Gamma ( 1+ \\nu /2 -\\tau ) \\Gamma ( 1- \\nu /2 -\\tau )\\Gamma ( \\tau +\\nu /2 ) \\Gamma ( \\tau - \\nu /2 )} d\\tau $ $= 2^{s-w+1} {\\cos (\\pi \\nu )\\over \\sqrt{\\pi }} \\frac{\\Gamma ( (1+s- w)/2) \\Gamma ( (w-s)/2) \\Gamma ( (w -s)/2 -\\nu ) \\Gamma ((w-s)/2+ \\nu )}{\\Gamma ( (1+ s)/2 ) \\Gamma ( (1-s)/2 )\\Gamma ((1+ w)/2 ) \\Gamma ( (1- w)/2)},$ which leads after simple substitution to the value of possibly new integral with the product of the associated Legendre functions of the second kind $\\int _1^\\infty \\left(t- 1\\right)^{(w- s)/2 - 1} Q^{-\\nu }_{- (w+1)/2} \\left( t\\right) Q^{-\\nu }_{(s-1)/2} \\left( t\\right) dt = 2^{(s-w)/2 -1} e^{-2i\\nu \\pi } {\\cos (\\pi \\nu )\\over \\sqrt{\\pi }} $ $\\times \\frac{\\Gamma ( (1+s- w)/2) \\Gamma ( (w-s)/2) \\Gamma ( (w -s)/2 -\\nu ) \\Gamma ((w-s)/2+ \\nu )}{ \\Gamma ( (1-s)/2 )\\Gamma ((1+ w)/2 ) }$ $\\times \\Gamma ( (1+ s)/2 - \\nu ) \\Gamma ( (1-w)/2- \\nu ), {\\rm Re} \\nu < {\\rm Re} \\left( {w -s\\over 2} \\right) < {1\\over 2},\\qquad \\mathrm {(18)}$ when $w$ is related to $s$ by the condition $- 1/2 > {\\rm Re} w > \\hbox{max} \\left( 2{\\rm Re} \\nu + {\\rm Re} s, -1\\right) $ .",
"Further, recalling (16) and substituting the value of the integral (18), we write it in the form $\\int _0^\\infty G_N(x) x^w dx = { \\cos (\\pi \\nu ) \\over 2 \\pi ^3 \\sqrt{\\pi }i} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\Gamma ( (1+s- w)/2) \\Gamma ( (w-s)/2)$ $\\times \\Gamma ( (w -s)/2 -\\nu ) \\Gamma ((w-s)/2+ \\nu ) a^{s-w} ds $ $- { 2^{1+w} e^{2 i\\nu \\pi } \\over \\pi ^3 i} \\frac{\\Gamma ((1+w)/2)}{ \\Gamma ( (1- 2\\nu -w)/2)}\\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} I_N(s,w) 2^{-s} ds,\\qquad \\mathrm {(19)}$ where $I_N(s,w)$ is defined by (17).",
"Meanwhile, the first term in the right-hand side of (19) can be treated by the Parseval equality (7) for the Mellin transform with the use of relation (8.4.20.35) in [3], Vol.",
"3, which gives the Mellin- Barnes integral representation for the kernel $J_\\nu ^2(ax)+ Y_\\nu ^2(ax)$ , namely $J_\\nu ^2(ax)+ Y_\\nu ^2(ax) = { \\cos (\\pi \\nu ) \\over 2 \\pi ^3 \\sqrt{\\pi }i} \\int _{\\gamma - i\\infty }^{\\gamma +i\\infty } \\Gamma ( (1-s)/2) \\Gamma ( s/2) \\Gamma ( s/2 -\\nu ) \\Gamma (s/2+ \\nu ) (ax)^{-s} ds,\\ 2 {\\rm Re} \\nu < \\gamma < 1.",
"$ Therefore, under conditions of the theorem we obtain $ { \\cos (\\pi \\nu ) \\over 2 \\pi ^3 \\sqrt{\\pi }i} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\Gamma ( (1+s- w)/2) \\Gamma ( (w-s)/2)$ $\\times \\Gamma ( (w -s)/2 -\\nu ) \\Gamma ((w-s)/2+ \\nu ) a^{s-w} ds = \\int _0^\\infty g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax) \\right] x^{w-1} dx.\\qquad \\mathrm {(20)}$ On the other hand, making use the same substitution, we write (17) as follows $ I_N(s,w) = 2^{s-w-1} a^{s-w} \\int _{1}^{(N^2+a^2)/ (N^2- a^2)} \\left(t- 1\\right)^{(w-s)/2- 1} Q^{-\\nu }_{- (w+1)/2} \\left( t\\right) Q^{-\\nu }_{(s-1)/2} \\left( t\\right) dt.\\qquad \\mathrm {(21)}$ The associated Legendre functions of the second kind $Q^{-\\nu }_{- (w+1)/2} \\left( t\\right),\\ Q^{-\\nu }_{(s-1)/2} \\left( t\\right)$ , in turn, can be represented in terms of the Euler integral for the Gauss hypergeometric function [1], Vol.",
"I and we find $Q^{-\\nu }_{- (w+1)/2} \\left( t\\right) = \\frac{ 2^{(w-1)/2} e^{-i\\nu \\pi } \\sqrt{\\pi }\\ \\Gamma ((1-w - 2\\nu )/2)}{ \\Gamma ( (3- w - 2\\nu )/4) \\Gamma ((1-w+2\\nu )/4)}\\ (t^2-1)^{-\\nu /2} $ $\\times \\int _0^1u^{- (w+1+2\\nu )/4} (1-u)^{ (2\\nu - w-3)/4} (t^2-u)^{ (2\\nu + w- 1)/4} du,\\qquad \\mathrm {(22)}$ $Q^{-\\nu }_{ (s-1)/2} \\left( t\\right) = \\frac{ 2^{-(s+1)/2} e^{-i\\nu \\pi } \\sqrt{\\pi }\\ \\Gamma ((s+1 - 2\\nu )/2)}{ \\Gamma ((3+ s -2 \\nu )/4) \\Gamma ((1+s+2\\nu )/4)}\\ (t^2-1)^{-\\nu /2} $ $\\times \\int _0^1u^{- (1-s +2\\nu )/4} (1-u)^{ (s-3+2\\nu )/4} (t^2-u)^{ (2\\nu - 1-s)/4} du, \\ {\\rm Re} s > 2 {\\rm Re} \\nu -1.\\qquad \\mathrm {(23)}$ Hence, we obtain the following uniform estimates for the associated Legendre functions of the second kind $\\left| Q^{-\\nu }_{- (w+1)/2} \\left( t\\right) \\right| \\le \\sqrt{\\pi }\\ 2^{( {\\rm Re} w-1)/2} (t^2-1)^{- {\\rm Re} \\nu /2} \\left| \\frac{ \\Gamma ((1-w - 2\\nu )/2)}{ \\Gamma ( (3- w - 2\\nu )/4) \\Gamma ((1-w+2\\nu )/4)}\\right|$ $\\times \\int _0^1u^{- {\\rm Re} (w+1+2\\nu )/4} (1-u)^{ {\\rm Re}\\nu -1} du = O \\left( |w|^{ - {\\rm Re} \\nu } (t^2-1)^{- {\\rm Re} \\nu /2} \\right),\\ t > 1,\\qquad \\mathrm {(24)}$ $\\left| Q^{-\\nu }_{ (s-1)/2} \\left( t\\right)\\right| \\le \\sqrt{\\pi }\\ 2^{-({\\rm Re} s+1)/2} (t^2-1)^{-{\\rm Re} \\nu /2} \\left| \\frac{ \\Gamma ((s+1 - 2\\nu )/2)}{ \\Gamma ((3+ s -2 \\nu )/4) \\Gamma ((1+s+2\\nu )/4)}\\right|\\ $ $\\times \\int _0^1u^{- {\\rm Re} (1-s +2\\nu )/4} (1-u)^{ {\\rm Re}\\nu -1} du= O \\left( |s|^{ - {\\rm Re} \\nu } (t^2-1)^{- {\\rm Re} \\nu /2} \\right),\\ t >1,\\qquad \\mathrm {(25)}$ since via Stirling's asymptotic formula for the gamma-function [1], Vol.",
"I $ \\left| \\frac{ \\Gamma ((1-w - 2\\nu )/2)}{ \\Gamma ( (3- w - 2\\nu )/4) \\Gamma ((1-w+2\\nu )/4)}\\right| = O \\left( |w|^{ - {\\rm Re} \\nu }\\right),\\ \\left|{\\rm Im} w\\right| \\rightarrow \\infty ,$ $ \\left| \\frac{ \\Gamma ((s+1 - 2\\nu )/2)}{ \\Gamma ((3+ s -2 \\nu )/4) \\Gamma ((1+s+2\\nu )/4)}\\right| = O \\left( |s|^{ - {\\rm Re} \\nu }\\right),\\ \\left|{\\rm Im} s\\right| \\rightarrow \\infty .$ Therefore, returning to (21), we have by the straightforward estimate $\\left| I_N(s,w) \\right| \\le C |w s|^{ - {\\rm Re} \\nu } \\int _{1}^{(N^2+a^2)/ (N^2- a^2)} \\left(t- 1\\right)^{{\\rm Re} (w-s-2\\nu )/2- 1} dt$ $ = O\\left( |w s|^{ - {\\rm Re} \\nu } N^{ {\\rm Re} (s+2\\nu - w)}\\right),\\ N\\rightarrow \\infty \\qquad \\mathrm {(26)}$ under condition (see (18)) ${\\rm Re} w > 2{\\rm Re} \\nu + {\\rm Re} s$ .",
"Further, substituting (20) into (19), using the obtained estimates and Stirling's asymptotic formula for the gamma-function, we find $\\left| \\int _0^\\infty \\left[ G_N(x)- g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right] x^{-1}\\right] x^w dx \\right| $ $\\le C |w |^{ - {\\rm Re} \\nu } N^{ {\\rm Re} (s+2\\nu - w)} \\left| \\frac{\\Gamma ((1+w)/2)}{ \\Gamma ( (1- 2\\nu -w)/2)}\\right| \\int _{\\mu - i\\infty }^{\\mu +i\\infty } \\left| g^*(s) \\right| |s|^{ - {\\rm Re} \\nu } \\left|\\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} ds\\right|$ $\\le C |w |^{ {\\rm Re} w} N^{ {\\rm Re} (s+2\\nu - w)} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } \\left| g^*(s) \\right| |s|^{ - \\mu } \\left| ds\\right| = O\\left( N^{ {\\rm Re} (s+2\\nu - w)} \\right),\\ N \\rightarrow \\infty .$ This is because $g \\in {\\mathcal {M}}_{0,1} ^{-1}(L_c)$ .",
"Thus for all $w$ from the strip $-1/2 > {\\rm Re} w > \\mu + 2 {\\rm Re} \\nu $ it yields the equality $\\lim _{N\\rightarrow \\infty } \\int _0^\\infty G_N(x) x^w dx = \\int _0^\\infty g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right] x^{w -1} dx.\\qquad \\mathrm {(27)}$ The latter equality means that the sequence of Mellin transforms (6) of variable $w$ of functions $x G_N(x)$ converges pointwisely to the Mellin transform of $g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right]$ .",
"On the other hand, recalling (19), (20) and the generalized Minkowski inequality, we estimate $L_2$ -norm of the difference of these functions.",
"In fact, using (26), we have $\\left( \\int _{-\\infty }^\\infty \\left| \\int _0^\\infty \\left[ x G_N(x) - g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right] \\right] x^{{\\rm Re w} +i\\tau -1} dx\\right|^2 d\\tau \\right)^{1/2} $ $= {2^{1+{\\rm Re} w}\\over \\pi ^{3} } \\left( \\int _{-\\infty }^\\infty \\left| \\frac{\\Gamma ((1+ {\\rm Re} w +i\\tau )/2)}{ \\Gamma ( (1- 2\\nu - {\\rm Re} w - i\\tau )/2)}\\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} I_N(s,w) 2^{-s} ds \\right|^2 d\\tau \\right)^{1/2} $ $\\le {2^{1+{\\rm Re} w- \\mu }\\over \\pi ^{3} } \\int _{\\mu - i\\infty }^{\\mu +i\\infty } \\left| g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)}\\right| \\left( \\int _{-\\infty }^\\infty \\left| \\frac{\\Gamma ((1+ {\\rm Re} w +i\\tau )/2)}{ \\Gamma ( (1- 2\\nu - {\\rm Re} w - i\\tau )/2)}\\right|^2 \\ \\left| I_N(s,w) \\right|^2 d\\tau \\right)^{1/2} | ds|$ $\\le C \\ N^{ \\mu + {\\rm Re} (2\\nu - w)} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } \\left| g^*(s) \\right| |s|^{-\\mu } |ds| \\left( \\int _{-\\infty }^\\infty \\left( ({\\rm Re} w)^2 + \\tau ^2 \\right)^{{\\rm Re} w} d\\tau \\right)^{1/2} = O\\left( N^{ \\mu + {\\rm Re} (2\\nu - w)}\\right) \\rightarrow 0, \\ N \\rightarrow \\infty $ under conditions of the theorem and the choice of $w$ from the strip $-1/2 > {\\rm Re} w > \\mu + 2 {\\rm Re} \\nu $ .",
"This estimates imply that the limit (27) exists also in the mean square sense.",
"However, an analog of the Plancherel theorem for the Mellin transform (see [5], Th.",
"71 ) and Parseval equality (7) say ${1\\over 2\\pi } \\int _{-\\infty }^\\infty \\left| \\int _0^\\infty \\left[ x G_N(x) - g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right] \\right] x^{{\\rm Re w} +i\\tau -1} dx\\right|^2 d\\tau $ $ = \\int _{0}^\\infty \\left| x G_N(x) - g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right] \\right|^2 x^{ 2 {\\rm Re w} -1} dx.",
"$ Consequently, the right-hand side of the latter equality tends to zero as well when $N \\rightarrow \\infty $ and we get the value of the limit $\\lim _{N\\rightarrow \\infty } x G_N(x) = g(x) \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right],\\ x >0, \\qquad \\mathrm {(28)}$ in the mean square sense.",
"Moreover, as it is known, this limit holds for almost all $x >0$ for some subsequence $G_{N_k}$ , namely, $\\lim _{k\\rightarrow \\infty } G_{N_k} (x) = g(x)\\ x^{-1} \\left[ J_\\nu ^2(ax)+ Y_\\nu ^2(ax)\\right].$ Therefore, in order to complete the proof, we need to show that the sequence $G_N$ is a Cauchy one.",
"Indeed, taking (14), we write for some positive big enough $M, N, \\ M > N$ and fixed $x >0$ $G_M(x)- G_N(x) = { e^{i\\nu \\pi } \\over \\pi ^2 i} \\int _N^M C_\\nu (xt, xa) t \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s)\\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt.$ Hence, appealing to (2), the asymptotic behavior of Bessel functions of the first and second kind at infinity and the estimate (25), we obtain $G_M(x)- G_N(x) = { e^{i\\nu \\pi } \\sqrt{2}\\over \\pi ^2 \\sqrt{\\pi a} i} \\left[ Y_\\nu (xa) \\int _N^M \\cos (xt - \\pi (2\\nu +1)/4) \\left(\\sqrt{t} + O\\left(t^{-1/2} \\right)\\right)\\right.",
"$ $\\times \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt$ $- J_\\nu (xa) \\int _N^M \\sin (xt - \\pi (2\\nu +1)/4) \\left(\\sqrt{t} + O\\left(t^{-1/2} \\right)\\right) $ $\\left.",
"\\times \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt\\right] $ $= { e^{i\\nu \\pi } \\sqrt{2}\\over \\pi ^2 \\sqrt{\\pi a} i} \\left[ Y_\\nu (xa) \\int _N^M \\cos (xt - \\pi (2\\nu +1)/4) \\sqrt{t} \\right.",
"$ $\\times \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt$ $- J_\\nu (xa) \\int _N^M \\sin (xt - \\pi (2\\nu +1)/4) \\sqrt{t} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\left.",
"\\times Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt\\right]+ O\\left( N^{\\mu + {\\rm Re}\\nu - 1/2}\\right).",
"\\qquad \\mathrm {(29)}$ Meanwhile, integrating by parts in the first integral by $t$ at the right-hand side of latter equality in (29) (the second one can be treated analogously) and using again (25), we find $\\int _N^M \\cos (xt - \\pi (2\\nu +1)/4) \\sqrt{t} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2)}{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt$ $= O\\left( N^{\\mu + {\\rm Re}\\nu - 1/2}\\right) - \\int _N^M \\sin (xt - \\pi (2\\nu +1)/4) t^{3/2} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2) (s-1) }{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right) 2^{-s} \\left(t^2- a^2\\right)^{(s-3)/2} ds dt$ $- \\int _N^M \\sin (xt - \\pi (2\\nu +1)/4) t^{1/2} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2) }{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times {d\\over dt} \\left[ Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right)\\right] 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt $ $= O\\left( N^{\\mu + {\\rm Re}\\nu - 1/2}\\right) - O\\left( N^{\\mu + {\\rm Re}\\nu - 3/2}\\right) $ $- \\int _N^M \\sin (xt - \\pi (2\\nu +1)/4) t^{1/2} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2) }{ \\Gamma ( (1- 2\\nu + s)/2)} $ $\\times {d\\over dt} \\left[ Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right)\\right] 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt, $ where the differentiation under the integral sign is motivated by the absolute and uniform convergence and will be clearly seen form the estimate of the derivative of the second kind associated Legendre function, being deduced below.",
"Indeed, recalling (23), we deduce ${d\\over dt} \\left[Q^{-\\nu }_{ (s-1)/2} \\left( t\\right)\\right] = \\frac{ 2^{-(s+1)/2} e^{-i\\nu \\pi } \\nu \\sqrt{\\pi }\\ \\Gamma ((s+1 - 2\\nu )/2)}{ \\Gamma ((3+ s -2 \\nu )/4) \\Gamma ((1+s+2\\nu )/4)}\\ t (t^2-1)^{-\\nu /2}\\left[ - {\\nu \\over t^2-1} \\right.",
"$ $\\times \\int _0^1u^{- (1-s +2\\nu )/4} (1-u)^{ (s-3+2\\nu )/4} (t^2-u)^{ (2\\nu - 1-s)/4} du$ $\\left.",
"+ { 2\\nu - 1-s\\over 2} \\int _0^1u^{- (1-s +2\\nu )/4} (1-u)^{ (s-3+2\\nu )/4} (t^2-u)^{ (2\\nu - 5-s)/4}du\\right],\\ t > 1,$ and, consequently, in the same manner as in (25), we obtain $\\left| {d\\over dt} \\left[Q^{-\\nu }_{ (s-1)/2} \\left( t\\right)\\right] \\right| = O\\left( |s|^{1- {\\rm Re} \\nu } t (t^2-1)^{-\\nu /2- 1}\\right),\\ t >1$ under the same condition $ {\\rm Re} s > 2 {\\rm Re} \\nu -1$ .",
"Hence, we easily get $\\left| {d\\over dt} \\left[ Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right)\\right] \\right| =O\\left( |s|^{1- {\\rm Re} \\nu } t (t^2-a^2)^{\\nu /2- 1}\\right),\\ t >a$ and, finally, $\\left| \\int _N^M \\sin (xt - \\pi (2\\nu +1)/4) t^{1/2} \\int _{\\mu - i\\infty }^{\\mu +i\\infty } g^*(s) \\frac{\\Gamma ((1-s)/2) }{ \\Gamma ( (1- 2\\nu + s)/2)} \\right.$ $\\left.",
"\\times {d\\over dt} \\left[ Q^{-\\nu }_{(s-1)/2} \\left( {t^2+a^2 \\over t^2-a^2}\\right)\\right] 2^{-s} \\left(t^2- a^2\\right)^{(s-1)/2} ds dt\\right| \\le C \\big \\vert \\big \\vert g \\big \\vert \\big \\vert _{{\\mathcal {M}}_{0,1}^{-1}(L_c)} \\int _N^M t^{ {\\rm Re} \\nu +\\mu -3/2} dt$ $= O\\left( N^{\\mu + {\\rm Re}\\nu - 1/2}\\right),\\ N \\rightarrow \\infty , \\ \\mu + {\\rm Re}\\nu < 1/2.",
"$ Thus, combining with previous estimates, we see that $G_N$ is a Cauchy sequence and $x G_N(x)$ has the same pointwise limit (28) for almost all positive x. Theorem is proved.",
"Acknowledgments The work was partially supported by CMUP [UID/MAT/00144/2013], which is funded by FCT(Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.",
"The author thanks Mark Craddock for pointing out the Weber integral equation to his attention."
]
] | 1612.05455 |
[
[
"Octal Games on Graphs: The game 0.33 on subdivided stars and bistars"
],
[
"Abstract Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore.",
"We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices.",
"Thus, an octal game on a path P\\_n is equivalent to playing the same octal game on a heap of n counters.",
"We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph.",
"We study this game on trees and give a complete resolution of this game on subdivided stars and bistars."
],
[
"Introduction",
"Combinatorial games are finite two-player games without chance, with perfect information and such that the last move alone determines which player wins the game.",
"Since the information is perfect and the game finite, there is always a winning strategy for one of the players.",
"A formal definition of combinatorial games and basic results will be given in Section .",
"For more details, the interested reader can refer to [1], [2] or [3].",
"A well-known family of combinatorial games is the family of subtraction games, which are played on a heap of counters.",
"A subtraction game is defined by a list of positive integers $\\displaystyle L$ and is denoted by $\\displaystyle Sub(L)$ .",
"A player is allowed to remove $\\displaystyle k$ counters from the heap if and only if $\\displaystyle k \\in L$ .",
"The first player unable to play loses the game.",
"For example, consider the game $\\displaystyle Sub(\\lbrace 1,2\\rbrace )$ .",
"In this game, both players take turns removing one or two counters from the heap, until the heap is empty.",
"If the initial number of counters is a multiple of 3, then the second player has a winning strategy: by playing in such a way that the first player always gets a multiple of 3, he will take the last counter and win the game.",
"A natural generalization of subtraction games is to allow the players to split a heap into two nonempty heaps after having removed counters.",
"This defines a much larger class of games, called octal games [1].",
"An octal game is represented by an octal code which entirely defines its rules.",
"As an example, $\\displaystyle Sub(\\lbrace 1,2\\rbrace )$ is defined as 0.33.",
"A precise definition will be given in Section .",
"Octal games have been extensively studied.",
"One of the most important questions [4] is the periodicity of these games.",
"Indeed, it seems that all finite octal games have a periodic behaviour in the following sense: the set of initial numbers of counters for which the first player has a winning strategy is ultimately periodic.",
"This is true for all subtraction games and for all finite octal games for which the study has been completed [5], [1].",
"Octal games can also be played by placing counters in a row.",
"Heaps are constituted by consecutive counters and only consecutive counters can be removed.",
"According to this representation, it seems natural to play octal games on more complex structures like graphs.",
"A position of the game is a graph and players remove vertices that induce a connected component which corresponds to consecutive counters.",
"The idea to extend the notion of octal games to graphs was already suggested in [6].",
"However, to our knowledge, this idea has not been further developed.",
"With our definition, playing the generalization of an octal game on a path is the same as playing the original octal game.",
"In the special case of subtraction games, players have to keep the graph connected.",
"As an example, playing 0.33 on a graph consists in removing one vertex or two adjacent vertices from the graph without disconnecting it.",
"This extension of octal games is in line with several take-away games on graphs such as Arc Kayles [7] and Grim [8].",
"However, it does not describe some other deletion games, such as the vertex and edge versions of the game geography [7], [9], vertex and edge deletion games with parity rules, considered in [10] and [11], or scoring deletion games such as Le Pic arête [12].",
"We will first give in Section basic definitions from combinatorial game theory as well as a formal definition of octal games on graphs.",
"We then focus on the game 0.33 which is one of the simplest octal games, and to its study on trees.",
"We first study subdivided stars in Section .",
"We prove that paths can be reduced modulo 3 which leads to a complete resolution, in contrast with the related studies on subdivided stars of Node Kayles [6] and Arc Kayles [13].",
"In Section , we extend our results to subdivided bistars (i.e.",
"trees with at most two vertices of degree at least 3) using a game operator similar to the sum of games.",
"Unfortunately, these results cannot be extended to all trees and not even to caterpillars.",
"In a forthcoming paper [14], some of our results are generalized to other subtraction games on subdivided stars.",
"Combinatorial games [1] are two-player games such that: The two players play alternately.",
"There is no chance.",
"The game is finite (there are finitely many positions and no position can be encountered twice during the game).",
"The information is perfect.",
"The last move alone determines the winner.",
"In normal play, the player who plays the last move wins the game.",
"In misère play, the player who plays the last move loses the game.",
"Impartial games are combinatorial games where at each turn the moves are the same for both players.",
"Hence the only distinction between the players is who plays the first move.",
"In this paper, we will only consider impartial games in normal play.",
"Positions in impartial games have exactly two possible outcomes: either the first player has a winning strategy, or the second player has a winning strategy.",
"If a game position falls into the first category, it is an $\\displaystyle \\mathcal {N}$ -position (for $\\displaystyle \\mathcal {N}$ ext player wins); otherwise, it is a $\\displaystyle \\mathcal {P}$ -position (for $\\displaystyle \\mathcal {P}$ revious player wins).",
"From a given position $\\displaystyle J$ of the game, the different positions that can be reached by playing a move from $\\displaystyle J$ are the options of $\\displaystyle J$ , and the set of options of $\\displaystyle J$ is denoted $\\displaystyle \\mathrm {opt}(J)$ .",
"If we know the outcomes of the positions in $\\displaystyle \\mathrm {opt}(J)$ we can deduce the outcome of $\\displaystyle J$ , using the following proposition: Proposition 1 Let $\\displaystyle J$ be a position of an impartial combinatorial game in normal play: If $\\displaystyle \\mathrm {opt}(J)=\\emptyset $ , then $\\displaystyle J$ is a $\\displaystyle \\mathcal {P}$ -position.",
"If there exists a $\\displaystyle \\mathcal {P}$ -position $\\displaystyle J^{\\prime }$ in $\\displaystyle \\mathrm {opt}(J)$ , then $\\displaystyle J$ is an $\\displaystyle \\mathcal {N}$ -position: a winning move consists in playing from $\\displaystyle J$ to $\\displaystyle J^{\\prime }$ .",
"If all the options of $\\displaystyle J$ are $\\displaystyle \\mathcal {N}$ -positions, then $\\displaystyle J$ is a $\\displaystyle \\mathcal {P}$ -position.",
"Every position $\\displaystyle J$ of a combinatorial game can be viewed as a combinatorial game with $\\displaystyle J$ as the initial position.",
"We therefore often consider positions as games.",
"Some games can be described as the union of smaller game positions.",
"In order to study them, we define the concept of the sum of games.",
"Given two games $\\displaystyle J_1$ and $\\displaystyle J_2$ , their disjoint sum, denoted by $\\displaystyle J_1+J_2$ , is defined as the game where, at their turn, each player plays a legal move on either $\\displaystyle J_1$ or $\\displaystyle J_2$ .",
"Once $\\displaystyle J_1$ (resp.",
"$\\displaystyle J_2$ ) is finished, the two players play exclusively on $\\displaystyle J_2$ (resp.",
"$\\displaystyle J_1$ ), until it is over.",
"The player who plays the last move wins the game.",
"The question is now whether we can determine the outcome of a disjoint sum $\\displaystyle J_1+J_2$ as a function of the outcomes of $\\displaystyle J_1$ and $\\displaystyle J_2$ .",
"If $\\displaystyle J_1$ is a $\\displaystyle \\mathcal {P}$ -position, then $\\displaystyle J_1+J_2$ has the same outcome as $\\displaystyle J_2$ : the winning player of $\\displaystyle J_2$ applies his strategy on $\\displaystyle J_2$ , and if the other player plays on $\\displaystyle J_1$ then he applies the winning strategy on $\\displaystyle J_1$ .",
"However, the disjoint sum of two $\\displaystyle \\mathcal {N}$ -positions cannot be determined so easily.",
"In order to study the disjoint sum of two $\\displaystyle \\mathcal {N}$ -positions, the equivalence of two games $\\displaystyle J_1$ and $\\displaystyle J_2$ is defined as follows: $\\displaystyle J_1 \\equiv J_2$ if and only if $\\displaystyle J_1+J_2$ is a $\\displaystyle \\mathcal {P}$ -position.",
"According to this relation, one can attribute to a game a value corresponding to its equivalence class, called the Grundy value.",
"The Grundy value of a game position $\\displaystyle P$ for a game $\\displaystyle J$ , denoted by $\\displaystyle \\mathcal {G}_J(P)$ , can be computed from the Grundy value of its options using the following formula: $\\mathcal {G}_J(P) = \\operatorname{mex}(\\mathcal {G}_J(P^{\\prime }) | P^{\\prime } \\in \\mathrm {opt}(P))$ where, for any set of integers $\\displaystyle S$ , $\\displaystyle \\operatorname{mex}(S)$ is the smallest nonnegative integer not in $\\displaystyle S$ .",
"In particular, $\\displaystyle P$ is a $\\displaystyle \\mathcal {P}$ -position if and only if $\\displaystyle \\mathcal {G}_J(P)=0$ .",
"Note that this is consistent with Proposition REF .",
"When the context is clear, we will denote $\\displaystyle \\mathcal {G}_J(P)$ as $\\displaystyle \\mathcal {G}(P)$ .",
"A fundamental result of Combinatorial Game Theory is the Sprague-Grundy Theorem that gives the Grundy values of the sum of games: Theorem 2 (Sprague-Grundy Theorem [15]) Let $\\displaystyle J_1$ and $\\displaystyle J_2$ be two game positions.",
"Then $\\displaystyle \\mathcal {G}(J_1+J_2)=\\mathcal {G}(J_1)\\oplus \\mathcal {G}(J_2)$ , where $\\displaystyle \\oplus $ , called the nim-sum, is the bitwise XOR applied to the two values written in base 2.",
"A direct application of this theorem is that for any game position $\\displaystyle J$ , we have $\\displaystyle \\mathcal {G}(J+J)=0$ .",
"Moreover, we can see that two games $\\displaystyle J_1$ and $\\displaystyle J_2$ have the same Grundy value if and only if their disjoint sum $\\displaystyle J_1+J_2$ is a $\\displaystyle \\mathcal {P}$ -position."
],
[
"Octal games",
"A well-known family of impartial games is the family of octal games, which are played on heaps of counters.",
"On their turn, each player removes some counters from one heap and may also divide the remaining counters of the heap into two nonempty heaps.",
"The rules of an octal game are encoded according to an octal number as follows: Definition 3 (Octal games [1]) Let $\\displaystyle u_1,u_2,\\ldots ,u_n,\\ldots $ be nonnegative integers such that for all $\\displaystyle i$ , $\\displaystyle u_i \\le 7$ .",
"In the octal game $\\displaystyle {\\bf 0.u_1u_2...u_n...}$ , a player can remove $\\displaystyle i$ counters from a heap if and only if $\\displaystyle u_i \\ne 0$ .",
"Moreover, if we write $\\displaystyle u_i$ as $\\displaystyle u_i = b^i_1 + 2 \\cdot b^i_2 + 4 \\cdot b^i_3$ with $\\displaystyle b^i_j \\in \\lbrace 0,1\\rbrace $ , then, the player can, when removing $\\displaystyle i$ counters from a heap: empty the heap if and only if $\\displaystyle b^i_1=1$ ; leave the heap nonempty if and only if $\\displaystyle b^i_2=1$ ; split the remaining heap in two nonempty heaps if and only if $\\displaystyle b^i_3=1$ .",
"An octal game is finite if it has a finite number of non-zero values.",
"In this case, we stop the code at the last non-zero $\\displaystyle u_i$ .",
"For example, $\\displaystyle {\\bf u_i}=3$ means that a player can remove $\\displaystyle i$ counters from a heap without splitting it.",
"Octal games with only $\\displaystyle {\\bf 0}$ and $\\displaystyle {\\bf 3}$ in their code correspond to subtraction games since the heap is never divided.",
"In particular, the game 0.33 is the game where one can remove one or two counters from a heap.",
"A value of $\\displaystyle {\\bf u_i=7}$ means that one can remove $\\displaystyle i$ counters from a heap, possibly dividing the heap in two heaps whereas $\\displaystyle {\\bf u_i=6}$ means that one can remove $\\displaystyle i$ counters from a heap except if the heap has exactly $\\displaystyle i$ counters, and possibly divide it into two heaps.",
"To study an octal game, it suffices to consider it on a single heap.",
"Indeed, using Theorem REF , one can obtain the Grundy value of any octal game by computing the nim-sum of its components.",
"The Grundy sequence of an octal game is the sequence of the Grundy values of the game on a heap of $\\displaystyle n$ counters with $\\displaystyle n=0,1,2,...$ .",
"For example, the Grundy sequence of 0.33 is $\\displaystyle 0,1,2,0,1,2,...$ since the Grundy value of the game 0.33 on a heap of size $\\displaystyle n$ is $\\displaystyle n \\bmod 3$ .",
"The Grundy sequence of 0.33 is periodic and one can prove that this is the case for all finite subtraction games [1].",
"Actually, all the octal games which have been completely studied have an ultimately periodic Grundy sequenceFor an up-to-date table of octal games, see http://wwwhomes.uni-bielefeld.de/achim/octal.html.",
"This led to the following conjecture, proposed by Guy: Conjecture 4 (Guy's conjecture [4]) All finite octal games have ultimately periodic Grundy sequences."
],
[
"Octal games on graphs",
"A natural question is whether this periodicity can be extended to more complex structures.",
"A relevant structure is graphs.",
"Indeed, as explained in the introduction, octal games are generally played with counters in a row.",
"Considering a row of counters as a path and replacing the notion of consecutive counters by connected components, we get the following definition of octal games on graphs: Definition 5 (Octal game on graphs) Let $\\displaystyle u_1,u_2,\\ldots ,u_n,\\ldots $ be nonnegative integers such that for all $\\displaystyle i$ , $\\displaystyle u_i \\le 7$ .",
"Let $\\displaystyle G$ be a graph.",
"In the octal game $\\displaystyle {\\bf 0.u_1u_2...u_n...}$ played on $\\displaystyle G$ , a player can remove a set $\\displaystyle X_i$ of $\\displaystyle i$ vertices of $\\displaystyle G$ if and only if $\\displaystyle u_i \\ne 0$ and $\\displaystyle X_i$ induces a connected graph.",
"Moreover, if we write $\\displaystyle { u_i}$ as $\\displaystyle { u_i} = b^i_1 + 2 \\cdot b^i_2 + 4 \\cdot b^i_3$ , with $\\displaystyle b^i_j\\in \\lbrace 0,1\\rbrace $ , and $\\displaystyle H$ is the connected component of $\\displaystyle G$ containing $\\displaystyle X_i$ , then: the player can remove $\\displaystyle H$ (i.e.",
"$\\displaystyle X_i=V(H)$ ) if and only if $\\displaystyle b^i_1=1$ ; the player can leave $\\displaystyle H$ connected with at least one vertex remaining (i.e $\\displaystyle H\\setminus \\lbrace X_i\\rbrace $ is nonempty and connected) if and only if $\\displaystyle b^i_2=1$ ; the player can disconnect $\\displaystyle H$ if and only if $\\displaystyle b^i_3=1$ .",
"If $\\displaystyle G$ is a path, then the game is equivalent to the corresponding standard octal game of Definition REF .",
"We now consider several examples.",
"The game $\\displaystyle {\\bf 0.33}$ on a connected graph corresponds to the game where one can take one vertex or two adjacent vertices without disconnecting it.",
"The game $\\displaystyle {\\bf 0.07}$ corresponds to the game where one can remove any two adjacent vertices of the graph.",
"That is exactly the well-known game Arc Kayles [7].",
"Recently, Adams et al.",
"[8] studied the game Grim that is exactly 0.6 on some graphs (players are allowed to remove any vertex of the graph, except if it is an isolated vertex).",
"A scoring version of $\\displaystyle {\\bf 0.6}$ is also currently studied [16].",
"Hence our definition is relevant with existing work.",
"Note that the well-known game Node Kayles cannot be seen as such an octal game even though on a path it is equivalent to $\\displaystyle {\\bf 0.137}$ .",
"Indeed, in Nodes Kayles, when four vertices can be removed, they cannot induce a $\\displaystyle P_4$ .",
"This cannot match our definition.",
"Remark 6 In the definition of octal games, if $\\displaystyle b^i_3=1$ , then the players can split a nonempty heap in exactly two nonempty heaps.",
"Generalizations of octal games may then be defined with $\\displaystyle b^i_j$ for $\\displaystyle j \\ge 4$ in order to allow the splitting of a nonempty heap into more than two nonempty heaps.",
"However, our extension of octal games on graphs do not make this distinction: if $\\displaystyle b^i_3=1$ , then the players can disconnect the graph and leave as many components as they like.",
"Thus this move is not a move that leaves a given number of components, but one which breaks the connectivity of a graph.",
"This is still coherent with the definition of octal games on a row of counters since the path graph can only be split in two components, and allows us to include previously defined vertex deletion games, such as Arc Kayles and Grim.",
"Remark 7 We ask for the $\\displaystyle i$ removed vertices to form a connected component for two reasons.",
"First, in traditional octal games, the counters are generally taken consecutively.",
"The second reason is that if we remove this condition, then all subtraction games on graphs will be trivial.",
"Indeed, it is always possible to remove a vertex of a connected graph and keep the graph connected.",
"Therefore it is also always possible to remove $\\displaystyle i$ vertices of the graph without disconnecting it if the vertices do not need to induce a connected graph.",
"Thus playing a subtraction game on a graph would be equivalent to playing the same game on a path with the same number of vertices and we lose the interest of considering more complex structures.",
"With our definition, subtraction games on graphs are not so straightforward.",
"In the rest of this paper, we focus on one octal game, namely $\\displaystyle {\\bf 0.33}$ , for which we provide a detailed analysis on subdivided stars and bistars: by proving lemmas about reducibility of paths, we provide an equivalence between families of stars and bistars which allows us to determine their Grundy value.",
"If $\\displaystyle n$ is an integer, we define the graph $\\displaystyle P_n$ as the path on $\\displaystyle n$ vertices, with $\\displaystyle n-1$ edges.",
"A subdivided star is the tree obtained by subdividing each edge of a star $\\displaystyle K_{1,k}$ (with $\\displaystyle k \\ge 0$ ) as many times as we want.",
"Each of the subdivided edges will be called a path.",
"A subdivided star is denoted by $\\displaystyle S_{\\ell _1,\\ldots ,\\ell _k}$ , where $\\displaystyle \\ell _i \\ge 1$ is the number of vertices of the $\\displaystyle i$ th path.",
"Figure REF shows an example of such a graph.",
"The standard definition of subdivided stars actually requires $\\displaystyle k \\ge 3$ and thus excludes the paths, however we will need to consider the paths as base cases for subdivided stars and bistars.",
"This is why we will consider the subdivided star $\\displaystyle S_{\\ell _1}$ (resp.",
"$\\displaystyle S_{\\ell _1,\\ell _2}$ ) which is isomorphic to $\\displaystyle P_{\\ell _1+1}$ (resp.",
"to $\\displaystyle P_{\\ell _1+\\ell _2+1}$ ).",
"Note that the star $\\displaystyle K_{1,0}$ is isomorphic to $\\displaystyle P_1$ .",
"For clarity, the notation as paths will be used whenever applicable.",
"Figure: The subdivided star S 1,1,3,4 \\displaystyle S_{1,1,3,4}.In the 0.33 game played on a graph, players can remove a vertex or two adjacent vertices from the graph, provided that they do not disconnect the graph.",
"Figure REF shows the moves that are available for the first player on a subdivided star.",
"Note that in every figure describing moves, the original position will be boxed.",
"Figure: The available moves for the first player in the 0.33 game played on the subdivided star S 1,1,3,4 \\displaystyle S_{1,1,3,4}.The 0.33 game on paths and cycles has the same nim-sequence as the 0.33 game on heaps of counters: Proposition 8 For any $\\displaystyle n \\ge 0$ , $\\displaystyle \\mathcal {G}(P_n) = \\mathcal {G}(C_n) = n \\bmod 3$ .",
"In this section, we will prove a similar result for subdivided stars: every path of length $\\displaystyle \\ell $ can be reduced to a path of length $\\displaystyle \\ell \\bmod 3$ without changing the Grundy value.",
"Theorem 9 For all $\\displaystyle \\ell _1,\\ldots ,\\ell _k$ , we have $\\displaystyle \\mathcal {G}($$\\displaystyle S_{\\ell _1,\\ldots ,\\ell _k}$$\\displaystyle )=\\mathcal {G}($$\\displaystyle S_{\\ell _1 \\bmod 3,\\ldots ,\\ell _k \\bmod 3}$$\\displaystyle )$ .",
"To prove this theorem, it suffices to prove that a $\\displaystyle P_3$ can be attached to the central vertex or attached to a leaf of a subdivided star without changing the Grundy value.",
"This will follow from a series of lemmas.",
"First we make an observation that will be useful for several proofs.",
"Observation 10 Let $\\displaystyle P_n$ be a path with $\\displaystyle n \\ge 4$ , and $\\displaystyle x$ a vertex of $\\displaystyle P_n$ .",
"Then a move in $\\displaystyle P_n$ that removes $\\displaystyle x$ has an equivalent move that does not remove $\\displaystyle x$ : removing the symmetric of $\\displaystyle x$ leads to the same position.",
"In particular, we will use this observation when $\\displaystyle x$ is the central vertex of a star with one or two paths.",
"Lemma 11 Let $\\displaystyle \\ell \\ge 0$ and $\\displaystyle S=$$\\displaystyle S_{1,1,\\ell }$ .",
"We have $\\displaystyle \\mathcal {G}(S)=|V(S)| \\bmod 3=\\ell \\bmod 3$ .",
"We use induction on $\\displaystyle \\ell $ .",
"First, suppose that one can remove the central vertex of $\\displaystyle S$ .",
"This is only possible if $\\displaystyle \\ell =0$ , thus $\\displaystyle S=P_3$ and we are done.",
"Now, if $\\displaystyle \\ell \\ge 1$ , then one cannot remove the central vertex of $\\displaystyle S$ .",
"In this case, up to three moves are available from $\\displaystyle S$ : Removing one of the two leaves, leaving $\\displaystyle P_{\\ell +2}$ whose Grundy value is $\\displaystyle (\\ell +2) \\bmod 3$ ; Removing one vertex from the path of length $\\displaystyle \\ell $ , leaving a star whose Grundy value is $\\displaystyle (\\ell +2) \\bmod 3$ by induction hypothesis; If $\\displaystyle \\ell \\ge 2$ , removing two vertices from the path of length $\\displaystyle \\ell $ , leaving a star whose Grundy value is $\\displaystyle (\\ell +1) \\bmod 3$ by induction hypothesis.",
"Thus, we have $\\displaystyle \\mathcal {G}(S)=\\operatorname{mex}((\\ell +1) \\bmod 3, (\\ell +2) \\bmod 3)=\\ell \\bmod 3$ .",
"Note that if $\\displaystyle \\ell =1$ then all moves are equivalent and leave $\\displaystyle P_3$ , thus $\\displaystyle \\mathcal {G}(S)=\\operatorname{mex}(\\mathcal {G}(P_3))=\\operatorname{mex}(0)=1$ .",
"Lemma 12 A $\\displaystyle P_3$ can be attached to any leaf or to the central vertex of a subdivided star without changing its Grundy value.",
"Let $\\displaystyle S$ be a subdivided star, and $\\displaystyle S^{\\prime }$ be the subdivided star obtained by attaching a $\\displaystyle P_3$ to any leaf or to the central vertex of $\\displaystyle S$ .",
"We show that $\\displaystyle S + S^{\\prime }$ is a $\\displaystyle \\mathcal {P}$ -position by proving that the second player can always play to a $\\displaystyle \\mathcal {P}$ -position after the first player's move.",
"The proof is by induction on $\\displaystyle |V(S)|$ .",
"Suppose first that the first player can remove the central vertex of $\\displaystyle S$ : If $\\displaystyle S$ is empty (resp.",
"$\\displaystyle S=P_1$ , $\\displaystyle S=P_2$ ), then $\\displaystyle S^{\\prime }=P_3$ (resp.",
"$\\displaystyle S^{\\prime }=P_4$ , $\\displaystyle S^{\\prime }=P_5$ ), and thus $\\displaystyle S+S^{\\prime }$ is a $\\displaystyle \\mathcal {P}$ -position since $\\displaystyle \\mathcal {G}(S)=\\mathcal {G}(S^{\\prime })$ ; If $\\displaystyle S=P_3$ , then either $\\displaystyle S^{\\prime }=P_6$ and we are done, or $\\displaystyle S^{\\prime }=$$\\displaystyle S_{1,1,3}$ and the result follows from Lemma REF .",
"If $\\displaystyle S=S_{\\ell }$ with $\\displaystyle \\ell \\ge 4$ or $\\displaystyle S=S_{1,\\ell }$ with $\\displaystyle \\ell \\ge 2$ , then, as stated in Observation REF , the second player will always be able to replicate the first player's move on $\\displaystyle S^{\\prime }$ , by playing the symmetrical move.",
"By induction hypothesis, the new position will be a $\\displaystyle \\mathcal {P}$ -position.",
"Suppose now that the first player cannot remove the central vertex of $\\displaystyle S$ : If the first player takes one vertex (resp.",
"two vertices) from the attached $\\displaystyle P_3$ in $\\displaystyle S^{\\prime }$ , then the second player takes two vertices (resp.",
"one vertex) from it, leaving $\\displaystyle S + S$ which is a $\\displaystyle \\mathcal {P}$ -position.",
"If the first player plays elsewhere on $\\displaystyle S^{\\prime }$ , the second player answers by playing the same move on $\\displaystyle S$ .",
"By induction hypothesis, the new position will be a $\\displaystyle \\mathcal {P}$ -position.",
"If $\\displaystyle S \\ne P_m$ , then the first player cannot remove the central vertex.",
"In this case, for every first player's move on $\\displaystyle S$ , the second player can replicate it on $\\displaystyle S^{\\prime }$ , allowing us to invoke the induction hypothesis.",
"Theorem REF then directly follows from Lemma REF .",
"Hence, all paths of length $\\displaystyle 3p$ can be removed, all paths of length $\\displaystyle 3p+1$ can be reduced to a single edge, and all paths of length $\\displaystyle 3p+2$ can be reduced to a path of length 2.",
"If we want to know the Grundy value of a given subdivided star, it then suffices to study the Grundy values of the subdivided stars with paths of length 1 and 2 attached to their central vertex.",
"We are able to build a table of positions and their options: the subdivided star in row $\\displaystyle i$ and column $\\displaystyle j$ , $\\displaystyle j \\le i$ , has $\\displaystyle i$ paths attached to its central vertex, $\\displaystyle j$ of them being of length 2.",
"Figure REF shows the first six rows of this table (the first two rows correspond to the empty graph and the subdivided star reduced to its central vertex, respectively): Figure: The first six rows of the table of positions and their options.Since the Grundy value of the empty graph is 0, we can deduce the Grundy value of every star by proceeding inductively from the top lines: Theorem 13 Figure REF shows the table of the Grundy values of subdivided stars after reduction of their paths modulo 3.",
"Except for the four first rows, the rows corresponding to stars with an odd number of paths are of the form $\\displaystyle 1203(12)^*$ whereas the rows corresponding to stars with an even number of paths are of the form $\\displaystyle 03120(30)^*$ .",
"Moreover, except for the four first columns, the columns with an even number of paths of length 2 are of the form $\\displaystyle (01)^*$ whereas the columns with an odd number of paths of length 2 are of the form $\\displaystyle (23)^*$ .",
"Figure: First six rows, and rows 2p\\displaystyle 2p and 2p+1\\displaystyle 2p+1, of the table of Grundy values for the subdivided stars."
],
[
"The game ",
"Let $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ be two subdivided stars.",
"The subdivided bistar S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle m$ ; S' is the graph obtained by joining the central vertices of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ by a path of $\\displaystyle m$ edges.",
"If $\\displaystyle m=0$ , then the subdivided bistar is a subdivided star.",
"Likewise, if $\\displaystyle m \\ge 1$ , $\\displaystyle S=$$\\displaystyle S_{\\ell _1,\\ldots ,\\ell _k}$ and $\\displaystyle S^{\\prime }=\\emptyset $ , then the subdivided bistar S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle m$ ; S' is the subdivided star $\\displaystyle S_{\\ell _1,\\ldots ,\\ell _k,m-1}$ .",
"Figure REF shows an example of a subdivided bistar.",
"For the sake of convenience, we will denote S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle 1$ ; S' by $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S^{\\prime }$ .",
"Figure: The subdivided bistar S 1,2,3 \\displaystyle S_{1,2,3}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] 2\\displaystyle 2;S 1,3,4 \\displaystyle S_{1,3,4}.We notice that playing the 0.33 game on a subdivided bistar is similar to playing the 0.33 game on the two subdivided stars composing it with an \"adjustment\" depending on the length of the path linking the two stars, except for some small cases where one of the stars can be emptied so that one can play on the middle path.",
"This section is divided in two parts.",
"In the first part, we will prove that every path of length $\\displaystyle \\ell $ in a subdivided bistar can be reduced to a path of lenth $\\displaystyle \\ell \\bmod 3$ without changing the Grundy value: Theorem 14 For all $\\displaystyle \\ell _1,\\ldots ,\\ell _k,\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }},m$ , we have: $\\mathcal {G}(S_{\\ell _1,\\ldots ,\\ell _k}\\begin{tikzpicture}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {m};\\end{tikzpicture}S_{\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }}})=\\mathcal {G}(S_{\\ell _1 \\bmod 3,\\ldots ,\\ell _k \\bmod 3}\\begin{tikzpicture}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {m\\bmod 3};\\end{tikzpicture}S_{\\ell ^{\\prime }_{1} \\bmod 3,\\ldots ,\\ell ^{\\prime }_{k^{\\prime }} \\bmod 3})$ In the second part, we compute the Grundy value of a subdivided bistar, depending on the Grundy values of each of its two subdivided stars."
],
[
"Reducing the paths of a subdivided bistar",
"In this section, we prove Theorem REF .",
"We begin by proving the result for the middle path, before proving it for the paths of the two subdivided stars composing the bistar.",
"Note that we allow the length of the middle path to be 0, in which case the subdivided bistar is simply a subdivided star.",
"Thus, if a subdivided bistar has a middle path of $\\displaystyle 3k$ edges, then it can be reduced to a subdivided star without changing its Grundy value.",
"Lemma 15 For all $\\displaystyle \\ell _1,\\ldots ,\\ell _k,\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }},m$ , we have: $\\mathcal {G}(S_{\\ell _1,\\ldots ,\\ell _k}\\begin{tikzpicture}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {m};\\end{tikzpicture}S_{\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }}})=\\mathcal {G}(S_{\\ell _1,\\ldots ,\\ell _k}\\begin{tikzpicture}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {m\\bmod 3};\\end{tikzpicture}S_{\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }}})$ It is enough to prove that adding three edges to the path does not change the Grundy value of the subdivided bistar.",
"Let $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ be two subdivided stars.",
"Let $\\displaystyle B=$ S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle m$ ; S' and $\\displaystyle B^{\\prime }=$ S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle m+3$ ; S', $\\displaystyle m \\ge 0$ .",
"We show that $\\displaystyle B + B^{\\prime }$ is a $\\displaystyle \\mathcal {P}$ -position by proving that for every first player's move, the second player always has an answer leading to a $\\displaystyle \\mathcal {P}$ -position.",
"We use induction on the size of $\\displaystyle B$ .",
"The first player can play on the middle path if and only if one of the two stars is either empty or reduced to a single vertex or a $\\displaystyle P_2$ .",
"In this case, $\\displaystyle B$ and $\\displaystyle B^{\\prime }$ are subdivided stars, and the result follows from Lemma REF .",
"Assume now that both $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ have at least two vertices.",
"Hence, the first player is unable to play on the middle path and can play either on $\\displaystyle S$ or $\\displaystyle S^{\\prime }$ .",
"The second player will replicate the same move on the other subdivided bistar.",
"By induction hypothesis, the result follows.",
"In order to prove that the paths of the stars can be reduced, we need a few technical lemmas.",
"Lemma 16 Let $\\displaystyle S$ be a subdivided star, and $\\displaystyle B=S$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] ;$\\displaystyle S_{1,1}$ .",
"We have $\\displaystyle \\mathcal {G}(S)=\\mathcal {G}(B)$ .",
"We show that $\\displaystyle S + B$ is a $\\displaystyle \\mathcal {P}$ -position by proving that for every first player's move, the second player always has an answer leading to a $\\displaystyle \\mathcal {P}$ -position.",
"We use induction on the size of $\\displaystyle S$ .",
"The cases where $\\displaystyle S$ is empty or the first player can remove its central vertex are: $\\displaystyle S$ is empty, thus $\\displaystyle B = P_3$ , which is a $\\displaystyle \\mathcal {P}$ -position; $\\displaystyle S$ is a single vertex, thus $\\displaystyle B=$$\\displaystyle S_{1,1,1}$ .",
"We know by Lemma REF that $\\displaystyle \\mathcal {G}(B)=1=\\mathcal {G}(S)$ ; $\\displaystyle S=P_2$ , thus $\\displaystyle B=$$\\displaystyle S_{1,1,2}$ .",
"Considering Figure REF , we get $\\displaystyle \\mathcal {G}(B)=2=\\mathcal {G}(S)$ ; $\\displaystyle S=P_3$ , and in that case, either $\\displaystyle S=$$\\displaystyle S_{1,1}$ or $\\displaystyle S=S_2$ : $\\displaystyle B=$$\\displaystyle S_{1,1}$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] ;$\\displaystyle S_{1,1}$ .",
"$\\displaystyle B$ is a $\\displaystyle \\mathcal {P}$ -position: the first player has only one available move, and from the resulting graph the second player can play to $\\displaystyle P_3$ which is a $\\displaystyle \\mathcal {P}$ -position.",
"Both $\\displaystyle B$ and $\\displaystyle S$ being $\\displaystyle \\mathcal {P}$ -positions, we have $\\displaystyle \\mathcal {G}(B)=\\mathcal {G}(S)$ .",
"$\\displaystyle B=$$\\displaystyle S_{1,1,3}$ .",
"By Theorem REF , $\\displaystyle \\mathcal {G}(S)=\\mathcal {G}(B)$ .",
"$\\displaystyle S=S_{\\ell }$ with $\\displaystyle \\ell \\ge 4$ or $\\displaystyle S=S_{1,\\ell }$ with $\\displaystyle \\ell \\ge 2$ .",
"By Observation REF , the second player will always be able to replicate the first player's move on $\\displaystyle B$ , by playing the symmetrical move.",
"By induction hypothesis, the new position is a $\\displaystyle \\mathcal {P}$ -position.",
"Figure REF depicts the cases where the first player does not take the central vertex of $\\displaystyle S$ , and completes the proof.",
"Figure: The inductive part of the proof of Lemma Let $\\displaystyle S$ be a subdivided star, we denote S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle 2$ ; $\\displaystyle \\emptyset $ by $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}$ .",
"We then have: Lemma 17 Let $\\displaystyle S$ be a subidvided star.",
"We have $\\displaystyle \\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};})=\\mathcal {G}(S$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle 2$ ;$\\displaystyle S_{1,1}$$\\displaystyle )$ .",
"Let $\\displaystyle B=S$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle 2$ ;$\\displaystyle S_{1,1}$ .",
"We show that $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}+ B$ is a $\\displaystyle \\mathcal {P}$ -position by proving that for every first player's move, the second player always has an answer leading to a $\\displaystyle \\mathcal {P}$ -position.",
"We use induction on the size of $\\displaystyle S$ .",
"The cases where $\\displaystyle S$ is empty, or where the first player can remove either the central vertex of $\\displaystyle S$ or both the central vertex of $\\displaystyle S$ and the vertex from the middle path of $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}$ are: $\\displaystyle S$ is empty, thus $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}= P_1$ and $\\displaystyle B=$$\\displaystyle S_{1,1,1}$ .",
"We know by Lemma REF that $\\displaystyle \\mathcal {G}(B)=1=\\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};})$ so $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}+B$ is a $\\displaystyle \\mathcal {P}$ -position; $\\displaystyle S$ is a single vertex, thus $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}= P_2$ and $\\displaystyle B=$$\\displaystyle S_{1,1,2}$ .",
"Considering Figure REF , we get $\\displaystyle \\mathcal {G}(B)=2=\\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};})$ so $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}+B$ is a $\\displaystyle \\mathcal {P}$ -position; $\\displaystyle S=P_2$ , thus $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}= P_3$ and $\\displaystyle B=$$\\displaystyle S_{1,1,3}$ which by Lemma REF has the same Grundy value as $\\displaystyle S_{1,1}$ , i.e.",
"as $\\displaystyle P_3$ .",
"Thus, $\\displaystyle \\mathcal {G}(B)=\\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};})$ so $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}+B$ is a $\\displaystyle \\mathcal {P}$ -position; $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}=$$\\displaystyle S_{1,1,1}$ , thus $\\displaystyle B=$$\\displaystyle S_{1,1}$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle 2$ ; $\\displaystyle S_{1,1}$ .",
"Considering the table in Figure REF , we get $\\displaystyle \\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};})=1$ .",
"It is easy to see that $\\displaystyle \\mathcal {G}(B)=1$ , since only one move is available for the first player (removing one leaf vertex), which leaves $\\displaystyle S_{1,1,3}$ which is a $\\displaystyle \\mathcal {P}$ -position.",
"Thus $\\displaystyle \\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};})=\\mathcal {G}(B)$ so $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}+B$ is a $\\displaystyle \\mathcal {P}$ -position.",
"$\\displaystyle S=S_{\\ell }$ with $\\displaystyle \\ell \\ge 4$ or $\\displaystyle S=S_{1,\\ell }$ with $\\displaystyle \\ell \\ge 2$ .",
"By Observation REF , the second player will always be able to replicate the first player's move on $\\displaystyle B$ , by playing the symmetrical move.",
"By induction hypothesis, the new position is a $\\displaystyle \\mathcal {P}$ -position.",
"Figure REF depicts the cases where the first player takes neither the central vertex of $\\displaystyle S$ nor both the central vertex of $\\displaystyle S$ and the vertex from the middle path of $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}$ , and completes the proof.",
"Figure: The inductive part of the proof of Lemma .We are now ready to prove that the paths of the two subdivided stars of a bistar can be reduced: Lemma 18 For all $\\displaystyle \\ell _1,\\ldots ,\\ell _k,\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }},m$ , we have: $\\mathcal {G}(S_{\\ell _1,\\ldots ,\\ell _k}\\begin{tikzpicture}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {m};\\end{tikzpicture}S_{\\ell ^{\\prime }_{1},\\ldots ,\\ell ^{\\prime }_{k^{\\prime }}})=\\mathcal {G}(S_{\\ell _1 \\bmod 3,\\ldots ,\\ell _k \\bmod 3}\\begin{tikzpicture}[baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {m};\\end{tikzpicture}S_{\\ell ^{\\prime }_{1} \\bmod 3,\\ldots ,\\ell ^{\\prime }_{k^{\\prime }} \\bmod 3})$ Thanks to Lemma REF , we only have to prove the result on the subdivided bistars with a middle path of length 1 or 2.",
"Let $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ be two subdivided stars, $\\displaystyle B=$ S [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle i$ ; S' with $\\displaystyle i \\in \\lbrace 1,2\\rbrace $ , and $\\displaystyle B^{\\prime }$ the subdivided bistar obtained by attaching a $\\displaystyle P_3$ to a leaf or to the central vertex of (without loss of generality) $\\displaystyle S^{\\prime }$ .",
"We prove by induction on $\\displaystyle |V(B)|$ that $\\displaystyle \\mathcal {G}(B)=\\mathcal {G}(B^{\\prime })$ .",
"First, we consider the cases where $\\displaystyle S^{\\prime }$ is empty or the first player can remove its central vertex: If either $\\displaystyle S$ or $\\displaystyle S^{\\prime }$ is empty (resp.",
"a single vertex), then $\\displaystyle B$ is a subdivided star, and the result holds by Lemma REF ; If $\\displaystyle B=S$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] ;$\\displaystyle P_2$ (resp.",
"$\\displaystyle B=S$ [baseline=-4](0,0) node[circle,fill=black,minimum size=0,inner sep=1] – (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] node[midway,above,scale=0.5] $\\displaystyle 2$ ;$\\displaystyle P_2$ ) and the first player empties $\\displaystyle S^{\\prime }$ on $\\displaystyle B$ , then the second player is unable to replicate the move on $\\displaystyle B^{\\prime }$ .",
"The strategy is then to take two vertices from the attached $\\displaystyle P_3$ .",
"By Lemma REF (resp.",
"Lemma REF ), we have $\\displaystyle \\mathcal {G}(B)=\\mathcal {G}(B^{\\prime })$ .",
"Now, we consider the cases where the first player cannot take the central vertex of $\\displaystyle S^{\\prime }$ : If $\\displaystyle S^{\\prime }$ is a path with more than two vertices, and the $\\displaystyle P_3$ is attached to its central vertex, then replicating the first player's move will always be possible and lead to a $\\displaystyle \\mathcal {P}$ -position by induction hypothesis.",
"If the first player takes one vertex (resp.",
"two vertices) from the attached $\\displaystyle P_3$ on $\\displaystyle B^{\\prime }$ , then the second player answers by taking two vertices (resp.",
"one vertex) from it, leaving $\\displaystyle B + B$ which is a $\\displaystyle \\mathcal {P}$ -position.",
"Otherwise, the first player plays either on $\\displaystyle S$ or on $\\displaystyle S^{\\prime }$ in either of the two bistars.",
"Note that the first player cannot remove the central vertex of $\\displaystyle S^{\\prime }$ , since this case has already been treated above.",
"The second player answers by replicating his move on the other bistar, which is always a legal move, allowing us to invoke the induction hypothesis.",
"Theorem REF then follows from Lemmas REF and REF .",
"As in the subdivided stars section, we are left with a limited number of bistars to study.",
"The next subsection presents the study of the Grundy value of a subdivided bistar depending on the Grundy values of its subdivided stars."
],
[
"Computing the Grundy value of a subdivided bistar",
"We will express the Grundy value of a subdivided star as a function of the Grundy values of its two stars.",
"By Lemma REF , it is enough to consider bistars whose middle path has length either 1 or 2.",
"We consider these two cases separately."
],
[
"When the middle path is of length 1",
"Playing on a subdivided bistar with a middle path of length 1 is almost equivalent to playing in the disjoint union of the two subdivided stars, except for small cases when some moves are not available in the bistar.",
"We will see in what follows that except for some small cases, the Grundy value of the bistar is indeed the nim-sum of the Grundy values of the two stars.",
"We refine the equivalence relation $\\displaystyle \\equiv $ for subdivided stars as follows.",
"Let $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ be two subdivided stars.",
"We say that $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are $\\displaystyle \\sim _1$ -equivalent, denoted $\\displaystyle S \\sim _1 S^{\\prime }$ , if and only if for any subdivided star $\\displaystyle \\hat{S}$ , $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} \\equiv S^{\\prime } [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ .",
"Note that the Grundy value of a bistar $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S^{\\prime }$ only depends of the equivalence class under $\\displaystyle \\sim _1$ of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ .",
"The equivalence $\\displaystyle \\sim _1$ is a refinement of $\\displaystyle \\equiv $ since taking $\\displaystyle \\hat{S}=\\emptyset $ we have $\\displaystyle S\\equiv S^{\\prime }$ .",
"By Lemma REF , we already know that $\\displaystyle P_3\\sim _1\\emptyset $ , and thus $\\displaystyle S_2 \\sim _1 \\emptyset $ and $\\displaystyle S_{1,1}$$\\displaystyle \\sim _1\\emptyset $ .",
"We will prove that there are actually eight equivalence classes for $\\displaystyle \\sim _1$ : $\\displaystyle \\mathcal {C}_1^*=\\lbrace P_1,$$\\displaystyle S_{2,1}$ ,$\\displaystyle S_{2,2,2}$$\\displaystyle \\rbrace $ (these stars have Grundy value 1); $\\displaystyle \\mathcal {C}_2^*=\\lbrace P_2$ ,$\\displaystyle S_{2,2}$$\\displaystyle \\rbrace $ (these stars have Grundy value 2); $\\displaystyle \\mathcal {C}_2^{\\Box }$ : subdivided stars $\\displaystyle S$ such that $\\displaystyle \\mathcal {G}(S)=2$ and $\\displaystyle S$ contains one or three paths of length $\\displaystyle 2$ ; $\\displaystyle \\mathcal {C}_3^{\\Box }$ : subdivided stars $\\displaystyle S$ such that $\\displaystyle \\mathcal {G}(S)=3$ and S contains one or three paths of length $\\displaystyle 2$ ; For $\\displaystyle i\\in \\lbrace 0,1,2,3\\rbrace $ , $\\displaystyle \\mathcal {C}_i$ : subdivided stars $\\displaystyle S$ with $\\displaystyle \\mathcal {G}(S)=i$ and $\\displaystyle S$ is not in a previous class.",
"Figure REF shows the equivalence classes of the subdivided stars.",
"Figure: First six rows, and rows 2p\\displaystyle 2p and 2p+1\\displaystyle 2p+1, of the table of equivalence classes for ∼ 1 \\displaystyle \\sim _1 of the subdivided stars.",
"Stars belonging to resp.",
"𝒞 1 * \\displaystyle \\mathcal {C}_1^*, 𝒞 2 * \\displaystyle \\mathcal {C}_2^*, 𝒞 2 □ \\displaystyle \\mathcal {C}_2^{\\Box }, 𝒞 3 □ \\displaystyle \\mathcal {C}_3^{\\Box } are depicted by resp.",
"1 * \\displaystyle 1^*, 2 * \\displaystyle 2^*, 2 □ \\displaystyle 2^\\Box , 3 □ \\displaystyle 3^\\Box , while the 𝒞 i \\displaystyle \\mathcal {C}_i's are depicted by i\\displaystyle i.Theorem 19 The equivalence classes for $\\displaystyle \\sim _1$ are exactly the sets $\\displaystyle \\mathcal {C}_0$ , $\\displaystyle \\mathcal {C}_1$ , $\\displaystyle \\mathcal {C}_1^*$ , $\\displaystyle \\mathcal {C}_2$ , $\\displaystyle \\mathcal {C}_2^*$ , $\\displaystyle \\mathcal {C}_2^{\\Box }$ , $\\displaystyle \\mathcal {C}_3$ and $\\displaystyle \\mathcal {C}_3^{\\Box }$ .",
"Moreover, Table REF describes how the Grundy value of $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S^{\\prime }$ can be computed depending on the equivalence class of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ .",
"Table: Computing the Grundy value of S[baseline=-4](0,0)node[circle,fill=black,minimumsize=0,innersep=1]--(0.3,0)node[circle,fill=black,minimumsize=0,innersep=1];S ' \\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S^{\\prime } depending on the equivalence class of S\\displaystyle S and S ' \\displaystyle S^{\\prime }.We will need some technical lemmas before proving the theorem: Lemma 20 We have: $\\displaystyle P_1 \\sim _1$ $\\displaystyle S_{2,1}$ $\\displaystyle P_2 \\sim _1$ $\\displaystyle S_{2,2}$ $\\displaystyle S_{1,1}$ $\\displaystyle \\sim _1$ $\\displaystyle S_{2,2,1}$ $\\displaystyle S_{2,1}$ $\\displaystyle \\sim _1$ $\\displaystyle S_{2,2,2}$ .",
"Therefore, any two elements in $\\displaystyle \\mathcal {C}_1^*$ (resp.",
"$\\displaystyle \\mathcal {C}_2^*$ ) are $\\displaystyle \\sim _1$ -equivalent.",
"Each of these equivalences will be proved in the same way: for an equivalence $\\displaystyle S \\sim _1 S^{\\prime }$ , we prove that for every subdivided star $\\displaystyle \\hat{S}$ , $\\displaystyle \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S^{\\prime })$ .",
"We will use induction on the size of $\\displaystyle \\hat{S}$ .",
"The base cases will be when $\\displaystyle |\\hat{S}| \\in \\lbrace 0,1,2\\rbrace $ , that is to say when the first player is able to take the central vertex of $\\displaystyle \\hat{S}$ .",
"Each of these cases corresponds to a subdivided star, whose Grundy value is given in Figure REF .",
"In the inductive part, we need to prove that for every move on $\\displaystyle \\hat{S}[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S + \\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S^{\\prime }$ by the first player, the second player has a move leading to a $\\displaystyle \\mathcal {P}$ -position.",
"In every case, if the first player plays on $\\displaystyle \\hat{S}$ , then the second player can replicate the move, allowing us to invoke the induction hypothesis.",
"Thus, we will only consider the moves on $\\displaystyle S$ or $\\displaystyle S^{\\prime }$ in each case.",
"Case 1 : $\\displaystyle P_1 \\sim _1$ $\\displaystyle S_{2,1}$ Figure REF shows the possible moves on $\\displaystyle P_1$ or $\\displaystyle S_{2,1}$ , and the answer leading to a $\\displaystyle \\mathcal {P}$ -position (for readability, we write $\\displaystyle S$ instead of $\\displaystyle \\hat{S}$ in the figure).",
"Figure: The inductive part of the proof that P 1 ∼ 1 \\displaystyle P_1 \\sim _1S 2,1 \\displaystyle S_{2,1}.Case 2 : $\\displaystyle P_2 \\sim _1$ $\\displaystyle S_{2,2}$ Figure REF shows the possible moves on $\\displaystyle P_2$ or $\\displaystyle S_{2,2}$ , and the answer leading to a $\\displaystyle \\mathcal {P}$ -position (for readability, we write $\\displaystyle S$ instead of $\\displaystyle \\hat{S}$ in the figure).",
"Figure: The inductive part of the proof that P 2 ∼ 1 \\displaystyle P_2 \\sim _1 S 2,2 \\displaystyle S_{2,2}.Case 3 : $\\displaystyle S_{1,1}$ $\\displaystyle \\sim _1$ $\\displaystyle S_{2,2,1}$ Figure REF shows the possible moves on $\\displaystyle S_{1,1}$ or $\\displaystyle S_{2,2,1}$ , and the answer leading to a $\\displaystyle \\mathcal {P}$ -position (for readability, we write $\\displaystyle S$ instead of $\\displaystyle \\hat{S}$ in the figure).",
"Figure: The inductive part of the proof that S 1,1 \\displaystyle S_{1,1} ∼ 1 \\displaystyle \\sim _1 S 2,2,1 \\displaystyle S_{2,2,1}.Case 4 : $\\displaystyle S_{2,1}$ $\\displaystyle \\sim _1$ $\\displaystyle S_{2,2,2}$ Figure REF shows the possible moves on $\\displaystyle S_{2,1}$ or $\\displaystyle S_{2,2,2}$ , and the answer leading to a $\\displaystyle \\mathcal {P}$ -position (for readability, we write $\\displaystyle S$ instead of $\\displaystyle \\hat{S}$ in the figure).",
"Figure: The inductive part of the proof that S 2,1 \\displaystyle S_{2,1} ∼ 1 \\displaystyle \\sim _1 S 2,2,2 \\displaystyle S_{2,2,2}.Lemma 21 Let $\\displaystyle S$ be a subdivided star not belonging to $\\displaystyle \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ .",
"Then $\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S \\equiv P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S$ .",
"We use induction on $\\displaystyle |S|$ .",
"The base cases are the subdivided stars having an option in $\\displaystyle \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ : $\\displaystyle S=\\emptyset $ .",
"In this case, $\\displaystyle \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle ) = 1 = \\mathcal {G}(P_1) = \\mathcal {G}(P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"$\\displaystyle S=$$\\displaystyle S_{1,1}$ .",
"In this case, $\\displaystyle \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle )$ (by Lemma REF ) $\\displaystyle = \\mathcal {G}(P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"$\\displaystyle S=$$\\displaystyle S_{2,1,1}$ .",
"In this case, $\\displaystyle \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = 3 = \\mathcal {G}($$\\displaystyle S_{2,1,1,1}$$\\displaystyle ) = \\mathcal {G}(P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"$\\displaystyle S=$$\\displaystyle S_{2,2,1}$ .",
"In this case, $\\displaystyle \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}$$\\displaystyle S_{1,1}$$\\displaystyle )$ (by Lemma REF ) $\\displaystyle = 1 = \\mathcal {G}($$\\displaystyle S_{2,2,1,1}$$\\displaystyle ) = \\mathcal {G}(P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"$\\displaystyle S=$$\\displaystyle S_{2,2,2,1}$ .",
"In this case, $\\displaystyle \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = 3 = \\mathcal {G}($$\\displaystyle S_{2,2,2,1,1}$$\\displaystyle ) = \\mathcal {G}(P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"$\\displaystyle S=$$\\displaystyle S_{2,2,2,2}$ .",
"In this case, $\\displaystyle \\mathcal {G}($$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S) = 1 = \\mathcal {G}($$\\displaystyle S_{2,2,2,2,1}$$\\displaystyle ) = \\mathcal {G}(P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"Although tedious, all these values can be computed by considering the Grundy values of the sets $\\displaystyle \\mathrm {opt}(S_{1,1,1}[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ and $\\displaystyle \\mathrm {opt}(P_1[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S)$ .",
"We now prove that if $\\displaystyle S$ is a subdivided star not belonging to $\\displaystyle \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ and not having an option in $\\displaystyle \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ , then $\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S \\equiv P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S$ .",
"We note that the first player can neither empty $\\displaystyle S$ nor take its central vertex.",
"We show that for every first player's move on $\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S + P_1 [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}S$ , the second player can always move to a $\\displaystyle \\mathcal {P}$ -position.",
"If the first player plays from $\\displaystyle S$ to $\\displaystyle S^{\\prime }$ , then $\\displaystyle S^{\\prime } \\notin \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ , thus if the second player replicates the move, we can invoke the induction hypothesis.",
"Figure REF shows the case where the first player does not play on $\\displaystyle S$ , completing the proof.",
"Figure: The inductive part of the proof for Lemma .",
"[Proof of Theorem REF ] We prove by induction on the total number of vertices of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ that if $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are in the same set $\\displaystyle \\mathcal {C}_0$ , $\\displaystyle \\mathcal {C}_1$ , $\\displaystyle \\mathcal {C}_1^*$ , $\\displaystyle \\mathcal {C}_2$ , $\\displaystyle \\mathcal {C}_2^*$ , $\\displaystyle \\mathcal {C}_2^{\\Box }$ , $\\displaystyle \\mathcal {C}_3$ or $\\displaystyle \\mathcal {C}_3^{\\Box }$ , then they are $\\displaystyle \\sim _1$ -equivalent.",
"By Lemma REF , this is true if $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are in $\\displaystyle \\mathcal {C}_1^*$ or in $\\displaystyle \\mathcal {C}_2^*$ .",
"This is also true if $\\displaystyle \\lbrace S,S^{\\prime }\\rbrace =\\lbrace \\emptyset ,$$\\displaystyle S_{1,1}$$\\displaystyle \\rbrace $ by Lemma REF or if $\\displaystyle \\lbrace S,S^{\\prime }\\rbrace =\\lbrace \\emptyset ,P_3\\rbrace $ since it is the same as attaching a $\\displaystyle P_3$ to the central vertex of a subdivided star.",
"Furthermore, one can check that the rows and columns for $\\displaystyle \\mathcal {C}_1^*$ and $\\displaystyle \\mathcal {C}_2^*$ in Table REF are correct.",
"For that, it suffices to prove it for one representant of $\\displaystyle \\mathcal {C}_1^*$ ($\\displaystyle P_1$ ) and one representant of $\\displaystyle \\mathcal {C}_2^*$ ($\\displaystyle P_2$ ).",
"Attaching $\\displaystyle P_1$ to any subdivided star $\\displaystyle \\hat{S}$ results in the subdivided star listed directly under $\\displaystyle \\hat{S}$ in Figure REF , while attaching $\\displaystyle P_2$ results in the subdivided star listed diagonally to the right and below.",
"For example, for $\\displaystyle S_{1,1,2,2}$ , attaching $\\displaystyle P_1$ results in $\\displaystyle S_{1,1,1,2,2}$ , while attaching $\\displaystyle P_2$ results in $\\displaystyle S_{1,1,2,2,2}$ .",
"Comparing the Grundy values of the individual stars and the resulting bistar, one can verify that the table columns for $\\displaystyle \\mathcal {C}_1^*$ and $\\displaystyle \\mathcal {C}_2^*$ are correct.",
"Identifying the elements of the various sets in Figure REF , we see that below any element of $\\displaystyle \\mathcal {C}_1^*$ is a bisected star with Grundy value 2, and similarly, below any element of $\\displaystyle \\mathcal {C}_2^*$ is a bisected star with Grundy value 0.",
"All other stars are either part of a 0-1 pattern (going down the columns), or part of a 2-3 pattern, which fits the computation of the Grundy values via the nim-sum, since $\\displaystyle 3 \\oplus 1 = 2$ .",
"This verifies the result for the $\\displaystyle \\mathcal {C}_1^*$ column.",
"Likewise, one can verify the column $\\displaystyle \\mathcal {C}_2^*$ .",
"Suppose now that $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ belong to the same set $\\displaystyle C$ , with $\\displaystyle C\\ne \\mathcal {C}_1^*$ and $\\displaystyle C\\ne \\mathcal {C}_2^*$ .",
"Thus both $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are either empty or not a path.",
"We prove by induction on the size of $\\displaystyle \\hat{S}$ that $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} \\equiv S^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ for any subdivided star $\\displaystyle \\hat{S}$ .",
"This is true if $\\displaystyle \\hat{S}=\\emptyset $ (since $\\displaystyle \\mathcal {G}(S)=\\mathcal {G}(S^{\\prime })$ ) or if $\\displaystyle \\hat{S}\\in \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ (as discussed before).",
"Hence we can assume that $\\displaystyle \\hat{S}\\notin \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ and $\\displaystyle \\hat{S}$ is not a path.",
"We will prove that $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} + S^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ is a $\\displaystyle \\mathcal {P}$ -position.",
"The first player cannot play both in $\\displaystyle S$ and $\\displaystyle \\hat{S}$ nor both in $\\displaystyle S^{\\prime }$ and $\\displaystyle \\hat{S}$ since $\\displaystyle \\hat{S}$ is not a path.",
"If the first player plays in $\\displaystyle \\hat{S}$ , leading to $\\displaystyle \\hat{S}^{\\prime }$ in one of the two games, the first player cannot take the central vertex (since $\\displaystyle \\hat{S}$ is not a path).",
"Hence the second player can reply to $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}^{\\prime } + S^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}^{\\prime }$ which is a $\\displaystyle \\mathcal {P}$ -position by induction hypothesis.",
"Otherwise, the first player plays in $\\displaystyle S$ or in $\\displaystyle S^{\\prime }$ .",
"By symmetry, we can assume that the first player plays in $\\displaystyle S$ , leading to a game $\\displaystyle T[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} + S^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ .",
"We have to find an answer from that game to a $\\displaystyle \\mathcal {P}$ -position.",
"If there is a move from $\\displaystyle T$ to $\\displaystyle T^{\\prime }$ with $\\displaystyle T^{\\prime }$ in the same set as $\\displaystyle S$ , then the second player plays to $\\displaystyle T^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} + S^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ (this is always possible since if the move from $\\displaystyle T$ to $\\displaystyle T^{\\prime }$ is taking the central vertex and $\\displaystyle T^{\\prime }$ is not empty, it means that $\\displaystyle T$ is a path which is neither $\\displaystyle P_3$ nor $\\displaystyle P_4$ , a contradiction).",
"By induction, $\\displaystyle T[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} + S^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ is a $\\displaystyle \\mathcal {P}$ -position.",
"If there is a move from $\\displaystyle S^{\\prime }$ to $\\displaystyle T^{\\prime }$ with $\\displaystyle T$ and $\\displaystyle T^{\\prime }$ in the same set, then the second player plays to $\\displaystyle T[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}+ T^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ (again, this is always possible since $\\displaystyle S^{\\prime }$ is not a path), which is a $\\displaystyle \\mathcal {P}$ -position by induction hypothesis.",
"Assume that none of these two cases occurs.",
"If $\\displaystyle \\mathcal {G}(S)=3$ then we are in case (ii).",
"If $\\displaystyle \\mathcal {G}(S)=0$ then we are in case (i).",
"Hence we have $\\displaystyle \\mathcal {G}(S)\\in \\lbrace 1,2\\rbrace $ .",
"If $\\displaystyle \\mathcal {G}(S)=1$ , then $\\displaystyle S,S^{\\prime }\\in \\mathcal {C}_1$ .",
"If $\\displaystyle \\mathcal {G}(T)=0$ then we are in case (ii).",
"Otherwise, $\\displaystyle \\mathcal {G}(T)>1$ , and there is always a move from $\\displaystyle T$ to $\\displaystyle T^{\\prime }\\in \\mathcal {C}_1$ and we are in case (ii).",
"Hence $\\displaystyle \\mathcal {G}(S)=2$ .",
"If $\\displaystyle \\mathcal {G}(T)=0$ or if $\\displaystyle T\\in \\mathcal {C}_1$ , then we are in case (ii).",
"If $\\displaystyle \\mathcal {G}(T)=3$ , we are in case (i).",
"Hence the only remaining case is $\\displaystyle T\\in \\mathcal {C}_1^*$ .",
"Then there is a move from $\\displaystyle S^{\\prime }$ to $\\displaystyle T^{\\prime }$ with $\\displaystyle T^{\\prime }\\in \\mathcal {C}_1$ .",
"By induction, $\\displaystyle T^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} \\equiv $$\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ (indeed, the number of vertices in $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ is strictly greater than the number of vertices in $\\displaystyle T^{\\prime }$ and $\\displaystyle S_{1,1,1}$ since $\\displaystyle S$ has at least five vertices).",
"By Lemma REF , $\\displaystyle S_{1,1,1}$$\\displaystyle [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} \\equiv P_1[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} \\equiv T[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ (since $\\displaystyle \\hat{S}\\notin \\mathcal {C}_1^*\\cup \\mathcal {C}_2^*$ ).",
"Thus $\\displaystyle T[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S} \\equiv T^{\\prime }[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ and $\\displaystyle T[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}+ T^{\\prime } [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}\\hat{S}$ is a $\\displaystyle \\mathcal {P}$ -position.",
"To compute Table REF , it is enough to consider one representant of each class, for instance $\\displaystyle \\emptyset $ , $\\displaystyle P_1$ , $\\displaystyle P_2$ , $\\displaystyle S_{1,1,1}$ , $\\displaystyle S_{2,1,1}$ , $\\displaystyle S_{2,1,1,1}$ , $\\displaystyle S_{2,2,2,2,2}$ , $\\displaystyle S_{2,2,2,2,2,1}$ , respectively, and compute their Grundy value."
],
[
"When the middle path is of length 2",
"The situation in that case will be more complicated than in the previous case.",
"We similarly define an equivalence relation $\\displaystyle \\sim _2$ .",
"Let $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ be two subdivided stars.",
"We say that $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are $\\displaystyle \\sim _2$ -equivalent, denoted $\\displaystyle S \\sim _2 S^{\\prime }$ , if and only if for any subdivided star $\\displaystyle \\hat{S}$ , $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}\\hat{S} \\equiv S^{\\prime } [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}\\hat{S}$ .",
"By Lemma REF , we already know that $\\displaystyle P_3\\sim _2\\emptyset $ , and thus $\\displaystyle S_2 \\sim _2 \\emptyset $ and $\\displaystyle S_{1,1}$$\\displaystyle \\sim _2\\emptyset $ .",
"We will prove that there are exactly ten equivalence classes for $\\displaystyle \\sim _2$ : $\\displaystyle \\mathcal {D}_0^*$ : subdivided stars $\\displaystyle S$ such that $\\displaystyle \\mathcal {G}(S)=0$ and $\\displaystyle S$ contains zero or two paths of length $\\displaystyle 2$ , plus $\\displaystyle S_{2}$ ; $\\displaystyle \\mathcal {D}_1^*=\\lbrace P_1,$$\\displaystyle S_{2,1}$ ,$\\displaystyle S_{2,2,2}$$\\displaystyle \\rbrace $ (these stars have Grundy value 1); $\\displaystyle \\mathcal {D}_1^{\\Box }$ : subdivided stars $\\displaystyle S$ such that $\\displaystyle \\mathcal {G}(S)=1$ , $\\displaystyle S$ contains zero or two paths of length $\\displaystyle 2$ and $\\displaystyle S \\ne P_1$ ; $\\displaystyle \\mathcal {D}_2^*=\\lbrace P_2$ ,$\\displaystyle S_{2,2}$$\\displaystyle \\rbrace $ (these stars have Grundy value 2); $\\displaystyle \\mathcal {D}_2^{\\Box }$ : subdivided stars $\\displaystyle S$ such that $\\displaystyle \\mathcal {G}(S)=2$ and $\\displaystyle S$ contains one or three paths of length $\\displaystyle 2$ ; $\\displaystyle \\mathcal {D}_3^{\\Box }$ : subdivided stars $\\displaystyle S$ such that $\\displaystyle \\mathcal {G}(S)=3$ and $\\displaystyle S$ contains one or three paths of length $\\displaystyle 2$ ; For $\\displaystyle i\\in \\lbrace 0,1,2,3\\rbrace $ , $\\displaystyle \\mathcal {D}_i$ : subdivided stars $\\displaystyle S$ with $\\displaystyle \\mathcal {G}(S)=i$ and $\\displaystyle S$ is not in a previous class.",
"Figure REF shows the equivalence classes of the subdivided stars.",
"Figure: First six rows, and rows 2p\\displaystyle 2p and 2p+1\\displaystyle 2p+1, of the table of equivalence classes for ∼ 2 \\displaystyle \\sim _2 of the subdivided stars.",
"Stars belonging to resp.",
"𝒟 0 * \\displaystyle \\mathcal {D}_0^*, 𝒟 1 * \\displaystyle \\mathcal {D}_1^*, 𝒟 1 □ \\displaystyle \\mathcal {D}_1^{\\Box }, 𝒟 2 * \\displaystyle \\mathcal {D}_2^*, 𝒟 2 □ \\displaystyle \\mathcal {D}_2^{\\Box }, 𝒟 3 □ \\displaystyle \\mathcal {D}_3^{\\Box } are depicted by resp.",
"0 * \\displaystyle 0^*, 1 * \\displaystyle 1^*, 1 □ \\displaystyle 1^\\Box , 2 * \\displaystyle 2^*, 2 □ \\displaystyle 2^\\Box , 3 □ \\displaystyle 3^\\Box , while the 𝒟 i \\displaystyle \\mathcal {D}_i's are depicted by i\\displaystyle i.Theorem 22 The equivalence classes for $\\displaystyle \\sim _2$ are exactly the sets $\\displaystyle \\mathcal {D}_0$ , $\\displaystyle \\mathcal {D}_0^*$ , $\\displaystyle \\mathcal {D}_1$ , $\\displaystyle \\mathcal {D}_1^*$ , $\\displaystyle \\mathcal {D}_1^{\\Box }$ , $\\displaystyle \\mathcal {D}_2$ , $\\displaystyle \\mathcal {D}_2^*$ , $\\displaystyle \\mathcal {D}_2^{\\Box }$ , $\\displaystyle \\mathcal {D}_3$ and $\\displaystyle \\mathcal {D}_3^{\\Box }$ .",
"Moreover, Table REF describes how the Grundy value of $\\displaystyle S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ can be computed depending on the equivalence class of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ .",
"Table: Computing the Grundy value of S[baseline=-4](0,0)node[circle,fill=black,minimumsize=0,innersep=1]--(0.3,0)node[circle,fill=black,minimumsize=0,innersep=1]node[midway,above,scale=0.5]2;S ' \\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime } depending on the equivalence class of S\\displaystyle S and S ' \\displaystyle S^{\\prime }.",
"Recall that ⊕\\displaystyle \\oplus denotes the nim-sum.",
"Moreover, x⊕ 1 y\\displaystyle x \\oplus _1 y stands for x⊕y⊕1\\displaystyle x \\oplus y \\oplus 1.Notice that when the two subdivided stars are of sufficiently large order, they are in the classes $\\displaystyle \\mathcal {D}_0,\\mathcal {D}_1,\\mathcal {D}_2,\\mathcal {D}_3$ , and the Grundy value of the bistar is given by the nim-sum of the Grundy values of the two stars.",
"For most of the smallest subdivided stars, $\\displaystyle \\mathcal {G}(S[baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }) = \\mathcal {G}(S) \\oplus \\mathcal {G}(S^{\\prime }) \\oplus 1$ .",
"The following lemma proves the equivalence for $\\displaystyle \\mathcal {D}_1^*$ and $\\displaystyle \\mathcal {D}_2^*$ .",
"Its proof is not included, since it is similar to the proof of Lemma REF .",
"Lemma 23 We have: $\\displaystyle P_1 \\sim _2$ $\\displaystyle S_{2,1}$ $\\displaystyle P_2 \\sim _2$ $\\displaystyle S_{2,2}$ $\\displaystyle S_{1,1}$ $\\displaystyle \\sim _2$ $\\displaystyle S_{2,2,1}$ $\\displaystyle S_{2,1}$ $\\displaystyle \\sim _2$ $\\displaystyle S_{2,2,2}$ .",
"Therefore, any two elements in $\\displaystyle \\mathcal {D}_1^*$ (resp.",
"$\\displaystyle \\mathcal {D}_2^*$ ) are $\\displaystyle \\sim _2$ -equivalent.",
"We can now prove Theorem REF : [Proof of Theorem REF ] Rather than proving the validity of equivalence classes and then deducing the table, we prove by induction on the total number of vertices in $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ that the Grundy value of $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ is given by Table REF .",
"One can check that the rows and columns for $\\displaystyle \\mathcal {D}_1^*$ and $\\displaystyle \\mathcal {D}_2^*$ in Table REF are correct: it suffices to prove it for one representant for $\\displaystyle \\mathcal {D}_1^*$ (say $\\displaystyle P_1$ ) and for $\\displaystyle \\mathcal {D}_2^*$ (say $\\displaystyle P_2$ ).",
"This is possible since if $\\displaystyle S,S^{\\prime } \\in \\mathcal {D}_1^*,\\mathcal {D}_2^*$ , then they are $\\displaystyle \\sim _2$ -equivalent by Lemma REF .",
"For any subdivided star $\\displaystyle \\hat{S}$ , $\\displaystyle \\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_1$ is $\\displaystyle \\hat{S}$ with a path of length 2 attached to its central vertex.",
"Thus, for every class, we only need to look at the value diagonally to the right and below in Figure REF .",
"One can check that if $\\displaystyle \\mathcal {G}(\\hat{S})=0$ , then $\\displaystyle \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_1)=2$ , if $\\displaystyle \\hat{S} \\in \\mathcal {D}_1^*,\\mathcal {D}_2, \\mathcal {D}_3^{\\Box }$ , then $\\displaystyle \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_1)=0$ , if $\\displaystyle \\hat{S} \\in \\mathcal {D}_1,\\mathcal {D}_1^{\\Box }$ , then $\\displaystyle \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_1)=3$ , if $\\displaystyle \\hat{S} \\in \\mathcal {D}_2^*,\\mathcal {D}_2^{\\Box },\\mathcal {D}_3$ , then $\\displaystyle \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_1)=1$ .",
"For any subdivided star $\\displaystyle \\hat{S}$ , $\\displaystyle \\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_2$ is $\\displaystyle \\hat{S}$ with a path of length 3 attached to its central vertex.",
"Thus, $\\displaystyle \\mathcal {G}(\\hat{S} [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}P_2) = \\mathcal {G}(\\hat{S})$ .",
"Now we study the Grundy value of $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ depending on the class of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ .",
"We can suppose that $\\displaystyle S,S^{\\prime } \\notin \\mathcal {D}_1^*,\\mathcal {D}_2^*$ , and that neither $\\displaystyle S$ nor $\\displaystyle S^{\\prime }$ are $\\displaystyle S_{1,1}$ or $\\displaystyle P_3$ (since, by Lemma REF , $\\displaystyle S_{1,2}$$\\displaystyle \\sim _2 \\emptyset $ ; and $\\displaystyle P_3 \\sim _2 \\emptyset $ by Lemma REF ).",
"We can find the Grundy values of the options of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ thanks to Figure REF .",
"None of the options of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ involves taking their central vertex.",
"We can verify Table REF by computing the Grundy value of $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ depending on the Grundy values of their options, by using the induction hypothesis: $\\displaystyle \\mathcal {G}(S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }) = \\operatorname{mex}( \\mathcal {G}( T [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime } ) , \\mathcal {G}( S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}T^{\\prime } ) | T$ option of $\\displaystyle S$ , $\\displaystyle T^{\\prime }$ option of $\\displaystyle S^{\\prime } )$ In order to prove that the equivalence classes are correct, we need to check that the Grundy value of $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ does not change with the classes of the options of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ .",
"Indeed, two subdivided stars belonging to the same class can have different options.",
"We will prove two cases, the other ones being similar.",
"Case 1: $\\displaystyle S \\in \\mathcal {D}_1$ and $\\displaystyle S^{\\prime } \\in \\mathcal {D}_3$ In this case, $\\displaystyle S$ always has three different options, but these options are not the same depending on $\\displaystyle S$ .",
"$\\displaystyle S$ always has an option in $\\displaystyle \\mathcal {D}_0$ , and it can have two options either in $\\displaystyle \\mathcal {D}_2^{\\Box }$ and $\\displaystyle \\mathcal {D}_3^{\\Box }$ or in $\\displaystyle \\mathcal {D}_2$ and $\\displaystyle \\mathcal {D}_3$ .",
"$\\displaystyle S^{\\prime }$ has three options, which are in $\\displaystyle \\mathcal {D}_1,\\mathcal {D}_2$ and $\\displaystyle \\mathcal {D}_3$ .",
"These possible options of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are shown in Figure REF .",
"On the left are the possible options of $\\displaystyle S$ , and on the right are the possible options of $\\displaystyle S^{\\prime }$ .",
"The notation $\\displaystyle \\mathcal {D}_i [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}\\mathcal {D}_j$ expresses the fact that the two subdivided stars $\\displaystyle T$ and $\\displaystyle S^{\\prime }$ (resp.",
"$\\displaystyle S$ and $\\displaystyle T^{\\prime }$ ) are in the classes $\\displaystyle \\mathcal {D}_i$ and $\\displaystyle \\mathcal {D}_j$ , and that the subdivided bistar is smaller than $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ , allowing us to invoke the induction hypothesis.",
"Figure: The possible options of S[baseline=-4](0,0)node[circle,fill=black,minimumsize=0,innersep=1]--(0.3,0)node[circle,fill=black,minimumsize=0,innersep=1]node[midway,above,scale=0.5]2;S ' \\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime } when S 1 ∈𝒞 1 \\displaystyle S_1 \\in \\mathcal {C}_1 and S 2 ∈𝒞 3 \\displaystyle S_2 \\in \\mathcal {C}_3.Now, we can compute the Grundy value of $\\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime }$ .",
"First, we compute this value in the case where the options of $\\displaystyle S$ are in $\\displaystyle \\mathcal {D}_2^{\\Box }$ and $\\displaystyle \\mathcal {D}_3^{\\Box }$ : Table: NO_CAPTIONNow, we compute this value in the case where the options of $\\displaystyle S$ are in $\\displaystyle \\mathcal {D}_2$ and $\\displaystyle \\mathcal {D}_3$ : Table: NO_CAPTIONThe Grundy value being the same in both cases, we can conclude that $\\displaystyle \\mathcal {G}(S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime })=2$ .",
"Case 2: $\\displaystyle S \\in \\mathcal {D}_0$ and $\\displaystyle S^{\\prime } \\in \\mathcal {D}_2$ In this case, the possible options of $\\displaystyle S$ and $\\displaystyle S^{\\prime }$ are shown in Figure REF .",
"On the left are the options of $\\displaystyle S$ , and on the right are the options of $\\displaystyle S^{\\prime }$ .",
"Below each possible bistar is the Grundy value of the bistar, thanks to the induction hypothesis.",
"By computing the $\\displaystyle \\operatorname{mex}$ value of each of the six sets of options, we always find the value 2.",
"Thus, $\\displaystyle \\mathcal {G}(S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime })=2$ .",
"Figure: The possible options of S[baseline=-4](0,0)node[circle,fill=black,minimumsize=0,innersep=1]--(0.3,0)node[circle,fill=black,minimumsize=0,innersep=1]node[midway,above,scale=0.5]2;S ' \\displaystyle S [baseline=-4]{(0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {2};}S^{\\prime } when S∈𝒟 0 \\displaystyle S \\in \\mathcal {D}_0 and S ' ∈𝒟 2 \\displaystyle S^{\\prime } \\in \\mathcal {D}_2.Overall, there are 36 cases to consider.",
"As they are all similar to the two we already considered, we only present the possible classes of the options of $\\displaystyle S$ in Figure REF .",
"The full proof can be found in [17].",
"Figure: The classes of the possible options of S\\displaystyle S depending on its class.Going through all the cases allows to prove the correctness of Table REF .",
"This concludes our study of subdivided bistars."
],
[
"Conclusion",
"In this paper, we introduced a general definition of octal games on graphs, capturing some existing take-away games on graphs.",
"We then focused on one of the simplest octal games, 0.33, on some subclasses of trees, namely subdivided stars and bistars.",
"We proved that for subdivided stars and bistars, as in paths, one can reduce the length of the paths to their length modulo 3.",
"Thanks to this result, we have computed the exact Grundy value of any subdivided star, and exihibited a periodic behaviour.",
"We have extended these results to bistars for which one can also reduce the lengths of any path modulo 3.",
"Using operators and equivalence classes similar to the nim-sum and Grundy classes, we could then compute the Grundy value of a subdivided bistar using values of the two stars composing it.",
"However, the reduction of paths modulo 3 cannot be generalized to trees: Observation 24 Attaching a $\\displaystyle P_3$ to a vertex of a bistar which is not one of the central vertices of the stars may change the Grundy value (and even the outcome) of the game.",
"Indeed, the bistar of Figure REF is an $\\displaystyle \\mathcal {N}$ -position, but attaching a $\\displaystyle P_3$ to $\\displaystyle u$ changes it into an $\\displaystyle \\mathcal {P}$ -position.",
"Figure: Counter-example for trees: attaching a P 3 \\displaystyle P_3 to u\\displaystyle u changes the outcome.The bistar is an $\\displaystyle \\mathcal {N}$ -position: removing $\\displaystyle u$ and the leaf attached to it leaves $\\displaystyle S_{1,1,3}$ which is equivalent to $\\displaystyle S_{1,1}$ by Theorem REF , which is a $\\displaystyle \\mathcal {P}$ -position.",
"Attaching a $\\displaystyle P_3$ to $\\displaystyle u$ changes the outcome: by a straightforward case analysis, one can check that every move leaves a $\\displaystyle \\mathcal {N}$ -position.",
"Actually, we conjecture that the Grundy value of trees for the 0.33 game is not even bounded.",
"Conjecture 25 For all $\\displaystyle n \\ge 4$ , there exists a tree $\\displaystyle T$ such that $\\displaystyle \\mathcal {G}_{{\\bf 0.33}}(T)=n$ .",
"This conjecture might even be true in the class of caterpillars.",
"A feeble argument to illustrate this intuition comes from our computations.",
"We may provide examples of caterpillars with Grundy values as large as 11.",
"Figure REF depicts a caterpillar with a Grundy value of 10 (checked by computer).",
"Figure: A caterpillar with a Grundy value of 10.Conjecture 26 For all $\\displaystyle n \\ge 4$ , there exists a caterpillar $\\displaystyle C$ such that $\\displaystyle \\mathcal {G}_{\\textbf {0.33}}(C)=n$ .",
"However, some of our results can be generalized to other octal games on subdivided stars, see [14].",
"Finally, we would like to mention that it would certainly be interesting to consider the misère version of the 0.33 game on graphs."
]
] | 1612.05772 |
[
[
"A User Simulator for Task-Completion Dialogues"
],
[
"Abstract Despite widespread interests in reinforcement-learning for task-oriented dialogue systems, several obstacles can frustrate research and development progress.",
"First, reinforcement learners typically require interaction with the environment, so conventional dialogue corpora cannot be used directly.",
"Second, each task presents specific challenges, requiring separate corpus of task-specific annotated data.",
"Third, collecting and annotating human-machine or human-human conversations for task-oriented dialogues requires extensive domain knowledge.",
"Because building an appropriate dataset can be both financially costly and time-consuming, one popular approach is to build a user simulator based upon a corpus of example dialogues.",
"Then, one can train reinforcement learning agents in an online fashion as they interact with the simulator.",
"Dialogue agents trained on these simulators can serve as an effective starting point.",
"Once agents master the simulator, they may be deployed in a real environment to interact with humans, and continue to be trained online.",
"To ease empirical algorithmic comparisons in dialogues, this paper introduces a new, publicly available simulation framework, where our simulator, designed for the movie-booking domain, leverages both rules and collected data.",
"The simulator supports two tasks: movie ticket booking and movie seeking.",
"Finally, we demonstrate several agents and detail the procedure to add and test your own agent in the proposed framework."
],
[
"Introduction",
"Practical dialogue systems consist of several components.",
"The natural language understanding (NLU) module maps free texts to structured semantic frames of utterances.",
"The natural language generation (NLG) module maps the structured representations back into a natural-language form.",
"Knowledge bases (KBs) and state trackers provide access to side information and track the evolving state of the dialogue, respectively.",
"The dialogue policy is a central component of the system that chooses an action given the current state of the dialogue.",
"In traditional systems, dialogue policies might be programmed explicitly with rules.",
"However, rule-based approaches have several weaknesses.",
"First, for complex systems, it may not be easy to design a reasonable rule-based policy.",
"Second, the optimal policy might change over time, as user behavior changes.",
"A rule-based system cannot cope with such non-stationarity.",
"Thus, reinforcement learning, in which policies are learned automatically from experience, offers an appealing alternative."
],
[
"Why Is User Simulation Needed?",
"Typically, researchers seek to optimize dialogue policies with either supervised learning (SL) or reinforcement learning (RL) methods.",
"In SL approaches, a policy is trained to imitate the observed actions of an expert.",
"Supervised learning approaches often require a large amount of expert-labeled data for training.",
"For task-specific domains, intensive domain knowledge is usually required for collecting and annotating actual human-human or human-machine conversations, and is often expensive and time-consuming.",
"Additionally, even with a large amount of training data, it is possible that some dialogue state spaces may not be explored sufficiently in the training data, preventing a supervised learner to find a good policy.",
"In contrast, RL approaches allow an agent to learn without any expert-generated example.",
"Given only a reward signal, the agent can optimize a dialogue policy through interaction with users.",
"Unfortunately, RL can require many samples from an environment, making learning from scratch with real users impractical.",
"To overcome this limitation, many researchers in the dialogue systems community train RL agents using simulated users [2], [4], [6], [11], [12], [15], [18].",
"The goal of user simulation is to generate natural and reasonable conversations, allowing the RL agent to explore the policy space.",
"The simulation-based approach allows an agent to explore trajectories which may not exist in previously observed data, overcoming a central limitation of imitation-based approaches.",
"Dialogue agents trained on these simulators can then serve as an effective starting point, after which they can be deployed against real humans to improve further via reinforcement learning."
],
[
"Related Work",
"Given the reliance of the research community on user simulations, it seems important to assess the quality of the simulator.",
"How best to assess a user simulator remains an open issue, and there is no universally accepted metric [13].",
"One important feature of a good user simulator requires coherent behavior throughout the dialogue; ideally, a good metric should measure the correlation between user simulation and real human behaviors, but it is hard to find a widely accepted metric.",
"Therefore, to the best of our knowledge, there is no standard way to build a user simulator.",
"Here, we summarize the literature of user simulation in different aspects: At the granularity level, the user simulator can operate either at the dialog-actHere, a dialog-act consists of one intent, as well as zero, one or multiple slot-value pairs.",
"In the rest of the paper, we will use dialog-acts and dialog actions interchangeably level, or at the utterance level [8].",
"At the methodology level, the user simulator could use a rule-based approach, or a model-based approach where the model is learned from training data.",
"Many models have been introduced for user modeling in different dialogue systems.",
"Early work [4], [9] employed a simple, naive bi-gram model $P(a_u|a_m)$ to predict the next user-act $a_u$ based on the last system-act $a_m$ .",
"The parameters of this model are simple, but it cannot produce coherent user behaviors, for two reasons: (1) this model can only look at the last system action, and (2) if the user changes its goal, this bi-gram model might produce some illogical behavior since it does not consider the user goal when generating the next user-act.",
"Much of the follow-up work on user simulators has tried to address these issues.",
"The first issue can be addressed by looking at longer dialogue histories to select the next user action [5], [6]; the second issue can be attacked by explicitly incorporating the user goal into user state modeling [19].",
"The recently proposed sequence-to-sequence approach [21] has inspired end-to-end trainable user simulators [1].",
"This approach treats user-turn dialogue to agent-turn dialogue as a source-to-target sequence generation problem, which might be suitable for chatbot-like systems, but may not work well for domain-specific, task-completion dialogue systems, which require the ability to interact with databases and aggregate useful information into the system responses.",
"The benefit of such model-based approaches is they do not need intensive feature engineering, but they typically require a large amount of labeled data to generalize well and deal with user states not included in the training data.",
"On the other hand, agenda-based user simulation [16] provides a convenient mechanism to explicitly encode the dialogue history and user goal.",
"The user goal consists of slot-value pairs describing the user's requests and constraints.",
"A stack-like format models the state transitions and user action generation as a sequence of simple push and pop operations, which ensures the consistency of user behavior over the course of conversation.",
"In this paper, we combine the benefits of both model-based and rule-based approaches.",
"Our user simulation for the task-completion dialogue setting follows an agenda-based approach at the dialog-act level, and a sequence-to-sequence natural language generation (NLG) component is used to convert the selected dialog-act into natural language."
],
[
"Dialogue Systems for Task-Completion",
"We consider a dialogue system for helping users to book movie tickets or to look up the movies they want, by interacting with them in natural language.",
"Over the course of conversation, the agent gathers information about the customer’s desires and ultimately books the movie tickets, or identify the movie of interest.",
"The environment then assesses a binary outcome (success or failure) at the end of the conversation, based on (1) whether a movie is booked, and (2) whether the movie satisfies the user’s constraints."
],
[
"Data:",
"The data we used in the paper was collected via Amazon Mechanical Turk, and the annotation was done internally using our own schema.",
"There are 11 intents (i.e., inform, request, confirm_question, confirm_answer, etc.",
"), and 29 slots (i.e., moviename, starttime, theater, numberofpeople, etc.).",
"Most of the slots are informable slots, which users can use to constrain the search, and some are requestable slots, of which users can ask values from the agent.",
"For example, numberofpeople cannot be a requestable slot, since arguably user knows how many tickets he or she wants to buy.",
"In total, we labeled 280 dialogues in the movie domain, and the average number of turns per dialogue is approximately 11."
],
[
"User Simulator",
"In this work, we follow the agenda-based user simulation approach [16], in which a stack-like representation of user state provides a convenient mechanism to explicitly encode the dialogue history and user's goal, and user state update (state transition and user action generation) can be modeled as sequences of push and pop operations with stacks.",
"Here, we describe the rule-based user simulator in detail."
],
[
"User Goal",
"In the task-oriented dialogue setting, the first step of user simulation is to generate a user goal; the agent knows nothing about the user goal but its objective is to help the user to accomplish this goal.",
"Hence, the entire conversation exchange is around this goal implicitly.",
"Generally, the definition of user goal contains two parts: inform_slots contain a number of slot value pairs which serve as constraints from the user.",
"request_slots contain a set of slots that user has no information about the values, but wants to get the values from the agent side during the conversation.",
"To make the user goal more realistic, we add some constraints in the user goal: Slots are split into two groups.",
"For movie-booking scenario, some of elements must appear in the user goal, we called these elements as Required slots, which includes moviename, theater, starttime, date, numberofpeople; the rest slots are Optional slots; ticket is a default slot which always appears in the request_slots part of user goal.",
"We generated the user goals from the labeled dataset, using two mechanisms.",
"One mechanism is to extract all the slots (known and unknown) from the first user turns (excluding the greeting user turn) in the data, since usually the first turn contains some or all the required information from user.",
"The other mechanism is to extract all the slots (known and unknown) that first appear in all the user turns, and then aggregate them into one user goal.",
"We dump these user goals into a file as the user-goal database for the simulator.",
"Every time when running a dialogue, we randomly sample one user goal from this user goal database."
],
[
"First user-act:",
"The work focuses on user-initiated dialogues, so we randomly generated a user goal as the first turn (a user turn).",
"To make the user-act more reasonable, we add further constraints in the generation process.",
"For example, the first user turn is usually a request turn; it has at least one informable slot; if the user knows the movie name, moviename will appear in the first user turn; etc.",
"During the course of a dialogue, the user simulator maintains a compact stack-like representation named as user agenda [16], where the user state $s_u$ is factored into an agenda $A$ and a goal $G$ , which consists of constraints $C$ and request $R$ .",
"At each time-step $t$ , the user simulator will generate the next user action $a_{u,t}$ based on the its current status $s_{u,t}$ and the last agent action $a_{m,t-1}$ , and then update the current status $s^{\\prime }_{u,t}$ .",
"Here, when training or testing a policy without natural language understanding (NLU), an error model [14] is introduced to simulate the noise from the NLU component, and noisy communication between the user and agent.",
"There are two types of noise channels in the error model: one is at the intent level, the other is slot level.",
"Furthermore, at the slot level, there are three kinds of possible noise: slot deletion: to simulate the scenario that the slot was not recognized by the NLU; incorrect slot value: to simulate the scenario that the slot name was recognized correctly, but the slot value was not recognized correctly, e.g., wrong word segmentation; incorrect slot: to simulate the scenario that both the slot and its value were not recognized correctly.",
"When training or testing a policy with natural language understanding (NLU), it is not necessary to use the error model because the NLU component itself introduces noise.",
"If the agent action is inform(taskcomplete), this is to inform that the agent has gathered all the information and is ready to book the movie ticket.",
"The user simulator will check whether the current stack is empty, and also conduct constraint checking to make sure that the agent is trying to book the right movie tickets.",
"This guarantees that the user behaves in a consistent, goal-oriented manner."
],
[
"Dialogue Status",
"There are three statuses for a dialogue: no_outcome_yet, success and failure.",
"The status is no_outcome_yet if the agent has not issued the inform(taskcomplete) action and if the number of turns of the conversation has not exceeded the maximum value; otherwise, the dialogue is finished with either a success or a failure outcome.",
"To be a success dialogue, the agent must answer all the questions (a.k.a.",
"requestable slots of the user) and book the right movie tickets finally, within the maximum number of turns.",
"All other cases are failure dialogues.",
"For example, the whole dialogue exceeds the limit of max turns, or the agent books the wrong movie tickets for the user.",
"There is a special case, where the user's constraints are not satisfiable in our movie database, and the agent correctly informs that no ticket can be booked.",
"One can argue this is a successful outcome, as the agent does what is correct.",
"Here, we choose to treat it as a failure, as no ticket is booked.",
"It should be noted that this choice does not affect algorithm comparison much."
],
[
"Natural Language Understanding (NLU)",
"The natural language understanding (NLU) component is a recurrent neural network model with long-short term memory (LSTM) cells.",
"This single NLU model [7] can do intent prediction, and slot filling simultaneously.",
"For joint modeling of intent and slots, the predicted tag set is a concatenated set of IOB-format slot tags and intent tags, and an additional token <EOS> is introduced at the end of each utterance, its supervised label is an intent tag, while the supervised label of all other preceding words is an IOB tag.",
"In this way, we can still use the sequence-to-sequence training approach, the last hidden layer of the sequence is supposed to be a condensed semantic representation of the whole input utterance, so that it can be utilized for intent prediction at the utterance level.",
"This model is trained using all available dialogue actions and utterance pairs in our labeled dataset."
],
[
"Natural Language Generation (NLG)",
"The user simulator is designed on dialog act level, but it can also work on utterance level, we provide a natural language generation (NLG) component in the framework.",
"Due to the limited labeled dataset, our empirical tests found that a pure model-based NLG might not generalize well, which will introduce a lot of noise for the policy training.",
"Thus, we use a hybrid approach which consists of: Template-based NLG: outputs some predefined rule-based templates for dialog acts Model-based NLG: is trained on our labeled dataset in a sequence-to-sequence fashion.",
"It takes dialog-acts as input, and generates template-like sentences with slot placeholders via an LSTM decoder.",
"Then, a post-processing scan is performed to replace the slot placeholders with their actual values [23], [22].",
"In the LSTM decoder, we apply beam search, which iteratively considers the top $k$ best sentences up to time step $t$ when generating the token of the time step $t+1$ .",
"For the sake of the trade-off between the speed and performance, we use the beam size of 3 in our experiments.",
"In our hybrid model, if the dialog act can be found in the predefined rule-based templates, we use the template-based NLG for generating the utterance; otherwise, the utterance is generated by the model-based NLG."
],
[
"Usages",
"We conduct experiments training agents with our user simulator for the following two tasks.",
"The first is a task-completion dialogue setting on the movie-booking domain [10].",
"Here, the agent's job is to engage with the user in a dialogue with the ultimate goal of helping the user to successfully book a movie.",
"To measure the quality of the agent, there are three metrics: {success rateSuccess rate is sometimes known as task completion rate — the fraction of dialoges that finish successfully., average reward, average turns}; each of them provides different information about the quality of agents.",
"There exists a strong correlation among them: generally, a good policy should have a higher success rate, higher average reward and lower average turns.",
"Here, we choose success rate as our major evaluation metric to report for the quality of agents.",
"In the appendix, Table REF demonstrates some example dialogues for this task.",
"The second task pertains to training an KB-InfoBot [3].",
"The setting is a simplified version of the previous goal-oriented dialogues, in which an agent and user communicate with only two intents (request and inform).",
"Accordingly, for this task the experiments in KB-InfoBot [3] engage a simplified version of the simulator described in this paper, using the two aforementioned intents and six slots.",
"In this paper, the knowledge-base is drawn from the IMDB dataset.",
"In the appendix, Table REF demonstrates some example dialogues for KB-InfoBot."
],
[
"Discussion",
"In this paper, we demonstrated that rule-based user simulation can be a safe way to train reinforcement learning agents for task-completion dialogues.",
"Since rule-based user simulation requires application-specific domain knowledge to curate these hand-crafted rules, it is usually a time-consuming process.",
"One improvement for the current user simulation in the task-completion dialogue setting is to include user goal changes which make the dialogue more complex, but also realistic.",
"Another potential direction for future improvement is model-based user simulation for task-completion dialogues.",
"The advantage of model-based user simulation is that it can be adapted to other domains easily as long as there are enough labeled data.",
"Since model-based user simulation is data-driven, one potential risk is that it asks for a large amount of labeled data to train a good simulator, and it might be risky to use the user simulator to train RL agents due to the uncertainty of the model.",
"When training reinforcement learning agents with such a user simulator, the RL agents can easily learn these errors or loopholes existing in the model-based user simulator and make the false dialogues “success”.",
"In this case, the quality of learned RL policy can be misleadingly high.",
"But model-based user simulator for task-completion dialogue setting is still a good direction to investigate."
],
[
"Acknowledgments",
"We thank Asli Celikyilmaz, Alex Marin, Paul Crook, Dilek Hakkani-Tür, Hisami Suzuki, Ricky Loynd and Li Deng for their insightful comments and discussion in the project."
],
[
"Recipes",
"This framework provides you a way to develop and compare different algorithms/models (i.e., agents in the dialogue setting).",
"The dialogue system consists of two parts: agent and user simulator.",
"Here, we walk through some examples to show how to build and plug in your own agents and user simulators."
],
[
"How to build your own agent?",
"For all the agents, they are inherited from the Agent class (agent.py) which provides some common interfaces for users to implement their agents.",
"In the agent_baseline.py file, five basic rule-based agents are implemented: InformAgent informs all the slots one by one in every turn; it cannot request any information/slot.",
"RequestAllAgent requests all the slots one by one in every turn; it cannot inform any information/slot.",
"RandomAgent requests any random request in every turn; it cannot inform any information/slot.",
"EchoAgent informs the slot in the request slots of last user action; it cannot request any information/slot.",
"RequestBasicsAgent requests all basic slots in a subset one by one, then chooses inform(taskcomplete) at the last turn; it cannot inform any information/slot.",
"All the agents just re-implement two functions: initialize_episode and state_to_action.",
"Here state_to_action function makes no assumption about the structure of the agent, it is an interface to implement the mapping from state to action, which is the core part in the agent.",
"Here is an example of RequestBasicsAgent: mystyle class RequestBasicsAgent(Agent): \"\"\" A simple agent to test the system. This agent should simply request all the basic slots and then issue: thanks(). \"\"\"",
"def initialize_episode(self): self.state = {} self.state['diaact'] = 'UNK' self.state['inform_slots'] = {} self.state['request_slots'] = {} self.state['turn'] = -1 self.current_slot_id = 0 self.request_set = ['moviename', 'starttime', 'city', 'date', 'theater', 'numberofpeople'] self.phase = 0 def state_to_action(self, state): \"\"\" Run current policy on state and produce an action \"\"\" self.state['turn'] += 2 if self.current_slot_id < len(self.request_set): slot = self.request_set[self.current_slot_id] self.current_slot_id += 1 act_slot_response = {} act_slot_response['diaact'] = \"request\" act_slot_response['inform_slots'] = {} act_slot_response['request_slots'] = {slot: \"UNK\"} act_slot_response['turn'] = self.state['turn'] elif self.phase == 0: act_slot_response = {'diaact': \"inform\", 'inform_slots': {'taskcomplete': \"PLACEHOLDER\"}, 'request_slots': {}, 'turn':self.state['turn']} self.phase += 1 elif self.phase == 1: act_slot_response = {'diaact': \"thanks\", 'inform_slots': {}, 'request_slots': {}, 'turn': self.state['turn']} else: raise Exception(\"THIS SHOULD NOT BE POSSIBLE (AGENT CALLED IN UNANTICIPATED WAY)\") return {'act_slot_response': act_slot_response, 'act_slot_value_response': None} All the above rule-based agents can support only either inform or request action, here you can practice to implement a sophisticated rule-based agent which can support multiple actions, including inform, request, confirm_question, confirm_answer, deny etc.",
"agent_dqn.py provides a RL agent (agt=9), which wraps a DQN model.",
"Besides the two above functions, there are two major functions in the RL agent: run_policy and train.",
"run_policy implements an $\\epsilon $ -greedy policy, and train calls the batch training function of DQN.",
"mystyle class AgentDQN(Agent): def run_policy(self, representation): \"\"\" epsilon-greedy policy \"\"\" if random.random() < self.epsilon: return random.randint(0, self.num_actions - 1) else: if self.warm_start == 1: if len(self.experience_replay_pool) > self.experience_replay_pool_size: self.warm_start = 2 return self.rule_policy() else: return self.dqn.predict(representation, {}, predict_model=True) def train(self, batch_size=1, num_batches=100): \"\"\" Train DQN with experience replay \"\"\" for iter_batch in range(num_batches): self.cur_bellman_err = 0 for iter in range(len(self.experience_replay_pool)/(batch_size)): batch = [random.choice(self.experience_replay_pool) for i in xrange(batch_size)] batch_struct = self.dqn.singleBatch(batch, {'gamma': self.gamma}, self.clone_dqn) agent_cmd.py provides a command line agent (agt=0), which you as an agent can interact with the user simulator.",
"The command line agent supports two types of input: natural language (cmd_input_mode=0) and dialog act(cmd_input_mode=1).",
"Listing shows an example of command line agent interacting with the user simulator via the natural language; Listing shows an example of command line agent interacting with the user simulator via dialog act form.",
"Note: When the last user turn is a request action, the system will show a line of suggested available answers in the database for the agent, like the turn 0 in the Listing .",
"Both rule-based agents and RL agent, they will answer the user with the slot values from the database.",
"Here a special case for command line agent is, human (as command line agent) might type any random answer to user's request, when the typed answer is not in the database, the state tracker will correct it, and force the agent to use the values from the database in the agent response.",
"For example, in turn 1 of the Listing , if you input inform(theater=amc pacific), the actual answer received by the user is inform(theater=carmike summit 16), because amc pacific doesn't exist in the database, to avoid this wired behavior that agent informs the user a unavailable value, we restrict the agent to use the values from the suggested list.",
"The last second turn of agent is usually an inform(taskcomplete) in dialog act form or something like “Okay, your tickets are booked.” in natural language, which is to inform the user simulator that the agent nearly completes the task, and is ready to book the movie tickets.",
"To end a conversation, the last turn of the agent is usually a thanks() in dialog act form or “thanks” in natural language.",
"mystyle python run.py --agt 0 --usr 1 --max_turn 40 --episodes 150 --movie_kb_path .\\deep_dialog\\data\\movie_kb.1k.json --goal_file_path .\\deep_dialog\\data\\user_goals_first_turn_template.part.movie.v1.p --intent_err_prob 0.00 --slot_err_prob 0.00 --episodes 500 --act_level 0 --run_mode 0 --cmd_input_mode 0 New episode, user goal: { \"request_slots\": { \"ticket\": \"UNK\" }, \"diaact\": \"request\", \"inform_slots\": { \"city\": \"seattle\", \"numberofpeople\": \"2\", \"theater\": \"amc pacific place 11 theater\", \"starttime\": \"9:00 pm\", \"date\": \"tomorrow\", \"moviename\": \"deadpool\" } } Turn 0 usr: Can I buy tickets for deadpool at seattle?",
"Turn 1 sys: Which city do you want to buy the ticket?",
"Turn 2 usr: I want to watch at seattle.",
"Turn 3 sys: Which theater do you want?",
"Turn 4 usr: I want to watch at amc pacific place 11 theater.",
"Turn 5 sys: What date would you like?",
"Turn 6 usr: I want to set it up tomorrow Turn 7 sys: And what start time do you like?",
"Turn 8 usr: I want to watch at 9:00 pm.",
"Turn 9 sys: How many tickets do you need?",
"Turn 10 usr: I want 2 tickets please!",
"Turn 11 sys: Okay, your tickets were booked.",
"Turn 12 usr: Thank you Turn 13 sys: thanks Successful Dialog!",
"mystyle python run.py --agt 0 --usr 1 --max_turn 40 --episodes 150 --movie_kb_path .\\deep_dialog\\data\\movie_kb.1k.json --goal_file_path .\\deep_dialog\\data\\user_goals_first_turn_template.part.movie.v1.p --intent_err_prob 0.00 --slot_err_prob 0.00 --episodes 500 --act_level 0 --run_mode 0 --cmd_input_mode 1 New episode, user goal: { \"request_slots\": { \"ticket\": \"UNK\", \"theater\": \"UNK\" }, \"diaact\": \"request\", \"inform_slots\": { \"city\": \"birmingham\", \"numberofpeople\": \"2\", \"state\": \"al\", \"starttime\": \"4 pm\", \"date\": \"today\", \"moviename\": \"deadpool\" } } Turn 0 usr: Which theater will play the deadpool at 4 pm?",
"(Suggested Values: {'theater': ['carmike summit 16']}) Turn 1 sys: inform(theater=carmike summit 16) Turn 2 usr: I need tickets at al.",
"Turn 3 sys: request(numberofpeople) Turn 4 usr: I want 2 tickets please!",
"Turn 5 sys: request(city) Turn 6 usr: I want to watch at birmingham.",
"Turn 7 sys: request(starttime) Turn 8 usr: I want to watch at 4 pm.",
"Turn 9 sys: request(date) Turn 10 usr: I want to set it up today Turn 11 sys: inform(taskcomplete) Turn 12 usr: Thank you Turn 13 sys: thanks() Successful Dialog!"
],
[
"How to build your own user simulator?",
"Similarly, there is one user simulator class (usersim.py) which provides a few common interfaces for users to implement their user simulators.",
"All the user simulators are inherited from this class, they should re-implement these two functions: initialize_episode and next.",
"The usersim_rule.py file implements a rule-based user simulator.",
"Here the next function implements all the rules and mechanism to issue the next user action based on the last agent action.",
"Here is the example of usersim_rule.py: mystyle def next(self, system_action): \"\"\" Generate next User Action based on last System Action \"\"\" self.state['turn'] += 2 self.episode_over = False self.dialog_status = dialog_config.NO_OUTCOME_YET sys_act = system_action['diaact'] if (self.max_turn > 0 and self.state['turn'] > self.max_turn): self.dialog_status = dialog_config.FAILED_DIALOG self.episode_over = True self.state['diaact'] = \"closing\" else: self.state['history_slots'].update(self.state['inform_slots']) self.state['inform_slots'].clear() if sys_act == \"inform\": self.response_inform(system_action) elif sys_act == \"multiple_choice\": self.response_multiple_choice(system_action) elif sys_act == \"request\": self.response_request(system_action) elif sys_act == \"thanks\": self.response_thanks(system_action) elif sys_act == \"confirm_answer\": self.response_confirm_answer(system_action) elif sys_act == \"closing\": self.episode_over = True self.state['diaact'] = \"thanks\" self.corrupt(self.state) response_action = {} response_action['diaact'] = self.state['diaact'] response_action['inform_slots'] = self.state['inform_slots'] response_action['request_slots'] = self.state['request_slots'] response_action['turn'] = self.state['turn'] response_action['nl'] = \"\" # add NL to dia_act self.add_nl_to_action(response_action) return response_action, self.episode_over, self.dialog_status"
],
[
"Training Details",
"To train a RL agent, you can either start with some rule policy experience tuples to initialize the experience replay buffer pool or start with an empty experience replay buffer pool.",
"We recommend to use some rule or supervised policy to initialize the experience replay buffer pool, many work [24], [20], [25], [10] have demonstrated the benefits of such strategy as a good initialization to speed up the RL training.",
"Here, we use a very simple rule-based policy to initialize the experience replay buffer pool.",
"The RL agent is a DQN network.",
"In the training, we use the $\\epsilon $ -greedy policy and a dynamic experience replay buffer pool.",
"The size of experience replay buffer pool is dynamic changing.",
"One important trick of DQN is to introduce the target network, which is updated slowly and used to compute the target value in a short period.",
"The training procedure goes like this way: at each simulation epoch, we simulate $N$ dialogues and add these state transition tuples ($s_t, a_t, r_t, s_{t+1}$ ) into experience replay buffer pool, train and update the current DQN network.",
"In one simulation epoch, the current DQN network will be updated multiple times, depending on the batch size and the current size of experience replay buffer, at the end of simulation epoch, the target network will be replaced by the current DQN network, the target DQN network is only updated for once in one simulation epoch.",
"The experience replay strategy is critic for the training [17].",
"Our experience reply buffer update strategy is as follows: at the beginning, we will accumulate all the experience tuples from the simulation and flush the experience reply buffer pool till the current RL agent reaches a success rate threshold (i.e.",
"success_rate_threshold = 0.30), then use the experience tuples from the current RL agent to re-fill the buffer.",
"The intuition behind is the initial performance of the DQN is not good to generate enough good experience replay tuples, thus we do not flush the experience replay pool till the current RL agent can reach a certain success rate which we think it is good, for example, the performance of a rule-based agent.",
"Then in the following training process, at every simulation epoch, we estimate the success rate of the current DQN agent, if the current DQN agent is better enough (i.e.",
"better than the target network), the experience replay buffer poll will be flushed and re-filled.",
"Figure REF shows a learning curve for RL agent without NLU and NLG, Figure REF is a learning curve for RL agent with NLU and NLG, it takes longer time to train the RL agent to adapt the errors and noise from NLU and NLG.",
"Figure: Learning curve for policy training, without NLU and NLG: Green line is a rule agent which we employ to initialize the experience replay buffer pool; the blue line is the learning curve for the RL agent; orange line is the optimal upper bound, which is computed by the ratio of the number of reachable user goals in the database of the agent to the total number of user goals.Figure: Learning curve for the end-to-end policy training, with NLU and NLG: Green line is a rule agent which we employ to initialize the experience replay buffer pool; the blue line is the learning curve for the RL agent; orange line is the optimal upper bound, which is computed by the ratio of the number of reachable user goals in the database of the agent to the total number of user goals."
],
[
"Task-Completion Bot",
"Table REF shows one success and one failure dialogue examples generated by the rule-based agent and RL agent interacting with user simulator in the movie-booking domain.",
"To be informative, we also explicitly show the user goal at the head of the dialogue, but the agent knows nothing about the user goal, its goal is to help the user to accomplish this goal and book the right movie tickets.",
"Table: Two sample dialogues generated by rule-based agent and RL agent with user simulator: Left column shows both rule and RL agents succeed; Right column shows that rule-based agent fails, while RL agent succeeds."
],
[
"KB-InfoBot",
"Table REF shows some sample dialogues between the user simulator and SimpleRL-SoftKB and End2End-RL agents [3].",
"Value of the critic_rating slot is a common source of error in the user simulator, and hence all learned policies tend to ask for this value multiple times.",
"Table: Sample dialogues between user simulator and SimpleRL-SoftKB and End2End-RL agents.",
"At the end of each dialogue, the agent informs top 5 results from the KB posterior.",
"User target, if informed, is in bold."
]
] | 1612.05688 |
[
[
"Discrete Knot Ejection from the Jet in a Nearby Low Luminosity Active\n Galactic Nucleus, M81*"
],
[
"Abstract Observational constraints of relativistic jets from black holes has largely come from the most powerful and extended jets\\cite{Jorstad05,Asada14}, leaving the nature of the low luminosity jets a mystery\\cite{Falcke04}.",
"M81* is one of the nearest low-luminosity jets, which underwent an extremely large radio flare in 2011, allowing us to study compact core emission with unprecedented sensitivity and linear resolution.",
"Utilizing a multi-wavelength campaign, we were able to track the flare as it re-brightened and became optically thick.",
"Simultaneous X-ray observations indicated the radio re-brightening was preceded by a low energy X-ray flare at least $t_{\\rm delay}>12\\ {\\rm days}$ prior.",
"Associating the time delay between the two bands as the cooling time in a synchrotron flare\\cite{Urry97,Bai03}, we find the magnetic field strength was $1.9<B<9.2\\ {\\rm G}$, which is consistent with magnetic field estimate from spectral-energy distribution modeling\\cite{Kellerman81}, $B<10.2\\ {\\rm G}$.",
"In addition, VLBA observations at 23 GHz clearly illustrate a discrete knot moving mildly relativistically at $v_{\\rm app}/c=0.51\\pm0.17$ associated with the initial radio flare.",
"The observations indicate radial jet motions for the first time in M81*.",
"This has profound implications for jet production, as it means radial motion can be observed in even the lowest-luminosity AGN, but at slower velocities and smaller radial extents ($\\approx10^4\\ R_{\\rm G}$)."
],
[
"Discrete Knot Ejection from the Jet in a Nearby Low Luminosity Active Galactic Nucleus, M81* Ashley L. King$^{1,2}$ , Jon M. Miller$^{3}$ , Michael Bietenholz$^{4,5}$ , Kayhan Gültekin$^{3}$ , Mark T. Reynolds$^{3}$ , Amy Mioduszewski$^{6}$ , Michael Rupen$^{7}$ , Norbert Bartel$^{4}$ Department of Physics, 382 Via Pueblo Mall, Stanford, CA 94305, [email protected] Einstein Fellow Department of Astronomy, University of Michigan, 1085 S. University Ave, Ann Arbor, MI 48109-1107, USA Department of Physics and Astronomy, York University, Toronto, M3J 1P3, Ontario, Canada Hartebeesthoek Radio Observatory, PO Box 443, Krugersdrop, 1740, South Africa National Radio Astronomical Observatory, P.O.",
"Box O, Socorro, NM, 87801, USA NRC Dominion Radio Astrophysical Observatory, Penticton, British Columbia V2A 6J9 Observational constraints of relativistic jets from black holes has largely come from the most powerful and extended jets[1], [2], leaving the nature of the low luminosity jets a mystery[3].",
"M81* is one of the nearest low-luminosity jets, which underwent an extremely large radio flare in 2011, allowing us to study compact core emission with unprecedented sensitivity and linear resolution.",
"Utilizing a multi-wavelength campaign, we were able to track the flare as it re-brightened and became optically thick.",
"Simultaneous X-ray observations indicated the radio re-brightening was preceded by a low energy X-ray flare at least $t_{\\rm delay}>12\\ {\\rm days}$ prior.",
"Associating the time delay between the two bands as the cooling time in a synchrotron flare[4], [5], we find the magnetic field strength was $1.9<B<9.2\\ {\\rm G}$ , which is consistent with magnetic field estimate from spectral-energy distribution modeling[6], $B<10.2\\ {\\rm G}$ .",
"In addition, VLBA observations at 23 GHz clearly illustrate a discrete knot moving mildly relativistically at $v_{\\rm app}/c=0.51\\pm 0.17$ associated with the initial radio flare.",
"The observations indicate radial jet motions for the first time in M81*.",
"This has profound implications for jet production, as it means radial motion can be observed in even the lowest-luminosity AGN, but at slower velocities and smaller radial extents ($\\approx 10^4\\ R_{\\rm G}$ ).",
"M81*, at a distance of $3.96\\pm 0.29$ Mpc[7], is one of the nearest low-luminosity active galactic nuclei (AGN), and has a black hole mass of $7^{+2}_{-1}\\times 10^7\\ M_\\odot $[8].",
"It has been well surveyed at many frequencies, including radio[9], [10], [11], [12] and X-ray[13], [14], [15].",
"Like many other low-luminosity AGN, the X-ray and radio luminosities of M81*, along with its mass, place it on the “fundamental plane of black hole activity\" [14], [3].",
"The existence of the this plane suggest a functional relationship between the jet production level, as indicated by the 5 GHz radio luminosity, the 2–10 keV X-ray emission, and the black hole mass[14], [3].",
"Although M81* fits on the fundamental plane within small fluctuations around its mean[16], it shows highly variable radio emission[17] at frequencies greater than 5 GHz.",
"At 15 GHz, M81* has an average flux density of 116 mJy[18] with typical flares reaching on order of 150 mJy [17].",
"Late in 2011, M81* exceeded its typical flare strength, increasing from a flux density of 140 mJy on 24 August 2011 (MJD 55797), to 261 mJy and 321 mJy on 27 August (MJD 55800) and 1 September (MJD 55805), respectively[18].",
"Observers assembled a campaign in both the radio and X-ray bands, to track this substantial brightening, never before observed in a low-luminosity active galactic nucleus of such proximity.",
"We targeted M81* with four epochs of radio observations, including both broadband frequency coverage with the Karl Jansky Very Large Array (VLA) and high resolution radio imaging with the Very Long Baseline Array (VLBA).",
"Figures 1a & 1b illustrate the VLA broadband radio frequency behavior in the unresolved compact core.",
"As indicated by the decrease in flux density between September 17 and September 21, the flare was already cooling nearly 20 days after the initial report of activity.",
"The radio spectral index, $\\alpha $ (where $S_\\nu \\propto \\nu ^\\alpha $ ), was negative, consistent with optically thin synchrotron emission.",
"However, by the end of the four epochs, the radio emission at higher frequencies had re-brightened and the spectral index has become consistent with optically thick synchrotron emission ($\\alpha >0$ ).",
"This indicates renewed flaring activity at very small radii, i.e., the core of the jet.",
"Very Long Baseline Array (VLBA) observations confirm this, and will be discussed below.",
"Compact radio core emission is generally modeled as self-absorbed synchrotron emission[19].",
"The peak in the spectrum occurs at the boundary between optically thick and optically thin emission, from which we can make an estimate of the magnetic field strength[6].",
"Our first epoch observations peak at $\\nu _{peak,1}=10.4^{+1.0}_{-2.5}\\ {\\rm GHz}$ with a flux density of $S_p =0.22\\pm 0.01\\ {\\rm Jy}$ .",
"Taking the restoring beam from the 8.4 GHz VLBA observations as an upper limit to the size of the core, we find the magnetic field strength to be $B\\lesssim 10.2\\ {\\rm G}$ .",
"See Supplementary Information section for more details.",
"During the second epoch, the second flare peaks at roughly $\\nu _p\\sim 28\\ {\\rm GHz}$ , however with only a few data points at higher frequencies, it is difficult to assess the degeneracy between the peak of the second flare, the flux density, and the contribution from the emission at lower frequencies.",
"However, higher frequencies reveal smaller spatial scales, closer to the base of the jet, indicating the second flare originated at smaller radii.",
"In addition, changes in the spectral slope moving from optically thin ($\\alpha <0$ ) to optically thick ($\\alpha >0$ ) emission at higher frequencies throughout the four epochs indicates increases in either the particle energy density or magnetic field strength (see Figure 1a & 1b).",
"Contemporaneous X-ray observations also showed increased emission, which we associate with the radio re-brightening.",
"In general, X-ray emission in low-luminosity AGN is modeled with a fiducial power-law attributed to either inverse Compton scattering of disk photons[20], synchrotron[21] or synchrotron self-Compton[15] emission from a jet.",
"Shown in Figures 2a & 2b are the 2011 absorption-corrected light curves in the 0.5-2 and 2–10 keV X-ray bands.",
"The low energy band shows strong evidence for a flare at MJD=55816–55838, while the high energy band does not show as obvious of a trend.",
"Due to lack of sampling prior to the start of the initial radio flare, it is unclear if a similar event preceded the initial radio flare.",
"However, analysis of all the archival M81* Swift data finds this event to be unique in the six years of archival data (2006–2011), indicating its rare nature.",
"See Supplementary Materials.",
"As M81* is known to have many radio flares, though less intense, the rare X-ray behavior associated with this flare may explain its peculiar behavior, including the first detection of a discrete knot in M81*, which we discus in more detail below.",
"Interestingly, the lack of a strong flare in the high energy band suggests that the “corona\", which is responsible for the production of the high-energy X-rays via inverse Comptonization in AGN[20], is not associated with this particular radio flare.",
"In addition, because the X-ray luminosity at 2–10 keV did not vary as dramatically as the emission at low frequencies, this suggest that even during this type of event, M81* would still lie on the “fundamental plane of black hole activity”.",
"Conversely, the low energy X-ray emission does appear to be associated with the radio flare at higher frequencies.",
"It precedes the radio re-brightening by at least $t_{\\rm delay}>12\\pm 1$ days, determined via a Z-transformed cross-correlation analysis.",
"We note that the core is not in a flaring mode nearly three months later on January 16, 2012 based on its flux density and optically thin spectral index, which also sets an upper limit on $t_{\\rm delay}$ of three months.",
"This last epoch will be discussed below.",
"Accelerated charged particles in the jet core will cool via synchrotron radiation, and therefore the spectra will peak at progressively lower frequencies at later times.",
"The time lag between the peak emission at different frequencies is dependent on the magnetic field strength, velocity of the flow, and viewing angle[4], [5], as long as adiabatic losses do not dominate[1].",
"As will be discussed below, we find a one-sided VLBA radio knot moving with an apparent velocity of $v_{\\rm app}/c=0.51\\pm 0.17$ and viewing angle to our line-of-sight of $\\theta <56^\\circ $ .",
"Assuming the de-projected velocity is close to the bulk flow velocity, we constrain the magnetic field strength, $B$ , from the time lag between the X-ray flare and radio re-brightening to be $1.6<B<9.2\\ {\\rm G}$ .",
"We assumed the angle is $\\theta =14^\\circ $[8] for the the upper limit, while the minimum flux ratio and maximum time delay give the lower limit.",
"This is consistent with the magnetic field strength derived from the peak of the radio spectral-energy distribution in the first epoch, $B<10.2\\ {\\rm G}$ .",
"See the Supplementary Materials for further modeling details.",
"Our measurement of the magnetic field is nearly two orders of magnitude greater than what is measured from the radio core-shift of M81*, which find a value on order of $B\\sim 34\\ {\\rm mG}$ at 0.21 milli-arcsec (1300 $R_{\\rm G}$ ) scales at 8.4 GHz[12].",
"Even if we extrapolate to 23 GHz, assuming the magnetic field is inversely correlated with radius, we find that the magnetic field strength measured from the core shifts is $B\\sim 90\\ {\\rm mG}$ at $\\sim $ 0.08 milli-arcsec ($\\sim $ 500 $R_{\\rm G}$ ) , which is still two orders of magnitude less than our measurements at much smaller radii.",
"Though this discrepancy could be due to measuring different components of the magnetic field via the two different methods, we are more likely measuring an increase in magnetic field associated with the large flare event.",
"The magnetic field scales as $B\\propto \\nu _p^5 S_p^{-2} \\theta _A^4$ , where $\\nu _p$ is the peak frequency, $S_p$ is the peak flux density, and $\\theta _A$ is the angular size[6].",
"Even with such a large increase in magnetic field strength, only a small increase in either the peak frequency or emitting region is required in order to explain the increase in flux density observed during the flares.",
"Both of which are reasonable assumptions, as we note the peak frequency does in fact increase in the re-brightening of the flare, and the emitting region extends, as evidenced by the detection of the knot discussed below.",
"We note that the radiation energy density determined via the radio luminosity of the knot is much less than the magnetic energy density as determined via synchrotron losses.",
"If the particle energy density is roughly in equipartition with the magnetic field, our results imply that the internal energy of particles and field dominate over the radiative energy, similar to what is observed in more massive AGN[22] and stellar-mass black hole Cygnus X-1[23].",
"In addition to having a strong magnetic field, we detect a discrete knot moving mildly relativistically for the first time in M81*.",
"Figure 3a shows the knot as it moves radially outward over the four VLBA epochs, taken at 23.7 GHz with a full-width half-maximum (FWHM) restoring beam of 1.2$\\times $ 0.55 milli-arcsecs with a position angle of $-23$ degrees.",
"Figure 3b shows the 8.4 GHz observations, which have a larger restoring beam of 3.1$\\times $ 1.4 milli-arcsecs with a position angle of 21 degrees that makes detection of the knot difficult in all but the last epoch.",
"Figure 3c shows the spectral index, $\\alpha $ , between 8.4 and 23.7 GHz.",
"At a distance of 3.96$\\pm 0.29$ Mpc[7], 1 milli-arcsec corresponds to $4.0\\times 10^3$ AU, or equivalently, $5.7\\times 10^3R_{\\rm G}$ , where $R_{\\rm G}=GM_{\\rm BH}/c^2$ and $M_{\\rm BH}=7^{+2}_{-1}\\times 10^7\\ M_\\odot $ , which indicates the knot has a projected distance of $R_{projected}\\sim 7500R_{\\rm G}$ .",
"The elongation of the nucleus at 8.4 GHz has been observed before[10], [24], [25], [11], [15], [12].",
"However, we are the first to measure the radial motion of the knot in M81*.",
"Fitting the knot detected at 23.7 GHz with a Gaussian component and determining its position with respect to the brightest component, i.e., the core, we find that the knot is moving radially from the center of the radio core with a projected velocity of $v_{\\rm app}/c=0.51\\pm 0.17$ (see Figure 3a&4a).",
"Assuming the jet is bipolar, the apparent velocity, together with the brightness ratio lower limit from the one sided detection, place a constraint on the viewing angle of $\\theta <56^\\circ $ to our line-of-sight.",
"See Figure 5.",
"This limit is broadly consistent with the angle of $\\theta =14^\\circ \\pm 2$ determined via optical modeling of the accretion disk on parsec scales[8].",
"However, high resolution radio studies have found evidence that the core emission in M81* bends in the plane of the sky on milli-arcsec scales[11] as well as precesses on a timescale of years[25], [11], [12], which suggests the inclination may change even on small scales, and long time periods.",
"Deeper observations during similar outburst events are needed to further constrain the inclination angle.",
"As the knot moves radially outward, its brightness also changes.",
"Figure 4b shows the variability observed in both the knot and core at 23.7 GHz, as well as the variability observed in the core at 8.4 GHz.",
"The core emission becomes optically thick by the end of the four epochs, which can also be seen in the spectral index map in Figure 3c.",
"Figure 3c suggests that the knot is optically thin throughout the four epochs.",
"The knot does however continue to brighten until the third epoch, and then drops in flux density by the fourth epoch.",
"See Figure 4b.",
"One can use the variability of the knot as it moves out along the jet to estimate the Doppler factor[1], $\\delta _{\\rm Doppler}$ , shown as the red curve in Figure 5.",
"We discuss this in more detail in the method section, and find the Doppler factor to be $\\delta _{\\rm Doppler}\\sim 1.5$ .",
"This corresponds to a viewing angle of $\\theta \\sim 44^\\circ $ , which is much larger then the angle inferred from optical modeling.",
"However, large uncertainties due to the poor temporal sampling of the knot light curve do not exclude a $\\theta =14^\\circ $ .",
"We also reduced archival VLBA observations of M81* on January 16, 2012 (MJD 55943), nearly three months later, which also show a discrete knot, in addition to the compact core emission at both 23 and 15 GHz (Figure 6a–e).",
"Again, due to the lack of temporal sampling, we can not distinguish between a separate, distinct knot ejection associated with the re-brightening of the core, stalling of the original knot, or a recollimation of the jet.",
"The January knot might have been launched during the radio re-brightening flare if the velocity is less than that observed from the first knot.",
"In contrast, the knot may have decelerated to the position in this fifth epoch.",
"Recollimation of jets and deceleration of knots is observed in other jets, like M87[2], but at much larger gravitational radii ($\\sim 10^6 R_{\\rm G}$ ) compared to what is observed in M81*.",
"The closeness of the bright knots to the core in M81*, $R<10^5\\ R_{\\rm G}$ (for all but the smallest inclinations, $\\theta \\lesssim 6^\\circ $ ) is peculiar as compared to other AGN of similar masses but which are radio loud, e.g., 3C 120 and 3C 111[26], [27].",
"These latter AGN produce jets that reach radial extents exceeding $R>10^6R_{\\rm G}$ .",
"The mildly relativistic knot velocity may partially explain the small radial extent of M81*, as the knot may not travel as far before dissipating.",
"However, this can not be the only solution, as M87 has both mildly relativistic knots at small radii and highly relativistic knots at large radii[2].",
"Therefore, high magnetic field strength, accretion rate, small black hole mass, or even a potentially low spin may be responsible for setting the small radial extent in M81*.",
"Current paradigms suggest that jet production in both stellar-mass and supermassive black holes at low mass accretion rates should be steady and continuous[3], while black holes with high mass accretion rates, $L_{\\rm X}\\sim L_{\\rm Edd}$ – produce jets with discrete, highly relativistic knots[28], [26], [27].",
"As M81* has a relatively low-mass supermassive black hole ($7\\times 10^7M_\\odot $[8]) and is accreting material at a very low rate ($L_{\\rm X}\\sim 10^{-5.8} L_{\\rm Edd}$ ), detecting discrete, mildly relativistic knots reveals the similarities between jet production at all masses and mass accretion rates.",
"The observations demonstrate that though the radial extent and knot velocities may differ, jets are intrinsically capable of producing the same knot-like structures.",
"In addition, the multi-wavelength analysis indicates that jet knot ejections are not only associated with radio flares, but also X-ray flares, and can be further utilized to measure the magnetic field strength very close to the black hole.",
"Author Contributions: A.L.K led the data reduction and analysis, with contributions from J.M.M., M.B., A.M..",
"K.G., M.T.R., M.R.",
"and N.B contributed to discussion and interpretation.",
"Corresponding author: Correspondence and requests for materials should be addressed to A.L.K.",
"([email protected]).",
"Acknowledgements: The authors would like to thank the referees for their invaluable comments.",
"ALK would like to thank the support for this work, which was provided by NASA through Einstein Postdoctoral Fellowship grant number PF4-150125 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060 Figure: VLBA 16 January 2012 Observations: (MJD 55943) This figure shows the 23.7 GHz, 15 GHz, 8.4 GHz, and spectral index maps for the fifth epoch.",
"Panels have contours of the σ rms ×(-3,3,6,\\sigma _{rms}\\times (-3, 3, 6, 12,24,48,96,192,384)×10 -3 Jy beam -1 12, 24, 48, 96, 192, 384)\\times 10^{-3} {\\rm Jy~beam}^{-1}, where σ rms \\sigma _{rms} is 0.32, 0.15 and 0.22 mJy beam -1 ^{-1}, for a, b and c respectively.",
"Panels c and d have contours of (-3,-2.5,-2,-1.5,-1,-0.1,0,0.5,1)(-3,-2.5,-2,-1.5,-1,-0.1,0,0.5,1).",
"The restoring beams are shown in the bottom right corner; a) 0.43×0.380.43\\times 0.38 milli-arcsec with a position angle of 36 ∘ ^\\circ , b) 0.62×0.560.62\\times 0.56 milli-arcsec with a position angle of -0.6 ∘ -0.6^\\circ , and c-d) 1.1×0.931.1\\times 0.93 milli-arcsec with a position angle of 21 ∘ ^\\circ .",
"As can be seen in a-c, the knot is extended much further from the core than in previous observations.",
"Panels d and e, show the core and knot are both optically thin (α<0\\alpha <0) indicating the flare has stopped.Supplementary Information Radio Reduction M81* exhibited a large radio flare on 23 August 2011[18], reaching 321 mJy by 1 September 2011[18].",
"The mean flux density during the previous 5 years was 116 mJy, at 15 GHz.",
"We triggered four epochs of observations with the VLA on 17 September, 21 September, 26 September, and 2 October 2011.",
"During the first two observations, the telescope array was in transition from its most extended configuration (A) to its most compact configuration (D).",
"The later two observations were both taken in the compact, D configuration.",
"We used radio bands centered on the frequencies of 1.2, 1.7, 8.5, 11.9, 12.6, 16.0, 21.5, 23.7, 29.0, and 37.0 GHz.",
"Each sub-band had a bandwidth of 1.024 GHz, except the 1.7 GHz which had a 512 MHz bandwidth.",
"3C 286 was used as the flux density calibrator with two minute integrations at each band, except 1.7 GHz, which had 3.5 minutes.",
"J0958+6533 was used as the primary phase calibrator, bracketing the integrations on M81*.",
"A total of seven minutes was spent on M81* at each frequency band, except at 1.7 GHz, which had 5.25 minutes on source.",
"The pointing offsets were calibrated before the high frequency observations.",
"The total observing time at each epoch was 1.5 hours.",
"Utilizing the CASA software package, version 3.4.0[30], the visibilities were examined for radio frequency interference, which primarily affected the low frequency bands.",
"Standard reduction techniques were used to transfer the flux density scale from the flux density calibrator to phase calibrator and M81*, and then bandpass and gain calibrations were done.",
"A Gaussian centered at the phase center was then fit to the uv visibilities.",
"A 5% error was added to the statistical fit error to account for the uncertainty in the absolute flux density scale.",
"Figures 1a & 1b show the resulting flux density measurements.",
"We also triggered observations at 8.4 and 23 GHz with the VLBA.",
"These observations were taken on the same days as the VLA observations, except for the second epoch which was taken on 20 September 2011.",
"We focus the following method sections on the 23 GHz data reduction, though the reduction was similar for 8.4 GHz data.",
"In each observation epoch, 90 minutes were spent on source, M81*, and 50 minutes on the phase calibrator, J0958$+$ 6533.",
"The observations were centered at 23.7 GHz, and had 8 spectral windows, each with 16 spectral channels.",
"Each spectral window had 8 MHz of bandwidth giving a total bandwidth of 64 MHz, and were taken with full polarizations.",
"Each epoch had 8 telescopes in the array, except on September 20, which had 6.",
"We reduced the data using NRAO's AIPS software, version 31DEC13.",
"We again used J0958+6533 as the phase calibrator.",
"A grid of 512$\\times $ 512 pixels with size of $4.5\\times 10^{-2}$ milli-arcsec was used to produce an image.",
"After the initial gain calibrations were transferred to M81*, we proceeded to use self-calibration to improve the images.",
"Initial self-calibration was phase-only.",
"Amplitude and phase calibrations were then carried out with solution intervals initially set to the entire observation, decreasing with each iteration.",
"Figure 3a shows the 23.7 GHz VLBI image of M81*.",
"The resulting root-mean-square (rms) of the images is given in Table 1.",
"Figures 3b and 3c show the 8.4 GHz VLBI images and spectral index map of M81*, respectively.",
"The emission at 23.7 GHz was modeled in the UV plane with OMFIT using four point sources (see Figure 4a, Table 1).",
"The first two adjacent point sources were used to fit the core emission, and two additional point sources were used to model the knot as well as the extended emission observed to the north-east in each image.",
"Though OMFIT produces an error associated with the measured positions, we calculated the position errors using the full-width at half-maximum of the beam divided by the component's signal-to-noise, which increased the errors, allowing a more conservative estimate fore the measured velocity.",
"The third component, between the core and the most extended emission, is the least constrained, likely due to the approximation of the irregular structure with Gaussian components.",
"After the positions had been measured, distances between the brightest knot and the brightest core component were computed.",
"The westernmost core component was always the brightest one.",
"We note that in the 8.4 GHz observations, the extended emission did not extend farther than 2 milli-arcsec ($1.1\\times 10^4R_{\\rm G}$ ).",
"See Figure 3b.",
"Utilizing the projected distance measurements for the discrete knot moving to the north-east, we made a linear fit to the distance as a function of time.",
"We found that the apparent velocity was $v_{\\rm app}/c=0.52\\pm 0.08$ ($v_{\\rm app}=8.2\\pm 1.0$ milli-arcsec year$^{-1}$ , $1\\sigma $ confidence errors).",
"When calculating the uncertainty in the apparent velocity in physical units, we include both the statistical errors from the linear fit to the distance in Figure 4a, as well as the distance uncertainty to M81* of 3.96$\\pm $ 0.29 Mpc[7].",
"There may be an intrinsic bias in the first observations when the knot is initially detected.",
"The knot emission may still overlap with the core emission, which would bias the measured position toward the core.",
"If only the first or second observations were effected, i.e., the observations where the knot is closest to the core, then this would artificially increase the measured velocity ,as it would artificially decrease the distance measured from the core in these epochs.",
"We therefore exclude the first epoch from our velocity fits, measuring a velocity of $v_{\\rm app}/c=0.51\\pm 0.17$ .",
"More conservatively, if we exclude both the first and second epochs, we measure a velocity of $v_{\\rm app}/c=0.55\\pm 0.18$ .",
"These velocities are still consistent with our initial measurement and $3\\sigma $ inconsistent with a zero velocity, indicating the knot is not stationary, and moving radially from the core.",
"Consequently, we adopt a value of $v_{\\rm app}/c=0.51\\pm 0.17$ as the measured velocity, as it is consistent with all the velocity fits, but likely best estimates the systematic uncertainties of the knot velocity measurement.",
"Finally, the spectral index maps displayed in Figures 3c, 6d and 6e were made by convolving the highest resolution restoring beams with the lowest resolution restoring beams (8GHz).",
"Then we utilized the AIPS tool COMB to find the spectral index between the two frequency across the image, making sure to clip the initial images below three times the noise level to ensure a significant detection of the spectral slope.",
"Some emission at higher frequencies could be resolved-out when comparing to the 8.4 GHz observation, making the spectral index a lower limit.",
"However, we have quantified the amount of flux density that could be resolved-out using the tool UVSUB to transfer the 8 GHz cleaned model components through the 23 GHz UV coverage.",
"After convolving the transformed 8 GHz visibilities with the initial 8.4 GHz restoring beam, we compared the resulting image to the original 8.4 GHz image.",
"We find that only 1.9% of the flux density is resolved-out during this last epoch.",
"X-ray Data Reduction The X-ray data were taken with the Swift satellite between September 7, 2011 to October 3, 2011, with a cadence of approximately two days.",
"Additional Swift data from 2011 were also included for comparison to the campaign during the radio flare.",
"The average count rate for all the observations in 2011 was 0.42 counts s$^{-1}$ with an average exposure time of 960 seconds.",
"The data were reduced using the xrtpipeline via FTOOLS software, version 6.16[31].",
"Exposure maps and background files were created using the standard tools xrtexpomap and xselect, respectively.",
"Extraction regions for the source were nominally an annulus with an outer radius of 118 arcsecs and an inner radius of 4.7 arcsec to avoid pile-up contamination.",
"The background region also used a annulus with outer radius of 213 arcsec and inner radius of 141 arcsec.",
"After the source and background spectra were extracted, we utilized XSPEC, version 12.8.2[32], to fit the data with a neutral absorption component modeled with tbabs, and a power-law component.",
"The absorption component was frozen at the Galactic hydrogen column density of $5\\times 10^{20}$ cm$^{-2}$ [33], while the power-law index and normalization were allowed to vary.",
"Figures 2a & 2b show the low-energy flux (0.5–2.0 keV), and high-energy flux (2-10 keV) during 2011.",
"The high energy mean for 2011 was $1.2\\times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ with a standard deviation of $0.2\\times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ .",
"In the low energy band (0.5–2 keV), excluding the time of the flare ($55790-55840$ MJD), the data fluctuate around a mean of $6.3\\times 10^{-12}$ ergs s$^{-1}$ cm$^{-2}$ with a standard deviation of $0.8\\times 10^{-12}$ ergs s$^{-1}$ cm$^{-2}$ .",
"The low-energy X-ray flaring data had a peak well above the mean at $1.4^{+0.4}_{-0.3}\\times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ .",
"We show that this flare is model independent by plotting the intensity across the entire band (0.5–10 keV) versus the ratio between the detected counts in the 0.5–2 keV and 2–10 keV bands in Supplementary Information Figure 1a.",
"The flare, depicted in black, moves well above all the archival Swift data to lower energies as the flare proceeds (clockwise).",
"Though the flare is not strictly unique in this behavior compared to other outliers in Supplementary Information Figure 1a, it does show unique behavior in terms of the overall intensity.",
"We preform a 2D Anderson-Darling test utilizing the python scipy (version 0.14.0) script anderson_ksamp to determine how unique the flare intensity distribution is compared to the rest of 2011 and to the entire six year period.",
"We find that the flare distribution is inconsistent with both the 2011 and the six year intensity distribution at the $4.3\\sigma $ and $4.0\\sigma $ level.",
"Moreover, we fit the data with a simple power-law model shown in Supplementary Information Figure 1b.",
"The average spectral index is $\\Gamma =1.72\\pm 0.14$ , while the peak spectral index is $\\Gamma =2.41\\pm 0.42$ .",
"We find this event to be rare, in that analysis of all the M81* Swift archival data (2005-2011) shows only this one event is associated with a statistically significant ($3\\sigma $ ) increase in the spectral index above the nominal $\\Gamma =1.7$ .",
"Interestingly, an increase in the spectral index to $\\Gamma \\simeq 2.5$ is also seen in the black hole at the center of the Milky Way, Sgr A*, during its X-ray flares[34].",
"An increase in spectral index and X-ray brightening also occurs in stellar-mass black holes before discrete knot ejections as well[35].",
"This suggest that this behavior is not unique to supermassive black holes but is ubiquitous in jet production in accreting black holes across the mass scale.",
"If we add a second neutral absorption component that is allowed to vary to account for intrinsic variable absorption in M81*, the results do not vary significantly.",
"A varying absorption component could easily explain the other outliers in Supplementary Information Figure 1a as absorption moves out of our line-of-sight, increasing the counts in the soft band, though not necessarily the spectral index.",
"This indicates that the low-energy X-ray flare is intrinsic to the source and not to neutral material serendipitously moving out of our line-of-sight.",
"Figure: X-ray Flare Characteristics: Panel a: This figure shows the count rate versus the count rate ratio between the 0.5–2 keV and 2–10 keV X-ray bands.",
"This shows that the flare (in black) is model independent, increases in count rate well above the typical count rate, while also becoming “softer\" dominated by the low energy band emission.",
"The flare moves clockwise in this diagram.",
"Panel b: The X-ray spectral index, measured in the 0.5-10 keV band, increases during the soft-energy X-ray flare, and then returns to its nominal value.",
"The dashed line indicates the average spectral index, excluding the flare from the fit.",
"The dotted lines are 1σ\\sigma standard deviations.",
"There is no other event that deviates from the average spectral index by over 3σ3\\sigma confidence in all of the Swift archival data.",
"Error bars are 1σ\\sigma confidence.Cross Correlation Analysis In order to estimate the time delay between the low-energy X-ray and radio flare, we utilized a Z-transformed discrete correlation function[36].",
"This analysis allowed for the cross-correlation between two light curves with sparse, uneven temporal sampling.",
"We required that at least three points be used in the the time lag bin.",
"Cross-correlation between the low-energy X-ray band (0.5-2.0 keV) and the three radio bands, 15, 23 and 37 GHz indicate a peak 12$\\pm $ 1 days in all three bands as we do not yet measure the peak of the flare in any of these three bands.",
"A positive lag means that the X-rays are leading the radio flare.",
"The uncertainty on this time delay is strictly statistical and does not take into account systematics effects that can result from under-sampling the light curves.",
"Likewise, as we do not measure a peak in the flux density, our measurement is a lower limit, $t_{\\rm delay}>12\\pm 1$ days.",
"However, we can place a large upper limit to the time delay, as we know the flare to have finished by the fifth epoch of observation with the VLBA on 16 January, 2012.",
"Therefore the time delay is between $12<t_{\\rm delay}<106$ days.",
"Magnetic field Estimates The magnetic field strength can be estimated using the time delay between the X-ray and Radio bands[5], [4], with the following equation.",
"$t_{\\rm delay}\\approx 2\\times 10^7 [(1+z)/\\delta _{\\rm Doppler}]^{-1/2} B^{-3/2} [{\\rm s}]$ where $t_{\\rm delay}$ is the time delay between X-ray and radio bands, $\\delta _{\\rm Doppler}= [\\gamma (1-\\beta \\cos \\theta )]^{-1}$ is the Doppler factor, $\\gamma $ is the Lorentz factor, $\\beta $ is the intrinsic velocity, $\\theta $ is the viewing angle, and $B$ is the magnetic field strength measured in Gauss.",
"This method assumes that both the X-ray and radio flares are produced in the same shock event, and both frequency bands are dominated by synchrotron emission.",
"However, if the X-rays are instead dominated by synchrotron self-Compton scattering[15], we would expect the X-rays to lag behind the radio flare[37].",
"We estimate the Doppler factor with three methods, all of which utilize measurements of the knot observed with the VLBA.",
"We caution that this knot was ejected during a previous flare, but is assumed to be representative of subsequent flares.",
"The first method of estimating the Doppler factor assumes the jet is bipolar, and uses the ratio of flux densities between the approaching and receding knots via the following relation, $J=\\left(\\frac{1+\\beta \\cos \\theta }{1-\\beta \\cos \\theta }\\right)^p$ where $J$ is the brightness ratio, $\\beta $ is the actual velocity $\\beta =\\beta _{\\rm app}/(\\beta _{\\rm app}\\cos \\theta +\\sin \\theta )$ , $\\theta $ is the angle to our line-of-sight, and $p=3-\\alpha $ , where $\\alpha $ is the radio spectral index.",
"We assumed the knot emission was optically thin and discrete, i.e., $\\alpha \\approx -0.7$ .",
"See spectral index maps Figures 3c.",
"Due to our lower limit on the brightness ratio, we place an upper limit on the inclination of $\\theta <56^\\circ $ , and a lower limit on the Doppler factor of $\\delta _{\\rm Doppler}>1.24$ .",
"See the black curve in Figure 5.",
"This also assumes that once the apparent velocity of the knot is de-projected to intrinsic velocity of the knot, it is the bulk flow speed.",
"The second method for estimating the Doppler factor utilizes the flux density variability observed in the knot, and assumes such variability occurs on the light travel time across the knot [1].",
"$\\delta _{\\rm var} = \\frac{s {\\rm D}}{c \\Delta t_{\\rm var} (1+z)}$ where $\\delta _{\\rm var}$ is the Doppler factor derived from variability, $s$ is the angular size of the component and we assumed 1.6$a$ , where $a$ is the full-width half-maximum of the Gaussian components.",
"Further, D is the distance to M81*, $c$ is the speed of light, and $\\Delta t_{\\rm var}$ is the variability timescale defined as $\\Delta t_{\\rm var}=dt/\\ln (S_{\\rm max}/S_{\\rm min})$ .",
"Here, $dt$ is the time between the maximum ($S_{\\rm max}$ ) and minimum ($S_{\\rm min}$ ) flux densities.",
"This methods assumes radiative losses, and we note that the knot we observe may be subject to adiabatic losses in addition to radiative losses.",
"However, variability studies of quasars show that radiative losses are the dominant cooling mechanism in similar knot structures[1].",
"As shown in Figure 4b, the knot peaks in flux density during the third epoch on 26 September 2011.",
"Almost a week later it shows a minimum in flux density on 2 October 2011.",
"This gives a brightness ratio of $S_{\\rm max}/S_{\\rm min} = 2.1\\pm 0.6$ and a time delay of $dt = 6$ days.",
"The brightest knot was fit with a Gaussian with full-width half-maximum of $0.76\\times 0.29$ milli-arcsecs with a position angle of $-16.4$ degrees.",
"The shortest light travel path is across the minor axis, thus we used $a=0.29$ milli-arcsec for the angular size of the knot, which gives an estimate of $\\delta _{\\rm var} = 1.33$ .",
"If we assume that the Doppler boosting observed from the knot is equal to the Doppler boosting set by the variance, this corresponds to a viewing angle of $\\theta = 44^\\circ $ , which is broadly consistent with the inclination inferred from the brightness ratio of approaching and receding knots.",
"See the red curve in Figure 5.",
"Unfortunately, the temporal sampling is sparse, resulting in a large uncertainty in both the time as well as the ratio between the maximum and minimum flux densities.",
"By definition, the observed ratio between the maximum and minimum is a lower limit, making $\\delta _{\\rm var}$ a lower limit as well.",
"However, the time between the maximum and minimum could both be shorter or longer, resulting in a larger or smaller $\\delta _{\\rm var}$ , respectively.",
"Therefore, estimating the uncertainty in our measurement of $\\delta _{\\rm var}$ is difficult and should be taken as an initial estimate only.",
"Finally, we utilize the viewing angle of $\\theta =14^\\circ \\pm 2$ , inferred from optical modeling of the M81* accretion disk [8], as well as the apparent knot velocity of $v_{\\rm app}/c=0.51\\pm 0.17$ as a third method to infer a Doppler factor of $\\delta _{\\rm Doppler}=2.2\\pm 0.4$ .",
"See blue curve in Figure 5.",
"This is also broadly consistent with the estimates inferred from our previous two methods.",
"However, we stress that the viewing angle is highly model dependent, and although it gives low statistical errors, this inclination measurement may also be subject to systematic errors that could bias it both higher or lower.",
"Moving forward, we assume the Doppler factor from the second flare is the same as the first, which is derived from the observed knot and its structure.",
"We use $\\delta _{\\rm Doppler}=2.2$ in the magnetic field calculation, as it is our largest estimated Doppler factor, while still consistent with all three methods.",
"As the time-delay is assumed to be a lower limit, due to the failure to sample the peak of the radio flare, both quantities, $\\delta _{\\rm Doppler}$ and $t_{\\rm delay}$ , and their limits make the estimate of the magnetic field an upper limit (i.e., $B\\propto \\delta _{\\rm Doppler}^{1/3} t_{\\rm delay}^{-2/3}$ ).",
"We estimate the magnetic field strength to be $B<9.2\\ {\\rm G}$ .",
"If we assume that the time delay is an order of magnitude larger, $t_{\\rm Delay}<106\\ days$ , which is roughly the time of our fifth epoch, where we do not see the core in a flaring state, and the Doppler factor has its minimum possible value of $\\delta _{\\rm Doppler}>1.4$ , we find the minimum magnetic field strength of $B>1.9\\ {\\rm G}$ .",
"The magnetic field strength can also be assessed utilizing both the angular size and spectral-energy distribution of the core emission.",
"This is done via the following relation[6], $B \\sim f^{-5}\\nu _p^{5}S_p^{-2}\\theta _A^{4}(1+z)^{-1} [G]$ where $f\\sim 8$ for an electron distribution with an index of $\\gamma =2$ , ($N(E)dE\\propto E^{-\\gamma }dE$ ), $S_p$ is the flux density in Jy at the peak frequency, $\\nu _p$ , measured in GHz.",
"To find the peak of the spectral energy distribution during our first epoch of observation we extrapolate the spectrum from both low and high frequencies.",
"We fit the spectrum between 1.8 and 8.4 GHz with a exponential fit of $S_\\nu \\propto \\nu ^\\alpha $ , where $\\alpha =0.27\\pm 0.04$ .",
"This is consistent with the slope measured just with the 8.4 GHz.",
"At higher frequencies, we find the best fit between 12 and 16 GHz with exponential fit has an exponent of $\\alpha =-(0.35\\pm 0.27)$ .",
"These fits intersect at $10.4^{+1.0}_{-2.5}$ with a peak flux density of 0.22$\\pm 0.01$ Jy.",
"Assuming the angular size at 10.4 GHz is smaller than the restoring beam which scales inversely with frequency, our smallest full-width half-maximum restoring beam at 8.4 GHz of 0.75 milli-arcsecs puts an upper limit of $B\\lesssim 10.2$ G on the magnetic field strength.",
"This is of the same order of magnitude as our previous estimate using the time delay and Doppler boosting of the knot.",
"c c | c c c c c | c c c c c 7 0pc VLBA Component Parameters 5c23 GHz 5c 8.4 GHz Date Comp S$_\\nu $ D $\\sigma _{\\rm rms}$ $\\theta _{\\rm FWHM}$ $\\theta _{\\rm PA}$ S$_\\nu $ D $\\sigma _{\\rm rms}$ $\\theta _{\\rm FWHM}$ $\\theta _{\\rm PA}$ (mJy) (marcsec) (mJy (marcsec) ($^\\circ $ ) (mJy) (marcsec) (mJy (marcsec) ($^\\circ $ ) beam$^{-1}$ ) beam$^{-1}$ ) Sept. 17 0.21 0.78$\\times $ 0.32 5.7$^\\circ $ 0.13 2.3$\\times $ 0.72 0.8$^\\circ $ 1 $ 62.4\\pm 1.1$ - $ 103.0\\pm 0.5$ - 2 $ 49.6\\pm 1.0$ $ 0.18\\pm 0.01$ $ 80.2\\pm 0.5$ $ 0.43\\pm 0.01$ 3 $ 16.8\\pm 0.5$ $ 0.54\\pm 0.01$ 4 $ 4.1\\pm 0.3$ $ 1.13\\pm 0.02$ Sept. 20 0.34 1.17$\\times $ 0.33 $-26.7^\\circ $ 0.14 2.8$\\times $ 0.96 $-29.8^\\circ $ 1 $ 42.8\\pm 0.7$ - $ 99.0\\pm 1.2$ - 2 $ 22.4\\pm 0.7$ $ 0.32\\pm 0.02$ $ 69.6\\pm 1.2$ $ 0.56\\pm 0.01$ 3 $ 0.6\\pm 0.7$ $ 0.87\\pm 0.63$ 4 $ 3.7\\pm 0.8$ $ 1.27\\pm 0.13$ Sept. 26 0.24 0.76$\\times $ 0.29 $-16.4^\\circ $ 0.10 2.4$\\times $ 0.75 $-23.3^\\circ $ 1 $ 77.9\\pm 0.6$ - $ 95.5\\pm 1.0$ - 2 $ 37.1\\pm 0.7$ $ 0.24\\pm 0.01$ $ 61.4\\pm 1.1$ $ 0.48\\pm 0.01$ 3 $ 5.0\\pm 0.4$ $ 0.83\\pm 0.03$ 4 $ 5.2\\pm 0.4$ $ 1.34\\pm 0.02$ Oct. 2 0.17 0.77$\\times $ 0.27 $-14.7^\\circ $ 0.16 2.3$\\times $ 0.74 $-13.2^\\circ $ 1 $ 159.4\\pm 0.7$ - $ 94.9\\pm 0.4$ - 2 $ 31.3\\pm 0.6$ $ 0.19\\pm 0.01$ $ 65.1\\pm 0.4$ $ 0.47\\pm 0.01$ 3 $ 8.0\\pm 0.4$ $ 0.46\\pm 0.02$ 4 $ 2.5\\pm 0.3$ $ 1.48\\pm 0.04$ Jan. 16 0.32 0.43$\\times $ 0.38 $36.0^\\circ $ 0.22 1.1$\\times $ 0.93 $20.8^\\circ $ 1 $ 47.8\\pm 0.8$ - $ 59.3\\pm 0.8$ - 2 $ 11.9\\pm 0.8$ $ 0.24\\pm 0.03$ $ 17.1\\pm 0.8$ $ 0.79\\pm 0.02$ 3 $ 1.7\\pm 0.4$ $ 1.24\\pm 0.10$ 4 $ 1.3\\pm 0.4$ $ 1.79\\pm 0.13$ 5c 15 GHz Jan. 16 0.15 0.62$\\times $ 0.56 $-0.6^\\circ $ 1 $ 48.1\\pm 0.7$ - 2 $ 17.3\\pm 0.7$ $ 0.29\\pm 0.02$ 3 $ 2.3\\pm 0.4$ $ 1.26\\pm 0.07$ 4 $ 1.9\\pm 0.4$ $ 1.67\\pm 0.08$ This table shows all the best fit parameters to the fit to the VLBA data.",
"We assumed point sources for each component, fitting in the UV plane.",
"The FWHM ($\\theta _{\\rm FWHM}$ ) and position angles ($\\theta _{\\rm PA}$ ) to that of the restoring beam are given for each observation along with the image rms ($\\sigma _{\\rm rms}$ ).",
"The 23 GHz and 15 GHz observations were fit with four components, while the 8.4 GHz were fit with two components.",
"All distances are measured relative to the brightest component, i.e., the core, and the component with the largest distance is taken as the “knot” in our analysis."
]
] | 1612.05654 |
[
[
"Unconditional construction of K3 surfaces over finite fields with given\n L-function in large characteristic"
],
[
"Abstract We give an unconditional construction of K3 surfaces over finite fields with given L-function, up to finite extensions of the base fields, under some mild restrictions on the characteristic.",
"Previously, such results were obtained by Taelman assuming semistable reduction.",
"The main contribution of this paper is to make Taelman's proof unconditional.",
"We use some results of Nikulin and Bayer-Fluckiger to construct an appropriate complex projective K3 surface with CM which admits an elliptic fibration with a section, or an ample line bundle of low degree.",
"Then using Saito's construction of strictly semistable models and applying a slight refinement of Matsumoto's good reduction criterion for K3 surfaces, we obtain a desired K3 surface over a finite field."
],
[
"Introduction",
"In this paper, we shall give an unconditional construction of $K3$ surfaces over finite fields with given $L$ -function, up to finite extensions of the base fields.",
"Previously, Taelman conditionally proved the existence of such $K3$ surfaces assuming a strong version of the existence of semistable reduction for $K3$ surfaces [22].",
"The main contribution of this paper is to make Taelman's proof unconditional.",
"Fix a prime number $p$ and an integer $m$ with $1\\le {m}\\le 10$ .",
"Let $q$ be a power of $p$ .",
"We fix an embedding $\\overline{\\mathbb {Q}} \\hookrightarrow \\overline{\\mathbb {Q}}_p$ , and let $\\nu _{q}\\colon \\overline{\\mathbb {Q}}_p\\rightarrow \\mathbb {Q}\\cup \\lbrace \\infty \\rbrace $ be the $p$ -adic valuation normalized by ${\\nu _q}(q)=1$ .",
"For a polynomial $Q(T)=\\prod _{j}(1-{{\\beta }_{j}}T) \\in \\mathbb {Q}[T]\\qquad (\\beta _{j}\\in \\overline{\\mathbb {Q}}),$ we put $Q_{<0}(T)=\\prod _{\\nu _{q}({\\beta }_{j})<0}(1-{{\\beta }_{j}}T) \\in \\mathbb {Q}_{p}[T].$ Consider a polynomial $L(T) \\in {1+T\\mathbb {Q}[T]}$ of degree $2m$ satisfying the following conditions.",
"Condition 1.1 All complex roots of $L(T)$ have absolute value one, no root of $L(T)$ is a root of unity, $L(T) \\in \\mathbb {Z}_{\\ell }[T]$ for all prime numbers ${\\ell }\\ne {p}$ , there exists a positive integer $h\\in \\mathbb {Z}$ with $1\\le {h}\\le {m}$ such that, if we denote the roots of $L(T)$ by $\\alpha _1, \\cdots , \\alpha _{2m}\\in \\overline{\\mathbb {Q}}$ , then, after permuting them, they satisfy ${\\left\\lbrace \\begin{array}{ll}\\nu _{q}({\\alpha }_i)=-1/h &\\quad \\ ({1}\\le {i}\\le {h})\\\\\\nu _{q}({\\alpha }_i)=0 &\\quad \\ ({h+1}\\le {i}\\le {2m-h})\\\\\\nu _{q}({\\alpha }_i)=1/h &\\quad \\ ({2m-h+1}\\le {i}\\le {2m})\\\\\\end{array}\\right.",
"}$ for a power $q$ of $p$ , $L(T)=Q(T)^e$ for some $e\\ge 1$ and some irreducible polynomial ${Q(T)} \\in {\\mathbb {Q}[T]}$ , and $Q_{<0}(T)$ is an irreducible polynomial in $\\mathbb {Q}_{p}[T].$ Recall that a $K3$ surface $X$ over a field is a projective smooth surface with trivial canonical bundle satisfying $H^1(X, {O}_X)=0$ .",
"It is well-known that if we write the $L$ -function of a non-supersingular $K3$ surface $X$ over a finite field $\\mathbb {F}_q$ in the following form $L(X/{\\mathbb {F}_q}, T):={\\rm {det}}(1-T{\\mathrm {Frob}}_{\\mathbb {F}_q}; H^2_{\\rm {\\acute{e}t}}(X_{\\overline{\\mathbb {F}}_q}, \\mathbb {Q}_{\\ell }{(1)}))=\\prod ^{22}_{i=1}(1-{\\gamma _i}T),$ then the polynomial $L_{\\mathrm {{trc}}}(X/{\\mathbb {F}_q}, T):=\\prod _{{\\gamma _i} \\notin \\mu _{\\infty }}(1-{\\gamma _i}T)$ satisfies Condition REF ; see [22].",
"Here, ${\\mathrm {Frob}}_{\\mathbb {F}_q}\\in {\\rm {Gal}}({\\overline{\\mathbb {F}}_q}/{\\mathbb {F}_q})$ is the geometric Frobenius element, and $\\mu _{\\infty }$ is the set of roots of unity.",
"We call $L_{\\mathrm {{trc}}}(X/{\\mathbb {F}_q}, T)$ the transcendental part of $L(X/{\\mathbb {F}_q}, T)$ .",
"In this paper, we shall prove the following conjecture due to Taelman under some mild restrictions on the characteristic.",
"Conjecture 1.2 Let $p$ be a prime number and $m$ an integer with $1 \\le m \\le 10$ .",
"For a polynomial $L(T)=\\prod ^{2m}_{i=1}(1-{\\gamma _i}T)\\in {1+T\\mathbb {Q}[T]}$ satisfying Condition REF for a power $q$ of $p$ , there exist a positive integer $n\\ge 1$ and a $K3$ surface $X$ over $\\mathbb {F}_{q^n}$ such that $L_{\\mathrm {{trc}}}(X/{\\mathbb {F}_{q^n}}, T)=\\prod ^{2m}_{i=1}(1-{\\gamma ^n_i}T).$ The main result of this paper is as follows.",
"Theorem 1.3 Let $p$ be a prime number and $m$ an integer with $1 \\le m \\le 10$ .",
"Let $L(T)=\\prod ^{2m}_{i=1}(1-{\\gamma _i}T)\\in {1+T\\mathbb {Q}[T]}$ be a polynomial satisfying Condition REF for a power $q$ of $p$ .",
"When $p\\ge 7$ , the assertion of Conjecture REF is true.",
"When $p = 5$ , the assertion of Conjecture REF is true if at least one of the following conditions holds: $1\\le {m}\\le 9$ .",
"The discriminant ${\\rm {disc}}(\\mathbb {Q}(\\gamma _1))$ of the number field $\\mathbb {Q}(\\gamma _1)$ is a square.",
"The integer $e$ of Condition REF is even.",
"Taelman proved that, assuming a strong version of the existence of semistable reduction for $K3$ surfaces in the sense of [11], then Conjecture REF is true [22].",
"Remark 1.4 For a $K3$ surface $X$ over $\\mathbb {F}_{q^n}$ as in Conjecture REF or Theorem REF , the height $h(X_{\\overline{\\mathbb {F}}_p})$ of the formal Brauer group associated with $X_{\\overline{\\mathbb {F}}_p}:=X\\otimes _{\\mathbb {F}_{q^n}}{\\overline{\\mathbb {F}}_p}$ is equal to the integer $h$ of Condition REF ; see [22].",
"We also have $\\rho (X_{\\overline{\\mathbb {F}}_p})=22-2m$ by the Tate conjecture [12], [14], [4], [10].",
"Here $\\rho (X_{\\overline{\\mathbb {F}}_p})$ is the Picard number of $X_{\\overline{\\mathbb {F}}_p}$ .",
"In fact, for $K3$ surfaces $X$ constructed in the proof of Theorem REF , the equality $\\rho (X_{\\overline{\\mathbb {F}}_p})=22-2m$ follows from the construction, and we do not need the Tate conjecture for this equality; see Section and Section for details.",
"Remark 1.5 It seems an interesting but difficult question to ask whether we can take $n=1$ in Conjecture REF or Theorem REF ; see [22].",
"Remark 1.6 Currently, by our methods, it seems difficult to prove Conjecture REF unconditionally in the remaining cases (i.e.",
"when $p\\le 3$ , or $p=5$ , $m=10$ , ${\\rm {disc}}(\\mathbb {Q}(\\gamma _1))$ is not a square, and $e$ is odd).",
"See Remark REF and Remark REF for explanations on technical difficulties.",
"In the proof of Theorem REF , we basically follow Taelman's strategy.",
"We shall construct a $K3$ surface over a finite field with given $L$ -function as the reduction modulo $p$ of an appropriate $K3$ surface with $CM$ .",
"To obtain unconditional results, we need to apply Matsumoto's good reduction criterion without a priori assuming a strong version of the existence of semistable reduction for $K3$ surfaces in the sense of [11].",
"The point is that we can construct an appropriate $K3$ surface with $CM$ so that it admits an elliptic fibration with a section or an ample line bundle of low degree.",
"Then we use Saito's results to construct a strictly semistable model and apply a slight refinement of Matsumoto's criterion.",
"The outline of this paper is as follows.",
"In Section , we recall Matsumoto's good reduction criterion for $K3$ surfaces.",
"We also give a slight refinement for $K3$ surfaces with an elliptic fibration with a section and apply these results to $K3$ surfaces with $CM$ .",
"In Section , we recall Bayer-Fluckiger's results on quadratic spaces and $CM$ fields.",
"Section is the technical heart of this paper.",
"In Section , we construct appropriate projective $K3$ surfaces over $\\mathbb {C}$ with $CM$ .",
"Previously, Taelman constructed for any $CM$ field $F$ with $[F:\\mathbb {Q}]\\le 20$ , a projective $K3$ surface over $\\mathbb {C}$ with $CM$ by $F$ by using Bayer-Fluckiger's results.",
"Improving his arguments and using Nikulin's results on primitive embeddings of lattices into the $K3$ lattice, we can construct a projective $K3$ surface over $\\mathbb {C}$ with $CM$ by $F$ so that it admits an elliptic fibration with a section or an ample line bundle of degree 2.",
"In Section , we prove Theorem REF .",
"In Section , as an application of Theorem REF , we construct $K3$ surfaces with given geometric Picard number and height over finite fields of characteristic $p\\ge 5$ ."
],
[
"Good reduction criterion for $K3$ surfaces",
"In this section, we recall the good reduction criterion for $K3$ surfaces due to Matsumoto [13].",
"We also give a slight refinement for $K3$ surfaces with an elliptic fibration with a section.",
"Then we apply these results to $K3$ surfaces with $CM$ .",
"First, we recall well-known facts about $K3$ surfaces.",
"Let $U$ be the hyperbolic plane, i.e.",
"$U$ is the lattice $U:=\\mathbb {Z}{e} \\oplus \\mathbb {Z}{f}$ such that its intersection pairing is given by $(e, e)=(f, f)=0$ and $(e, f)=1$ .",
"The following result is well-known; see the proof of [7] for example.",
"Although it is stated and proved only for complex $K3$ surfaces in [7], the same argument works for $K3$ surfaces over an algebraically closed field of characteristic different from $2, 3$ .",
"Proposition 2.1 Let $X$ be a $K3$ surface over an algebraically closed field $k$ of characteristic different from $2, 3$ .",
"Then the following assertions are equivalent: There exists an embedding of lattices $U \\hookrightarrow \\mathop {\\mathrm {Pic}}\\nolimits (X),$ where $U$ is the hyperbolic plane.",
"$X$ admits an elliptic fibration with a section.",
"Remark 2.2 If the assertions of Proposition REF hold, the $K3$ surface $X$ is isomorphic to the Jacobian of an elliptic fibration of $X$ with a section; see [7] for details.",
"However, we will not use this fact in this paper.",
"Proposition 2.3 ([7]) Let $X$ be a $K3$ surface over an algebraically closed field $k$ of characteristic different from $2, 3$ .",
"Assume that the Picard number of $X$ satisfies $12 \\le \\rho (X) \\le 20$ .",
"Then $X$ admits an elliptic fibration with a section.",
"See [7].",
"Note that although [7] is stated and proved only for $K3$ surfaces over $\\mathbb {C}$ , it also holds in characteristic $p\\ge 5$ .",
"Indeed, the $K3$ surface $X$ is not supersingular by the Tate conjecture [12], [14], [4], [10], and we can lift the Picard group to characteristic 0; see the proof of [16] and the proof of [16].",
"Then we apply Proposition REF .",
"Next, we recall Saito's results on the construction of strictly semistable models [19].",
"We state his results in the form suitable for our purposes.",
"For an extension of fields $L/K$ and a scheme $X$ over $K$ , we put $X_{L}:=X\\otimes _{K}{L}$ .",
"See also the proof of (c) in [13].",
"Proposition 2.4 (Saito [19]) Let $K$ be a discrete valuation field whose residue field is of characteristic $p \\ge 5$ .",
"Let $X$ be a $K3$ surface over $K$ .",
"Assume that $X$ admits an elliptic fibration with a section.",
"Then there exist a finite extension $L/K$ and a regular scheme ${Y}$ projective flat over the valuation ring $\\mathop {{O}}\\nolimits _L$ whose special fiber is a reduced simple normal crossing divisor and generic fiber is birational to $X_{L}$ .",
"The following theorem is essentially proved in Matsumoto's paper; see [13] and the proof of (c) in [13].",
"See also [11].",
"Theorem 2.5 (Matsumoto [13]) Let $K$ be a henselian discrete valuation field with perfect residue field of characteristic $p>0$ .",
"Let $X$ be a $K3$ surface over $K$ satisfying at least one of the following conditions: The $\\ell $ -adic $\\acute{e}$ tale cohomology $H^2_{\\mathrm {{\\acute{e}t}}}(X_{\\overline{K}}, {{\\mathbb {Q}}_{\\ell }})$ is unramified for some prime number $\\ell \\ne {p}$ .",
"$K$ is complete of characteristic 0 and the $p$ -adic $\\acute{e}$ tale cohomology $H^2_{\\mathrm {{\\acute{e}t}}}(X_{\\overline{K}}, {{\\mathbb {Q}}_{p}})$ is crystalline.",
"Moreover, assume that at least one of the following conditions holds: $X$ admits an ample line bundle ${L}$ with $p>({L})^2+4$ .",
"$p\\ge 5$ and $X$ admits an elliptic fibration with a section.",
"Then there exist a finite extension $K^{\\prime }/K$ and an algebraic space ${X}$ proper smooth over the valuation ring ${O}_{K^{\\prime }}$ such that ${X}\\otimes _{{O}_{K^{\\prime }}}{K^{\\prime }}\\simeq {X_{K^{\\prime }}}$ .",
"When $X$ admits an ample line bundle ${L}$ with $p>({L})^2+4$ , the assertion is proved in [13].",
"Precisely, the base field $K$ is assumed to be complete in [13].",
"But exactly the same proof works for henselian discrete valuation fields.",
"See also [11] where the results are stated over henselian discrete valuation fields.",
"As explained in [13] and [11], if we can construct a strictly semistable model, we achieve the theorem.",
"More precisely, if we have a finite extension $L/K$ and a regular scheme ${Y}$ proper flat over the valuation ring ${O}_L$ whose special fiber is a reduced simple normal crossing divisor and generic fiber is birational to $X_{L}$ , then $X$ satisfies [11] by [11].",
"Here we need $p \\ge 5$ .",
"Then, we achieve the theorem by [11].",
"When $p \\ge 5$ and $X$ admits an elliptic fibration with a section, by Proposition REF , we achieve the theorem as above.",
"Remark 2.6 It is not always possible to construct ${X}$ as a scheme in Theorem REF ; see [13].",
"We recall the definition of a complex projective $K3$ surface with $CM$ .",
"Let $X$ be a projective $K3$ surface over $\\mathbb {C}$ .",
"Let $T_{X}:=\\mathop {\\mathrm {Pic}}\\nolimits (X)^{\\perp }_{\\mathbb {Q}} \\subset H^{2}(X, \\mathbb {Q}(1))$ be the transcendental part of the cohomology, which has the $\\mathbb {Q}$ -Hodge structure coming from $H^{2}(X, \\mathbb {Q}(1))$ .",
"Let $E_{X}:={\\rm {End}}_{\\rm {Hdg}}(T_{X})$ be the $\\mathbb {Q}$ -algebra of $\\mathbb {Q}$ -linear endomorphisms on $T_{X}$ preserving the $\\mathbb {Q}$ -Hodge structure on it.",
"We say $X$ has complex multiplication $(CM)$ if $E_{X}$ is $CM$ and ${\\rm {dim}}_{E_{X}}(T_{X})=1$ .",
"Here a number field is called $CM$ if it is a purely imaginary quadratic extension of a totally real number field.",
"Pjateckiĭ-Šapiro and Šafarevič showed that every $K3$ surface with $CM$ is defined over a number field [17].",
"Rizov showed that, for a $CM$ field $F$ , a $K3$ surface $X$ over $\\mathbb {C}$ with $CM$ by $F$ is defined over an abelian extension of $F$ [18].",
"However we will not use this fact in this paper.",
"As an application of Theorem REF , we have the following results on the reduction of $K3$ surfaces with $CM$ ; see also [13].",
"Corollary 2.7 Let $X$ be a projective $K3$ surface over $\\mathbb {C}$ with $CM$ .",
"Let $K$ be a number field embedded in $\\mathbb {C}$ such that $X$ is defined over $K$ .",
"Let $v$ be a finite place of $K$ above a prime number $p$ .",
"Assume that at least one of the following conditions holds: $X$ admits an ample line bundle ${L}$ with $p>({L})^2+4$ .",
"$p\\ge 5$ and $X$ admits an elliptic fibration with a section.",
"Then there exist a finite extension $K^{\\prime }/K_v$ and an algebraic space ${X}$ proper smooth over the valuation ring ${O}_{K^{\\prime }}$ such that ${X}\\otimes _{{O}_{K^{\\prime }}}{K^{\\prime }}\\simeq {X_{K^{\\prime }}}$ .",
"Here $K_v$ is the completion of $K$ at $v$ .",
"Since $X$ has $CM$ , after replacing $K_v$ by a finite extension of it, we may assume $H^2_{\\mathrm {{\\acute{e}t}}}(X_{\\overline{K}_v}, {{\\mathbb {Q}}_{\\ell }})$ is unramified for some prime number $\\ell \\ne {p}$ ; see the proof of [13].",
"Then we apply Theorem REF ."
],
[
"Quadratic spaces and $CM$ fields",
"In this section, we recall the results of Bayer-Fluckiger on embeddings of quadratic spaces over $\\mathbb {Q}$ arising from a $CM$ field into a given quadratic space over $\\mathbb {Q}$ .",
"First, we recall some definitions on quadratic spaces.",
"Let $k$ be a field of characteristic different from 2 and $(V, q)$ a quadratic space over $k$ .",
"This means that $V$ is a finite dimensional $k$ -vector space equipped with a nondegenerate symmetric bilinear form $q\\colon {V} \\times {V} \\longrightarrow {k}.$ The determinant ${\\rm {det}}(V)\\in {k^*}/{({k^*}^2)}$ and the Hasse invariant $w(V)\\in \\mathop {\\mathrm {Br}}\\nolimits (k)[2]$ are defined as follows.",
"We put $m:={\\rm {dim}}_{k}V$ .",
"There exists an isomorphism of quadratic spaces $V\\simeq \\langle {a_1, \\cdots , a_m}\\rangle ,$ where $a_1, \\cdots , a_m\\in {k}^*$ and $\\langle {a_1, \\cdots , a_m}\\rangle $ is a diagonal form.",
"Then we define ${\\rm {det}}(V)&:=\\prod ^{m}_{i=1}a_i\\in {{k^*}/{({k^*}^2)}},\\\\w(V)&:=\\sum _{i<j}(a_i,a_j)\\in \\mathop {\\mathrm {Br}}\\nolimits (k)[2],$ where $(\\ ,\\ )$ denotes the Hilbert symbol and $\\mathop {\\mathrm {Br}}\\nolimits (k)[2]$ denotes the subgroup of elements of the Brauer group $\\mathop {\\mathrm {Br}}\\nolimits (k)$ killed by 2.",
"Then ${\\rm {det}}(V)$ and $w(V)$ do not depend on the choice of an isomorphism $V\\simeq \\langle {a_1, \\cdots , a_m}\\rangle $ ; see [20] for details.",
"Let $F$ be a $CM$ field of degree $[F:\\mathbb {Q}]=2m$ .",
"The maximal totally real subfield of $F$ is denoted by $F_0$ .",
"The discriminant of $F$ is denoted by ${\\rm {disc}}(F) \\in \\mathbb {Z}$ .",
"It is easy to see that $(-1)^m{\\rm {disc}}(F)$ is a positive integer.",
"For each $\\lambda \\in {F^*_0}$ , we define the quadratic space $(F, q_{\\lambda })$ over $\\mathbb {Q}$ by $q_{\\lambda }\\colon {F} \\times {F} &\\longrightarrow {\\mathbb {Q}}\\\\({x},{y}) &\\longrightarrow {{\\mathrm {Tr}}_{F/{\\mathbb {Q}}}}(\\lambda {x}{i(y)}),$ where $i\\colon F\\rightarrow {F}$ is the involution defined by the nontrivial element of ${\\rm {Gal}}(F/{F_0})$ .",
"Let $(V,q)$ be a quadratic space over $\\mathbb {Q}$ of dimension $2m$ and $F$ a $CM$ field of degree $[F:\\mathbb {Q}]=2m$ .",
"We say the hyperbolicity condition holds for $F$ and $V$ if, for every prime number $p$ such that all the places of $F_0$ above $p$ split in $F$ , we have $w(V_{\\mathbb {Q}_p})=w(U^{\\oplus {m}}_{\\mathbb {Q}_p}) \\in \\mathop {\\mathrm {Br}}\\nolimits (\\mathbb {Q}_p)[2],$ where we put $V_{\\mathbb {Q}_p}:=V\\otimes _{\\mathbb {Q}}{\\mathbb {Q}_p}$ and $U_{\\mathbb {Q}_p}:=U\\otimes _{\\mathbb {Z}}{\\mathbb {Q}_p}$ is the hyperbolic plane over $\\mathbb {Q}_p$ .",
"In [3], Bayer-Fluckiger showed general results on embeddings of an algebraic torus into a given orthogonal group over a global field of characteristic different from 2.",
"In terms of quadratic spaces, her results are stated as follows.",
"See [3].",
"Theorem 3.1 (Bayer-Fluckiger [3]) Let $F$ be a $CM$ field of degree $[F:\\mathbb {Q}]=2m$ , and $(V, q)$ a quadratic space over $\\mathbb {Q}$ of dimension $2m$ .",
"Then there exists $\\lambda \\in {F^*_0}$ such that $(V, q)\\simeq (F, q_{\\lambda })$ if and only if all of the following conditions hold: ${\\rm {det}}(V)=(-1)^{m}{\\rm {disc}}(F)$ in ${{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2})}$ .",
"The signature of $V$ is even.",
"The hyperbolicity condition holds for $F$ and $V$ ."
],
[
"Construction of $K3$ surfaces with {{formula:5540aca6-3169-44f8-805c-b81d2b60ebe1}} with additional conditions",
"In this section, we shall construct projective $K3$ surfaces over $\\mathbb {C}$ with $CM$ by a given $CM$ field.",
"Here the point is that, in order to make Taelman's argument unconditional, we require the $K3$ surfaces constructed in this section satisfy some additional conditions; see Proposition REF .",
"First, we make some preparations.",
"Lemma 4.1 Let $F$ be a $CM$ field with maximal totally real subfield $F_0$ .",
"Then there exist infinitely many prime numbers $p$ such that there exists a place of $F_0$ above $p$ which does not split in $F$ .",
"See the proof of [3].",
"Lemma 4.2 Let $p$ be an odd prime number.",
"Let $x$ be an integer not divisible by $p$ .",
"Then there exists an odd prime number $q$ such that $q\\equiv x \\pmod {p}$ and $q\\equiv 3 \\pmod {4}$ .",
"By the Chinese remainder theorem, there exists an integer $a\\in \\mathbb {Z}$ satisfying $a \\equiv x \\pmod {p}$ and $a \\equiv 3 \\pmod {4}$ .",
"By Dirichlet's theorem on arithmetic progressions, there exists a prime number $q$ satisfying $q \\equiv a \\pmod {4p}$ .",
"The prime number $q$ is odd, and satisfies $q\\equiv x \\pmod {p}$ and $q\\equiv 3 \\pmod {4}$ .",
"Let $U=\\mathbb {Z}{e} \\oplus \\mathbb {Z}{f}$ be the hyperbolic plane as in Section .",
"We define the $K3$ lattice $\\Lambda _{K3}:={(-E_8)}^{\\oplus {2}}\\oplus {U}^{\\oplus {3}},$ which is an even unimodular lattice of signature $(3, 19)$ over $\\mathbb {Z}$ ; see [22].",
"Lemma 4.3 Let $m$ be an integer with $6\\le {m}\\le 10$ , and $F$ a $CM$ field of degree $[F:\\mathbb {Q}]=2m$ with maximal totally real subfield $F_0$ .",
"There exist an even lattice $N$ of rank $22-2m$ and signature $(1, 21-2m)$ , and a primitive embedding $N \\hookrightarrow \\Lambda _{K3}$ satisfying the following conditions: For the orthogonal complement $T:=N^{\\perp }\\subset \\Lambda _{K3}$ , the quadratic space $(T_{\\mathbb {Q}}, \\langle , \\rangle )$ is isomorphic to $(F, q_{\\lambda })$ for some ${\\lambda }\\in {F_0}$ .",
"If $6 \\le m \\le 9$ , or $m=10$ and ${\\rm {disc}}(F)$ is a square, there exists an embedding $U \\hookrightarrow N$ .",
"If $m=10$ and ${\\rm {disc}}(F)$ is not a square, there exists an element $x \\in N$ with $(x)^2=2$ and no element $y \\in N$ satisfies $(y)^2=-2$ .",
"Here, an embedding $N \\hookrightarrow \\Lambda _{K3}$ is called primitive if the cokernel $\\Lambda _{K3}/N$ is torsion-free.",
"We put $T_{\\mathbb {Q}}:=T\\otimes _{\\mathbb {Z}}{\\mathbb {Q}}$ and $\\langle , \\rangle $ is the pairing induced from the pairing on $\\Lambda _{K3}$ .",
"Since $F$ is a $CM$ field of degree $2m$ , we can write ${\\rm {disc}}(F)={(-1)^m}n\\in \\mathbb {Z}$ for some positive integer $n\\ge 1$ .",
"By Lemma REF , there exists an odd prime number $p_1$ such that there exists a place of $F_0$ above $p_1$ which does not split in $F$ .",
"By Lemma REF , there exists an odd prime number ${p_2}\\ne {p_1}$ such that $p_2 \\equiv 3 \\pmod {4}$ and $\\bigg ( \\frac{p_2}{p_1} \\bigg )={\\left\\lbrace \\begin{array}{ll}1 &\\quad ({p_1}\\equiv 3\\ ({\\rm {mod}}\\ 4))\\\\-1 &\\quad ({p_1}\\equiv 1\\ ({\\rm {mod}}\\ 4)),\\\\\\end{array}\\right.",
"}$ where $\\displaystyle \\bigg ( \\frac{p_2}{p_1} \\bigg )$ is the Legendre symbol.",
"Let us consider the following lattice $N$ .",
"When $6 \\le m \\le 9$ , we put $N:={\\left\\lbrace \\begin{array}{ll}U \\oplus \\langle {-4n}\\rangle \\oplus {\\langle {-4}\\rangle }^{\\oplus {19-2m}} & (m=6, 9)\\\\U \\oplus \\langle {-4n}\\rangle \\oplus {{\\langle {-4}\\rangle }^{\\oplus 2}}\\oplus \\langle {-4p_1}\\rangle \\oplus \\langle {-4p_2}\\rangle \\oplus \\langle {-4{p_1}{p_2}}\\rangle & (m=7)\\\\U \\oplus \\langle {-4n}\\rangle \\oplus \\langle {-4p_1}\\rangle \\oplus \\langle {-4p_2}\\rangle \\oplus \\langle {-4{p_1}{p_2}}\\rangle & (m=8).\\\\\\end{array}\\right.",
"}$ When $m=10$ and ${\\rm {disc}}(F)$ is a square, we put $N:=U.$ When $m=10$ and ${\\rm {disc}}(F)$ is not a square, we put $N:=\\langle {2}\\rangle \\oplus \\langle {-8n}\\rangle .$ The lattice $N$ is an even lattice of rank $22-2m$ and signature $(1, {21-2m})$ .",
"Since $22-2m\\le 10$ , we have a primitive embedding $N \\hookrightarrow \\Lambda _{K3}$ by Nikulin's theorem [15], [7].",
"Let $T:=N^{\\perp } \\subset \\Lambda _{K3}$ be the orthogonal complement of $N$ in $\\Lambda _{K3}$ .",
"For the $K3$ lattice $\\Lambda _{K3}$ , we have: ${\\rm {sign}}(\\Lambda _{K3, \\mathbb {Q}})&=(3,\\ 19),\\\\{\\rm {det}}(\\Lambda _{K3, \\mathbb {Q}})&=-1\\in {{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2})},\\\\w(\\Lambda _{K3, \\mathbb {Q}})&=(-1,\\ -1) \\in \\mathop {\\mathrm {Br}}\\nolimits (\\mathbb {Q})[2].$ Hence we have: ${\\rm {sign}}{(T_{\\mathbb {Q}})}&=(2, 2m-2),\\\\{\\rm {det}}(T_{\\mathbb {Q}})&=-{\\rm {det}}(N_{\\mathbb {Q}})=n\\in {{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2})},\\\\w(T_{\\mathbb {Q}})&=(-1, -1)+w(N_{\\mathbb {Q}})+(-n, n)=(-1, -1)+w(N_{\\mathbb {Q}})\\in \\mathop {\\mathrm {Br}}\\nolimits (\\mathbb {Q})[2].$ See [22] for the calculation of these invariants.",
"Here $(-n, n)\\in \\mathop {\\mathrm {Br}}\\nolimits (\\mathbb {Q})[2]$ is trivial since the equation $-nX^2+nY^2=Z^2$ has a non-trivial solution such as $(X, Y, Z)=(1, 1, 0)$ .",
"We shall show that the hyperbolicity condition (see Section ) holds for $F$ and $T_{\\mathbb {Q}}$ .",
"Namely, for any prime number $q$ such that all the places in $F_0$ above $q$ split in $F$ , we shall show $w(U^{\\oplus {m}}_{\\mathbb {Q}_q})=w(T_{\\mathbb {Q}_q}) \\in \\mathop {\\mathrm {Br}}\\nolimits (\\mathbb {Q}_q)[2] \\simeq \\mathbb {Z}/{2\\mathbb {Z}}.$ Note that we have $w(U^{\\oplus {m}}_{\\mathbb {Q}_q})=((-1)^{m(m-1)/2}, -1)_q.$ When $m=6, 9$ , we have $N_{\\mathbb {Q}}\\simeq \\langle {1}\\rangle \\oplus {{\\langle {-1}\\rangle }^{\\oplus 20-2m}}\\oplus \\langle {-n}\\rangle .$ Hence we have $w(T_{\\mathbb {Q}_q})={\\left\\lbrace \\begin{array}{ll}(-1, -1)_q & (m=6)\\\\0 & (m=9).\\end{array}\\right.",
"}$ Hence we have $w(U^{\\oplus {m}}_{\\mathbb {Q}_q})=w(T_{\\mathbb {Q}_q})$ for any prime number $q$ .",
"When $m=7, 8$ , we have $N_{\\mathbb {Q}}\\simeq {\\left\\lbrace \\begin{array}{ll}\\langle {1}\\rangle \\oplus {{\\langle {-1}\\rangle }^{\\oplus 3}}\\oplus \\langle {-n}\\rangle \\oplus \\langle {-p_1}\\rangle \\oplus \\langle {-p_2}\\rangle \\oplus \\langle {-p_1p_2}\\rangle & (m=7)\\\\\\langle {1}\\rangle \\oplus {{\\langle {-1}\\rangle }}\\oplus \\langle {-n}\\rangle \\oplus \\langle {-p_1}\\rangle \\oplus \\langle {-p_2}\\rangle \\oplus \\langle {-p_1p_2}\\rangle & (m=8).\\end{array}\\right.",
"}$ Hence we have $w(T_{\\mathbb {Q}_q})={\\left\\lbrace \\begin{array}{ll}(-1, -1)_{q}+(-p_1, -p_2)_{q} & (m=7)\\\\(-p_1, -p_2)_{q} & (m=8).\\end{array}\\right.",
"}$ Let $q$ be a prime number such that all the places in $F_0$ above $q$ split in $F$ .",
"Then we have $q \\ne p_1$ .",
"By the above calculation, it is enough to show $(-p_1, -p_2)_{q}=0$ for any prime number $q \\ne p_1$ .",
"When $q \\ne 2, p_1, p_2$ , we have $(-p_1, -p_2)_{q}=0$ .",
"When $q=2$ , since $-p_2 \\equiv 1 \\pmod {4}$ , we have $(-p_1, -p_2)_{2}=0$ ; see [20].",
"When $q=p_2$ , it is enough to prove $\\bigg ( \\frac{-p_1}{p_2} \\bigg )=1.$ See [20].",
"By the Quadratic Reciprocity Law [20], we have $\\bigg (\\frac{-p_1}{p_2}\\bigg )=\\bigg (\\frac{-1}{p_2}\\bigg )\\bigg (\\frac{p_1}{p_2}\\bigg )=-(-1)^{(p_1-1)(p_2-1)/4}\\bigg (\\frac{p_2}{p_1}\\bigg )=1.$ When $m=10$ and ${\\rm {disc}}(F)$ is a square, we have $N_{\\mathbb {Q}}\\simeq \\langle {1}\\rangle \\oplus {\\langle {-1}\\rangle }.$ Hence we have $w(T_{\\mathbb {Q}_q})=(-1, -1)_{q}.$ Therefore we have $w(U^{\\oplus {m}}_{\\mathbb {Q}_q})=w(T_{\\mathbb {Q}_q})$ for any prime number $q$ .",
"When $m=10$ and ${\\rm {disc}}(F)$ is not a square, we have $N_{\\mathbb {Q}}\\simeq \\langle {2}\\rangle \\oplus {\\langle {-2n}\\rangle }.$ Hence we have $w(T_{\\mathbb {Q}_q})=(-1, -1)_{q}+(2, -2n)_{q}.$ Let $q$ be a prime number such that all the places in $F_0$ above $q$ split in $F$ .",
"Then we have ${\\rm {disc}}(F)=1\\in {{{\\mathbb {Q}}^*_q}}/({{{\\mathbb {Q}}^{*2}_q}}).$ Since ${\\rm {disc}}(F)=(-1)^{10}n=n$ , we have $(2, -2n)_{q}=(2, -2)=0.$ Hence we have $w(U^{\\oplus {10}}_{\\mathbb {Q}_q})=(-1, -1)_q=w(T_{\\mathbb {Q}_q}).$ By Theorem REF , the quadratic space $(T_{\\mathbb {Q}}, \\langle , \\rangle )$ is isomorphic to $(F, q_{\\lambda })$ for some ${\\lambda }\\in {F^*_0}$ .",
"If $m=10$ and ${\\rm {disc}}(F)$ is not a square, it is easy to see that no element $y\\in {N}$ satisfies $(y, y)=-2$ .",
"Indeed, if there exist $a, b\\in \\mathbb {Z}$ such that $2a^2-8nb^2=-2,$ then $a^2+1$ is divisible by 4, which is absurd.",
"Now, we shall show the main results of this section.",
"Compare with Taelman's construction in the proof of [22].",
"Proposition 4.4 Let $m$ be an integer with $1\\le {m}\\le 10$ , and $F$ a $CM$ field of degree $[F:\\mathbb {Q}]=2m$ .",
"Then there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ satisfying the following conditions: $\\rho (Z)=22-2m$ .",
"If $1\\le {m}\\le 9$ , or $m=10$ and ${\\rm {disc}}(F)$ is a square, the $K3$ surface $Z$ admits an elliptic fibration with a section.",
"If $m=10$ and ${\\rm {disc}}(F)$ is not a square, the $K3$ surface $Z$ admits an ample line bundle of degree 2.",
"If $1\\le {m}\\le 5$ , by [22], there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F.$ Since $\\rho (Z)=22-2m\\ge 12$ , by Proposition REF , the $K3$ surface $Z$ admits an elliptic fibration with a section.",
"If $6\\le m \\le 10$ , by Lemma REF , there exist an even lattice $N$ of rank $22-2m$ and signature $(1, 21-2m)$ , an element ${\\lambda }\\in {F^*_0}$ , and a primitive embedding $N \\hookrightarrow \\Lambda _{K3}$ satisfying $(T_{\\mathbb {Q}}, \\langle , \\rangle )\\simeq (F, q_{\\lambda })$ , where we put $T:=N^{\\perp } \\subset \\Lambda _{K3}$ , $T_{\\mathbb {Q}}:=T\\otimes _{\\mathbb {Z}}{\\mathbb {Q}}$ , and $\\langle , \\rangle $ is the pairing induced from the pairing on $\\Lambda _{K3}$ .",
"Following Taelman's arguments [22], we shall show that there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ satisfying $N\\simeq \\mathop {\\mathrm {Pic}}\\nolimits ({Z}).$ We fix an isomorphism $(T_{\\mathbb {Q}}, \\langle , \\rangle )\\simeq (F, q_{\\lambda })$ .",
"Take an embedding $\\epsilon \\colon F \\hookrightarrow \\mathbb {C}$ with $\\epsilon (\\lambda )>0$ .",
"Such an embedding exists because the signature of $T$ is $(2, 2m-2)$ .",
"Let $\\overline{\\epsilon }$ be the complex conjugate of $\\epsilon $ .",
"Then we have a natural decomposition $F \\otimes _{\\mathbb {Q}}\\mathbb {C}\\simeq \\mathbb {C}_{\\epsilon } \\times \\mathbb {C}_{\\overline{\\epsilon }} \\times \\prod _{f \\ne \\epsilon , \\overline{\\epsilon }} \\mathbb {C}_f,$ where, for an embedding $f \\colon F \\hookrightarrow \\mathbb {C}$ , we denote $\\mathbb {C}$ by $\\mathbb {C}_f$ if we consider $\\mathbb {C}$ as an $F$ -algebra by $f$ .",
"This decomposition gives a $\\mathbb {Z}$ -Hodge structure of weight 0 on $\\Lambda _{K3}$ such that ${\\left\\lbrace \\begin{array}{ll}\\Lambda _{K3}^{1, -1}=\\mathbb {C}_{\\epsilon }, \\\\\\Lambda _{K3}^{-1, 1}=\\mathbb {C}_{\\overline{\\epsilon }}, \\\\\\Lambda _{K3}^{0, 0}=N_{\\mathbb {C}} \\oplus \\prod _{f \\ne \\epsilon , \\overline{\\epsilon }} \\mathbb {C}_f.\\end{array}\\right.",
"}$ By the surjectivity of the period map for $K3$ surfaces over $\\mathbb {C}$ [23], there is a complex analytic $K3$ surface $Z$ over $\\mathbb {C}$ and a Hodge isometry $\\Lambda _{K3} \\simeq H^2(Z, \\mathbb {Z}(1)).$ See also [7].",
"Since $F \\cap \\prod _{f \\ne \\epsilon , \\overline{\\epsilon }} \\mathbb {C}_f=0$ , a natural embedding $N \\hookrightarrow \\Lambda _{K3}^{0, 0} \\cap \\Lambda _{K3}$ is an isomorphism after tensoring with $\\otimes _{\\mathbb {Z}}\\mathbb {Q}$ .",
"Since $N \\hookrightarrow \\Lambda _{K3}$ is a primitive embedding, it follows that $N \\simeq \\Lambda _{K3}^{0, 0} \\cap \\Lambda _{K3}.$ By [2], the complex analytic $K3$ surface $Z$ is projective.",
"Now, the claim follows from the Lefschetz theorem on $(1, 1)$ -classes.",
"Moreover, if $6\\le m \\le 9$ , or $m=10$ and ${\\rm {disc}}(F)$ is a square, there exists an embedding $U\\hookrightarrow N \\simeq \\mathop {\\mathrm {Pic}}\\nolimits (Z).$ By Proposition REF , the $K3$ surface $Z$ admits an elliptic fibration with a section.",
"If $m=10$ and ${\\rm {disc}}(F)$ is not a square, no element $y\\in {N} \\simeq \\mathop {\\mathrm {Pic}}\\nolimits (X)$ satisfies $(y)^2=-2$ , hence the $K3$ surface $Z$ does not contain smooth rational curves.",
"Therefore, any line bundle on $Z$ in the positive cone of $\\mathop {\\mathrm {Pic}}\\nolimits (Z)_{\\mathbb {R}}$ is ample; see [7].",
"In particular, for any element $x \\in N \\simeq \\mathop {\\mathrm {Pic}}\\nolimits (Z)$ with $(x)^2=2$ , we see that $x$ or $-x$ corresponds to an ample line bundle of degree 2.",
"The statement of Proposition REF on the existence of an elliptic fibration with a section on a $K3$ surface with $CM$ is optimal.",
"Indeed, we have the following proposition.",
"Proposition 4.5 Let $F$ be a $CM$ field with $[F:\\mathbb {Q}]=20$ .",
"Then the following assertions are equivalent: ${\\rm {disc}}(F)$ is a square.",
"There exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ which admits an elliptic fibration with a section.",
"If ${\\rm {disc}}(F)$ is a square, by Proposition REF , there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ which admits an elliptic fibration with a section.",
"Next, we assume that there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ which admits an elliptic fibration with a section.",
"Let $F_0$ be the maximal totally real subfield of $F$ .",
"It is well-known that the quadratic space $(T_{Z}, \\langle , \\rangle )$ is isomorphic to $(F, q_{\\lambda })$ for some element $\\lambda \\in F^*_0$ ; see [5] for example.",
"Here, the lattice $T_{Z}$ is the transcendental part of $H^{2}(Z, \\mathbb {Q}(1))$ and $\\langle , \\rangle $ is the intersection pairing; see Section .",
"We briefly recall the argument.",
"Since ${\\rm {dim}}_{F}(T_{Z})=1$ , there exists an element $\\alpha \\in T_{Z}$ with $T_{Z}\\simeq F\\alpha $ .",
"Consider the bilinear form $\\Phi \\colon T_{Z} \\times T_{Z} \\longrightarrow F$ defined by $\\langle ex, y \\rangle = {\\rm {Tr}}_{F/{\\mathbb {Q}}}(e\\Phi (x, y)),$ for all $e\\in F$ and $x, y \\in T_{Z}$ .",
"We put $\\lambda :=\\Phi (\\alpha , \\alpha )$ .",
"Then we have $\\langle a\\alpha , b\\alpha \\rangle =\\langle ai(b)\\alpha , \\alpha \\rangle ={\\rm {Tr}}_{F/\\mathbb {Q}}(ai(b)\\lambda )=q_{\\lambda }(a, b)$ for any $a, b \\in F$ ; see [24] for the first equality.",
"We also have $\\langle a\\alpha , b\\alpha \\rangle =\\langle b\\alpha , a\\alpha \\rangle ={\\rm {Tr}}_{F/\\mathbb {Q}}(bi(a)\\lambda )={\\rm {Tr}}_{F/\\mathbb {Q}}(ai(b)i(\\lambda )).$ Hence $\\lambda \\in F^*_0$ , and we have $(T_{Z}, \\langle , \\rangle )\\simeq (F, q_{\\lambda })$ .",
"Since $\\rho (Z)=2$ and $Z$ has an elliptic fibration with a section, we have $\\mathop {\\mathrm {Pic}}\\nolimits (Z) \\simeq U$ .",
"In particular, we have ${\\rm {det}}(\\mathop {\\mathrm {Pic}}\\nolimits (Z)_{\\mathbb {Q}})=-1$ .",
"Since ${\\rm {det}}(\\Lambda _{K3, \\mathbb {Q}})=-1 \\in {{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2}})$ and $\\Lambda _{K3, \\mathbb {Q}} \\simeq \\mathop {\\mathrm {Pic}}\\nolimits (Z)_{\\mathbb {Q}}\\oplus T_{Z}$ , we have ${\\rm {disc}}(F)=(-1)^{10}{\\rm {det}}(T_{Z})=1 \\in {{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2}})$ by [3]."
],
[
"Proof of Theorem ",
"In this section, we shall prove Theorem REF .",
"First, we prove the following lemma.",
"Lemma 5.1 For a finite extension of $CM$ fields $F/E$ with $[F:E]=e$ , we have ${\\rm {disc}}(F)={\\rm {disc}}(E)^e \\in {{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2})}.$ We put $[E:\\mathbb {Q}]=2m$ .",
"Then the sign of ${\\rm {disc}}(E)$ is $(-1)^m$ .",
"Similarly, the sign of ${\\rm {disc}}(F)$ is $(-1)^{em}$ .",
"Hence ${\\rm {disc}}(F)$ and ${\\rm {disc}}(E)^e$ have the same sign.",
"Therefore, to prove Lemma REF , it suffices to prove that ${\\rm {disc}}(F)\\mathbb {Z}$ is equal to ${\\rm {disc}}(E)^e\\mathbb {Z}$ in the group of fractional ideals of $\\mathbb {Q}$ modulo squares.",
"For a number field $K$ , we denote the group of fractional ideals of $K$ by $I_K$ .",
"For an extension of number fields $K_2/K_1$ , we denote the relative discriminant of $K_2/K_1$ by $\\mathfrak {d}_{K_2/K_1}$ .",
"Let $F_0$ (resp.",
"$E_0$ ) be the maximal totally real subfield of $F$ (resp.",
"$E$ ).",
"By [21], we have the following equalities in $I_{\\mathbb {Q}}/I^2_{\\mathbb {Q}}$ ${\\rm {disc}}(F)\\mathbb {Z}&={\\mathrm {N}}_{F_0/\\mathbb {Q}}(\\mathfrak {d}_{F/F_0}),\\\\{\\rm {disc}}(E)\\mathbb {Z}&={\\mathrm {N}}_{E_0/\\mathbb {Q}}(\\mathfrak {d}_{E/E_0}).$ We put $E=E_0(\\sqrt{\\beta })$ for some $\\beta \\in E_0$ .",
"Then we have $\\mathfrak {d}_{E/E_0} = \\beta {O}_{E_0}$ in $I_{E_0}/I^2_{E_0}$ ; see [21].",
"Here ${O}_{E_0}$ is the ring of integers of $E_0$ .",
"We have ${\\mathrm {N}}_{E_0/\\mathbb {Q}}(\\mathfrak {d}_{E/E_0})={\\mathrm {N}}_{E_0/\\mathbb {Q}}(\\beta )\\mathbb {Z}$ in $I_{\\mathbb {Q}}/I^2_{\\mathbb {Q}}$ .",
"Similarly, since $F = F_0(\\sqrt{\\beta })$ , we have $\\mathfrak {d}_{F/F_0} = \\beta {O}_{F_0}$ in $I_{F_0}/I^2_{F_0}$ and we have ${\\rm {N}}_{F_0/\\mathbb {Q}}(\\mathfrak {d}_{F/F_0})={\\rm {N}}_{E_0/\\mathbb {Q}}(\\beta )^e\\mathbb {Z}$ in $I_{\\mathbb {Q}}/I^2_{\\mathbb {Q}}$ .",
"Therefore, we have the following equality in $I_{\\mathbb {Q}}/I^2_{\\mathbb {Q}}$ $ {\\rm {disc}}(F) \\mathbb {Z}= {\\rm {disc}}(E)^e\\mathbb {Z}.$ We briefly recall Taelman's argument; see [22] for details.",
"Consider a polynomial $L(T)=\\prod ^{2m}_{i=1}(1-{\\gamma _i}T) \\in {1+T\\mathbb {Q}[T]}$ of degree $2m$ satisfying Condition REF .",
"We put $E:={\\mathbb {Q}}(\\gamma _1)$ .",
"By Condition REF , the number field $E$ is a $CM$ field which has a unique finite place $v$ satisfying $v(\\gamma _1)<0$ above $p$ .",
"Take a finite extension $F/E$ of $CM$ fields such that $[F:{\\mathbb {Q}}]=2m$ and $F$ has a unique finite place above $v$ ; see [22].",
"By Lemma REF , ${\\rm {disc}}(F)$ is a square if ${\\rm {disc}}(E)$ is a square or $e$ is even.",
"Assume that at least one of the following conditions is satisfied: $p\\ge 5$ and $1\\le m \\le 9$ .",
"$p\\ge 5$ , $m=10$ , and ${\\rm {disc}}(E)$ is a square.",
"$p\\ge 5$ , $m=10$ , and $e$ is even.",
"Then, by Proposition REF , there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ which admits an elliptic fibration with a section.",
"Assume $p\\ge 7$ , $m=10$ , ${\\rm {disc}}(E)$ is not a square, and $e$ is odd.",
"By Proposition REF , there exists a projective $K3$ surface $Z$ over $\\mathbb {C}$ with $CM$ by $F$ which admits an ample line bundle ${L}$ of degree 2.",
"In any of these cases, by the theorem of Pjateckiĭ-Šapiro and Šafarevič [17], the $K3$ surface $Z$ is defined over a finite extension $L$ of $F$ .",
"By Corollary REF , after replacing $L$ by a finite extension of it, there is an algebraic space ${Z}$ which is proper and smooth over the valuation ring ${O}_{L_{v^{\\prime }}}$ , where $v^{\\prime }$ is a finite place of $L$ above $v$ .",
"(Here, we use the inequality $p>({L})^2+4=6$ when $p\\ge 7$ , $m=10$ , ${\\rm {disc}}(E)$ is not a square, and $e$ is odd.)",
"We denote the special fiber of ${Z}$ by ${Z}_{s}$ , which is a $K3$ surface over the residue field $k(s)$ .",
"Rizov also proved the main theorem of complex multiplication for $K3$ surfaces [18].",
"Taelman used Rizov's results to calculate the zeta function of the reduction modulo $p$ of a $K3$ surface with $CM$ if it has good reduction; see the proof of [22].",
"As explained in [22], it follows that there exists a finite extension $k^{\\prime }/k(s)$ such that the transcendental part of the $L$ -function of ${Z}_{s}\\otimes _{k(s)}{k^{\\prime }}$ satisfies $L_{\\mathrm {trc}}(({Z}_{s}\\otimes _{k(s)}{k^{\\prime }})/{k^{\\prime }}, T)=\\prod ^{2m}_{i=1}(1-{\\gamma ^{[k^{\\prime } : \\mathbb {F}_{q}]}_i}T).$ The proof of Theorem REF is complete.",
"Remark 5.2 It seems difficult to weaken the assumption of Theorem REF by our methods.",
"Currently, in order to apply Matsumoto's good reduction criterion for $K3$ surfaces or its variants, we always need $p\\ge 5$ ; see Theorem REF and [13], [11].",
"The main reason is, in Matsumoto's proof, Saito's construction of strictly semistable models [19] and Kawamata's results on minimal models [8], [9] are crucially used, and both results require $p\\ge 5$ .",
"(See [9] for explanations why $p\\ge 5$ is necessary to use Kawamata's results.)",
"When $p=5$ , we cannot treat the remaining case (i.e.",
"when $m=10$ , ${\\rm {disc}}(E)$ is not a square, and $e$ is odd) by our methods.",
"Indeed, in this case, for any $CM$ field $F$ containing $E$ with $[F:E]=e$ , the discriminant ${\\rm {disc}}(F)$ is not a square by Lemma REF .",
"By Proposition REF , no $K3$ surface with $CM$ by $F$ admits an elliptic fibration with a section.",
"Hence we cannot apply Theorem REF for such $K3$ surfaces.",
"See also Remark REF ."
],
[
"Construction of K3 surfaces over finite fields with given geometric Picard number and height",
"In [1], Artin proved that, for a $K3$ surface $X$ over an algebraically closed field of characteristic $p>0$ , the Picard number $\\rho (X)$ and the height $h(X)$ of the formal Brauer group satisfy the inequality $\\rho (X)\\le 22-2h(X).$ In this section, as an application of Theorem REF , we shall construct $K3$ surfaces with given geometric Picard number and height over finite fields of characteristic $p\\ge 5$ .",
"Lemma 6.1 Let $p$ be a prime number, and $h^{\\prime }, m^{\\prime }$ integers with $1 \\le h^{\\prime } \\le m^{\\prime } \\le 10$ .",
"Then there exists a polynomial $L(T) \\in 1+T\\mathbb {Q}[T]$ of degree $2m^{\\prime }$ satisfying Condition REF for a power $q$ of $p$ , $h=h^{\\prime }$ , and $m=m^{\\prime }$ .",
"Let $S \\subset \\mathbb {C}$ be the set of roots of unity whose minimal polynomials over $\\mathbb {Q}$ have degree less than or equal to 20.",
"We put $S^{\\prime } := \\lbrace \\, x \\in \\mathbb {C}\\mid x=t+1/t\\ {\\rm {for\\ some}}\\ t \\in S\\,\\rbrace .$ We can take a monic polynomial $F(T)=T^{m^{\\prime }}+c_{m^{\\prime }-1}T^{m^{\\prime }-1}+\\cdots +c_{1}T+c_0 \\in \\mathbb {Q}[T]$ of degree $m^{\\prime }$ with roots $\\alpha _1,\\cdots ,\\alpha _{m^{\\prime }}\\in \\overline{\\mathbb {Q}} \\subset \\mathbb {C}$ such that all of the following conditions are satisfied: $\\alpha _1,\\cdots ,\\alpha _{m^{\\prime }}\\in \\mathbb {R}$ .",
"$\\vert \\alpha _i\\vert <2$ .",
"$\\alpha _i \\notin S^{\\prime }$ for any $1 \\le i \\le m^{\\prime }$ .",
"$F(T) \\in \\mathbb {Z}_{\\ell }[T]$ for all prime numbers ${\\ell }\\ne {p}$ .",
"$\\nu _p(c_{m^{\\prime }-h^{\\prime }})=-a$ for some positive integer $a\\ge 1$ which is coprime to $h^{\\prime }$ .",
"$c_{i} \\in \\mathbb {Z}$ for any $i \\ne m^{\\prime }-h^{\\prime }$ .",
"For example, we can find such polynomials as follows.",
"We put: $f_0(T):=T-1$ , $f_i(T):=T^2-i$ $(i=1, 2, 3),$ $f_4(T):=T^3-3T+1,$ $f_5(T):=T^4-4T^2+1.$ As a product of $f_0,\\cdots , f_5$ , we can find a monic polynomial $F_0(T) \\in \\mathbb {Z}[T]$ of degree $m^{\\prime }$ which has $m^{\\prime }$ distinct real roots $\\beta _1, \\cdots , \\beta _{m^{\\prime }} \\ne 0$ with $\\vert \\beta _i\\vert <2$ for any $i$ (for example, take $F_0(T)$ as in Table 1 below).",
"Table: NO_CAPTION Table 1.",
"Since $S^{\\prime }$ is a finite set, if we take a sufficiently large positive integer $a$ which is coprime to $h^{\\prime }$ , the polynomial $F(T)=F_0(T)+{p^{-a}}T^{m^{\\prime }-h^{\\prime }}$ satisfies the above conditions.",
"Then the polynomial $L(T)=T^{m^{\\prime }}F(T+1/T)$ satisfies Condition REF .",
"The first three conditions can be checked easily.",
"To check the last three conditions, it is enough to show that $L(T)$ is irreducible in $\\mathbb {Q}[T]$ and that $L_{<0}(T)$ is irreducible in $\\mathbb {Q}_p[T]$ .",
"If we write $L(T)$ in the following form $L(T)=T^{2m^{\\prime }}+d_{2m^{\\prime }-1}T^{2m^{\\prime }-1}+\\cdots +d_{1}T+1,$ we have ${\\left\\lbrace \\begin{array}{ll}\\nu _{p}(d_i)\\ge 0 &\\quad \\ (1\\le {i}\\le {h^{\\prime }-1},\\ {2m^{\\prime }-h^{\\prime }+1}\\le {i}\\le {2m^{\\prime }-1}),\\\\\\nu _{p}(d_i)=-a &\\quad \\ (i=h^{\\prime }, 2m^{\\prime }-h^{\\prime }),\\\\\\nu _{p}(d_i)\\ge {-a} &\\quad \\ ({h^{\\prime }+1}\\le {i}\\le {2m^{\\prime }-h^{\\prime }-1}).\\\\\\end{array}\\right.",
"}$ It follows that the degree of $L_{<0}(T) \\in \\mathbb {Q}_p[T]$ is $h^{\\prime }$ and the inverse $\\gamma $ of every root of $L_{<0}(T)$ satisfies $\\nu _p(\\gamma )=-a/h^{\\prime }.$ Hence $L(T)$ satisfies the third condition of Condition REF for $q=p^a$ and $h=h^{\\prime }$ .",
"Since $a$ is coprime to $h^{\\prime }$ , it follows that $L_{<0}(T)$ is irreducible in $\\mathbb {Q}_p[T]$ .",
"Assume that $L(T)$ is not irreducible in $\\mathbb {Q}[T]$ .",
"Since $L_{<0}(T)$ is irreducible in $\\mathbb {Q}_p[T]$ , $L(T)$ is divisible by two different irreducible monic polynomials $L_1(T), L_2(T) \\in \\mathbb {Q}[T]$ .",
"Since they do not have common roots and $L_{<0}(T)$ is irreducible in $\\mathbb {Q}_p[T]$ , we may assume that every root $\\gamma $ of $L_1(T)$ satisfies $\\nu _p(\\gamma )=0$ .",
"Since $L(T) \\in \\mathbb {Z}_{\\ell }[T]$ for all prime numbers ${\\ell }\\ne {p}$ , it follows that $L_1(T) \\in \\mathbb {Z}[T]$ .",
"Since every root of $L(T)$ has complex absolute value one, it follows that $L_1(T)$ is a cyclotomic polynomial by Kronecker's theorem.",
"This contradicts to our construction of $L(T)$ .",
"Hence $L(T)$ is irreducible in $\\mathbb {Q}[T]$ .",
"Lemma 6.2 Let $p$ be a prime number, and $h^{\\prime }$ an integer with $1\\le h^{\\prime } \\le 10$ .",
"Then there exists a polynomial $L(T)=\\prod ^{20}_{i=1}(1-{\\gamma _i}T) \\in 1+T\\mathbb {Q}[T]$ of degree 20 such that Condition REF is satisfied for a power $q$ of $p$ , $h=h^{\\prime }$ , and $m=10$ , and at least one of the following conditions is satisfied: ${\\rm {disc}}(\\mathbb {Q}(\\gamma _1))$ is a square.",
"The integer $e$ of Condition REF is even.",
"First, we assume $h^{\\prime }$ is even.",
"We write $h^{\\prime }=2h^{\\prime \\prime }$ .",
"By Lemma REF , there exists a polynomial $L(T) \\in 1+T\\mathbb {Q}[T]$ of degree 10 satisfying Condition REF for a power $q$ of $p$ , $h=h^{\\prime \\prime }$ , and $m=5$ .",
"Then the square $L(T)^2$ satisfies our conditions for $q^2$ , $h=h^{\\prime }=2h^{\\prime \\prime }$ , and $m=10$ .",
"Next, we assume $h^{\\prime }=1, 3, 7$ , or 9.",
"We take a totally real field $M_0$ with $[M_0:\\mathbb {Q}]=10$ such that $M_0$ has a finite place $v$ above $p$ with ramification index $e_v=h^{\\prime }$ and residue degree $f_v=1$ .",
"For example, we can take $M_0:=\\mathbb {Q}[T]/(F(T))$ for the polynomial $F(T)$ constructed in the proof of Lemma REF .",
"Choose a positive integer $n\\ge 1$ such that $v$ splits into the $CM$ field $M:=M_0(\\sqrt{-n})$ .",
"As in the proof of Lemma REF , we have ${\\rm {disc}}(M)={\\rm {N}}_{M_0/\\mathbb {Q}}(-n)=(-n)^{10} \\in {{{\\mathbb {Q}}^*}/{({{\\mathbb {Q}}^*}^2})}.$ Hence ${\\rm {disc}}(M)$ is a square.",
"Let $v_1, v_2$ be the finite places of $M$ above $v$ .",
"Since the class number of $M$ is finite, there exists an element $\\alpha \\in M^*$ with $v_1(\\alpha )>0$ and $v^{\\prime }(\\alpha )=0$ for every finite place $v^{\\prime } \\ne v_1$ of $M$ .",
"Let $i$ be the nontrivial element of ${\\rm {Gal}}(M/M_0)$ .",
"Let $\\beta :=\\alpha /i(\\alpha )$ .",
"We shall show $\\mathbb {Q}(\\beta )=M$ by a similar argument as in the proof of [6].",
"Let $u$ be a place of $\\mathbb {Q}(\\beta )$ under $v_1$ .",
"From our construction, $v_1$ is the unique place above $u$ .",
"Since $e_{v_1}=h^{\\prime }$ and $f_{v_1}=1$ , the extension degree $[M:\\mathbb {Q}(\\beta )]$ divides $h^{\\prime }$ .",
"Since $h^{\\prime }$ is coprime to $[M:\\mathbb {Q}]=20$ , we have $[M:\\mathbb {Q}(\\beta )]=1$ .",
"Hence the minimal polynomial $L(T)$ of $\\beta $ over $\\mathbb {Q}$ has degree 20, and it satisfies our conditions for a power $q$ of $p$ , $h=h^{\\prime }$ , and $m=10$ .",
"Finally, we assume $h^{\\prime }=5$ .",
"By the above discussion, we can construct a polynomial $L(T)=\\prod ^{4}_{i=1}(1-{\\gamma _i}T) \\in 1+T\\mathbb {Q}[T]$ of degree 4 satisfying Condition REF for a power $q$ of $p$ , $h=1$ , and $m=2$ such that ${\\rm {disc}}(\\mathbb {Q}(\\gamma _1))$ is a square.",
"Then the fifth power $L(T)^5$ satisfies our conditions for $q^5$ , $h=5$ , and $m=10$ .",
"Remark 6.3 For $p=5$ , there exists a polynomial $L(T)=\\prod ^{20}_{i=1}(1-{\\gamma _i}T) \\in 1+T\\mathbb {Q}[T]$ of degree 20 satisfying Condition REF for a power $q$ of 5, $h=1$ , and $m=10$ such that ${\\rm {disc}}(\\mathbb {Q}(\\gamma _1))$ is not a square.",
"Then, as explained in Remark REF , we cannot show the assertion of Theorem REF for $L(T)$ by our methods.",
"For example, such a polynomial $L(T)$ can be constructed as follows.",
"Let $M_0$ be a totally real field with $[M_0:\\mathbb {Q}]=10$ such that 5 splits completely in $M_0$ .",
"Let $v_1,\\cdots , v_{10}$ be the places of $M_0$ above 5.",
"We take a $CM$ field $M/M_0$ such that $[M:M_0]=2$ , $v_1$ splits completely in $M$ , $v_2$ is ramified in $M$ , $v_i$ is unramified in $M$ for $3\\le i \\le 10$ .",
"Then ${\\rm {disc}}(M)$ is not a square since $\\nu _5({\\rm {disc}}(M))=1$ .",
"Let $v^{\\prime }_1, v^{\\prime }_2$ be the finite places of $M$ above $v_1$ .",
"Take an element $\\alpha \\in M^*$ with $v^{\\prime }_1(\\alpha )>0$ and $v^{\\prime }(\\alpha )=0$ for every finite place $v^{\\prime } \\ne v^{\\prime }_1$ of $M$ .",
"We put $\\beta :=\\alpha /i(\\alpha )$ , where $i$ is the nontrivial element of ${\\rm {Gal}}(M/M_0)$ .",
"Then, by the same argument as in the proof of Lemma REF , we have $\\mathbb {Q}(\\beta )=M$ , and the minimal polynomial of $\\beta $ over $\\mathbb {Q}$ satisfies the above conditions.",
"Recall that the Picard number $\\rho (X)$ of a $K3$ surface $X$ over $\\overline{\\mathbb {F}}_p$ is a positive even integer by the Tate conjecture [12], [14], [4], [10]; see [7].",
"Theorem 6.4 Let $p$ be a prime number with $p\\ge 5$ .",
"Let $\\rho \\in (2\\mathbb {Z})_{>0}$ be a positive even integer and $h^{\\prime }\\in \\mathbb {Z}_{>0}$ a positive integer.",
"Then the following assertions are equivalent: $\\rho $ and $h^{\\prime }$ satisfy $\\rho \\le 22-2h^{\\prime }.$ There exists a $K3$ surface $X$ over $\\overline{\\mathbb {F}}_p$ such that $\\rho (X)=\\rho $ and $h(X)=h^{\\prime }$ .",
"(2) $\\Rightarrow $ (1): This follows from [1].",
"(1) $\\Rightarrow $ (2): We put $m:=11-\\rho /2$ .",
"When $\\rho \\ge 4$ , by Lemma REF , there exists a polynomial $L(T)=\\prod ^{2m}_{i=1}(1-{\\gamma _i}T)\\in {1+T\\mathbb {Q}[T]},$ satisfying Condition REF for a power $q$ of $p$ and $h=h^{\\prime }$ .",
"When $\\rho =2$ , by Lemma REF , we may assume ${\\rm {disc}}(\\mathbb {Q}(\\gamma _1))$ is a square or $e$ is even.",
"By Theorem REF , there exist a positive integer $n\\ge 1$ and a $K3$ surface $X_0$ over $\\mathbb {F}_{q^n}$ such that $L_{\\mathrm {{trc}}}(X_0/{\\mathbb {F}_{q^n}}, T)=\\prod ^{2m}_{i=1}(1-{\\gamma ^n_i}T).$ The $K3$ surface $X:={X_0}\\otimes _{\\mathbb {F}_{q^n}}{\\overline{\\mathbb {F}}_p}$ satisfies $\\rho (X)=\\rho $ and $h(X)=h^{\\prime }$ as required; see Remark REF ."
],
[
"Acknowledgements",
"The author is deeply grateful to my advisor, Tetsushi Ito, for his kindness, support, and advice.",
"He gave me a lot of invaluable suggestions.",
"The author also would like to thank Yuya Matsumoto and Seidai Yasuda for helpful suggestions and comments.",
"Moreover the author would like to thank the anonymous referees for sincere remarks and comments.",
"The work of the author was supported by JSPS Research Fellowships for Young Scientists KAKENHI Grant Number 18J22191."
]
] | 1612.05382 |
[
[
"On the crucial impact of the coupling projector-backprojector in\n iterative tomographic reconstruction"
],
[
"Abstract The performance of an iterative reconstruction algorithm for X-ray tomography is strongly determined by the features of the used forward and backprojector.",
"For this reason, a large number of studies has focused on the to design of projectors with increasingly higher accuracy and speed.",
"To what extent the accuracy of an iterative algorithm is affected by the mathematical affinity and the similarity between the actual implementation of the forward and backprojection, referred here as \"coupling projector-backprojector\", has been an overlooked aspect so far.",
"The experimental study presented here shows that the reconstruction quality and the convergence of an iterative algorithm greatly rely on a good matching between the implementation of the tomographic operators.",
"In comparison, other aspects like the accuracy of the standalone operators, the usage of physical constraints or the choice of stopping criteria may even play a less relevant role."
],
[
"[block]I.1em"
],
[
"[block]i.1em [subtable]position=bottom [table]position=bottom -4"
],
[
"Introduction",
"[nindent=0em,lines=3]Iterative reconstruction for X-ray tomography has been studied since the introduction of the first CT scans in the mid 70s [1].",
"Differently from the filtered backprojection (FBP) algorithm [2], iterative methods are non-linear and less computationally efficient, as the forward projector and its adjoint operator, the backprojector, are generally called few times per iteration.",
"In contrast to FBP, iterative methods can, however, provide high quality reconstructions of tomographic underconstrained datasets, characterized by poor signal-to-noise ratio (SNR), little number of views and/or missing data.",
"In general, iterative algorithms consist of the following elements: a solver for the cost function, physical constraints, a regularization scheme linked to the a-priori-knowledge regarding the specimen under study and tomographic projectors.",
"Four main families of solvers can be identified for iterative reconstruction.",
"Algebraic reconstruction techniques like ART [3], SIRT [4] and SART [5] handle the tomographic problem as a system of equations, which is solved by means of the Kaczmarz method [6].",
"Statistical methods as the maximum likelihood expectation maximization (MLEM) [7], the separable paraboloidal surrogate [8] and the penalized weighted least square method (PWLS) [9], [10] incorporate the statistical model ruling the signal formation at the detector.",
"Recently, modern techniques for convex optimization like the split Bregman method [11] and the alternate direction method of multipliers (ADMM) [12] have also been applied to tomographic reconstruction [13], [14], [15].",
"Finally, the projection-onto-convex-sets method [16] has been mainly used to address the interior tomography problem.",
"Physical constraints enforce at each iteration strict conditions in the image domain.",
"Setting to zero all negative pixel values and those falling outside the reconstruction circle is a typical example of broadly exploited physical constraints.",
"Regularization schemes often utilized by iterative algorithms are Tikhonov [17], Huber and total variation (TV) .",
"In particular, a Huber or TV term can steer the cost function towards a piecewise-constant solution, while preserving the spatial resolution.",
"Several implementations of the tomographic projectors have been proposed since the 70s.",
"The pixel-driven [2], , , ray-driven [2], , , distance-driven , and slant-stacking , approaches are different methods to approximate the Radon transform in real domain.",
"Since the listed approaches feature a complexity of $\\mathcal {O}(N^{3})$ , their implementation on GPUs is a must to make iterative reconstructions computationally feasible , , , .",
"Tomographic projectors with complexity $\\mathcal {O}(N^{2}\\log _{2}N)$ , based on hierarchical-decomposition , the non-uniform Fourier transform or the gridding method , are, instead, fast enough to not necessarily require a GPU architecture.",
"So far, research in iterative reconstruction algorithms has mainly addressed the design of regularization schemes leading to a better SNR-spatial resolution tradeoff and the development of tomographic projectors with increasingly higher accuracy and speed.",
"An aspect that has been generally neglected is the role of the coupling projector-backprojector, i.e., the level of mathematical affinity and matching between the actual implementation of the forward projector and its adjoint operator.",
"This work is an empirical investigation of the role played by this aspect on the performance of iterative reconstruction algorithms for X-ray tomography.",
"Ad-hoc experiments with state-of-the-art implementations of different tomographic operators (pixel-driven, ray-driven, distance-driven, slant-stacking, gridding method) have been designed for this purpose.",
"Reconstructions have been performed with both analytical (FBP) and iterative (ADMM, PWLS, MLEM, SIRT) schemes.",
"Results show that the coupling projector-backprojector substantially affects accuracy and convergence of an iterative algorithm.",
"In some cases, the degree of matching between the tomographic projectors can even play a more decisive role for the performance of the iterative method than other factors, like physical constraints, stopping criteria or the accuracy of the standalone projectors.",
"A mathematical justification of the presented experimental results is not straightforward and is outside the scope of this work.",
"The aim of this study is, instead, to provide convincing experimental evidence that a well-tuned coupling projector-backprojector is an absolute “must” for iterative tomographic reconstruction schemes to avoid systematically sub-accurate results.",
"A practical strategy for measuring the coupling degree is also proposed: this tool could be very useful for users and developers of software packages for iterative tomographic reconstruction to assess and validate the quality of the proposed projector pairs."
],
[
"Tomographic projectors",
"The Radon transform, $\\operatorname{\\mathcal {R}}$ , integrates a function $f(\\mathbf {x})\\::\\,\\mathbb {R}^{n} \\longrightarrow \\mathbb {R}$ over an hyperplane $HY(\\mathbf {n},t) = \\lbrace \\mathbf {x} \\in \\mathbb {R}^{n}\\:|\\: \\mathbf {x}\\cdot \\mathbf {n} = t\\rbrace $ , where $\\mathbf {n}$ is a unit vector and $t \\in \\mathbb {R}$ is the signed distance from the origin : $\\begin{split}\\operatorname{\\mathcal {R}}\\lbrace f\\rbrace (\\mathbf {n},t) &:= \\int \\limits _{HY}d\\mathbf {x}\\:f(\\mathbf {x})= \\int \\limits _{\\mathbb {R}^{n}}d\\mathbf {x}\\:\\delta (t-\\mathbf {x}\\cdot \\mathbf {n})f(\\mathbf {x})\\\\&= \\int \\limits _{\\mathbf {n}^{\\bot }}d\\mathbf {x}\\:f(t\\mathbf {n} + \\mathbf {x}) \\hspace{8.5359pt}.\\end{split}$ $\\delta $ is the Dirac function and $\\mathbf {n}^{\\bot } = \\lbrace \\mathbf {x} \\in \\mathbb {R}^{n}\\:|\\:\\mathbf {x}\\cdot \\mathbf {n} = 0\\rbrace $ is the subspace orthogonal to $\\mathbf {n}$ .",
"For $n=2$ , $f(\\mathbf {x}) = f(x_{1},x_{2})$ , $\\mathbf {n} = (\\cos \\theta ,\\sin \\theta )$ , $HY$ is a line of equation $\\mathbf {x}\\cdot \\mathbf {n} = x_{1}\\cos \\theta + x_{2}\\sin \\theta = t$ , thus, $\\operatorname{\\mathcal {R}}$ integrates $f$ along lines.",
"The second definition in (REF ) simplifies to: $\\operatorname{\\mathcal {R}}\\lbrace f\\rbrace (\\theta ,t) := \\int \\limits _{-\\infty }^{+\\infty }dx_{1}\\int \\limits _{-\\infty }^{+\\infty }dx_{2}\\:f(x_{1},x_{2})\\:\\delta (x_{1}\\cos \\theta + x_{2}\\sin \\theta - t) \\hspace{8.5359pt}.$ $\\operatorname{\\mathcal {R}}$ is also called forward projector and $\\theta $ is, here, the angle formed by the detector line and the positive $x$ -semiaxis.",
"The dual transform, i.e., the adjoint of the Radon transform, $\\operatorname{\\mathcal {R}}^{*}$ , is called backprojection.",
"For $n=2$ and given a generic function $g(\\mathbf {x}) = g(x_{1},x_{2})$ , $\\operatorname{\\mathcal {R}}^{*}$ is defined as : $\\operatorname{\\mathcal {R}}^{*}\\lbrace g\\rbrace (\\mathbf {x}) = \\frac{1}{2\\pi } \\int \\limits _{0}^{2\\pi } d\\theta \\: g\\left( \\theta , x_{1}\\cos \\theta + x_{2}\\sin \\theta \\right) \\hspace{14.22636pt}.$ The six implementations of $\\operatorname{\\mathcal {R}}$ and $\\operatorname{\\mathcal {R}}^{*}$ used in this work are for parallel beam geometry and a brief description is given in the following.",
"The pixel-driven (PD) approach [2], , Figure: Schematic representation of the different mechanisms characterizing the pixel-driven,ray-driven, distance-driven and slant-stacking approach for forward projection (and backprojection).works by connecting the source point to the selected pixel center until intersection with the detector line, as displayed in Fig.REF .",
"The pixel value is assigned on the basis of a linear interpolation scheme to the two detector cells that enclose the ray end point (they are indicated with a cross in Fig.REF ).",
"The ray-driven (RD) approach [2], , connects the source to the center of a selected detector cell as shown in Fig.REF .",
"The Siddon algorithm is used to compute the intersection points of the ray with the image grid (black dots in Fig.REF ).",
"Each pixel contributes to the selected detector cell according to the ray path length.",
"The distance-driven (DD) approach , in Fig.REF projects the pixel boundaries (black dots) of each image row/column and the detector cell boundaries (white dots) onto a common axis (in Fig.REF , the black squares are projected pixel boundaries, the white squares projected detector cells).",
"The overlap between the interval defined by the projected boundaries of an image pixel and the one defined by the projected boundaries of a detector cell weights the contribution of the selected image pixel to the selected detector cell (and viceversa).",
"The slant stacking (SS) method connects the source to each detector cell and divides the interval $[0,\\pi ]$ in two regions: one for nearly-vertical lines $0 \\le \\theta \\le \\pi /4$ and $3\\pi /4 \\le \\theta \\le \\pi $ ; one for nearly-horizontal lines $\\pi /4 \\le \\theta \\le 3\\pi /4$ ($\\theta $ is the angle formed by the detector line and the positive $x$ -semiaxis).",
"Figure REF shows how SS works with a nearly-vertical line: the abscissas of the black dots are obtained by using the ray equation and the $x_{2}$ coordinates (white dots) of all image pixels.",
"The computed points $(x_{1},x_{2})$ contribute to the selected detector cell according to a linear interpolation scheme.",
"The same approach is used for nearly-horizontal lines.",
"The gridding projectors , are implementations of $\\operatorname{\\mathcal {R}}$ and $\\operatorname{\\mathcal {R}}^{*}$ in the Fourier domain and are based on the Fourier slice theorem (FSM) .",
"For the forward operation, the input image grid is, first, multiplied with the deapodization matrix and, then, Fourier transformed (FFT-2D).",
"The Fourier Cartesian grid is convolved with a compact kernel to obtain Fourier samples on a polar grid.",
"According to the FSM, the inverse Fourier transform (IFFT-1D) of a polar slice at angle $\\theta $ corresponds to the object projection acquired at angle $\\theta $ .",
"The accuracy and efficiency of gridding projectors rely entirely on the choice of the convolving kernel (that also determines the deapodizer) and the oversampling ratio, $\\alpha $ , used for the Fourier grid.",
"In this work, two slightly different implementations are considered : one using a prolate spheroidal wavefunctions kernel and $\\alpha =2$ (abbreviated with WF); the other using a Kaiser-Bessel kernel and $\\alpha = 1.5$ (abbreviated with KB)."
],
[
"Degree of coupling projector-backprojector",
"Given a generic linear operator $\\operatorname{\\mathcal {A}}\\::\\:\\mathbb {C}^{n_{1}}\\longrightarrow \\mathbb {C}^{n_{2}}$ , the adjoint, $\\operatorname{\\mathcal {A}}^{*}$ , is defined as follows: $\\begin{split}&\\operatorname{\\mathcal {A}}^{*}\\::\\:\\mathbb {C}^{n_{2}}\\longrightarrow \\mathbb {C}^{n_{1}} \\hspace{14.22636pt}\\text{such that}\\hspace{14.22636pt}\\left< \\mathbf {y} , \\operatorname{\\mathcal {A}}(\\mathbf {x}) \\right> = \\left< \\operatorname{\\mathcal {A}}^{*}(\\mathbf {y}) , \\mathbf {x} \\right>\\\\&\\hspace{85.35826pt} \\forall \\:\\mathbf {x} \\in \\mathbb {C}^{n_{1}} \\,,\\, \\forall \\:\\mathbf {y} \\in \\mathbb {C}^{n_{2}}\\hspace{14.22636pt},\\end{split}$ where $<...>$ is the notation for the inner product.",
"Definition (REF ) can be used to measure how well a computer implementation of $\\operatorname{\\mathcal {A}}$ matches the computer implementation of $\\operatorname{\\mathcal {A}}^{*}$ .",
"The two inner products in (REF ) are numerically evaluated with $\\mathbf {x}$ and $\\mathbf {y}$ being vectors of randomly generated numbers.",
"If the ratio $r = \\left< \\operatorname{\\mathcal {A}}^{*}(\\mathbf {y}) , \\mathbf {x} \\right>/\\left< \\mathbf {y} , \\operatorname{\\mathcal {A}}(\\mathbf {x}) \\right>$ matches 1 up to a reasonably sufficient numerical precision, the implementations of $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {A}}^{*}$ can be considered well coupled.",
"For the tomographic case, a good coupling is achieved when the backprojector foresees the same exact operations of the forward projector, but in reverse order and switching the roles of input/output arrays for object and sinogram.",
"The coupled implementations of $\\operatorname{\\mathcal {R}}$ and $\\operatorname{\\mathcal {R}}^{*}$ listed in (REF ) feature $r = 1$ up to the 7th digit.",
"When not coupled, $r = 1$ at most up to the 4th digit."
],
[
"Reconstruction algorithms",
"Analytical reconstructions are here performed with filtered backprojection (FBP) [2], that inverts the Radon transform by applying the linear operator $\\operatorname{\\mathcal {R}}^{*} \\circ \\: \\Delta $ , where $\\Delta $ is the ramp or Ram-Lak filter.",
"The tradeoff between SNR and spatial resolution of FBP reconstructions depends on the type of window superimposed to the Ram-Lak filter .",
"For this reason, FBP is used here with four different filters : a pure Ram-Lak filter that provides the highest spatial resolution and poorest SNR (abbr.",
"RAMP); a Ram-Lak filter combined with a Shepp-Logan window (abbr.",
"SHLO); a Ram-Lak filter combined with a Hanning window (abbr.",
"HANN); a Ram-Lak filter combined with a Parzen window that provides the poorest spatial resolution and highest SNR (abbr.",
"PARZ).",
"Four different iterative reconstruction algorithms have been selected for this study: the alternate direction method of multipliers (ADMM) with TV regularization [14], the penalized weighted least square (PWLS) with Huber penalty [9], [10], the maximum-likelihood expectation maximization (MLEM) [7] and the simultaneous iterative reconstruction technique (SIRT) [3].",
"The number of iterations is set to around 100, when studying the algorithm convergence.",
"For the other experiments, the stopping criterion and regularization strength are optimized according to the characteristics of the considered dataset.",
"Iterative reconstructions are run with a range of different stopping criteria and weights of the penalty term.",
"We define the optimal number of iterations and regularization strength as those providing the best reconstruction accuracy, after appropriate exploration of the parameter space.",
"Nevertheless, it is important to point out that the presented trends in the performance of the iterative algorithms as a function of the coupling projector-backprojector are independent from the choice of the regularization parameters and confirmed also in case of a suboptimal selection."
],
[
"Dataset and image quality assessment",
"The Shepp-Logan (SL) phantom is used to create the simulated datasets for this study.",
"Since this phantom consists exclusively of roto-translated ellipses, its forward projection can be computed analytically .",
"An analytical forward projection can be used in two ways: (i) as reference when measuring the accuracy of a projector; (ii) as tomographic dataset not coupled to a specific operator used within the selected reconstruction algorithm.",
"A selection of experiments presented in Section and were also performed with different simulated objects and real datasets: the observed trends are comparable to those obtained with the SL phantom and are, therefore, independent from the chosen object.",
"The discretized forward projection of an object is also called sinogram, which corresponds to a matrix $\\, \\in \\mathbb {R}^{M\\times N}$ ; $M$ is the number of views and $N$ the number of detector cells.",
"In this study, projections are always homogeneously distributed in $[0,\\pi )$ .",
"A sinogram in parallel beam geometry is considered undersampled, when $M < N \\pi /2$ .",
"FBP reconstructions of undersampled datasets are affected by radially arranged line artifacts .",
"To simulate projections with a low photon statistics, Poisson noise with variance $\\sigma $ is added to the computed forward projection.",
"Poissonian statistics accounts for the shot noise affecting real projection data, whereas it neglects other sources of signal corruption, e.g., roundoff errors and electrical noise, not considered here.",
"Four different analytical forward projections of the SL phantom are used in the experimental sections: a well-sampled, noiseless SL sinogram with 402 views $\\times $ 256 pixels, abbreviated as SL-FULL; an undersampled, noiseless SL sinogram with 50 views $\\times $ 256 pixels, abbreviated as SL-UNDER; a well-sampled, noisy SL sinogram with 402 views $\\times $ 256 pixels and additional Poisson noise with $\\sigma =3\\%$ of the SL-FULL mean value, abbreviated as SL-NOISE; an undersampled noisy sinogram with 75 views $\\times $ 256 pixels and additional Poisson noise with the same $\\sigma $ of the SL-NOISE, abbreviated as SL-UCONSTR.",
"The image quality is measured by the peak-signal-to-noise ratio (PSNR) , defined as: $\\text{PSNR} = 10\\:\\log _{10}\\left( \\frac{\\max \\lbrace \\mathbf {r}\\rbrace ^{2}}{\\text{MSE}} \\right) =20\\:\\log _{10}\\left( \\frac{|\\max \\lbrace \\mathbf {r}\\rbrace |}{\\sqrt{\\text{MSE}}} \\right) \\hspace{14.22636pt},$ where the mean squared error (MSE) is: $\\text{MSE} = \\frac{1}{PQ}\\sum \\limits _{i = 0}^{P-1}\\sum \\limits _{j = 0}^{Q-1}\\left( f[i,j] - r[i,j] \\right)^{2} \\hspace{14.22636pt}.$ $\\mathbf {r}, \\mathbf {f} \\in \\mathbb {R}^{P\\times Q}$ are the reference and the image to be evaluated, respectively.",
"The PSNR is preferable over the MSE because more sensitive: as $( f[i,j] - r[i,j] )^{2}$ appears at the denominator, even small differences can elicit non negligible variations of the PSNR value.",
"In this study, the reference is either SL or its analytical forward projection.",
"When comparing an analytical or iterative reconstruction to SL, the PSNR is computed within the reconstruction circle."
],
[
"Operator coupling in analytical reconstruction",
"The following FBP tests provide a first indication of the role played by the coupling projector-backprojector in iterative tomographic reconstruction.",
"Reconstructed slices are not displayed here, because differences are usually not detectable at visual inspection.",
"The accuracy of the standalone forward projectors DD, KB, PD, RD, SS and WF with respect to SL-FULL is reported in Tab.REF .",
"The standalone backprojectors are used to perform FBP reconstructions with different filters of SL-FULL, SL-UNDER and SL-UCONSTRResults with SL-NOISE show the same trends characterizing the reconstruction of SL-FULL, SL-UNDER and SL-UCONSTR and therefore are not shown.",
"and the corresponding results are illustrated in Fig.REF .",
"The analysis in Tab.REF and Fig.REF suggest two facts.",
"(i) The accuracy of the standalone projector is not a good predictor of the accuracy of the standalone backprojector in analytical reconstruction: e.g., KB has the lowest PSNR value in Tab.REF , but it provides higher quality reconstruction of SL-FULL than PD and DD (Fig.REF ).",
"(ii) The performance of a backprojector is highly dependent on the characteristics of the dataset: e.g., SS has the highest PSNR score in Tab.REF and the best reconstruction quality for SL-FULL (Fig.REF ), but it performs poorly when reconstructing underconstrained datasets (SL-UNDER and SL-UCONSTR in Fig.REF and REF ).",
"The experiment in Fig.REF evaluates the reconstruction accuracy of well-sampled noiseless sinograms created by DD (Fig.REF ), KB (Fig.REF ) and PD (Fig.REF ).",
"Table: Accuracy of the standalone forward projectors with respect to SL-FULL.Figure: Accuracy of the standalone backprojectors in performing FBP reconstruction with different filters (RAMP, SHLO, HANN, PARZ) of SL analytical sinograms.Reconstruction of (a) SL-FULL, (b) SL-UNDER and (c) SL-UCONSTR.Figure: FBP reconstructions of sinograms with 402 views ×\\times 256 pixels created by the DD, KB and PD forward projectors.The reconstructions are perfomed withdifferent filters by the DD, KB, PD, RD, SS and WF backprojectors.Figure: FBP reconstructions of (a) an undersampled sinogram with 100 views ×\\times 256 pixels created by the DD,(b) a noisy sinogram with 402 views ×\\times 256 pixel and additional Poisson noise (σ=2%\\sigma = 2\\% of SL-FULL mean) created by KB and(c) an underconstrained sinogram with 100 views ×\\times 256 and additional Poisson noise (σ=2%\\sigma = 2\\% of SL-FULL mean)created by PD.",
"The reconstructions are perfomed withdifferent filters by the DD, KB, PD, RD, SS and WF backprojectors.The effect of the coupling projector-backprojector is clear: regardless of the filter choice, the best reconstruction quality is achieved when the backprojector matches the operator used to compute the input sinogram.",
"The weaker the action of the filter, the more pronounced the impact of the coupling on the reconstruction accuracy.",
"The results of the FBP reconstructions in Fig.REF show that the role of the coupling remains important even when dealing with undersampled (Fig.REF ), noisy (Fig.REF ) or underconstrained (Fig.REF ) datasets.",
"Considering that the performance of the standalone backprojectors can strongly vary as a function of the dataset (Fig.REF ), it is remarkable that undersampling and noise fail at breaking the effect of the coupling projector-backprojector.",
"The important role of the coupling projector-backprojector is also clear when the sinograms are computed by RD, SS and WF (not shown)."
],
[
"Operator coupling in iterative reconstruction",
"To study the coupling effect on the convergence of iterative algorithms, SL-FULL is reconstructed with ADMM, PWLS, MLEM and SIRT.",
"In each test, a different pair of forward and backward operators is used (Fig.",
"5-7).",
"Only results for selected combinations of tomographic operators are shown in this section for illustration.",
"The observed trends are however confirmed by all combinations.",
"ADMM converges and reaches the lowest value of the cost function when the backprojector matches the forward operator (Fig.REF ).",
"When the backprojector does not match the forward operator, three different scenarios are observed.",
"(i) ADMM converges but the cost function does not reach the minimum value (SS and PD curves in Fig.REF ).",
"(ii) ADMM simply does not converge (KB and WF curves in Fig.REF ).",
"(iii) ADMM reaches the lowest value of the cost function before diverging (DD curve in Fig.REF ).",
"Differently from ADMM, the convergence of PWLS is not endangered by a mismatch between tomographic operators.",
"Nevertheless, the cost function curve of PWLS with coupled operators is the lowest at each point after few initial iterations.",
"This is visible in the insets of Fig.REF and REF .",
"MLEM and SIRT behave similarly to PWLS: the matching between forward projector and backprojector is not essential to guarantee convergence, but is required to obtain the lowest cost function curve at each point, as shown in the insets of Fig.REF and REF .",
"Despite this similarity to PWLS, MLEM and SIRT can, instead, easily “explode” with an undersampled or noisy dataset if the operators are not coupled.",
"For this reason, no reconstruction of underconstrained datasets done by SIRT and only few cases with MLEM are shown in the following.",
"SIRT and MLEM share a common aspect: the computation of the diagonal matrix $\\mathbf {C} = \\lbrace c_{jj} = 1/\\sum _{i}a_{ij}\\rbrace $ is necessary, where $\\lbrace a_{ij}\\rbrace $ are the elements of the matrix representation of $\\operatorname{\\mathcal {R}}$ .",
"The $c_{jj}$ 's can be efficiently calculated as $\\operatorname{\\mathcal {R}}^{*}({1})\\,, \\,\\mathbb {R}^{M\\times N} \\ni {1} = \\lbrace ({1})_{ij} = 1 \\,\\,\\,\\forall \\:i,j\\rbrace $ .",
"This computation can be rather sensitive and produce very high values at the image boudaries, compromising the stability of the iterative procedure especially when using uncoupled projectors.",
"On the other hand, since ADMM and PWLS do not involve potentially sensitive computations, tests of these algorithms were not restricted to specific datasets or projector pairs.",
"The results in Fig.REF , REF and REF clearly illustrate the influence of the coupling projector-backprojector on the convergence of all considered iterative procedures: the best performance is achieved only when the operators match.",
"The level of the cost function of an iterative algorithm after a certain amount of iterations is not completely related to the reconstruction accuracy, or, in other words, reaching the minimum of the cost function does not necessarily mean reaching the closest possible approximation to the original phantom.",
"Additional experiments focusing on the reconstruction accuracy have been performed.",
"Reconstructions are displayed when differences can be perceived at visual inspection.",
"Table REF presents the results of ADMM reconstructions of SL-UNDER with the PD forward projector.",
"The best quality is achieved when the PD backprojector is used.",
"Nevertheless, differences are relatively small and the reconstructions look very similar.",
"The coupling has a much stronger effect when reconstructing SL-NOISE, as shown in Fig.REF : the best ADMM reconstruction is obtained when the operators match (SS, in this case) and differences in PSNR are up to 3.6 dB.",
"At visual inspection, reconstructions in Fig.REF , REF and REF are slightly more degraded than in Fig.REF , as suggested by the PSNR score.",
"Results in Fig.REF show once again the great impact of the coupling effect on the reconstruction accuracy in presence of noise.",
"Since KB and WF are both based on the gridding method and are highly coupled (as also resulting from the previous analysis), the reconstruction in Fig.REF is nearly identical to the one performed with matching operators in Fig.REF .",
"The combination of a noisy underconstrained dataset and poorly coupled operators leads, instead, to strongly degraded ADMM reconstructions (Fig.REF and Fig.REF ).",
"The PSNR values in Tab.REF (a) and REF (b) correspond, respectively, to PWLS reconstruction of SL-UNDER using the KB forward projector with KB, DD, SS and WF backprojectors and of SL-NOISE using the SS forward projector with SS, DD, WF and PD backprojectors.",
"Figure: Study of convergence of the ADMM, using RD (Fig.)",
"or DD (Fig.)",
"as forward projectorscombined to all six backprojectors considered in this study.Figure: Study of convergence of the PWLS, using WF (Fig.)",
"or SS (Fig.)",
"as forward projectorscombined to all six backprojectors considered in this study.Figure: Study of convergence of the MLEM and SIRT, both using PD as forward projectorcombined to all six backprojectors considered in this study.For PWLS, the coupling projector-backprojector has slightly more impact in presence of undersampled data than of purely noisy data: the spread of PSNR values in Tab.REF (a) is, indeed, a bit larger than for the values in Tab.REF (b).",
"Similarly to the results of Fig.REF , the PWLS reconstruction of underconstrained datasets with coupled projectors has the highest accuracy (Fig.REF ), whereas severe artifacts can occur when reconstructing an underconstrained dataset with uncoupled operators (Fig.REF ).",
"Reconstructions with MLEM and SIRT are very sensitive to the coupling effect with both undersampled and noisy datasets.",
"Several reconstruction attempts for SL-UNDER, SL-NOISE and SL-UCONSTR using these algorithms with non-matching operators failed, as the procedure quickly diverges after few iterations.",
"Figure REF shows an experiment with MLEM, PD forward projector and PD, RD and KB backprojectors: the reconstruction with coupled operators (Fig.REF ) is once again characterized by the highest accuracy.",
"Table: PSNR scores of ADMM reconstructions of SL-UNDER using PD as forward projector and PD, KB, RD, WF as backprojectors.Figure: ADMM reconstructions of SL-NOISE using SS as forward projector and SS, DD, KB, RD as backprojectors.Figure: ADMM reconstructions of SL-UCONSTR using KB as forward projector and KB, PD, RD, WF as backprojectors.The last experiment is designed to roughly estimate the impact of the coupling projector-backprojector on the reconstruction quality with respect to other two fundamental components: physical constraints (i.e., setting to zero all negative pixels at each iteration) and optimal number of iterations.",
"As example we show here the results for SL-UCONSTR and the ADMM.",
"The highest PSNR in Tab.REF corresponds to case (1), where all three components (coupling, constraints, optimal number of iterations) are present.",
"The interesting result is that case (2), that relies only on coupled operators, achieves a better reconstruction quality than case (3), where constraints and optimal number of iterations are kept, but the operators are not matching.",
"This experiment gives a hint of the fact that, in some cases, the coupling projector-backprojector could even play a more decisive role than other crucial factors on the accuracy of an iterative algorithm.",
"To validate the generality of these last results, further in-depth analysis is required, subject of future work.",
"Table: PSNR scores of PSWS reconstructions of SL-UNDER using PD as forward projector (left) and SL-NOISEusing SS as forward projector (right).Figure: PWLS reconstructions of SL-UCONSTR using KB as forward projector and KB, SS, DD, PD as backprojectors.Figure: MLEM reconstructions of SL-UNDER using PD as forward projector and PD, RD, KB as backprojectors.Table: Three different ADMM reconstructions of SL-UCONSTR.",
"Case (1): coupled operators + constraints + optimal number of iterations.Case (2): coupled operators.",
"Case (3): constraints + optimal number of iterations."
],
[
"Conclusion",
"This work is an experimental study on the impact of the coupling projector-backprojector in iterative reconstruction schemes.",
"Since iterative algorithms call the tomographic operators few times per iteration, it can be expected that the level of matching between the actual implementation of the forward projector and backprojector can deeply affect the performance of the entire iterative procedure.",
"A framework consisting of four iterative methods (the alternate direction method of multipliers, the penalized weighted least squares, the maximum-likelihood expectation maximization and the simultaneous iterative algebraic technique) working with six different projectors (distance-driven, pixel-driven, ray-driven, slant-stacking and two gridding methods) has been conceived to test the aforementioned hypothesis.",
"All iterative experiments on simulated data clearly show that the performance of every selected method is deeply affected by the coupling projector-backprojector in terms of convergence and accuracy.",
"The best convergence behaviour and the highest reconstruction quality are systematically obtained when the tomographic operators match.",
"This conclusion holds regardless of the nature of the input tomographic dataset in terms of angular sampling or SNR.",
"Moreover, there is indication that the coupling projector-backprojector may represent one of the major players determining the performance of an iterative algorithm, even with respect to physical constraints or optimal number of iterations.",
"The results of this study indicate that it would be strongly advisable for users and developers of software packages for iterative tomographic reconstructions to always select projector pairs with a high mathematical affinity and to carefully assess and validate the degree of coupling of the used implementations.",
"This strategy is important to avoid results systematically characterized by suboptimal accuracy."
]
] | 1612.05515 |
[
[
"Stability properties of a two-dimensional system involving one Caputo\n derivative and applications to the investigation of a fractional-order\n Morris-Lecar neuronal model"
],
[
"Abstract Necessary and sufficient conditions are given for the asymptotic stability and instability of a two-dimensional incommensurate order autonomous linear system, which consists of a differential equation with a Caputo-type fractional order derivative and a classical first order differential equation.",
"These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional order of the Caputo derivative.",
"In this setting, we obtain a generalization of the well known Routh-Hurwitz conditions.",
"These theoretical results are then applied to the analysis of a two-dimensional fractional-order Morris-Lecar neuronal model, focusing on stability and instability properties.",
"This fractional order model is built up taking into account the dimensional consistency of the resulting system of differential equations.",
"The occurrence of Hopf bifurcations is also discussed.",
"Numerical simulations exemplify the theoretical results, revealing rich spiking behavior.",
"The obtained results are also compared to similar ones obtained for the classical integer-order Morris-Lecar neuronal model."
],
[
"Introduction",
"In many real world applications, generalizations of dynamical systems using fractional-order differential equations instead of classical integer-order differential equations have proven to be more accurate, as fractional-order derivatives provide a good tool for the description of memory and hereditary properties.",
"Phenomenological description of colored noise [5], electromagnetic waves [12], diffusion and wave propagation [15], [31], viscoelastic liquids [16], fractional kinetics [29] and hereditary effects in nonlinear acoustic waves [40] are just a few areas where fractional-order derivatives have been successfully applied.",
"In addition to straightforward similarities that can be drawn between fractional- and integer-order derivatives and corresponding dynamical systems, it is important to realize that qualitative differences may also arise.",
"For instance, the fractional-order derivative of a non-constant periodic function cannot be a periodic function of the same period [20], which is in contrast with the integer-order case.",
"As a consequence, periodic solutions do not exist in a wide class of fractional-order systems.",
"Due to these qualitative differences, which cannot be addressed by simple generalizations of the properties that are available in the integer-order case, the theory of fractional-order systems is a very promising field of research.",
"With a multitude of practical applications, stability analysis is one of the most important research topics of the qualitative theory of fractional-order systems.",
"Comprehensive surveys of stability properties of fractional differential equations and fractional-order systems have been recently published in [25], [37].",
"When it comes to the stability of linear autonomous commensurate fractional order systems, the most important starting point is Matignon's theorem [30], which has been recently generalized in [38].",
"Linearization theorems (or analogues of the classical Hartman-Grobman theorem) for fractional-order systems have been proved in [24], [46].",
"Incommensurate order systems have not received as much attention as their commensurate order counterparts.",
"Some stability results for linear incommensurate fractional order systems with rational orders have been obtained in [34].",
"Oscillations in two-dimensional incommensurate fractional order systems have been investigated in [6], [36].",
"BIBO stability in systems with irrational transfer functions has been recently investigated in [43].",
"In the first of this paper, our aim is to explore necessary and sufficient conditions for the asymptotic stability and instability of a two-dimensional linear incommensurate fractional order system, which consists of a differential equation with a Caputo-type fractional order derivative and a classical first order differential equation.",
"In the second part of this paper, we propose and analyze a two-dimensional fractional-order Morris-Lecar neuronal model, by replacing the integer-order derivative from the equation describing the dynamics of the membrane potential by a Caputo fractional-order derivative, with careful treatment of the dimensional consistency problem of the resulting system.",
"This fractional-order formulation is justified by experimental results concerning biological neurons [1].",
"In [27], it has been underlined that \"fractional differentiation provides neurons with a fundamental and general computation ability that can contribute to efficient information processing, stimulus anticipation and frequency-independent phase shifts of oscillatory neuronal firing\", emphasizing the importance of developing and analyzing fractional-order models of neuronal activity.",
"The Grünwald-Letnikov derivative, the Riemann-Liouville derivative and the Caputo derivative are the most widely used types of fractional derivatives, which are generally not equivalent.",
"In this paper, we restrict our attention to the Caputo derivative, as it is more applicable to real world problems, given that it only requires initial conditions expressed in terms of integer-order derivatives, which represent well-understood features of physical situations.",
"We refer to [8], [21], [23], [35] for an introduction to fractional calculus and the qualitative analysis of fractional-order dynamical systems.",
"The Caputo fractional-order derivative of an absolutely continuous function $f$ on a real interval $[a,b]$ is $^cD^q f(x)=\\frac{1}{\\Gamma (1-q)}\\int _ 0^x(x-t)^{-q}f^{\\prime }(t)dt~,$ where the gamma function is defined, as usual, as: $\\Gamma (z)=\\int _0^\\infty e^{-t}t^{z-1}dt~.$ The Caputo derivative of a function $f$ can be expressed as $^c\\!D^qf(x)=(k\\ast f^{\\prime })(x),$ where $k(x)=\\frac{x^{-q}}{\\Gamma (1-q)}$ and $\\ast $ denotes the convolution operation.",
"The Laplace transform of the function $k(x)$ is $\\mathcal {L}(k)(s)=s^{q-1},$ where, according to [10] (example 8 on page 8), the principal value (first branch) of the complex power function has to be taken into account.",
"Therefore, the Laplace transform of the Caputo derivative is deduced in the following way: $\\mathcal {L}(^c\\!D^q f)(s)&=\\mathcal {L}(k\\ast f^{\\prime })(s)=\\mathcal {L}(k)(s)\\cdot \\mathcal {L}(f^{\\prime })(s)=\\\\&=s^{q-1}(s\\mathcal {L}(f)(s)-f(0))=\\\\&=s^{q}\\mathcal {L}(f)(s)-s^{q-1}f(0).$ In the following, we give an elementary result that will be useful in the theoretical analysis of the Morris-Lecar neuronal model.",
"For completeness, the proof is included in Appendix A.",
"Let $f$ and $g$ be two functions such that $g(x)=f(ax)$ , with $a\\ne 0$ .",
"Then $^c\\!D^qg(x)=a^q\\cdot ^c\\!\\!D^q f(ax)$"
],
[
"Stability of fractional-order systems",
"Let us consider the $n$ -dimensional fractional-order system $^c\\!D^\\mathbf {q}\\mathbf {x}(t)=f(t,\\mathbf {x})$ where $\\mathbf {q}=(q_1,q_2,...,q_n)\\in (0,1)^n$ and $f:[0,\\infty )\\times \\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ is continuous on the whole domain of definition and Lipschitz-continuous with respect to the second variable, such that $f(t,0)=0\\quad \\textrm {for any }t\\ge 0.$ Let $\\varphi (t,x_0)$ denote the unique solution of (REF ) which satisfies the initial condition $x(0)=x_0$ (it is important to note that the conditions on the function $f$ given above guarantee the existence and uniqueness of such a solution [8]).",
"It is well-known that in general, the asymptotic stability of the trivial solution of system (REF ) is not of exponential type [3], [14], because of the presence of the memory effect.",
"Due to this observation, a special type of non-exponential asymptotic stability concept has been defined for fractional-order differential equations [26], called Mittag-Leffler stability.",
"In this paper, we are concerned with $\\mathcal {O}(t^{-\\alpha })$ -asymptotical stability, which reflects the algebraic decay of the solutions.",
"The trivial solution of (REF ) is called stable if for any $\\varepsilon >0$ there exists $\\delta =\\delta (\\varepsilon )>0$ such that for every $x_0\\in \\mathbb {R}^n$ satisfying $\\Vert x_0\\Vert <\\delta $ we have $\\Vert \\varphi (t,x_0)\\Vert \\le \\varepsilon $ for any $t\\ge 0$ .",
"The trivial solution of (REF ) is called asymptotically stable if it is stable and there exists $\\rho >0$ such that $\\lim \\limits _{t\\rightarrow \\infty }\\varphi (t,x_0)=0$ whenever $\\Vert x_0\\Vert <\\rho $ .",
"Let $\\alpha >0$ .",
"The trivial solution of (REF ) is called $\\mathcal {O}(t^{-\\alpha })$ -asymptotically stable if it is stable and there exists $\\rho >0$ such that for any $\\Vert x_0\\Vert <\\rho $ one has: $\\Vert \\varphi (t,x_0)\\Vert =\\mathcal {O}(t^{-\\alpha })\\quad \\textrm {as }t\\rightarrow \\infty .$ It is important to remark that $\\mathcal {O}(t^{-\\alpha })$ -asymptotic stability, as defined above, clearly implies asymptotic stability."
],
[
"Stability results for a linear system involving one Caputo derivative",
"We will first investigate the stability properties of the following linear system: $\\begin{bmatrix}^c\\!D^qx(t)\\\\\\dot{y}(t)\\end{bmatrix}=A\\cdot \\begin{bmatrix}x(t)\\\\y(t)\\end{bmatrix}$ where $A=(a_{ij})$ is a real 2-dimensional matrix and $q\\in (0,1)$ .",
"We may assume $a_{12}\\ne 0$ .",
"Otherwise, the first equation of system (REF ) would be decoupled from the second equation.",
"Applying the Laplace transform to system (REF ), we obtain: $\\begin{bmatrix}s^qX(s)-s^{q-1}x(0)\\\\sY(s)-y(0)\\end{bmatrix}=A\\cdot \\begin{bmatrix}X(s)\\\\Y(s)\\end{bmatrix},$ where $X(s)=\\mathcal {L}(x)(s)$ and $Y(s)=\\mathcal {L}(y)(s)$ denote the Laplace transforms of the functions $x$ and $y$ respectively, and $s^q$ represents the principal value (first branch) of the complex power function [10].",
"Therefore: $\\left(\\text{diag}(s^q,s)-A\\right)\\cdot \\begin{bmatrix}X(s)\\\\Y(s)\\end{bmatrix}=\\begin{bmatrix}s^{q-1} x(0)\\\\ y(0)\\end{bmatrix}$ In the following, we will denote: $\\Delta _A(s)&=\\det \\left(\\text{diag}(s^q,s)-A\\right)=\\\\&=s^{q+1}-a_{11}s-a_{22}s^q+\\det (A).$ We can easily express $\\nonumber X(s)&=\\frac{x(0)s^{q}(s-a_{22})+a_{12}y(0)s}{s\\Delta _A(s)}\\\\Y(s)&=\\frac{a_{21}x(0)s^q+y(0)s(s^q-a_{11})}{s\\Delta _A(s)}$ With the aim of proving sufficient conditions for the stability of linear systems of fractional differential equations, several authors have exploited the Final Value Theorem of the Laplace Transform [7], [33].",
"For the sake of completeness, we state the following result: System (REF ) is $\\mathcal {O}(t^{-q})$ -globally asymptotically stable if and only if all the roots of $\\Delta _A(s)$ are in the open left half-plane ($\\Re (s)<0$ ).",
"If $\\det (A)\\ne 0$ and $\\Delta _A(s)$ has at least one root in the open right half-plane ($\\Re (s)>0$ ), then system (REF ) is unstable.",
"Part 1 - Necessity.",
"Assume that system (REF ) is $\\mathcal {O}(t^{-q})$ -globally asymptotically stable and let $(x(t),y(t))$ denote the solution of system (REF ) which satisfies the initial condition $(x(0),y(0))=(x_0,y_0)\\in \\mathbb {R}^2$ .",
"We may choose $x_0,y_0\\ne 0$ .",
"It follows that there exist $M>0$ and $T>0$ such that $|x(t)|\\le \\Vert (x(t),y(t))\\Vert \\le Mt^{-q}\\quad \\textrm {for any }t\\ge T.$ We obtain that the Laplace transform $X(s)$ is absolutely continuous and holomorphic in the open right half-plane ($\\Re (s)>0$ ) [10].",
"Therefore, $X(s)$ does not have any poles in the open right half-plane.",
"From (REF ), we remark that $X(s)=\\frac{x_0s^{q}(s-a_{22})+a_{12}y_0s}{s\\Delta _A(s)}$ and the function from the numerator is holomorphic on $\\mathbb {C}\\setminus \\mathbb {R}_{-}$ .",
"So far, we have obtained: $\\Delta _A(s)\\ne 0\\quad \\textrm {for any }s\\in \\mathbb {C},~\\Re (s)>0.$ We now argue that $\\Delta _A(0)\\ne 0$ .",
"Indeed, assuming that $\\Delta _A(0)=0$ , it follows that $\\det (A)=0$ and $\\Delta _A(s)=s^{q+1}-a_{11}s-a_{22}s^q.$ Therefore: $\\lim \\limits _{s\\rightarrow 0}sX(s)&=\\lim \\limits _{s\\rightarrow 0}\\frac{x_0s^{q}(s-a_{22})+a_{12}y_0s}{\\Delta _A(s)}=\\\\&=\\lim \\limits _{s\\rightarrow 0}\\frac{x_0(s-a_{22})+a_{12}y_0s^{1-q}}{s-a_{11}s^{1-q}-a_{22}}=\\\\&=\\left\\lbrace \\begin{array}{ll}x_0, & \\textrm {if }a_{22}\\ne 0 \\\\-\\frac{a_{12}}{a_{11}}y_0, & \\textrm {if }a_{22}=0\\end{array}\\right.\\quad \\ne 0,$ which contradicts the Final Value Theorem for the Laplace transform $X(s)$ (since $x(t)\\rightarrow 0$ as $t\\rightarrow \\infty $ ).",
"Hence, $\\Delta _A(0)\\ne 0$ .",
"Now and consider the solution $(x(t),y(t))$ of system (REF ) which satisfies the initial condition $(x(0),y(0))=(0,\\frac{1}{a_{12}})$ .",
"For $x(t)$ we obtain the Laplace transform $X(s)=\\Delta _A(s)^{-1}$ .",
"Assuming that $\\Delta _A(s)$ has a root on the imaginary axis (but not at the origin), it follows that $X(s)$ has a pole on the imaginary axis, which implies that $x(t)$ has persistent oscillations, contradicting the convergence of $x(t)$ to the limit 0, as $t\\rightarrow \\infty $ .",
"Therefore, we obtain $\\Delta _A(s)\\ne 0$ , for any $s\\in \\mathbb {C}$ , $\\Re (s)\\ge 0$ .",
"Part 1 - Sufficiency.",
"Let $(x(t),y(t))$ denote the solution of system (REF ) which satisfies the initial condition $(x(0),y(0))=(x_0,y_0)\\in \\mathbb {R}^2$ .",
"Assuming that all the roots of $\\Delta _A(s)$ are in the open left half-plane, it follows that all the poles of the Laplace transforms functions $X(s)$ and $Y(s)$ given by (REF ) are either in the open left half-plane or at the origin, and $X(s)$ and $Y(s)$ have at most a single pole at the origin (in fact, only $X(s)$ has a simple pole at the origin).",
"A simple application of the Final Value Theorem of the Laplace transform [4] yields $\\lim _{t\\rightarrow \\infty }x(t)&=\\lim _{s\\rightarrow 0}sX(s)=\\lim _{s\\rightarrow 0}\\frac{x_0s^{q}(s-a_{22})+a_{12}y_0s}{\\Delta _A(s)}=0;\\\\\\lim _{t\\rightarrow \\infty }y(t)&=\\lim _{s\\rightarrow 0}sY(s)=\\lim _{s\\rightarrow 0}\\frac{a_{21}x_0s^{q}+y_0s(s^q-a_{11})}{\\Delta _A(s)}=0.$ Moreover, the Laplace transform $X(s)$ is holomorphic in the left half-plane, except at the origin and has the asymptotic expansion $X(s)\\sim \\sum _{n=0}^\\infty c_n s^{\\lambda _n},\\quad \\textrm {as } s\\rightarrow 0, $ where $\\lambda _0=q-1<\\lambda _1<...<\\lambda _n<...$ .",
"Using Theorem 37.1 from [10], the following asymptotic expansion is obtained: $x(t)\\sim \\sum _{n=0}^\\infty \\frac{c_n}{\\Gamma (-\\lambda _n)}\\frac{1}{t^{\\lambda _n+1}} ,\\quad \\textrm {as } t\\rightarrow \\infty , $ where $\\Gamma $ represents the Gamma function with the understanding that $\\frac{1}{\\Gamma (-\\lambda _n)}=0\\quad \\textrm {if }\\lambda _n\\in \\mathbb {Z}_+.$ As $\\lambda _0+1=q$ , it follows that $x(t)$ converges to 0 as $t^{-q}$ .",
"On the other hand, the Laplace transform $Y(s)$ is holomorphic in the whole left half-plane and has a similar asymptotic expansion as $X(s)$ .",
"As above, it follows that $y(t)$ converges to 0 as $t^{-q}$ .",
"Combining the convergence results for the two components $x(t)$ and $y(t)$ , it follows, based on Definition REF that system (REF ) is $\\mathcal {O}(t^{-q})$ -globally asymptotically stable.",
"Part 2.",
"Assume that $\\det (A)\\ne 0$ , which is equivalent to $\\Delta _A(0)\\ne 0$ .",
"Consider the solution of $(x(t),y(t))$ of system (REF ) which satisfies the initial condition $(x(0),y(0))=(0,y_0)$ , with an arbitrary $y_0\\in \\mathbb {R}^\\star $ .",
"This solution has the Laplace transform $X(s)=a_{12}y_0\\Delta _A(s)^{-1}$ .",
"Based on Proposition 3.1 from [2], it follows that $\\Delta _A(s)$ has a finite number of roots in $\\mathbb {C}\\setminus \\mathbb {R}_{-}$ , and in particular, in the open right half-plane.",
"Obviously, the Laplace transform $X(s)$ is analytic in $\\mathbb {C}\\setminus \\mathbb {R}_{-}$ , except at the poles given by the roots of $\\Delta _A(s)$ .",
"If $\\Delta _A(s)$ has at least one root in the open right half-plane, let us denote by $\\rho >0$ the real part of a dominant pole of $X(s)$ , i.e.",
"$\\rho =\\max \\lbrace \\Re (s):\\Delta _A(s)=0\\rbrace $ , and by $\\nu \\ge 1$ the largest order of a dominant pole.",
"Following Theorem 35.1 from [10], we obtain that $|x(t)|$ is asymptotically equal to $k~t^{\\nu -1} e^{\\rho t}$ (with $k>0$ ) as $t\\rightarrow \\infty $ .",
"Hence, $x(t)$ is unbounded and therefore, system (REF ) is unstable.",
"$\\Box $ Taking into account the special form of the characteristic function $\\Delta _A(s)$ given above, we prove the following result: Consider the complex-valued function $\\Delta (s)=s^{q+1}+as+bs^q+c,$ where $q\\in (0,1)$ , $a,b,c\\in \\mathbb {R}$ , $b>0$ , and $s^q$ represents the principal value (first branch) of the complex power function.",
"If $c<0$ , then $\\Delta (s)$ has at least one positive real root.",
"$\\Delta (0)=0$ if and only if $c=0$ .",
"Assume $c>0$ .",
"If $a\\ge 0$ then all roots of $\\Delta (s)$ satisfy $\\Re (s)<0$ .",
"The function $\\Delta (s)$ has a pair of pure imaginary roots if and only if $a&=a^\\star (b,c,q)=\\\\ \\nonumber &=\\!\\!-b^q\\!\\left(\\!h_q^{-1}\\Big (\\frac{c}{b^{q+1}}\\Big )\\!\\right)^{\\!\\!q-1}\\!\\!\\!\\!\\Big (\\!h_q^{-1}\\!\\!\\Big (\\!\\frac{c}{b^{q+1}}\\!\\Big )\\cos \\!\\frac{q\\pi }{2}+\\sin \\!\\frac{q\\pi }{2}\\Big )$ where $h_q:\\Big (\\cot \\frac{q\\pi }{2},\\infty \\Big )\\rightarrow (0,\\infty )$ is the function defined by $h_q(\\omega )=\\omega ^q\\Big (\\omega \\sin \\frac{q\\pi }{2}-\\cos \\frac{q\\pi }{2}\\Big ).$ If $s(a,b,c,q)$ is a root of $\\Delta (s)$ such that $\\Re (s(a^\\star ,b,c,q))=0,$ where $a^\\star =a^\\star (b,c,q)$ , the following transversality condition holds: $\\frac{\\partial \\Re (s)}{\\partial a}\\Big |_{a=a^*}<0.$ All roots of $\\Delta (s)$ are in the left half-plane if and only if $a>a^\\star (b,c,q)$ .",
"$\\Delta (s)$ has a pair of roots in the right half-plane if and only if $a<a^\\star (b,c,q)$ .",
"For any $q\\in (0,1)$ , the following inequalities hold: $a^\\star (b,c,q)\\le -b^q\\le -\\min \\lbrace b,1\\rbrace .$ $ $ 1.",
"We have $\\Delta (0)=c<0$ and $\\Delta (\\infty )=\\infty $ , and therefore, due to the continuity of the function $\\Delta (s)$ on $(0,\\infty )$ , it follows that it has at least one positive real root.",
"2.",
"The proof is trivial as $\\Delta (0)=c$ .",
"3.",
"(a) Let $a\\ge 0$ and $c>0$ .",
"Assuming, by contradiction, that $\\Delta (s)$ has a root $s$ with $\\Re (s)\\ge 0$ , it follows that $|\\arg (s)|\\le \\frac{\\pi }{2}\\Rightarrow |\\arg (s^q)|=q\\cdot |\\arg (s)|\\le \\frac{q\\pi }{2}<\\frac{\\pi }{2}.$ Therefore, $\\Re (s^q)>0$ .",
"On the other hand, we have: $s^{q+1}+as+bs^q+c=0~~\\Leftrightarrow ~~ s^q=\\frac{-as-c}{s+b}$ and hence $ \\Re (s^q)&=\\Re \\Big (\\frac{-as-c}{s+b}\\Big )=\\Re \\left[\\frac{(-as-c)(\\bar{s}+b)}{|s+b|^2}\\right]\\\\&=\\frac{\\Re \\big [(-as-c)(\\bar{s}+b)\\big ]}{|s+b|^2}\\\\&=\\frac{\\Re (-as-c)\\Re (\\bar{s}+b)-\\Im (-as-c)\\Im (\\bar{s}+b)}{|s+b|^2}\\\\&=\\frac{(-a\\Re (s)-c)(\\Re (s)+b)+a\\Im (s)(-\\Im (s))}{|s+b|^2}\\\\&=\\frac{-a|s|^2-(ab+c)\\Re (s)-bc }{|s+b|^2}.$ As $a\\ge 0$ , $b>0$ , $c\\ge 0$ , then $-a|s|^2-(ab+c)\\Re (s)-bc\\le 0$ and so $\\Re (s^q)\\le 0$ , which contradicts $\\Re (s^q)>0$ .",
"We conclude that the equation $\\Delta (s)=0$ does not have any roots with $\\Re (s)\\ge 0$ .",
"3.",
"(b) Let $a<0$ and $c>0$ .",
"Assuming that $\\Delta (s)$ has a pair of pure imaginary roots, there exists $\\omega >0$ such that $s=ib\\omega $ is a root of $\\Delta (s)$ .",
"From $\\Delta (ib\\omega )=0$ we have: $\\nonumber b^{q+1}&\\omega ^{q+1}\\Big (-\\sin \\frac{q\\pi }{2}+i\\cos \\frac{q\\pi }{2}\\Big )+iab\\omega +\\\\&+b^{q+1}\\Big (\\cos \\frac{q\\pi }{2}+i\\sin \\frac{q\\pi }{2}\\Big )\\omega ^q+c=0$ Taking the real and the imaginary parts in this equation, we obtain: $& -b^{q+1}\\omega ^{q+1}\\sin \\frac{q\\pi }{2}+b^{q+1}\\omega ^q\\cos \\frac{q\\pi }{2}+c=0\\\\& b^{q+1}\\omega ^{q+1}\\cos \\frac{q\\pi }{2}+ab\\omega +b^{q+1}\\omega ^q\\sin \\frac{q\\pi }{2}=0$ which is equivalent to ${\\left\\lbrace \\begin{array}{ll}& a= -b^{q}\\omega ^{q-1}\\Big (\\omega \\cos \\frac{q\\pi }{2}+\\sin \\frac{q\\pi }{2}\\Big )\\\\& c=b^{q+1}\\omega ^q\\Big (\\omega \\sin \\frac{q\\pi }{2}-\\cos \\frac{q\\pi }{2}\\Big )\\end{array}\\right.",
"}$ As $c>0$ , it results that $\\omega \\sin \\frac{q\\pi }{2}>\\cos \\frac{q\\pi }{2}$ , which leads to $\\omega >\\cot \\frac{q\\pi }{2}$ .",
"Since $\\frac{c}{b^{q+1}}=\\omega ^q\\Big (\\omega \\sin \\frac{q\\pi }{2}-\\cos \\frac{q\\pi }{2}\\Big ),$ we consider the function $h_q:\\Big (\\cot \\frac{q\\pi }{2},\\infty \\Big )\\rightarrow (0,\\infty )$ defined by $h_q(\\omega )=\\omega ^q\\Big (\\omega \\sin \\frac{q\\pi }{2}-\\cos \\frac{q\\pi }{2}\\Big ).$ It is easy to see that for any $\\omega > \\cot \\frac{q\\pi }{2}$ , we have: ${h_q}^{\\prime }(\\omega )=q\\omega ^{q-1}\\Big (\\omega \\sin \\frac{q\\pi }{2}-\\cos \\frac{q\\pi }{2}\\Big )+\\omega ^q\\sin \\frac{q\\pi }{2}> 0.",
"$ Hence, $h_q$ is an increasing function on the interval $\\Big (\\cot \\frac{q\\pi }{2},\\infty \\Big )$ and therefore $h_q$ is invertible, with the inverse denoted by $h_q^{-1}:(0,\\infty )\\rightarrow (\\cot \\frac{q\\pi }{2},\\infty )$ .",
"Hence, from (REF ) we obtain: $\\omega =h_q^{-1}\\Big (\\frac{c}{b^{q+1}}\\Big ).$ From the first equation of (REF ), we conclude that $a=a^\\star (b,c,q)$ .",
"3.",
"(c) Let $s(a,b,c,q)$ denote the root of $\\Delta (s)$ with the property $s(a^\\star ,b,c,q)=ib\\omega ,$ as in 3.",
"(b), where $a^\\star =a(b,c,q)$ .",
"Differentiating with respect to $a$ in the equation: $s^{q+1}+as+bs^q+c=0$ we obtain: $(q+1)s^q\\cdot \\frac{\\partial s}{\\partial a}+s+a\\cdot \\frac{\\partial s}{\\partial a}+b\\cdot q\\cdot s^{q-1}\\cdot \\frac{\\partial s}{\\partial a}=0,$ and hence: $\\frac{\\partial s}{\\partial a}=\\frac{-s}{(q+1)s^q+qbs^{q-1}+a}.$ Taking the real part of this equation, we obtain: $\\frac{\\partial \\Re (s)}{\\partial a}=\\Re \\Big (\\frac{\\partial s}{\\partial a}\\Big )=\\Re \\Big (\\frac{-s}{(q+1)s^q+qbs^{q-1}+a}\\Big ).$ Therefore: $\\frac{\\partial \\Re (s)}{\\partial a}\\Big |_{a=a^\\star }=\\Re \\Big (\\frac{-ib\\omega }{(q+1)(ib\\omega )^q+qb(ib\\omega )^{q-1}+a^\\star }\\Big ).$ Denoting $P(\\omega )=(q+1)(ib\\omega )^q+qb(ib\\omega )^{q-1}+a^\\star $ , we obtain: $\\frac{\\partial \\Re (s)}{\\partial a}\\Big |_{a=a^\\star }&=\\Re \\Big ( \\frac{-ib\\omega }{P(\\omega )}\\Big )=b\\omega \\cdot \\Re \\Big (\\frac{-i\\overline{P(\\omega )}}{|P(\\omega )|^2}\\Big )\\\\&=\\frac{b\\omega }{|P(\\omega )|^2}\\cdot (-\\Im (P(\\omega )))$ As $P(\\omega )=&(q+1)b^q\\omega ^q\\Big (\\cos \\frac{q\\pi }{2}+i\\sin \\frac{q\\pi }{2}\\Big )+\\\\&+qb^q\\omega ^{q-1}(-i)\\Big (\\cos \\frac{q\\pi }{2}+i\\sin \\frac{q\\pi }{2}\\Big )+a^\\star ,$ we have: $\\Im (P(\\omega ))=(q+1)b^q\\omega ^q\\sin \\frac{q\\pi }{2}-qb^q\\omega ^{q-1}\\cos \\frac{q\\pi }{2}.$ As $\\omega >\\cot \\left(\\frac{q\\pi }{2}\\right)$ , it results that $\\frac{\\partial \\Re (s)}{\\partial a}\\Big |_{a=a^\\star }=\\frac{b^{q+1}\\omega ^q}{|P(\\omega )|^2}\\Big (q\\cos \\frac{q\\pi }{2}-(q+1)\\omega \\sin \\frac{q\\pi }{2}\\Big )<0.$ 3.",
"(d,e) The transversality condition obtained above shows that $\\Re (s)$ is decreasing in a neighborhood of $a^\\star $ , and therefore, when $a$ decreases below the critical value $a^\\star =a^\\star (b,c,q)$ , the pair of complex conjugated roots $(s,\\overline{s})$ cross the imaginary axis from the left half-plane into the right half-plane.",
"Combined with 3.",
"(a), we obtain the desired conclusions.",
"3.",
"(f) We will first prove that for any $q\\in (0,1)$ and any $\\omega >0$ , the following inequality holds: $\\omega \\cos \\frac{q\\pi }{2}+\\sin \\frac{q\\pi }{2}\\ge \\omega ^{1-q}.$ Indeed, let us denote $\\theta =\\arctan \\omega \\in \\left(0,\\frac{\\pi }{2}\\right)$ .",
"Inequality (REF ) is equivalent to $\\tan \\theta \\cos \\frac{q\\pi }{2}+\\sin \\frac{q\\pi }{2}\\ge (\\tan \\theta )^{1-q}$ which can be rewritten as $\\sin \\left(\\theta +\\frac{q\\pi }{2}\\right)\\ge (\\sin \\theta )^{1-q}(\\cos \\theta )^q.$ As the natural logarithm is an increasing function and $\\cos (\\theta )=\\sin \\left(\\theta +\\frac{\\pi }{2}\\right)$ , this inequality is further equivalent to: $\\ln \\left(\\sin \\left(\\theta +\\frac{q\\pi }{2}\\right)\\right)\\ge (1-q)\\ln \\left(\\sin \\theta \\right)+q\\ln \\left(\\sin \\left(\\theta +\\frac{\\pi }{2}\\right)\\right).$ This last inequality easily follows from the fact that the function $f(x)=\\ln (\\sin (x))$ , defined on the interval $(0,\\pi )$ , is a concave function (Jensen's inequality).",
"Therefore, inequality (REF ) holds and based on the definition of $a^\\star (b,c,q)$ , it leads to $a^\\star (b,c,q)\\le -b^q$ , for any $b>0$ , $c>0$ and $q\\in (0,1)$ .",
"The second inequality easily follows from the properties of the function $b^x$ , where $b>0$ and $x\\in [0,1]$ .",
"If $b\\in (0,1)$ then $b^x$ is decreasing on $[0,1]$ and therefore, for any $q\\in [0,1]$ , we have $b^q\\ge b^1=b=\\min \\lbrace b,1\\rbrace $ .",
"On the other hand, if $b\\ge 1$ , then $b^x$ is increasing on $[0,1]$ and hence, for any $q\\in [0,1]$ , we obtain $b^q\\ge b^0=1=\\min \\lbrace b,1\\rbrace $ .",
"We conclude that $b^q\\ge \\min \\lbrace b,1\\rbrace $ , for any $b>0$ and $q\\in [0,1]$ .$\\Box $ With the aim of deducing sufficient stability conditions which do not depend on the fractional order $q$ , we state the following: Let $b>0$ , $c>0$ , and consider the complex-valued function $\\Delta (s)$ defined as in Proposition .",
"If $a>-\\min \\lbrace b,1\\rbrace $ then all roots of $\\Delta (s)$ are in the open left-half plane, regardless of $q$ .",
"Let $a\\le -\\min \\lbrace b,1\\rbrace $ .",
"If either of the following conditions hold $a+b+c+1\\le 0$ ; $0<a+b+c+1<(\\sqrt{c}-1)^2$ and $c>1$ ; then $\\Delta (s)$ has at least one positive real root, regardless of $q$ .",
"1.",
"Let us consider an arbitrary $q\\in (0,1)$ and $a>-\\min \\lbrace b,1\\rbrace $ .",
"From Proposition .",
"(f) we have $a> -\\min \\lbrace b,1\\rbrace \\ge -b^q\\ge a^\\star (b,c,q).$ Hence, based on Proposition .",
"(d) it follows that all the roots of $\\Delta (s)$ are in the open left half-plane.",
"2.",
"We have: $\\Delta (s)\\le {\\left\\lbrace \\begin{array}{ll}& (1+a)s+b+c, \\quad \\textrm {if }s\\in (0,1),\\\\& s^2+(a+b)s+c,\\quad \\textrm {if } s\\ge 1,\\end{array}\\right.",
"}$ and we denote $(1+a)s+b+c=p_1(s)$ and $s^2+(a+b)s+c=p_2(s)$ .",
"It is easy to see that if either $p_1$ or $p_2$ have positive real roots, so does $\\Delta (s)$ .",
"(a) If $a+b+c+1\\le 0$ then $\\Delta (1)=1+a+b+c\\le 0$ and $\\Delta (\\infty )=\\infty $ .",
"So $\\Delta (s)$ has at least one real root belonging to the interval $[1,\\infty )$ .",
"(b) If $a+b+c+1>0$ then $p_1(s)>0$ for every $s\\in (0,1)$ .",
"Since $c>0$ , elementary calculus shows that necessary and sufficient conditions for $p_2(s)$ to take negative values in a subinterval of $[1,\\infty )$ are: discriminant $\\delta =(a+b)^2-4c\\ge 0$ and $-\\frac{a+b}{2}>1$ (i.e.",
"the minimum point of $p_2$ is larger than 1).",
"In turn, these are equivalent to $c>1$ and $-(c+1)<a+b<-2\\sqrt{c}$ , which lead to the desired conclusion.",
"$\\Box $ Based on Theorem and Propositions and , we obtain the following conditions for the stability of system (REF ), with respect to its coefficients and the fractional order $q$ : Consider the linear system (REF ) with the fractional order $q\\in (0,1)$ .",
"Denoting $a=-a_{11}$ , $b=-a_{22}$ , $c=\\det (A)$ and assuming $b>0$ , it follows that: If $c<0$ , system (REF ) is unstable, regardless of the fractional order $q$ .",
"Assume that $c>0$ .",
"(a) System (REF ) is $\\mathcal {O}(t^{-q})$ -asymptotically stable if and only if $a>a^\\star (b,c,q)$ , where $a^\\star (b,c,q)$ is defined by (REF ).",
"(b) If $a>-\\min \\lbrace b,1\\rbrace $ , system (REF ) is asymptotically stable, regardless of the fractional order $q$ .",
"(c) System (REF ) is unstable if $a<a^\\star (b,c,q)$ , where $a^\\star (b,c,q)$ is defined by (REF ).",
"(d) If $a\\le -\\min \\lbrace b,1\\rbrace $ and either of the following conditions hold $a+b+c+1\\le 0$ ; $0<a+b+c+1<(\\sqrt{c}-1)^2$ and $c>1$ ; then system (REF ) is unstable, regardless of the fractional order $q$ .",
"Condition 2.",
"(a) from Corollary is a generalization of the well known Routh-Hurwitz conditions for two dimensional systems of first order differential equations.",
"Indeed, if $A$ is the system's matrix, the Routh-Hurwitz conditions provide that the (first-order) system is asymptotically (exponentially) stable if and only if $\\text{trace}(A)<0$ and $\\det (A)>0$ .",
"In our setting, it can easily be seen that for $q=1$ , equation (REF ) gives $a^\\star (b,c,1)=-b$ .",
"Hence, with the notations $a=-a_{11}$ , $b=-a_{22}$ and $c=\\det (A)$ , condition 2.",
"(a) for the particular case $q=1$ is equivalent to the Routh-Hurwitz conditions.",
"Throughout this section, we have considered the assumption $b=-a_{22}>0$ .",
"A similar lengthy reasoning can also be applied in the case $b\\le 0$ , to deduce necessary and sufficient conditions for the asymptotic stability or instability of system (REF ).",
"However, as we will see in the following section, in the stability analysis of the equilibrium states of a fractional-order Morris-Lecar neuronal model we have a positive coefficient $b$ , which explains the restriction of our analysis to the case $b>0$ ."
],
[
"Construction of the fractional-order model",
"Neuronal activity of biological neurons has been typically modeled using the classical Hodgkin-Huxley mathematical model [18], dating back to 1952, including four nonlinear differential equations for the membrane potential and gating variables of ionic currents.",
"Several lower dimensional simplified versions of the Hodgkin-Huxley model have been introduced in 1962 by Fitzhugh and Nagumo [13], in 1981 by Morris and Lecar [32] and in 1982 by Hindmarsh and Rose [17].",
"These simplified models have an important advantage: while still being relatively simple, they allow for a good qualitative description of many different patterns of the membrane potential observed in experiments.",
"The classical Morris-Lecar neuronal model [32] describes the oscillatory voltage patterns of Barnacle muscle fibers.",
"Mathematically, the Morris-Lecar model is described by the following system of differential equations: $\\left\\lbrace \\begin{array}{rl}C_m\\frac{\\mathrm {d}V}{\\mathrm {d}t}=&g_{Ca}M_{\\infty }(V)(V_{Ca}-V)+g_K N(V_K-V)+\\\\&+g_L(V_L-V)+I\\\\\\frac{\\mathrm {d}N}{\\mathrm {d}t}=&\\overline{\\lambda _N}\\cdot \\lambda (V)(N_{\\infty }(V)-N)\\end{array}\\right.$ where $V$ is the membrane potential, $N$ is the gating variable for $K^+$ , $C_m$ is the membrane capacitance, $I$ represents the externally applied current, $V_{Ca}$ , $V_K$ and $V_L$ denote the equilibrium potentials for $Ca^{2+}$ , $K^+$ the leak current and $g_{Ca}$ , $g_K$ and $g_L$ are positive constants representing the maximum conductances of the corresponding ionic currents, and $\\overline{\\lambda _N}$ is the maximum rate constant for the $K^+$ channel opening.",
"The following assumptions are usually taken into consideration: $M_\\infty $ and $N_\\infty $ are increasing functions of class $C^1$ defined on $\\mathbb {R}$ with values in $(0,1)$ ; $\\lambda $ is a positive function of class $C^1$ on $\\mathbb {R}$ ; $V_K<V_L<0<V_{Ca}$ .",
"The conductances of both $Ca^{2+}$ and $K^+$ are sigmoid functions with respect to the membrane voltage $V$ .",
"Particular functions considered previously in the literature, which satisfy the above assumptions, are: $M_{\\infty }(V) &=\\frac{1}{2}\\left( 1+\\tanh \\left( \\frac{V-V_1}{V_2} \\right) \\right)\\\\N_{\\infty }(V) &=\\frac{1}{2}\\left( 1+\\tanh \\left( \\frac{V-V_3}{V_4} \\right) \\right)\\\\\\lambda (V) &=\\cosh \\left( \\frac{V-V_3}{V_4} \\right)$ where $V_i$ are positive constants, $i\\in \\lbrace 1,2,3,4\\rbrace $ .",
"In electrophysiological experiments, the neuronal membrane is considered to be equivalent to a resistor-capacitor circuit.",
"In this context, based on experimental observations concerning biological neurons [1], [27], the fractional-order capacitor proposed by Westerlund and Ekstam [48] has an utmost importance.",
"They showed that Jacques Curie's empirical law for the current through capacitors and dielectrics leads to the following capacitive current-voltage relationship for a non-ideal capacitor: $I_c^{\\alpha }=C_m^{\\alpha }\\frac{d^{\\alpha }V_c}{dt^{\\alpha }}$ where $0<\\alpha <1$ , the fractional-order capacitance with units (amp/volt)sec$^\\alpha $ is denoted by $C_m^{\\alpha }$ , and $\\frac{d^{\\alpha }}{dt^{\\alpha }}$ represents a fractional-order differential operator [47].",
"Several types of fractional-order neuronal models have been investigated in the recent years: fractional leaky integrate-and-fire model [41], fractional-order Hindmarsh-Rose model [19], three-dimensional slow-fast fractional-order Morris-Lecar models [39], [45] and fractional-order Hodgkin-Huxley models [42], [47].",
"Starting from system (REF ), we first consider the following general fractional-order Morris-Lecar neuronal model with two fractional orders $p,q\\in (0,1)$ : $\\left\\lbrace \\begin{array}{rl}C_m(q)\\cdot ^c\\!\\!D^qV(t)=&g_{Ca}M_{\\infty }(V)(V_{Ca}-V)+g_KN(V_K-V)+\\\\&+g_L(V_L-V)+I\\\\^cD^pN(t)=&\\overline{\\lambda _N}^p\\cdot \\lambda (V)(N_\\infty (V)-N)\\end{array}\\right.$ where $C_m(q)=\\frac{\\tau ^q}{R_m}$ is the membrane capacitance [47], $R_m$ is the membrane resistance, $\\tau $ is the time constant.",
"It is important to note that for $q=1$ we obtain the classical formula for the classical integer-order capacitance.",
"We emphasize that the inclusion of the fractional-order capacitance to the left hand-side of the first equation is a straightforward answer to the dimensional consistency problem (units of measurement consistency) of system (REF ).",
"For the same reason, in right side of the second equation we introduce the term $\\overline{\\lambda _N}^p$ (see for example [9] for a similar approach), and in this way, the dimensions of both sides coincide, and are (seconds)$^{-p}$ .",
"For the theoretical investigation of the fractional order Morris-Lecar neuronal model, we nondimensionalize the system (REF ) as in Appendix B, with the substitutions: $v(t)=\\frac{V(\\tau t)}{V_{Ca}}\\quad ,\\quad n(t)=N(\\tau t).$ Considering the following dimensionless constants: $&v_K=\\frac{V_K}{V_{Ca}}, \\quad v_L=\\frac{V_L}{V_{Ca}},\\quad v_i=\\frac{V_i}{V_{Ca}},~~i\\in \\lbrace 1,2,3,4\\rbrace \\\\&\\gamma _x=R_m\\cdot g_x,~~x\\in \\lbrace Ca,K,L\\rbrace ,\\quad \\tilde{I}=R_m\\cdot \\frac{I}{V_{Ca}},$ and the functions $m_\\infty (v)=&M_\\infty (V_{Ca}v)=\\frac{1}{2}\\Big (1+\\tanh \\left(\\frac{v-v_1}{v_2}\\right)\\Big ),\\\\n_{\\infty }(v)=&N_\\infty (V_{Ca}v)=\\frac{1}{2}\\left( 1+\\tanh \\left( \\frac{v-v_3}{v_4} \\right) \\right),\\\\\\ell (v)=&\\lambda (V_{Ca}v)=\\cosh \\Big (\\frac{v-v_3}{2v_4}\\Big ),$ we obtain the following nondimensional fractional-order system $\\left\\lbrace \\begin{array}{rl}^c\\!D^qv(t)=&\\gamma _{Ca}m_\\infty (v)(1-v)+\\gamma _K\\cdot n(v_K-v)+\\\\&+\\gamma _L(v_L-v)+\\tilde{I}\\\\^c\\!D^pn(t)=&(\\tau \\overline{\\lambda _N})^p\\cdot \\ell (v)(n_\\infty (v)-n)\\end{array}\\right.$ It is important to notice that, based on this procedure, the nondimensional system also involves the term $(\\tau \\overline{\\lambda _N})^p$ in the right hand side of the second equation.",
"Therefore, the correct version of the fractional-order variant of the Morris-Lecar neuronal model is quite different from the nondimensional version considered in [39], where the fractional-order capacitance $C_m(q)$ appearing in the dimensional system has not been taken into account (nor the dimensional consistency problem) and equal fractional orders have been considered for both equations (i.e.",
"$p=q$ ).",
"In fact, it appears that the nondimensionalization process which had to be carried out to obtain the nondimensional system in [39] did not take into account the simple property presented in Proposition REF .",
"In fact, we have to remark that there is no known biological reason to consider a fractional order derivative for the gating variable, and therefore, in what follows, we consider $p=1$ as in [47], obtaining the following nondimensional system: $\\left\\lbrace \\begin{array}{rl}^c\\!D^qv(t)=&\\gamma _{Ca}m_\\infty (v)(1-v)+\\gamma _K\\cdot n(v_K-v)+\\\\&+\\gamma _L(v_L-v)+\\tilde{I}\\\\\\dot{n}(t)=&\\phi \\ell (v)(n_\\infty (v)-n)\\end{array}\\right.$ where $\\phi =\\tau \\cdot \\overline{\\lambda _N}$ ."
],
[
"Existence of equilibrium states",
"System (REF ) is a particular case of the following generic two-dimensional fractional-order conductance-based neuronal model: $\\left\\lbrace \\begin{array}{rl}^c\\!D^qv(t)&=I-I(v,n)\\\\\\dot{n}(t)&=\\phi \\ell (v)(n_\\infty (v)-n)\\end{array}\\right.$ where $v$ and $n$ represent the membrane potential and the gating variable of the neuron, $I$ is an externally applied current, $I(v,n)$ represents the ionic current, $\\ell (v)$ and $n_\\infty (v)$ are the rate constant for opening ionic channels and the fraction of open ionic channels at steady state, respectively.",
"In particular, for the Morris-Lecar fractional neuronal model (REF ), we have: $I(v,n)=\\gamma _{Ca}m_\\infty (v)(v-1)+\\gamma _K\\cdot n(v-v_K)+\\gamma _L(v-v_L).$ The equilibrium states of system (REF ) are the solutions of the following algebraic system: ${\\left\\lbrace \\begin{array}{ll}&I=I(v,n)\\\\&n=n_\\infty (v)\\end{array}\\right.",
"},$ which is equivalent to ${\\left\\lbrace \\begin{array}{ll}&I=I(v,n_\\infty (v)):=I_\\infty (v)\\\\&n=n_\\infty (v)\\end{array}\\right.",
"}.$ In the following, we assume that the function $I_\\infty (v)$ satisfies the following properties: (A1) $I_\\infty \\in C^1(\\mathbb {R})$ ; (A2) $\\displaystyle \\lim \\limits _{v\\rightarrow -\\infty }I_\\infty (v)=-\\infty $ and $\\displaystyle \\lim \\limits _{v\\rightarrow \\infty }I_\\infty (v)=\\infty $ ; (A3) $I^{\\prime }_{\\infty }$ has exactly two real roots $v_\\alpha <v_\\beta $ .",
"It is important to underline that these properties are satisfied in the particular case of the Morris-Lecar neuronal model with the function $I(v,n)$ given by (REF ).",
"We denote $I_{max}=I_\\infty (v_\\alpha )$ , $I_{min}=I_\\infty (v_\\beta )$ .",
"Then $I_\\infty $ is increasing on the intervals $(-\\infty , v_\\alpha ]$ and $[v_\\beta , \\infty )$ and decreasing on the interval $(v_\\alpha , v_\\beta )$ .",
"As $I_\\infty :(-\\infty , v_\\alpha ]\\rightarrow (-\\infty , I_{max}]$ is increasing and continuous, it follows that it is bijective.",
"We denote $I_1=I_\\infty |_{(-\\infty , v_\\alpha ]}$ the restriction of $I_\\infty $ to the interval $(-\\infty ,v_\\alpha ]$ and consider its inverse: $v_1:(-\\infty , I_{max}]\\rightarrow (-\\infty , v_\\alpha ],\\quad v_1(I)=I_1^{-1}(I).$ Therefore, $(v_1(I),n_\\infty (v_1(I)))$ , with $I<I_{max}$ , represents the first branch of equilibrium states of system (REF ).",
"We obtain the other two branches of equilibrium states in a similar way: $I_2=I_\\infty |_{(v_\\alpha ,v_\\beta )},\\quad v_2:(I_{min},I_{max})\\rightarrow (v_\\alpha , v_\\beta ),\\quad v_2(I)=I_2^{-1}(I)$ $I_3=I_\\infty |_{[v_\\beta , \\infty )},\\quad v_3:[I_{min}, \\infty )\\rightarrow [v_\\beta , \\infty ),\\quad v_3(I)=I_3^{-1}(I).$ We have the following situations: If $I<I_{min}$ or if $I>I_{max}$ , then system (REF ) has an unique equilibrium state.",
"If $I=I_{min}$ or if $I=I_{max}$ , then system (REF ) has two equilibrium states.",
"If $I\\in (I_{min},I_{max})$ , then system (REF ) has three equilibrium states."
],
[
"Stability of equilibrium states",
"The Jacobian matrix associated to the system (REF ) at an equilibrium state $(v^\\star ,n^\\star )=(v^\\star , n_\\infty (v^\\star ))$ is: $J=\\begin{bmatrix}-I_v(v^\\star ,n^\\star )& -I_n(v^\\star ,n^\\star )\\\\\\phi \\ell ^{\\prime }(v^\\star )[n_\\infty (v^\\star )-n^\\star ]+\\phi \\ell (v^\\star )n^{\\prime }_\\infty (v^\\star )& -\\phi \\ell (v^\\star ))\\end{bmatrix}$ Since $n_{\\infty }(v^\\star )=n^\\star $ , we have: $J=\\begin{bmatrix}-I_v(v^\\star ,n_{\\infty }(v^\\star ))& -I_n(v^\\star ,n_{\\infty }(v^\\star ))\\\\\\phi \\ell (v^\\star )n^{\\prime }_\\infty (v^\\star )& -\\phi \\ell (v^\\star ))\\end{bmatrix}$ Based on the considerations from section 3, the characteristic equation at the equilibrium state $(v^\\star ,n^\\star )$ is $s^{q+1}+a(v^\\star )s+b(v^\\star )s^q+c(v^\\star )=0$ where $a(v^\\star )&=I_v(v^\\star ,n_\\infty (v^\\star )),\\\\b(v^\\star )&=\\phi \\ell (v^\\star )>0,\\\\c(v^\\star )&=\\det (J)=\\\\&=\\phi \\ell (v^\\star )[I_v(v^\\star , n_\\infty (v^\\star ))+n^{\\prime }_\\infty (v^\\star )\\cdot I_n(v^\\star , n_\\infty (v^\\star ))]=\\\\&=\\phi \\ell (v^\\star )I^{\\prime }_\\infty (v^\\star ).$ The equilibrium point $(v^\\star ,n^\\star )=(v^\\star , n_\\infty (v^\\star ))$ is asymptotically stable if and only if all the roots of the characteristic equation (REF ) are in the left half-plane (i.e.",
"$Re(s)<0$ ).",
"In what follows, we will show how the theoretical results presented in section 3 can be applied to analyze the stability of the steady states of systems (REF ) and (REF ).",
"The second branch of equilibrium points $(v_2(I),n_\\infty (v_2(I)))$ (with $I\\in (I_{min},I_{max})$ ) of system (REF ) is unstable, regardless of the fractional order $q$ .",
"Let $I\\in (I_{min},I_{max})$ and $v^\\star =v_2(I)\\in (v_\\alpha ,v_\\beta )$ .",
"It follows that $I^{\\prime }_\\infty (v^\\star )<0$ and hence $c(v^\\star )<0$ .",
"Based on Proposition , it follows that the characteristic equation REF has at least one positive real root.",
"Hence, the equilibrium point $(v^\\star ,n^\\star )=(v_2(I),n_\\infty (v_2(I)))$ is unstable, regardless of the fractional order $q$ .$\\Box $ If $(v^\\star ,n^\\star )=(v^\\star , n_\\infty (v^\\star ))$ is an equilibrium point belonging to either the first or the third branch, such that $v^\\star \\notin \\lbrace v_\\alpha ,v_\\beta \\rbrace $ and $a(v^\\star )>-\\min \\lbrace b(v^\\star ),1\\rbrace ,$ then $(v^\\star ,n^\\star )$ is asymptotically stable, regardless of the fractional order $q$ .",
"If $(v^\\star ,n^\\star )$ belongs to the first or third branch and $v^\\star \\notin \\lbrace v_\\alpha ,v_\\beta \\rbrace $ , we have $I^{\\prime }_\\infty (v^\\star )> 0$ which is equivalent to $c(v^\\star )> 0$ .",
"Point 2.",
"(b) of Corollary implies that $(v^\\star , n^\\star )$ is asymptotically stable, regardless of the fractional order $q$ .$\\Box $ We note that saddle-node bifurcations may take place in the generic system (REF ) if and only if $s=0$ is a root of the characteristic equation (REF ), which is equivalent to $c(v^\\star )=0$ , which in turn, means that $v^\\star \\in \\lbrace v_\\alpha ,v_\\beta \\rbrace $ .",
"This obviously corresponds to the collision of two branches of equilibrium states (first two branches at $v_\\alpha $ and last two branches at $v_\\beta $ , respectively).",
"It is important to emphasize that Propositions REF and REF have been obtained for the whole class of generic fractional-order conductance-based neuronal models (REF ).",
"In what follows, we restrict ourselves to the Morris-Lecar model (REF ).",
"Additional information on the three branches of equilibrium states is given in Fig.",
"REF For the particular case of the Morris-Lecar model (REF ), with the function $I(v,n)$ given by (REF ), assuming that: $v_K<v_\\alpha <v_\\beta <1,$ we have: Any equilibrium state $(v^\\star , n^\\star )$ of system (REF ) belonging to the first branch, with $v^\\star \\le v_{K}$ , is asymptotically stable, regardless of the fractional order $q$ .",
"Any equilibrium state $(v^\\star , n^\\star )$ of system (REF ) belonging to the third branch, with $v^\\star \\ge 1$ , is asymptotically stable, regardless of the fractional order $q$ .",
"Any equilibrium state $(v^\\star , n^\\star )$ of system (REF ) belonging to the second branch is unstable, regardless of the fractional order $q$ .",
"Let $(v^\\star , n^\\star )=(v^\\star ,n_\\infty (v^\\star ))$ be an equilibrium state of system (REF ) belonging to the third branch, with $v^\\star \\ge 1$ .",
"As $I(v,n)$ is given by (REF ), we have: $I_v(v,n)=g_{Ca}[m^{\\prime }_{\\infty }(v)(v-1)+m_{\\infty }(v)]+g_k\\cdot n+g_L$ and hence, we obtain: $a(v^\\star )&=I_v(v^\\star ,n^\\star )=\\\\&=g_{Ca}[m^{\\prime }_{\\infty }(v)(v^\\star -1)+m_{\\infty }(v^\\star )]+g_k\\cdot n_\\infty (v^\\star )+g_L\\ge 0.$ Based on Proposition REF , it follows that $(v^\\star , n^\\star )$ is asymptotically stable.",
"On the other hand, if $(v^\\star , n^\\star )=(v^\\star ,n_\\infty (v^\\star ))$ is an equilibrium state of system (REF ) belonging to the first branch, with $v^\\star \\le v_{K}$ , we first compute: $I^{\\prime }_{\\infty }(v)=\\frac{\\mathrm {d}}{\\mathrm {d}v}I(v,n_\\infty (v))=I_v(v,n_\\infty (v))+n_\\infty ^{\\prime }(v)\\cdot I_n(v,n_\\infty (v)),$ and therefore: $a(v^\\star )=I_v(v^\\star ,n_\\infty (v^\\star ))=n_\\infty ^{\\prime }(v^\\star )\\cdot g_k\\cdot (v_K-v^\\star )+I^{\\prime }_{\\infty }(v^\\star )\\ge 0,$ due to the fact that $n_\\infty $ is increasing on the whole real line and $I_\\infty $ is increasing on $(-\\infty ,v_\\alpha ]$ .",
"Hence, based on Proposition REF , it follows that $(v^\\star , n^\\star )$ is asymptotically stable.",
"The last part of the Proposition follows directly from Proposition REF .$\\Box $ In the following, we will discuss the stability of equilibrium states $(v^\\star ,n^\\star )$ belonging to the first or third branch, with $v^\\star \\in (v_K,v_\\alpha )$ or $v^\\star \\in (v_\\beta ,1)$ , respectively.",
"Assume that $\\phi $ is small (i.e.",
"$\\phi \\ll 1$ ) and that $\\ell (v)<\\phi ^{-1}$ , for any $v\\in (v_K,v_\\alpha )\\cup (v_\\beta ,1)$ (these are true in the case of numerical values considered in the literature).",
"In this case, we have $b(v)=\\phi \\ell (v)<1$ , for any $v\\in (v_K,v_\\alpha )\\cup (v_\\beta ,1)$ .",
"Moreover, from the last part of the proof of Corollary REF , we have $a(v)=n_\\infty ^{\\prime }(v)\\cdot g_k\\cdot (v_K-v)+I^{\\prime }_{\\infty }(v)$ and hence, we can easily see that $a(v_\\alpha )<0$ and $a(v_\\beta )<0$ (as $v_\\alpha $ and $v_\\beta $ are the roots of $I^{\\prime }_{\\infty }$ ).",
"On the other hand, from the proof of Corollary REF , we know that $a(v_K)>0$ and $a(1)>0$ , and therefore, the function $a(v)$ changes its sign on the intervals $(v_K,v_{\\alpha })$ and $(v_{\\beta },1)$ , respectively.",
"According to our assumption that $\\phi $ is small, it follows that the function $a(v)+b(v)$ also changes its sign on each of the intervals $(v_K,v_{\\alpha })$ and $(v_{\\beta },1)$ .",
"Therefore, there exist two roots $v^{\\prime }\\in (v_K,v_{\\alpha }) $ and $v^{\\prime \\prime }\\in (v_{\\beta },1)$ of the function $a(v)+b(v)$ .",
"We will further assume that these roots are unique, which is in accordance with the numerical data.",
"Based on Proposition REF , we deduce that an equilibrium state $(v^\\star ,n^\\star )$ belonging to the first branch or third branch with $v^\\star <v^{\\prime }$ or $v^\\star >v^{\\prime \\prime }$ , respectively is asymptotically stable, regardless of the fractional order $q$ (see Fig.",
"REF ).",
"The stability of an equilibrium state $(v^\\star ,n^\\star )$ belonging to the first branch with $v^\\star \\in [v^{\\prime },v_\\alpha )$ depends on the fractional order $q$ .",
"Indeed, according to , $(v^\\star ,n^\\star )$ is $\\mathcal {O}(t^{-q})$ -asymptotically stable if and only if $a(v^\\star )>a^\\star (b(v^\\star ),c(v^\\star ),q),$ where the function $a^\\star $ is defined by (REF ) (see Fig REF ).",
"At the critical value $q^\\star $ defined implicitly by the equality $a(v^\\star )=a^\\star (b(v^\\star ),c(v^\\star ),q^\\star ),$ a Hopf bifurcation is expected to occur, as it can be deduced from Proposition , points 3.(b,c).",
"We emphasize that even though bifurcation theory in integer-order dynamical systems has been widely and rigorously studied (see for example [22]), at this time, in the case of fractional-order systems, very few theoretical results are known regarding bifurcation phenomena.",
"In [11], some conditions for the occurrence of Hopf bifurcations have been formulated, based on observations arising from numerical simulations.",
"Moreover, a center manifold theorem has been recently obtained in [28].",
"However, the complete theoretical characterization of the Hopf bifurcation in fractional-order systems are still open questions.",
"This is the reason why we rely on numerical simulations to assess the qualitative behavior of fractional order systems near a Hopf bifurcation point, as well as the stability of the resulting limit cycle.",
"Bifurcations in the classical integer-order Morris-Lecar neuronal model are well-understood and have been thoroughly investigated in [44].",
"On the third branch, when analyzing the stability of an equilibrium state $(v^\\star ,n^\\star )$ with $v^\\star \\in (v_\\beta ,v^{\\prime \\prime }]$ , according to the numerical data, two situations may occur.",
"Let us denote by $v^{\\prime \\prime \\prime }\\in (v_\\beta ,v^{\\prime \\prime })$ the root of the equation $a(v)+b(v)+c(v)+1=0.$ If $v^\\star \\in (v_\\beta ,v^{\\prime \\prime \\prime }]$ , we have $a(v^\\star )+b(v^\\star )<0$ and $a(v^\\star )+b(v^\\star )+c(v^\\star )+1\\le 0$ , and therefore, from Corollary point 2.",
"(d), we obtain that $(v^\\star ,n^\\star )$ is unstable, regardless of the fractional order $q$ .",
"If $v^\\star \\in (v^{\\prime \\prime \\prime },v^{\\prime \\prime }]$ , the equilibrium state $(v^\\star ,n^\\star )$ is $\\mathcal {O}(t^{-q})$ -asymptotically stable if and only if $a(v^\\star )>a^\\star (b(v^\\star ),c(v^\\star ),q),$ where the function $a^\\star $ is defined by (REF ) (see Fig REF ).",
"As in the case of the first branch, at the critical value defined $q^\\star $ defined implicitly by the equality $a(v^\\star )=a^\\star (b(v^\\star ),c(v^\\star ),q^\\star ),$ a Hopf bifurcation is expected to occur.",
"Figure: Branches of equilibrium states for the Morris-Lecar model (), with the parameter values given by Table .",
"Here, V ' =-31.403V^{\\prime }=-31.403, V α =-29.568V_\\alpha =-29.568, V β =-3.5774V_\\beta =-3.5774, V ''' =5.28457V^{\\prime \\prime \\prime }=5.28457 and V '' =9.82288V^{\\prime \\prime }=9.82288.",
"The three branches coexist if and only if I∈(-14.4204,39.6935)I\\in (-14.4204,39.6935).Figure: Stability of equilibrium states (V ☆ ,N ☆ )(V^\\star ,N^\\star ) from branch B1 of the Morris-Lecar model () (with p=1p=1) with V ☆ ∈[V ' ,V α )=[-31.403,-29.568)V^\\star \\in [V^{\\prime },V_\\alpha )=[-31.403,-29.568) depends on the fractional order qq.",
"The blue curve represents the critical values of qq for which Hopf bifurcations may occur in a neighborhood of the corresponding equilibrium state (V ☆ ,N ☆ )(V^\\star ,N^\\star ).",
"The asymptotic stability region is the shaded area below the curve.Figure: Stability of equilibrium states (V ☆ ,N ☆ )(V^\\star ,N^\\star ) from branch B3 of the Morris-Lecar model () (with p=1p=1) with V ☆ ∈(V ''' ,V '' ]=(5.28457,9.82288]V^\\star \\in (V^{\\prime \\prime \\prime },V^{\\prime \\prime }]=(5.28457,9.82288] depends on the fractional order qq.",
"The blue curve represents the critical values of qq for which Hopf bifurcations may occur in a neighborhood of the corresponding equilibrium state (V ☆ ,N ☆ )(V^\\star ,N^\\star ).",
"The asymptotic stability region is the shaded area below the curve."
],
[
"Further numerical simulations",
"In the numerical simulations, we use the numerical values given in Table REF for the parameters of system (REF ), corresponding to a type-I neuron [32], [44].",
"Table: Numerical values and significance of the parameters used in the simulations.Interesting spiking behavior can be observed by numerical simulations for the externally applied current of $I=40$ ($\\mu A$ ) and different values of the fractional order $q$ (see Figs.",
"REF and REF ).",
"At $I=I_{max}=39.6935$ , the first two branches of equilibrium states collide, at the saddle-node bifurcation point with abscissa $V_\\alpha =-29.568$ , and disappear for $I>I_{max}$ .",
"When $I$ crosses the value $I_{max}$ , the corresponding equilibrium states of branch B3 (with the abscissa slightly larger than $V^{\\prime \\prime \\prime }$ ) are unstable for most values of the fractional order $q$ and asymptotically stable only for very small values of $q$ (which may be unrealistic from biologic point of view), as shown in Fig.",
"REF .",
"However, for $q$ large enough, a stable limit cycle exists in a neighborhood of each equilibrium state of branch B3 corresponding to $I$ slightly larger than $I_{max}$ (see green part of B3 in Fig.",
"REF ).",
"Fig.",
"REF shows that for the same value of the externally applied current $I=40$ ($\\mu A$ ), as the fractional order $q$ of the system decreases, the number of spikes over the same time interval increases, which may correspond to a better reflection of the biological properties by the fractional order model.",
"Figure: Limit cycles for the fractional-order Morris-Lecar model () with various values of the fractional order qq, when I=40I=40.Figure: Evolution of VV with respect to time for the fractional-order Morris-Lecar model () with various values of the fractional order qq, when I=40I=40.",
"The considered initial condition is the resting state."
],
[
"Conclusions",
"In this paper, we have obtained necessary and sufficient conditions for the asymptotic stability of a two-dimensional incommensurate order linear autonomous system with one fractional-order derivative and one first-order derivative.",
"These theoretical results have been successfully applied to the investigation of the equilibrium states of a fractional-order Morris-Lecar neuronal model.",
"The extension of the methods presented in the first part of the paper to more complicated (higher dimensional) incommensurate order linear fractional order systems represents a direction for future research, possibly leading to more extensive generalizations of the classical Routh-Hurwitz stability conditions.",
"A potential application of such results concerns the analysis of neuronal networks composed of several neurons of Morris-Lecar type."
],
[
"Proof of Proposition ",
"We compute: $^c\\!D^qg(x)&=\\frac{1}{\\Gamma (-q)}\\int \\limits _{0}^{x}(x-t)^{-q-1}[g(t)-g(0)]dt\\\\&=\\frac{1}{\\Gamma (-q)}\\int \\limits _{0}^{x}(x-t)^{-q-1}[f(at)-f(0)]dt\\\\&=\\frac{1}{a\\Gamma (-q)}\\int \\limits _{0}^{ax}\\Big (x-\\frac{s}{a}\\Big )^{-q-1}[f(s)-f(0)]dt\\\\&=\\frac{1}{a\\Gamma (-q)}\\int \\limits _{0}^{ax}a^{q+1}(ax-s)^{-q-1}[f(s)-f(0)]dt\\\\&=\\frac{a^{q}}{\\Gamma (-q)}\\int \\limits _{0}^{ax}(ax-s)^{-q-1}[f(s)-f(0)]dt\\\\&=a^q (^c\\!D^qf)(ax)$ It follows that: $^c\\!D^q g(x)=a^q\\cdot ^c\\!\\!D^qf(ax),\\quad \\textrm {for any }a\\ne 0,$ which completes the proof.$\\Box $"
],
[
"Deduction of the nondimensional system (",
"Starting from system (REF ), we consider the substitutions $v(t)=kV(\\alpha t)\\quad ,\\quad n(t)=N(\\alpha t),$ where $\\alpha $ and $k$ will be deduced in the following.",
"Applying Proposition , we have: $^c\\!D^qv(t)=&k\\cdot ^c\\!D^q[V(\\alpha t)]\\\\=&k\\alpha ^q(^c\\!D^qV)(\\alpha t)\\\\=&k\\alpha ^q\\frac{1}{C_m(q)}\\Big [g_{Ca}M_\\infty (V(\\alpha t))(V_{Ca}-V(\\alpha t))+\\\\&+g_KN(\\alpha t)(V_K-V(\\alpha t))+g_L(V_L-V(\\alpha t))+I\\Big ]\\\\=&k\\alpha ^q\\frac{1}{C_m(q)}\\Big [g_{Ca}M_\\infty \\Big (\\frac{v(t)}{k}\\Big )\\Big (V_{Ca}-\\frac{v(t)}{k}\\Big )+\\\\&+g_K n(t)\\Big (V_K-\\frac{v(t)}{k}\\Big )+g_L\\Big (V_L-\\frac{v(t)}{k}\\Big )+I\\Big ]\\\\=&\\frac{\\alpha ^q}{C_m(q)}\\Big [g_{Ca}m_\\infty (v)(kV_{Ca}-v)+\\\\&+g_Kn(kV_K-v)+g_L(kV_L-v)+kI\\Big ]$ and therefore, it makes sense to choose $k=\\frac{1}{V_{Ca}}$ .",
"Furthermore, with the notations from section 3, we obtain: $^c\\!D^qv(t)=R_m\\Big (\\frac{\\alpha }{\\tau }\\Big )^q\\Big [&g_{Ca}m_\\infty (v)(1-v)+g_K n(v_K-v)+\\\\&+g_L(v_L-v)+\\tilde{I}.$ At this step, it is easy to see that it makes sense to consider $\\alpha =\\tau $ , which leads to: $^c\\!D^qv(t)&=R_m\\Big [g_{Ca}m_\\infty (v)(1-v)+g_K n(v_K-v)+g_L(v_L-v)+\\frac{I}{V_{Ca}}\\Big ]\\\\&=\\gamma _{Ca}m_\\infty (v)(1-v)+\\gamma _K\\cdot n(v_K-v)+\\gamma _L(v_L-v)+\\tilde{I}.$ As for the second equation, applying Proposition , and taking into account that $\\alpha =\\tau $ , we obtain: $^c\\!D^pn(t)&=\\alpha ^p(^c\\!D^p)(\\alpha t)\\\\&=(\\alpha \\overline{\\lambda _N})^p\\lambda (V(\\alpha t))[N_\\infty (V(\\alpha t))-N(\\alpha t)]\\\\&=(\\tau \\overline{\\lambda _N})^p\\lambda \\Big (\\frac{v(t)}{k}\\Big )\\Big [N_\\infty \\Big (\\frac{v(t)}{k}\\Big )-n(t)\\Big ]\\\\&=(\\tau \\overline{\\lambda _N})^p\\cdot \\ell (v)[n_\\infty (v)-n],$ Therefore, the nondimensional system (REF ) is found."
]
] | 1612.05389 |
[
[
"In-situ Adaptive Encoding for Asymmetric Quantum Error Correcting Codes"
],
[
"Abstract We present techniques that improve the performance of asymmetric stabilizer codes in the presence of unital channels with unknown parameters.",
"Our method estimates the channel parameters using information recovered from syndrome measurements during standard stabilizer quantum error correction and adaptively realigns the codespace to minimize the uncorrectable error rate.",
"We find that for dephasing channels parametrized by a single angle our scheme yields lifetimes dominated by the bit-flip error rate for which the asymmetric code has an improved distance.",
"In the case of general unital channels we are able to learn and exploit orientations of the channel that yield a constant improvement to the code lifetime.",
"In both cases, since our method is adaptive and online, we are able to model the effect of drift in the channel parameters."
],
[
"Introduction",
"The last two decades of research in quantum computing have yielded remarkable advances in quantum error correction and fault-tolerant quantum computing.",
"Error correction is a necessary component for any quantum device exposed to an environment which, by necessity of interacting with the quantum system, will introduce noise and decoherence.",
"Error correction is considered successful when the action of the environment is minimized in some way, either by encoding information in a subspace (as is the case with decoherence-free subspace methods [24]), or by actively countering it via recovery operations as is the case with stabilizer quantum error correction [16].",
"An $[[n, k, d]]$ quantum stabilizer code protects $k$ logical qubits by encoding them into $n$ physical qubits.",
"Let ${P}_n$ be the group of all tensor products of Pauli operators $X$ , $Y$ , and $Z$ and the identity $I$ on $n$ qubits (possibly with overall factors of $\\pm 1$ or $\\pm i$ ).",
"The code itself is defined by an Abelian subgroup $S \\subset {P}_n$ such that $S$ is generated by a set of $n-k$ commuting stabilizer generators $g_s $ and $S$ does not contain $-I$ .",
"We say a quantum state ${\\psi }$ is in the codespace of the code if $g_s {\\psi } = {\\psi }$ for all $g_s$ .",
"A quantum state can accumulate an error by interacting with the environment.",
"By measuring all of the stabilizer generators $g_s$ , we can project the system to a state $E {\\psi }$ , where $E$ is an element of ${P}_n$ .",
"The results of these ${n-k}$ measurements collectively form the syndrome of $E$ .",
"Many different errors $E$ can result in the same syndrome and we denote by $d$ the distance of the code if it can unambiguously correct errors of weight $\\lfloor (d-1)/2 \\rfloor $ where the weight of the error is the number of non-identity elements in $E \\in {P}_n$ .",
"For a general noise process, the state may be projected into a mixture of states with the same error syndrome; in a well-designed code, this mixture will be dominated by the most likely error, which is used for the correction.",
"Generally, a stabilizer code is characterized only in terms of the weights of its correctable errors, and not in terms of the specific types of errors produced by a quantum channel.",
"It is natural therefore to consider error correction schemes that are designed to be effective for a specific type of error, rather than for all or errors below a given weight.",
"Many methods for doing this have already been considered including designing new error correction procedures through direct optimization [35], concatenating repetition codes with Calderbank-Shor-Steane (CSS) codes in the presence of dominant dephasing noise [1], and using asymmetric quantum codes to combat asymmetric noise [12], [19], [31], [3], [23], [9], [8].",
"Our focus in this paper will be this last case and we will design a system that exploits asymmetric quantum codes in the presence of sources of noise that exhibit a bias towards one type of error ($X$ ) over the other ($Z$ ).",
"(The reverse bias can be easily dealt with by switching the roles of the $X$ and $Z$ bases.)",
"[Asymmetric quantum stabilizer code [19]] The asymmetric quantum stabilizer code denoted by $[[n, k, d_X/d_Z]]$ encodes $k$ logical qubits into $n$ physical qubits and can correct errors with up to $t_x = \\lfloor (d_x - 1)/2 \\rfloor $ $X$ operators and $t_z = \\lfloor (d_z - 1)/2 \\rfloor $ $Z$ operators.",
"We will use as our test case the $[[15, 1, 7/3]]$ shortened Reed-Muller code [4] which is the smallest asymmetric CSS code, and has the added benefit of allowing transversal non-Clifford operations [7].",
"Another CSS construction [2] uses Bose-Chaudhuri-Hocquenghem (BCH) codes to form a $[[31, 6, 7/5]]$ asymmetric code which we will study in the second part of this work as well.",
"It should be noted that our method is compatible with all asymmetric stabilizer codes, and has the advantage that our feedback control scheme is only a function of the noise channel and the code distances.",
"As such, the analysis contained in this work applies to all codes that express asymmetric $X$ and $Z$ distances (though we only consider CSS codes in our examples)."
],
[
"Quantum channels",
"In general, a quantum operation $\\Lambda $ is any map on the set of bounded positive operators on a Hilbert space that is completely positive i.e.",
": $\\Lambda I_n$ is positive for identity maps of any dimension $n$ .",
"Every such operation can be written in terms of Kraus operators $A_i$ $\\Lambda \\rho = \\sum _i A_i^{\\dag } \\rho A_i.$ Additionally, if the operation preserves the trace, that is to say $\\Lambda \\rho = \\rho $ for any state $\\rho $ , then it is known as a quantum channel.",
"[Unital quantum channel] A unital quantum channel $\\Lambda $ is a completely positive trace-preserving map on quantum states for which the maximally mixed state is a fixed point, that is $\\Lambda (I / d) = I / d$ when $\\Lambda $ acts on states in a Hilbert space of dimension $d$ .",
"A unital channel can equivalently be expressed as the convex combination of unitary channels [28] and for this reason they often appear as the result of a partial trace after a joint unitary evolution over a quantum state and an environment.",
"A common and simple restriction to unital channels is the Pauli channel.",
"[Pauli channel] A Pauli channel is a channel over $n$ qubits that can be written in the form ${E} \\rho = (1-p) \\rho + \\sum _i p_i A_i \\rho A_i$ for $\\sum _i p_i = p$ and $A_i \\in I, X, Y, Z ^{n}$ .",
"In this work we will consider $n$ identical single-qubit channels acting on $n$ distinct qubits.",
"The state of each qubit can be represented as a vector $\\vec{r} \\in ^3$ contained inside the Bloch sphere [26].",
"We write the qubit state $\\rho $ as $\\rho = \\frac{I + \\vec{r} \\cdot \\vec{}}{2}$ where $\\vec{} = X, Y, Z $ .",
"Unital channels that act on single qubits can be described by their action on the vector $\\vec{r}$ $\\Lambda \\rho = \\frac{I + (M \\vec{r}) \\cdot \\vec{}}{2}$ where the Bloch matrix $M$ is any real $3 \\times 3$ matrix that respects the Fujiwara-Algoet conditions [14], [5], [6].",
"We now introduce a subset of unital channels that will be the focus of our analysis.",
"[Oriented Pauli channel] An oriented Pauli channel $M$ is the result of a unitary channel $Q_U$ , followed by a Pauli channel $D$ , and the inverse unitary channel $Q_U^T$ .",
"We are motivated to use this definition due to the fact that any unital channel Bloch matrix $M$ can always be decomposed using the polar decomposition $M = Q_VP$ where $Q_V \\in SO(3)$ and $P$ is positive semi-definite.",
"When the action of the channel can be described entirely by $P$ (i.e.",
": $Q_V=I$ ), we have the result of Lemma found the Appendix, $M = (1-2p) I + 2p Q_U^T \\begin{array}{ccc}k_1 & 0 & 0 \\\\0 & k_2 & 0 \\\\0 & 0 & k_3\\end{array} Q_U .$ where $Q_U \\in SO(3)$ .",
"In this picture, $p$ is the probability of acting non-identically on the qubit and $k_i = p_i / \\sum _j p_j$ where $p_i$ is are the $X$ , $Y$ , and $Z$ error rates in some non-standard basis $Q_U$ .",
"Note also that when $Q_U=I$ and $p<1/2$ , this reduces to a single-qubit Pauli channel.",
"Definition also lends itself a description in terms of CPTP maps.",
"In other words, an oriented Pauli channel represented by the map $\\Lambda _M$ is the result of a unitary channel $\\Lambda _U$ , followed by a Pauli channel $\\Lambda _D$ , and the inverse unitary channel $\\Lambda _{U^{\\dag }}$ .",
"Additionally, we note that the polar decomposition $M = Q_VP$ shows that any general unital channel can be written as the composition of an oriented Pauli channel with a unitary channel $\\Lambda _V$ .",
"Characterizing quantum channels is routinely done by quantum tomography which has yielded powerful methods for faithfully reconstructing process matrices with a tractable number of samples [30], [13].",
"Tomography methods such as compressed sensing require the preparation of resource-intensive randomized quantum states and measurements.",
"A full reconstruction of all the Kraus operators constituting a quantum operation might not be needed for every task however.",
"The goal of more recent work done in modeling error channels has been to allow for efficient simulation of fault-tolerance [20], [17], [29], [25] by approximating physical error models under the Pauli twirl operation.",
"In [11], the authors perform a coarse-grained averaging of the qubits by applying random Clifford operators and qubit permutations which allow for efficient extraction of channel parameters such as the probability of phase-flip errors of any given weight.",
"We specifically consider characterizing and exploiting the asymmetry of a quantum channel in-situ.",
"We will estimate channel parameters by using the existing error correction apparatus, which has the benefit of not causing interruptions to ongoing computations.",
"It also allows us to account for noise channels that may be changing over time.",
"The authors of [10] already do this with great success in the presence of dephasing noise followed by an unknown $X$ -axis rotation.",
"The recent experiments of [21] demonstrate a similar method in a system of nine physical qubits implementing the 5-bit repetition code.",
"Again, the authors are able to compensate for time-varying parameters in the noise channel but without the restriction that the channel, or the corrective controls, be identical on all qubits.",
"Like these works, our goal in learning the channel parameters will be to improve the performance of error correction."
],
[
"Control scheme",
"We now present a model for reducing the rate of uncorrectable errors in two different parameterized noise channels.",
"Our model of feedback control will be to apply a coherent rotation $\\hat{U}_t$ to the physical qubits and to modify our stabilizer measurements by the same unitary, as shown in Figure REF .",
"Specifically, for stabilizer generators $g_s$ , $g_s \\rightarrow \\hat{U}_t ^{n} g_s \\hat{U}_t^{\\dag } ^{n} .$ In the single-parameter case we describe below, we will use this construction to effectively rotate a dephasing channel into a bit-flip channel.",
"Later, when we examine a multi-parameter model, this same construction will counteract the unitary rotation $\\Lambda _U$ of an oriented Pauli channel.",
"Figure: Our model of feedback control for optimal use of the asymmetric code uses a real-time estimate of the parametrized noise channel Λ θ \\Lambda _{\\theta } to modify the codespace and stabilizers.",
"(A) We rotate all qubits in the codespace by U ^ t \\hat{U}_t.",
"(B) Stabilizer error correction using the modified stabilizers of Eq. ().",
"(C) The estimator updates the form of the stabilizer measurements.",
"Following the syndrome extraction step, the estimator also updates the parameter estimate, symbolized by the two directions of the flow of information.",
"Quantum information and classical information are distinguished in this circuit with single and double lines, respectively.",
"We note, however, that the classical information of θ ^ t \\hat{\\theta }_t is actually a complete encoding of the estimator of θ t \\theta _t.",
"This can mean, as it does in our case, that the estimator is encoded as a point-wise approximation of the Bayesian prior distribution of θ t \\theta _t.",
"Additionally, A, B, C, are error-free.In this paper we will approximate the rate of uncorrectable errors by counting all errors whose weight exceeds the minimum number of correctable errors of the code.",
"(In practice, it may be possible to improve on our results by considering correctable errors of all weights, since most codes can correct many higher-weight errors as well.)",
"Thus, the uncorrectable error rate for an $[[n, k, d_X/d_Z]]$ asymmetric code can be found by summing the probability of all possible errors that violate the distances $d_X$ and $d_Z$ yet act on the fewest number of qubits.",
"We will denote by ${T}$ the set of error weights which violate the constraints in this way.",
"The individual qubit errors can occur on any of the $n$ qubits, which we account for by the multinomial coefficient ${n w_x, w_y, w_z} = \\frac{n!}{(n-w_x-w_y-w_z)!",
"w_x!",
"w_y!",
"w_z!",
"}$ where $w_x+w_y+w_z = w \\le n$ .",
"Altogether, $p_{\\text{fail}} = \\sum _{{T}} {n w_x, w_y, w_z} p_x^{w_x}p_y^{w_y}p_z^{w_z} (1-p)^{n-w}$ where ${T} = w_x, w_y, w_z \\; : \\; w_x+w_y \\le t_x, w_z+w_y \\le t_z $ , $t_i = \\lfloor (d_i - 1)/2\\rfloor $ and $p=p_x+p_y+p_z$ ."
],
[
"Single-parameter estimation",
"We begin with an oriented Pauli channel $\\Lambda _{\\theta }$ which is a single-parameter generalization of dephasing noise.",
"The action of the channel on one qubit can be described by a single Kraus operator, $A \\theta $ $\\Lambda _{\\theta } \\rho = 1 - p \\rho + p A \\theta \\rho A \\theta .$ where $A(\\theta ) = e^{-i \\theta Y} Z e^{i \\theta Y} .$ We do not know a priori the parameter $\\theta $ and thus the effective channel, as perceived by our measurements of the stabilizer generators, is equivalent to that of $\\Lambda _{\\theta }$ under the Pauli twirl approximation [20]: ${P} \\Lambda _{\\theta } \\rho = 1 - p \\rho + p \\cos ^2 \\theta X \\rho X + p \\sin ^2 \\theta Z \\rho Z.$ This same expression can easily be derived by noting that $e^{-i\\theta Y} Z e^{i\\theta Y} = Z \\sin (2\\theta ) + X \\cos (2\\theta )$ then expanding the formula for $\\Lambda _{\\theta }$ and measuring the $X$ and $Z$ operators.",
"The goal of the protocol we describe below is to recover the parameter $\\theta $ and substitute for the stabilizer operators in the standard basis $g_s$ new stabilizers that are aligned to exploit the asymmetry of the noise channel and maximize the lifetime of the code.",
"This means applying $\\hat{U}_t = e^{-i \\hat{\\theta } Y}$ to the stabilizers and to the code space (by physically applying a rotation to each qubit of the codeword), where $\\hat{\\theta }$ is our estimate of $\\theta $ recovered after each round of error correction."
],
[
"Fixed dephasing angle",
"Our first case, and the one where we will demonstrate the greatest gain in code lifetime, is when the angle $\\theta $ is fixed in time.",
"We use a Bayesian estimator [33] that begins with uniform belief about the orientation channel $P_0(\\theta ; \\theta _0) = 1 / \\pi $ for $\\theta \\in [0, \\pi )$ .",
"Every round of stabilizer error correction will diagnose and correct a total of $w_x$ bit-flip errors and $w_z$ phase-flip errors.",
"Our knowledge of the dephasing angle at time $t+1$ includes information learned from the error weights, the previous configuration of the estimator, and the true value of $\\theta _0$ .",
"Thus, we can express our belief about $\\theta $ at time $t+1$ using Bayes' rule: Pt+1(| wx, wz; t, 0) = Pt+1(wx, wz | ; t, 0) Pt+1(; t, 0)Pt+1(wx, wz) = Pt+1(wx, wz | ; t, 0) Pt+1(; t, 0)Pt+1(wx, wz | ; t, 0) Pt+1(; t, 0) d where $\\theta $ is the random variable resulting from the update, $\\hat{\\theta }_t$ is the angle to which the stabilizers were configured in the previous round of error correction, and Pt+1(wx, wz | ; t, 0) = n wx, wz pwx + wz (1-p)n - wx - wz 2wx - t - 0 2wz - t - 0 .",
"After each update, we choose the new alignment of the stabilizers to be the maximum likelihood estimator of $\\theta $ : $\\hat{\\theta }_t = \\underset{\\theta }{\\mathrm {argmax}} \\;\\; P_t(\\theta | w_x, w_z; \\hat{\\theta }_{t-1}, \\theta _0) .$ This update rule has a particular advantage we can exploit to reduce the complexity of our estimator going forward.",
"Note that when $w_x \\ge 0$ and $w_z = 0$ , $\\hat{\\theta }_t = \\hat{\\theta }_{t-1}$ , which follows from the fact that $\\cos ^2 \\theta - \\hat{\\theta }_t - \\theta _0 $ has a single maximum on the interval $0, \\pi $ and is symmetric around it.",
"Thus, we can perform our updates to the distribution of $\\theta $ in bulk whenever we encounter a correctable $Z$ error by multiplying the distribution by the function ft+1(, nx; t, 0) = 2nx - t - 0 2 - t - 0 .",
"where $n_x$ is the number of $X$ errors observed since the previous update.",
"Each such step is followed by a renormalization of the posterior probability distribution $P_{t+1}(\\theta | w_x, w_z; \\hat{\\theta }_t, \\theta _0)$ .",
"This yields an advantage for our numerical simulations as well.",
"Since we need only update $\\hat{\\theta }_t$ on single $Z$ errors (as the $[[15, 1, 7/3]]$ code fails for $w_z > 1$ ), we can first sample the time until the next $Z$ error, then retroactively sample the number of $X$ errors to have occurred in the intermediate times, all the while accounting for the possibility of uncorrectable $X$ errors.",
"This proves especially useful when our estimator is close to the optimal value $\\theta $ .",
"In this case, $Z$ errors happen very infrequently, and the error rate is dominated by $O(p^4)$ terms, leading to many idle cycles of the simulation.",
"An important complication is that the distribution $P_t(\\theta )$ cannot be stored in memory exactly and must be discretized.",
"Let $P_t(\\theta )$ be defined for the midpoints of the cells $0, \\pi = \\bigcup _{j=1}^{N-1} \\frac{j \\pi }{N}, \\frac{(j+1) \\pi }{N} .$ Thus, the difference between the true value $\\theta _0$ and the maximum likelihood estimate $\\hat{\\theta }_t$ cannot necessarily be reduced to zero even at very long times, but may be as large as $\\pi /2N$ .",
"This means that the rate of $Z$ errors can be, at best, suppressed to $p\\sin ^2(\\pi /2N)$ .",
"Recall, however, that we do not need fully suppress $Z$ errors in order to take advantage of an asymmetric code.",
"For very low effective $Z$ error-rates, the uncorrectable error rate in Eq.",
"(REF ) is dominated by uncorrectable $X$ errors.",
"If $p_z = cp_x^{t_x/t_z}$ for some $c > 0$ , then $p_{\\text{fail}} = c{n t_z+1} + {n t_x+1} p_x^{t_x} + O p_x^{t_x+1} ,$ where $t_x=3$ and $t_z=1$ for the $[[15, 1, 7/3]]$ code.",
"We can expect this very low rate only when $p_z = c p_x^2$ which is possible when $\\sin ^2(\\pi /2N) = O(p)$ or $N = O(1/p)$ .",
"Thus, we can only make optimal use of our estimate of $\\theta $ when we take at least $N=O(1/p)$ partitions of $0, \\pi $ .",
"The results of our simulations in Figure REF show that for a fixed dephasing angle, our technique not only improves the code lifetime by a constant factor, but effectively increases the code distance.",
"The mean lifetime of our adaptive $[[15, 1, 7/3]]$ code agrees with a power law of $p_{\\text{fail}} = Op^{3.99} $ yielding an “effective” code distance of $6.98$ .",
"The next smallest CSS code that could yield a similar scaling is the $[[23, 1, 7]]$ code, but our adaptive technique outperforms even this code by a constant factor, thanks to the smaller number of physical qubits used."
],
[
"Drifting dephasing angle",
"We now address the version of the previous problem with a parameter $\\theta $ that is drifting in time.",
"Consider a dephasing angle that at each time-step evolves via random Brownian motion: t+1 = t + u with u N(0, 2).",
"We can incorporate our knowledge of this drift into the estimator from the fixed angle case by convolving, at each time-step, our belief about $P_t(\\theta _t)$ with the distribution of one step of the Brownian motion.",
"That is, following every update in Eq.",
"(REF ) we also apply $P_{t+1}^{\\prime }(\\theta _{t+1}) = P_{t+1}(\\theta _{t+1}) * \\frac{1}{\\kappa } \\exp ^{-\\theta _{t+1}^2 / 2 \\kappa ^2} ,$ where $*$ denotes a convolution of the two functions defined as $f(t) * g(t) = \\int f(s)g(t-s)ds.$ We then take $\\hat{\\theta }_{t+1}$ to be the maximum likelihood estimate with respect to $P_{t+1}^{\\prime }(\\theta _{t+1})$ instead.",
"Unlike in our simulations of the fixed angle case, we cannot make use of the retroactive $X$ error sampling that allowed us to simulate code lifetimes up to $10^{17}$ cycles.",
"We must therefore simulate every update straightforwardly.",
"However, as Figure REF demonstrates, our estimator retains the property that the maximum likelihood estimate $\\hat{\\theta }_t$ only moves following $Z$ errors.",
"Figure: Performance of the adaptive stabilizers for the [[15,1,7/3]][[15, 1, 7/3]] shortened Reed-Muller code in the presence of one-parameter generalized dephasing noise.",
"(A) Mean lifetime of 5000 samples (100 samples for p=10 -5 p=10^{-5}) measured in terms of error correction cycles.",
"(B) Mean lifetime of 200 samples in the presence of drifting dephasing noise (κ 2 =0.01\\kappa ^2=0.01) measured in terms of error correction cycles.",
"(C) Mode of the lifetime of the [[15,1,7/3]][[15, 1, 7/3]] code without adaptive stabilizers.",
"(D) Optimal lifetime of the [[15,1,7/3]][[15, 1, 7/3]] code given perfect a priori knowledge of θ\\theta .",
"(E) Expected lifetime of the [[23,1,7]][[23, 1, 7]] code.",
"Error bars in this plot are sufficiently small to be disregarded.Figure: One run of our Bayesian estimator for θ\\theta drifting according to Brownian motion κ 2 =0.03\\kappa ^2 = 0.03 and an error rate p=0.003p=0.003.Figure REF shows that using this modified estimator still yields a constant factor improvement to the lifetime of the $[[15, 1, 7/3]]$ code, but the lifetimes no longer scale with $O(1/p^4)$ , the rate of uncorrectable $X$ errors.",
"Instead, the drift in the channel effectively causes a constant but suppressed $Z$ error rate.",
"This is because the the drift in $\\theta _t$ implies that we cannot stay in the optimal configuration for very long.",
"This fact is reflected in our estimator by the convolution step, which widens any otherwise “sharp\" (i.e.",
": certain) belief about $\\hat{\\theta }_t$ .",
"From our simulations, the constant factor gain to the code lifetime for $\\kappa ^2=0.01$ is $6.33$ ."
],
[
"Multi-parameter estimation",
"Although the single-angle dephasing model from the previous section is useful for studying the limits to which we can exploit a code's asymmetry, it is not very general, and we now turn our attention to the case of oriented Pauli channels.",
"Recall that the Bloch matrix for such a channel can be written as a contraction in some non-standard basis: $M = (1-2p) I + 2p Q^T \\begin{array}{ccc}k_1 & 0 & 0 \\\\0 & k_2 & 0 \\\\0 & 0 & k_3\\end{array} Q ,$ for some $Q \\in SO(3)$ and its transpose $Q^T$ .",
"We choose this form for the Bloch matrix in order to highlight that our true objective in estimating $M$ is to find a rotation of the codespace and stabilizers such that the channel appears Pauli and allows for optimal use of the asymmetric code.",
"Let $k_i$ be the eccentricities of the oriented Pauli channel such that $k_1+k_2+k_3=1$ , $p$ be the total error rate, and $Q$ be the orientation of the channel.",
"Note that this channel, unlike the one in the previous section, can give rise to $Y$ errors.",
"These can be corrected by any CSS code as simultaneous $X$ and $Z$ errors on the same qubit.",
"Finally, we let $A$ be the matrix such that $M = (1-2p)I + 2pA$ .",
"Note that we do not need to know $M$ to design the control unitary $\\hat{U}_t$ for this scheme; knowledge of $A$ is sufficient to align the largest value of $k_i$ with the $X$ axis.",
"In other words, the optimal choice of $\\hat{U}_t$ does not depend on the total error rate $p$ (although estimating $p$ can also be done efficiently [15]).",
"If we let $\\hat{Q}_t$ be the real orthonormal matrix associated with the unitary $\\hat{U}_t$ , then the effect of this counter-rotation on the Bloch matrix is $M = (1-2p) I + 2p \\hat{Q}_t A \\hat{Q}_t^T$ , and it is clear that we should choose $\\hat{Q}_t$ to diagonalize $A$ and order the eigenvalues such that $k_1 \\ge k_3 \\ge k_2$ .",
"This last details ensures that $Z$ errors are favored over $Y$ errors and $X$ errors are favored over both.",
"The performance of asymmetric codes against oriented Pauli channels with all $k_i > 0$ is limited, since even with optimal orientation, they exhibit non-zero $Z$ and $Y$ error rates.",
"These channels cannot yield improvements to the effective code distance like those in the dephasing case, but they can yield improvements to the coefficient of the uncorrectable error rate, and thereby to the code lifetime.",
"Let $C_{\\text{opt}}$ be the constant factor improvement to the lifetime yielded by the $[[15, 1, 7/3]]$ code for an oriented Pauli channel, given perfect a priori knowledge of the optimal orientation $Q_{\\text{opt}}$ .",
"$C_{\\text{opt}}$ then is given by the ratio of the expected lifetimes $C_{\\text{opt}} \\simeq \\frac{t(Q_{\\text{opt}}) }{\\int t(Q) dQ}$ where $t(Q) $ is the lifetime at orientation $Q$ given by $1/p_{\\text{fail}}(Q)$ , the uncorrectable error rate.",
"We take $\\int dQ$ to be an integral over the Haar measure for $SO(3)$ [32].",
"For $k_1, k_2, k_3 \\gg p$ , the uncorrectable error is dominated by $ZZ$ , $YY$ , and $ZY$ errors.",
"The approximation becomes pfail(Q) = 15 0, 2, 0 py2 + 15 0, 0, 2 pz2 + 15 0, 1, 1 pypz , = 105(py + pz)2 , = 105(p-px)2 .",
"The $X$ error rate is $p_x = p \\vec{e}_1^T QDQ^T \\vec{e}_1$ where $\\vec{e}_i$ are the standard basis vectors and $D$ is the diagonal matrix with diagonal elements $k_i$ .",
"Note that for $Q$ with columns $\\vec{q}_i$ , the $X$ error rate reduces to $p \\vec{q}_1^TD\\vec{q}_1$ , where $\\vec{q}_1$ follows a uniform distribution on $SO(3)$ given by $\\vec{q} = \\begin{array}{c}u \\\\\\sqrt{1-u^2} \\sin 2 \\pi v \\\\\\sqrt{1-u^2} \\cos 2 \\pi v \\end{array} \\hspace{18.06749pt}\\begin{array}{l}u \\sim \\text{unif} -1, 1 , \\\\v \\sim \\text{unif} 0, 1 .\\end{array} $ We can thus derive the distribution on $k_x$ : $k_x = k_1u^2 + k_2(1-u^2)\\sin ^2 2 \\pi v + k_3(1-u^2)\\cos ^2 2 \\pi v .$ To take the Haar integral needed for the average lifetime, we must calculate t(Q) dQ = -11 01 1105p2(1-kx)2 dv du , -11 1105p2 1-k1u2 - k3(1-u2) 2 du , = -11 1105p2 (1-k3)-(k1-k3)u2 2 du , = -1 (k1-k3)/(1-k3) 210p2(1-k3)(1-k3)(k1-k3) + 1210p2(1-k3)(1-k1) .",
"where in the second line we've used that $k_3 \\le k_2$ .",
"For the channel of eccentricities $(0.7, 0.2, 0.1)$ , the optimal and average lifetimes become t(Qopt) 11.11105p2 , t(Q) dQ 2.71105p2 .",
"Thus, the optimal improvement to the code lifetime that we can expect is $C_{\\text{opt}} \\gtrsim 4.01$ .",
"A similar calculation for the $[[31, 6, 7/5]]$ code yields $C_{\\text{opt}} \\gtrsim 9.34$ ."
],
[
"Non-degenerate grids for Bayesian inference",
"We again will use Bayesian inference to estimate the relevant parameters of the matrix $A$ .",
"Note, however, that were we to learn the rates of $X$ , $Y$ and $Z$ errors, we could only determine the diagonal elements of matrix $A$ .",
"This is a problem acknowledged by [27] and [10].",
"In [27] the authors propose “toggling\" the codespace to estimate off-diagonal elements of the Bloch matrix by rotating the codespace and pre-processing the encoded state.",
"Although their method is applicable to much more general multi-qubit process matrices, it cannot be performed in-situ without incurring significant costs by moving the codespace away from the optimum.",
"The solution we propose is to sample the space of Bloch matrices randomly, and track the posterior probabilities for a field of $N$ sampling points in the parameter space of $A$ .",
"This method of approximating the likelihood function is similar to our point-wise estimate in section REF which was supported on regularly-spaced sampling points in $$ .",
"We sample each point $X$ in our randomized grid as follows.",
"First, we will sample eccentricities $x_1$ , $x_2$ , $x_3$ according to the distribution x1 = 1 , for x1 0, 1 , x2 |x1 = 1/(1-x1) , for x2 0, 1-x1 , and $x_3 = 1 - x_1 - x_2$ .",
"We take $D_X$ to be the diagonal matrix with diagonal elements $x_1$ , $x_2$ , and $x_3$ .",
"Note that the average channel in our grid will have eccentricities $(1/2, 1/4, 1/4)$ , which means that our prior distribution assumes some asymmetry in the channel.",
"Next, we sample an orthonormal basis $Q_X$ from the Haar measure over $SO(3)$ (for which efficient methods exist [22]).",
"Finally, we let our sampling point be $X = Q_X^T D_X Q_X .$ We now relate the number of sampling points $N$ to the optimal performance of our asymmetric codes.",
"We do this in two steps: first, by bounding the minimum distance of a fixed channel $A$ to the closest element in a randomized grid, and second, by bounding the rate of uncorrectable errors using the minimum distance of the grid.",
"The first step is summarized in the following Lemma, the proof of which can be found in the Appendix.",
"[Number of sampling points versus minimum distance] Let $X_i$ be $N$ i.i.d.",
"copies of randomly oriented Pauli channels according to (REF ), and let $A$ be a fixed channel with eccentricities $(a_1, a_2, a_3)$ .",
"Then we have that $\\Pr Z < \\ge 1 - 1 - \\frac{^5}{2^{12}\\sqrt{2}3^3 a_1^2a_2a_3} ^N,$ where $Z=\\min _i \\left\\Vert X_i - A \\right\\Vert _2$ and $\\left\\Vert Y \\right\\Vert _2 = \\sqrt{Y^T Y }$ is the Frobenius norm.",
"For $\\ll 1$ the bound in Lemma REF scales as $\\Pr \\min _i \\left\\Vert X_i - A \\right\\Vert _2 < \\ge N \\frac{^5}{2^{12}\\sqrt{2}3^3 a_1^2a_2a_3}.$ Thus, to ensure that a channel in our random grid approximates $A$ to precision $$ in the Frobenius norm with probability $p_{A}$ , we need to populate our random grid with a very large number of sampling points: $N = \\frac{2^{12}\\sqrt{2}3^3 a_1^2a_2a_3}{^5p_{A}}.$ We provide this estimate for the explicit purpose of characterizing the computational resources required to implement our scheme.",
"If the adaptive encoding procedure outlined in this text is to be implemented on classical computing architecture operating in the very low-latency setting of stabilizer quantum error correction, then it must be possible to achieve the gains we describe without unreasonable overhead.",
"We can also use this grid size spacing to estimate code performance.",
"For the oriented Pauli channel with eccentricities $k_1 = 0.7$ , $k_2 = 0.2$ , $k_3 = 0.1$ , and with $30,000$ sampling points, we can expect $\\Pr Z < \\ge 6.92 ^5.$ The estimate we found for the code performance in Eq.",
"(REF ) assumes complete and exact knowledge of the channel matrix $A$ .",
"What we've found in the derivation above, however, is a bound on our ability to learn an approximation $X$ of the matrix $A$ .",
"As such, the lifetime of our code in practice will not be bounded by the optimal performance of Eq.",
"(REF ), but rather by the quality of this approximation.",
"We now derive the relationship between the Frobenius norm distance between $X$ and $A$ and the code performance.",
"Let $x_{11}$ be the first diagonal element of $X$ .",
"Then we know that | x11 - k1 | X - A 2 | 1- x11 | | 1 - k1 | + X - A 2 .",
"We identify the lifetime coefficient $C$ as the ratio of $p_{\\text{fail}}$ for random and optimal orientations, which can be bounded from below by $C \\ge \\frac{(2/3)^{t_z}}{(1-k_1 + \\left\\Vert X - A \\right\\Vert _2)^{t_z}}.$ We call this lower bound the expected code performance, and will make use of it as a baseline against which to compare the numerical results of the following section."
],
[
"Numerical results",
"In Figures REF and REF we graph the results of 1000 trials of our adaptive estimation and error correction method for the $[[15, 1, 7/3]]$ and the $[[ 31, 6, 7/5 ]]$ codes, at $N=30,000$ sampling points.",
"We also plot the expected code performance as a function of the Frobenius norm using our bound derived above.",
"Recall that the expected code performance serves as a lower bound, and our results indicate that it is reasonably tight.",
"When our adaptive technique is applied, the distribution of lifetimes for the $[[31, 6, 7/5]]$ code changes from the usual geometric distribution to a more uniform distribution with a high proportion of trials lasting from 2 to 10 times the expected unadaptive lifetime.",
"We explain this behavior in the following way: the 31-qubit code fails at a rate of $p^3$ for weight 3 $Z$ errors, and thus the average number of updates to the posterior at the expected time of failure scales with $1/p^3$ .",
"The 15-qubit code, on the other hand, fails at a rate of $p^2$ and so has, on average, learned a worse estimate of $A$ than the 31-qubit code at the expected time of failure.",
"This suggests that the code distance has an additional effect beyond simply decreasing the baseline rate of uncorrectable errors.",
"The data in Figures REF and REF suggest that larger codes benefit proportionally more from our adaptive technique, as they are able to learn the channel parameters more accurately before the probability of an uncorrectable error becomes appreciably high.",
"Figure: Performance (as defined in Eq.",
"()) of the adaptive stabilizers for the [[15,1,7/3]][[15, 1, 7/3]] shortened Reed-Muller code with 30,00030,000 sampling points in the presence of a fixed unital channel of eccentricities (0.7,0.2,0.1)(0.7, 0.2, 0.1) and an unknown orientation.",
"300,000300,000 error correction simulations were performed to obtain this graph.",
"In each trial, the adaptive correction procedure is performed until an uncorrectable error is encountered.",
"The MLE (maximum likelihood estimator) error is taken as the Frobenius distance between the Bloch matrix of the channel given by the MLE and the target matrix at the time of the uncorrectable error.",
"For each error rate, the size of the marker represents the number of trials that yielded the given normalized lifetime.",
"The distribution of both of these trial properties is also summarized in the top and left histograms.Figure: Performance (as defined in Eq.",
"()) of the adaptive stabilizers for the [[31,6,7/5]][[ 31, 6, 7/5 ]] code with 30,00030,000 sampling points in the presence of a fixed unital channel of eccentricities (0.7,0.2,0.1)(0.7, 0.2, 0.1) and an unknown orientation.",
"The xx and yy axes, as well as the size of the plotted points, are defined as in Figure .In Table REF we summarize the improvement to the lifetime and the Frobenius error at the time of failure of each code.",
"In Figure REF we illustrate that the minimum Frobenius distance of a random channel does indeed follow a power law of $N^{1/5}$ , but our bound from Eq.",
"(REF ) overestimates the true grid spacing by a significant factor.",
"In the same figure we demonstrate the decay of the improvement to the code lifetime with decreasing number of sampling points $N$ .",
"Surprisingly, our method still yields some benefit even with only 10 sampling points.",
"This is likely due to the fact that when sampling random channels and random grid points, the probability of generating a grid point close to $A$ is still high enough that some trials have this property and inherit disproportionately long lifetimes, thereby raising the average.",
"Table: Average gains made to the code lifetime from numerical simulations of 1000 samples at each error rate.",
"The Frobenius error is calculated at the time of code failure.Figure: Effect of number of sampling points on lifetime factor for the [[15,1,7/3]][[15, 1, 7/3]] and [[31,6,7/5]][[31, 6, 7/5]] codes, and on the average Frobenius norm distance to the nearest grid element, for a random channel of eccentricity (0.7,0.2,0.1)(0.7, 0.2, 0.1).",
"The dotted lines represent the standard error around the mean."
],
[
"Discussion",
"By adaptively changing the codespace and stabilizers of asymmetric error correcting codes, we've demonstrated an in-situ method for boosting their performance in the presence of asymmetric noise channels.",
"Our method provides a constant factor improvement to the code lifetime in all cases, and in the case of fixed dephasing noise along an unknown axis even results in a higher “effective\" code distance.",
"Since our control unitaries on the physical qubits are functions only of the syndrome statistics, our method is also able to track noise parameters that drift in time.",
"It is interesting to note that in the case of oriented Pauli channels, the performance of the code is dominated by our ability to learn the direction of the largest channel eccentricity.",
"This allows us to align our code in a way that matches this eccentricity to $X$ errors, which our code is best equipped to correct.",
"The gains we make by learning the orientation of the second largest eccentricity are much less significant, since the code's ability to correct $Y$ errors is only slightly worse than for $Z$ errors.",
"In addition to our method of adaptively rotating the codespace and stabilizers, one might also consider re-encoding the logical quantum state as information about the noise channel becomes more refined.",
"For example, if the noise channel is discovered to be highly asymmetric for a sustained period of time, then one could convert the stabilizer code into one that tolerates fewer $Z$ and $Y$ errors and thereby save on the total number of physical qubits and stabilizer measurements.",
"Several works [34], [4], [18] already demonstrate how to convert between these codes fault-tolerantly.",
"Furthermore, one might consider correcting for some types of errors less often, as is done in [12] in the presence of a strongly biased channel.",
"It would seem that our method could also extend to unital channels in general.",
"In this case, we would model the process matrix $M$ not with an eigenvalue decomposition but with a singular value decomposition and our task would be to estimate both the left and right bases of $M$ .",
"Equivalently, we could retain our oriented Pauli channel estimator and concatenate it with another unitary channel.",
"This is similar to [10] where the authors consider a Pauli channel in the standard basis concatenated with a unitary rotation.",
"In both cases, a straightforward application of our randomized grid would incur a further $^3$ scaling to the bound on the number of sampling points in Eq.",
"(REF ), significantly driving up the minimum number of sampling points needed to yield lifetime improvements.",
"It might also be possible to go beyond unital channels, but this might require new methods to estimate the “displacement” of the noise channel.",
"Finally, a very natural extension of our method would be to consider separate noise channels and separate control unitaries for each qubit in the code.",
"This case could be addressed by simply running parallel channel estimators for each qubit, with updates applied only for errors that occur on that qubit.",
"Note that in this case the learning rate drops by $1/n$ on average, and each estimator converges much more slowly.",
"However, it is likely that the channels would be correlated based on the qubit topology, an assumption which could be used to decrease the number of free parameters needed.",
"The authors acknowledge useful conversations with Kung-Chuan Hsu, José Raul Gonzalez Alonso, Shengshi Pang and Christopher Cantwell.",
"This research was supported in part by the ARO MURI under Grant No.",
"W911NF-11-1-0268; NSF Grant No.",
"CCF-1421078; and an IBM Einstein Fellowship at the Institute for Advanced Study."
],
[
"Appendix",
"[Oriented Pauli channels as contractions] Given the oriented Pauli channel $\\Lambda _M$ (Def.",
"), we can always write the process matrix $M$ as $M = (1-2p) I + 2p Q^T_U \\begin{array}{ccc}k_1 & 0 & 0 \\\\0 & k_2 & 0 \\\\0 & 0 & k_3\\end{array} Q_U.$ Recall that $\\Lambda _M = \\Lambda _{U^{\\dag }} \\circ \\Lambda _D \\circ \\Lambda _{U}$ .",
"Consider the action of $\\Lambda _D$ on a qubit state: D() = (1-p) + px X X + py Y Y + pz Z Z = (1-p) I + r 2 + px I + Xr X2 + py I + Yr Y2 + pz I + Zr Z2 = I2 + (1-p) r 2 + (px - py - pz) r1 x2 + (-px + py - pz) r2 y2 + (-px - py + pz) r3 z2 = I2 + (1-2p) r 2 + 2pxr1 x + 2pyr2 y + 2pzr3 z2 = I + A r 2, where $A = (1-2p) I + 2p \\begin{array}{ccc}k_1 & 0 & 0 \\\\0 & k_2 & 0 \\\\0 & 0 & k_3\\end{array} $ where $p_x = pk_1$ , $p_y=pk_2$ and $p_z=pk_3$ .",
"It remains only to identify that $\\Lambda _U$ and $\\Lambda _{U^{\\dag }}$ act as transformations $Q_U$ and $Q_{U^{\\dag }}$ on $\\vec{r}$ before and after the channel $\\Lambda _D$ to complete the proof.",
"Proof of Lemma 1 1][Number of sampling points versus minimum distance] Let $X_i$ be $N$ i.i.d.",
"copies of randomly oriented Pauli channels according to (REF ), and let $A$ be a fixed channel with eccentricities $(a_1, a_2, a_3)$ .",
"Then we have that $\\Pr Z < \\ge 1 - 1 - \\frac{^5}{2^{12}\\sqrt{2}3^3 a_1^2a_2a_3} ^N,$ where $Z=\\min _i \\left\\Vert X_i - A \\right\\Vert _2$ and $\\left\\Vert Y \\right\\Vert _2 = \\sqrt{Y^T Y }$ is the Frobenius norm.",
"Consider the variable $Z \\sim \\min _i \\left\\Vert X_i - A \\right\\Vert _2$ .",
"The cumulative distribution function of $Z$ is Z < = 1 - i 1 - Xi - A 2 < = 1 - 1 - X - A 2 < N , and let (X) = X - A 2 = QATXQA - DA 2 = QXATDXQXA - DA 2, where in the second line we've conjugated both matrices by $Q_A$ and $Q_{XA} = Q_XQ_A$ is also Haar random [32], since the Haar measure is invariant under left or right conjugation.",
"Let $E_D = \\left| D_X - D_A \\right| \\le _D $ where $\\left| \\cdot \\right|$ is the absolute value, i.e., the event in which the eigenvalues of $X$ and $A$ are similar.",
"Conditioning on this event we see that, (X) = (X) | ED ED + (X) | ED ED (X) | ED ED .",
"Thus, we can consider only those channels $X$ for which $E_D$ is true.",
"Let $\\lbrace \\vec{q}_i\\rbrace $ be the column vectors of $Q_{XA}$ , and let $x_i$ and $a_i$ be the diagonals of $D_X$ and $D_A$ , respectively.",
"From $E_D$ we know that $|x_i - a_i| \\le _D$ .",
"The Frobenius norm can then be simplified to be QXATDXQXA - DA 2 = i=13 xi qi qiT - ai ei eiT 2 i=13 xi qi qiT - ai ei eiT 2 i=13 ai qi qiT - ei eiT 2 + D i=13 2 ai qi - ei 2 + D .",
"where in the second line we've used the convexity of the Frobenius norm, in the third line we've used previous our bounds on the distance between $x_i$ and $a_i$ , and in the fourth line we've used the result from Lemma .",
"Let $A_i = \\sqrt{2} a_i \\left\\Vert \\vec{q}_i - \\vec{e}_i \\right\\Vert _2 + _D < /3 $ .",
"Then we must have (X) < | ED A1 A2 A3 = A1 A2 | A1 A3 | A1 A2 .",
"Note that the marginal distribution of each $\\vec{q}_i$ is uniform over the unit sphere and can be constructed using [22], $\\vec{q} = \\begin{array}{c}u \\\\\\sqrt{1-u^2} \\sin 2 \\pi v \\\\\\sqrt{1-u^2} \\cos 2 \\pi v \\end{array} \\hspace{10.84006pt}\\begin{array}{l}u \\sim \\text{unif} -1, 1 , \\\\v \\sim \\text{unif} 0, 1 .\\end{array} $ Thus, for $\\vec{q}_1$ to lie within $$ of $\\vec{e}_1$ in the Frobenius norm, we need only that $u \\in 1 - ^2/2, 1 $ , and for $u$ uniformly distributed over $-1, 1 $ this means that $\\Pr \\left\\Vert \\vec{q}_1 - \\vec{e}_1 \\right\\Vert _2 \\le = ^2/2 .$ Next, the vector $\\vec{q}_2$ is sampled uniformly from the equator of vectors orthogonal to $\\vec{q}_1$ .",
"Given that $\\left\\Vert \\vec{q}_1 - \\vec{e}_1 \\right\\Vert _2 \\le $ , we are guaranteed that $\\vec{e}_2$ lies in the band of latitude $$ around this equator.",
"Thus, the probability of $\\vec{q}_2$ lying within $$ of $\\vec{e}_2$ is given by the ratio of the area of the circle of radius $$ on the unit sphere to that of the band.",
"For $\\ll 1$ this is, $\\Pr \\left\\Vert \\vec{q}_2 - \\vec{e}_2 \\right\\Vert _2 \\le = \\frac{\\int _0^{} 2\\pi r dr}{\\int _{1 - }^{1 + } 2\\pi r dr} = \\frac{}{2} .",
"\\\\$ Finally, given that $\\vec{q}_1$ and $\\vec{q}_2$ each lie within $$ of $\\vec{e}_1$ and $\\vec{e}_2$ respectively, the vector $\\vec{q}_3$ lies within $$ of $\\vec{e}_3$ with probability $1/4$ .",
"Altogether, q1 - e1 2 = 2/2 A1 = 18a12/3 - D 2, q2 - e2 2 = /2 A2 | A1 = 14a2 /3 - D , q3 - e3 2 = 1/4 A3 | A1 A2 = 12a3 1/4 - D .",
"If we choose $_D = /6$ , then we have altogether that $\\Pr \\mu (X) < | E_D \\ge \\frac{^3}{2^{10}\\sqrt{2}3^3 a_1^2a_2a_3}.$ It now remains to bound $\\Pr E_D $ to finish the proof.",
"The event $E_D$ is characterized by the probabilities, | X1 - a1 | , | X2 - a2 | | | X1 - a1 | .",
"Finally, if $|X_1 - a_1| \\le $ and $|X_2 - a_2 | \\le $ , then we are guaranteed that $|X_3 - a_3| \\le 2$ .",
"Putting this all together, we can use the chain rule to see that $\\Pr \\left| D_X - D_A \\right| \\le \\ge (/2)^2 .$ Thus, we have our bound on the spacing of a random channel $X$ : $\\Pr \\mu (X) < \\ge \\frac{^5}{2^{10}\\sqrt{2}3^3 a_1^2a_2a_3}.$ In our bound on the minimum distance Eq.",
"() this becomes $\\Pr Z < \\ge 1 - 1 - \\frac{^5}{2^{10}\\sqrt{2}3^3 a_1^2a_2a_3} ^N.$ [Frobenius distance of outer products] Given $\\vec{x}, \\vec{y} \\in ^n$ , where $\\vec{x}$ and $\\vec{y}$ are unit vectors, we can bound the Frobenius distance of their outer product by the Frobenius distance of the vectors themselves, $\\left\\Vert \\vec{x} \\vec{x}^T - \\vec{y} \\vec{y}^T \\right\\Vert _2 \\le 2 \\left\\Vert \\vec{x} - \\vec{y} \\right\\Vert _2 .$ Recall that for any real matrix $A$ , $\\left\\Vert A \\right\\Vert _2 = \\sqrt{A A^T }$ , thus x xT - y yT 22 = x xT x xT - x xT y yT .",
".",
"- y yT x xT + y yT y yT = x, x 2 + y, y 2 - 2 x, y 2 = 2 1 - x, y 2 22 1 - x, y = 2 x - y 22 .",
"[Bound on diagonal matrix elements] Let $D_A$ and $D_B$ be diagonal matrices of entries $a_i$ and $b_i$ respectively.",
"Let $X = Q^TD_AQ$ for some orthogonal matrix $Q$ with columns $\\vec{q}_i$ .",
"We then have that $\\left| X _{ii} - b_i \\right| \\le \\left\\Vert X - D_B \\right\\Vert _2 .$ First, recall the form of the Frobenius norm for any matrix $Y$ : $\\left\\Vert Y \\right\\Vert _2 = \\sqrt{\\displaystyle \\sum _{i=1}^n \\displaystyle \\sum _{j=1}^n \\left| y_{ij} \\right|^2} .$ In our case, this becomes X - DB 2 = i=1n j=1n | X ij-biij |2 | X ii-bi |2 = | X ii - bi | ."
]
] | 1612.05823 |
[
[
"Supervised Quantum Learning without Measurements"
],
[
"Abstract We propose a quantum machine learning algorithm for efficiently solving a class of problems encoded in quantum controlled unitary operations.",
"The central physical mechanism of the protocol is the iteration of a quantum time-delayed equation that introduces feedback in the dynamics and eliminates the necessity of intermediate measurements.",
"The performance of the quantum algorithm is analyzed by comparing the results obtained in numerical simulations with the outcome of classical machine learning methods for the same problem.",
"The use of time-delayed equations enhances the toolbox of the field of quantum machine learning, which may enable unprecedented applications in quantum technologies."
],
[
"Introduction",
"One of the main consequences of the revolution in computation sciences, started by Alan Turing, Konrad Zuse and John Von Neumann, among others [1], [2], is that computers are capable of substituting us and improving our performance in an increasing number of tasks.",
"This is due to the advances in the development of complex algorithms and the technological refinement allowing for faster processing and larger storage.",
"One of the goals in this area, in the frame of bio-inspired technologies, is the design of algorithms that provide computers human-like capacities such as image and speech recognition, as well as preliminary steps in some aspects related to creativity.",
"These achievements would enable us to interact with computers in a more efficient manner.",
"This research, together with other similar projects, is carried out in the field of artificial intelligence [3].",
"In particular, researchers in the area of machine learning (ML) inside artificial intelligence are devoted to the design of algorithms responsible of training the machine with data, such that it is able to find a given optimal relation according to specified criteria [4].",
"More precisely, ML is divided in three main lines depending on the nature of the protocol.",
"In supervised learning, the goal is to teach the machine a known function without explicitly introducing it in its code.",
"In unsupervised learning, the goal is that the machine develops the ability to classify data by grouping it in different subsets depending on its characteristics.",
"In reinforcement learning, the goal is that the machine selects a sequence of actions depending on its interaction with an environment for an optimal transition from the initial to the final state.",
"The previous ML techniques have also been studied in the quantum regime in a field called quantum machine learning [5], [6], [7], [8], [9], [10], [11], [12] with two main motivations.",
"The first one is to exploit the promised speedup of quantum protocols for improving the already existing classical ones.",
"The second one is to develop unique quantum machine learning protocols for combining them with other quantum computational tasks.",
"Apart from quantum machine learning, fields like quantum neural networks, or the more general quantum artificial intelligence, have also addressed similar problems [13], [14], [15], [16], [17].",
"Here, we introduce a quantum machine learning algorithm for finding the optimal control state of a multitask controlled unitary operation.",
"It is based on a sequentially-applied time-delayed equation that allows one to implement feedback-driven dynamics without the need of intermediate measurements.",
"The purely quantum encoding permits to speedup the training process by evaluating all possible choices in parallel.",
"Finally, we analyze the performance of the algorithm comparing the ideal solution with the one obtained by the algorithm.",
"The first step in the description of the algorithm is the definition of the concept of multitask controlled unitary operations $U$ .",
"In essence, these do not differ from ordinary controlled operations, but the multitask label is selected to emphasize that more than two operations are in principle possible.",
"$U$ acts on $\\left|\\psi \\right\\rangle $ , being $\\left|\\psi \\right\\rangle \\in \\mathbb {C}^d \\otimes \\mathbb {C}^d$ , a quantum state belonging to the tensor product of the control, $\\mathcal {H}_c \\subset \\mathbb {C}^d$ , and target, $\\mathcal {H}_t \\subset \\mathbb {C}^d$ , Hilbert spaces.",
"The dimension of both subspaces is the same, $d$ , and depends on the particular problem to address.",
"Mathematically, we define $U$ as $U=\\sum ^{d}_{i=1}\\left|c_i\\right\\rangle \\left\\langle c_i\\right|\\otimes s_i,$ where $\\left|c_i\\right\\rangle $ denotes the control state, and $s_i$ is the reduced or effective unitary operation that $U$ performs on the target subspace when the control is on $\\left|c_i\\right\\rangle $ .",
"The goal of our algorithm is to explore the control subspace $\\mathcal {H}_c$ and find the control state that maximizes the implementation of a known $s$ , $s : \\mathcal {H}_t \\rightarrow \\mathcal {H}_t$ , which is given in terms of the $\\left|\\text{in}\\right\\rangle $ and $\\left|\\text{out}\\right\\rangle $ states as $s \\left|\\text{in}\\right\\rangle = \\left|\\text{out}\\right\\rangle $ .",
"Therefore, our algorithm is appropriate when $U$ is experimentally implementable but its internal structure, the relation between $\\left|c_i\\right\\rangle $ and $s_i$ , is unknown.",
"In other words, our algorithm enables the training of the control subspace $\\mathcal {H}_c$ by providing data about the target subspace $\\mathcal {H}_t$ , in order to achieve that the complete system implements the desired $s$ operation in the target subspace $\\mathcal {H}_t$ .",
"Our inspirations for the model of controlled unitary operations are supervised learning protocols, in which the goal is that the system is able to learn a given known function.",
"Here, the control subspace plays the role of the memory of the system.",
"This control, or memory, is the mechanism by which the system is able to store the information about the operation that it has to implement.",
"The idea of our algorithm is that the user transmits the information of the operation the system has to make.",
"Therefore, the goal is not to perform a given gate, but to save this information in the system.",
"The protocol consists in sequentially reapplying the same dynamics in such a way that the initial state in the target subspace is always $\\left|in\\right\\rangle $ , while the initial state in the control subspace is the output of the previous cycle.",
"The equation modeling the dynamics is $\\frac{d}{dt} \\left|\\psi (t)\\right\\rangle = -i \\left[ \\theta (t-t_i)\\theta (t_f -t)\\kappa _1 H_1 \\left|\\psi (t)\\right\\rangle + \\kappa _2 H_2 \\left( \\left|\\psi (t)\\right\\rangle - \\left|\\psi (t-\\delta )\\right\\rangle \\right) \\right].$ In this equation $\\theta $ is the Heaviside function, $H_1$ is the Hamiltonian giving rise to $U$ with $U=e^{-i \\kappa _{1} H_{1}(t_f -t_i) }$ , and $H_2$ is the Hamiltonian connecting the input and output states, with $\\kappa _1$ and $\\kappa _2$ the coupling constants of each Hamiltonian.",
"We point out that this evolution cannot be realized with ordinary unitary or dissipative techniques.",
"Nevertheless, recent studies in time delayed equations provide all the ingredients for the implementation of this kind of processes [18], [19], [20], [21].",
"Up to future experimental analyses involving the scalability of the presented examples, the inclusion of time delayed terms in the evolution equation is a realistic approach in the technological framework provided by current quantum platforms.",
"Another important feature of Eq.",
"(REF ), which is related with the delayed term, is that it only acquires physical meaning once the output is normalized.",
"Regarding the behavior of the equation, each term has a specific role in the learning algorithm.",
"The mechanism is inspired in the most intuitive classical technique for solving this problem, which is the comparison between the input and output states together with the correspondent modification of the control state.",
"Here, the first Hamiltonian produces $U$ while the second Hamiltonian produces the reward by populating the control states responsible of the desired modification of the target subspace.",
"The structure of $H_2$ guarantees that only the population in the control $\\left|c_i\\right\\rangle $ associated with the optimal $s_i$ is increased, $H_2=\\mathbb {1}\\otimes \\left( -i \\left|\\text{in}\\right\\rangle \\left\\langle \\text{out}\\right|+i\\left|\\text{out}\\right\\rangle \\left\\langle \\text{in}\\right| \\right).$ Notice that while this Hamiltonian does not contain explicit information about $\\left|c_i\\right\\rangle $ , the solution of the problem, its multiplication with the feedback term, $ \\left|\\psi (t)\\right\\rangle - \\left|\\psi (t-\\delta )\\right\\rangle $ , is responsible for introducing the reward as an intrinsic part of the dynamics.",
"This is a convenient approach because it eliminates the measurements required during the training phase.",
"In this case where we employ a single pair of $\\lbrace \\left|\\text{in}\\right\\rangle $ , $\\left|\\text{out}\\right\\rangle $ } target states, $H_2$ is fixed and time independent.",
"However, this could change in a more complex situation of $p$ pairs of $\\lbrace \\left|\\text{in}\\right\\rangle $ , $\\left|\\text{out}\\right\\rangle \\rbrace $ target states, such that $s=\\sum ^{p}_{j} \\left|\\text{out}\\right\\rangle _j \\otimes \\left\\langle \\text{in}\\right|_j $ , where $H_2$ would also be time independent but different in each episode.",
"Even if this generalization is not included in this article, it points out a promising direction for enhancing the protocol.",
"We would also like to remark the similarity existing between the effect of the delay term in our quantum evolution and gradient ascent techniques in algorithms for optimization problems [3].",
"A possible strategy to perform the learning protocol would be to feed the system with random control states, measure each result, and combine them to obtain the final solution.",
"However, we have discovered that it suffices to initialize the control subspace in a superposition of the elements of the basis.",
"We would like to remark that this purely quantum feature reduces significantly the required resources, because a single initial state replaces a set of random states large enough to cover all possible solutions.",
"We have numerically tested our proposed algorithm in a selection of examples covering the cases with unique or multiple solutions, as well as higher-dimensional systems.",
"We consider as a figure of merit the fidelity function given by the trace of the product between the control state obtained by the algorithm and the ideal control state.",
"In order to recover the solution of the problem we need to trace out the target degrees of freedom, obtaining a density matrix.",
"Therefore, the iteration of the protocol would require solving Eq.",
"(REF ) written for density matrices.",
"This turns out to be a nontrivial task given the non-local cross terms of the generalized master equation, that reads, ddt $\\left|\\psi (t)\\right\\rangle $$\\left\\langle \\psi (t)\\right|$ =-i [ (t-ti)(tf -t)1 H1 + 2 H2, $\\left|\\psi (t)\\right\\rangle $$\\left\\langle \\psi (t)\\right|$ ] +i 2(H2 $\\left|\\psi (t-\\delta )\\right\\rangle $$\\left\\langle \\psi (t)\\right|$ -$\\left|\\psi (t)\\right\\rangle $$\\left\\langle \\psi (t-\\delta )\\right|$ H2).",
"To achieve the solution in the most efficient way, we have decomposed each density matrix in a convex sum of pure states and solved the vector equation in Eq.",
"(REF ) for each of them separately, retrieving the total solution as a linear convex superposition of the individual ones.",
"This method is consistent due to the linearity of Eq.",
"(REF ).",
"Figure: Node line networks.",
"We plot the graphical representation of every control state in the two, a), and three, b), node line networks.",
"The circles around the nodes denote the control being in the open state.",
"The effective operation that the control performs on the target subspace is the s ij s_{ij} SWAP gate between nodes ii and jj.A first specific example we address in this manuscript is given by the excitation transport produced by the controlled SWAP gate.",
"In this scenario, the complete system is an $n$ -node network, where each node is composed by a control and a target qubit.",
"Therefore, the control and target subspaces are defined as $\\mathcal {H}_c \\subset (\\mathbb {C}^2)^{\\otimes n}$ and $\\mathcal {H}_t \\subset (\\mathbb {C}^2)^{\\otimes n}$ .",
"The excitations in this system belong to the target subspace and are exchanged between two nodes, when both nodes are in a particular state of the control subspace.",
"The control states are in a superposition of open and close, $\\left|o\\right\\rangle $ and $\\left|c\\right\\rangle $ , while the target qubits are written in the standard $\\lbrace \\left|0\\right\\rangle , \\left|1\\right\\rangle \\rbrace $ basis denoting the absence or presence of excitations.",
"We define $U$ , the multitask controlled unitary operation, to implement the SWAP gate between connected nodes only if all the controls of the corresponding nodes are in the open state, $\\left|o\\right\\rangle $ .",
"See Fig.",
"REF for a graphical representation of the most simple cases, the two and three node line networks.",
"The explicit formula for $U_2$ is given by U2= ( $\\left|cc\\right\\rangle $$\\left\\langle cc\\right|$ +$\\left|co\\right\\rangle $$\\left\\langle co\\right|$ +$\\left|oc\\right\\rangle $$\\left\\langle oc\\right|$ ) 1 + $\\left|oo\\right\\rangle $$\\left\\langle oo\\right|$ s12 where $s_{ij}$ represents the SWAP gate between qubits $i$ and $j$ .",
"Here, the first two qubits represent the control subspace and the last two represent the target subspace.",
"Although we have employed unitary operations for illustration purposes, the equation requires the translation to Hamiltonians.",
"In order to do so, we first select $\\kappa _1(t_f - t_i)$ to be $\\pi /2$ and calculate the matrix logarithm, which yields the result for $H_1$ in Eq.",
"(REF ), $H_1=(\\left|oo\\right\\rangle \\left\\langle oo\\right|)\\otimes h_{12}$ .",
"Denoting with $\\sigma _k$ the Pauli matrices, $h_{ij}$ for $i<j$ reads $h_{ij}=\\frac{1}{2}\\left(\\sum ^{3}_{k=1}\\mathbb {1}^{\\otimes i-1}\\otimes \\sigma _k \\otimes \\mathbb {1}^{\\otimes j-i-1}\\otimes \\sigma _k\\otimes \\mathbb {1}^{\\otimes n-j} -\\mathbb {1}^{\\otimes n} \\right).$ Figure: Learning curves for single solutions.",
"a) We plot the fidelity of the learning process as a function of the number of episodes for the first examples of nn-node line networks.",
"We have selected the open state, o=1\\left|o\\right\\rangle =\\left|1\\right\\rangle of the {0,1}\\lbrace \\left|0\\right\\rangle , \\left|1\\right\\rangle \\rbrace basis.",
"b) We plot the fidelity for a different selection of o\\left|o\\right\\rangle in the n=3n=3 case.",
"Here, we have rotated the control states with the goal of testing the algorithm for an arbitrary basis.",
"The solution, ooo\\left|ooo\\right\\rangle , is given by 1 2[0+1]⊗1⊗[cos(π/3)0+sin(π/3)1]\\frac{1}{\\sqrt{2}}[\\left|0\\right\\rangle +\\left|1\\right\\rangle ]\\otimes \\left|1\\right\\rangle \\otimes [\\cos {(\\pi /3)}\\left|0\\right\\rangle +\\sin {(\\pi /3)}\\left|1\\right\\rangle ].The first family of problems we address is n-node line networks, in which the nodes are located in a unidimensional array and are only connected with their closest neighbors.",
"The goal is to find the control state that allows transmitting an excitation from the first to the last node of the network, which in this case requires that all intermediate connections are active.",
"The pair of $\\lbrace \\left|in\\right\\rangle , \\left|out\\right\\rangle \\rbrace $ is determined by these constrains as $\\left|\\text{in}\\right\\rangle =\\left|1\\right\\rangle \\left|0\\right\\rangle ^{\\otimes n-1}$ and $\\left|\\text{out}\\right\\rangle =\\left|0\\right\\rangle ^{\\otimes n-1}\\left|1\\right\\rangle $ .",
"Accordingly the problem has a unique solution, given by the control state with all the nodes open, $\\left|o\\right\\rangle ^{\\otimes n}$ .",
"The parameters we have selected are $\\delta =1$ , $\\kappa _1=100$ , $\\kappa _2=10$ and $T=2$ , where $T$ represents the total duration of each episode.",
"In Fig.",
"REF we plot the results together with the required resources.",
"These examples show how the algorithm is properly working for this family of problems independently of the natural basis of $U$ .",
"The $H_1$ Hamiltonians employed in the simulations for $n=2,3,4$ are given by H21=|oooo|h12, H31=|oooooo|h13 + |oocooc|h12 + |coocoo| h23, H41=|oooooooo| h14 + |ooocoooc|h13 + ( |ooccoocc| + |oocoooco| ) h12 + (|ccooccoo|+|ocooocoo|)h34 + |cooccooc|h23 + |cooocooo| h24.",
"Figure: Networks with two solutions.",
"We schematize the A, B, and C networks in a), b) and c), respectively.",
"In each of them, we write the pair of solution control states that corresponds to the control performing the s 13 s_{13} (a) and s 14 s_{14} (b,c) gates in the target subspace.We address now a set of more complicated networks which will allow us to clarify how the algorithm performs when solving problems with multiple solutions.",
"These are the A network for three nodes and the B and C networks for four nodes, depicted in Fig.",
"REF .",
"The goal of the algorithm is the same as in the previous case, i.e., to find the control state able of sending an excitation from the first to the last node.",
"The difference is that these networks accept two pure states and their superpositions as solutions, a feature that is reflected in the result obtained with the algorithm.",
"The asymptotic state achieved under the feedback induced quantum learning equation is a quantum superposition of both solutions, see Fig.",
"REF a for the numerical simulations.",
"In this case, the previous definition of the fidelity is not valid.",
"Therefore, we provide a new one in terms of the $\\left|\\text{in}\\right\\rangle $ and $\\left|\\text{out}\\right\\rangle $ states of the target space and the Hamiltonian $H_1$ .",
"The new fidelity corresponds to the trace of the product between the ideal output $\\left|\\text{out}\\right\\rangle $ , and the output obtained with the control state achieved by the algorithm after acting on $\\left|\\text{in}\\right\\rangle $ .",
"Both ideal and real outputs belong to the target subspace.",
"While the {$\\left|\\text{in}\\right\\rangle $ , $\\left|\\text{out}\\right\\rangle $ } pair is the same as in the previous case, the $H_1$ Hamiltonians change their definition to HA1=(|oooooo|+|ocooco|) h13 + |oocooc| h12 + |coocoo| h23, HB1=(|oooooooo|+|oocoooco|)h14 + |cooocooo|h34 + |ooocoooc| h13 + |ooccoocc| h12 + |cooccooc| h23 + |cocococo| h24, HC1=|ooccoocc| h12 + |cooccooc| h23 + (|cocococo| + |cooocooo|) h24 + |ooocoooc| h13 + (|ocooocoo| + |ccooccoo| )h34 + (|oocoooco| + |oooooooo| )h14.",
"Figure: Learning curves for two solutions and qutrit problems.",
"a) We depict the learning curve for the A, B, and C networks as a function of the number of episodes.",
"Notice that the curves for the B and C networks are identical.",
"b) We depict the learning curve for the multitask controlled unitary operation acting on two qutrits as a function of the number of episodes.",
"Here, in=0\\left|\\text{in}\\right\\rangle =\\left|0\\right\\rangle , out=2\\left|\\text{out}\\right\\rangle =\\left|2\\right\\rangle and the solution is given by c 2 =1\\left|c_2\\right\\rangle =\\left|1\\right\\rangle , where the control states coincide with the basis of the qutrit space.For the cases studied, the complete set of solutions is obtained encoded in the outcome of the algorithm.",
"This is convenient because it allows one to design a protocol to select a specific optimal solution according to given criteria.",
"In the networks we are analyzing, one might want to obtain the most efficient solution, defining efficiency as achieving the transmission of the excitation while minimizing the number of open nodes.",
"In order to accomplish this task a dissipative term has to be included in the evolution equation, in order to filter out the undesired solutions.",
"We point out that a control-dependent dissipation affects the target subspace, modifying the protocol in the required manner.",
"We explicitly write the Lindblad operators $\\sigma _i$ and dissipation constants $\\gamma _i$ for a two-node case, as follows $\\nonumber &&\\sigma _1=\\left|co01\\right\\rangle \\left\\langle co11\\right|+\\left|co00\\right\\rangle \\left\\langle co10\\right|, \\hspace{7.11317pt} \\sigma _2=\\left|co00\\right\\rangle \\left\\langle co01\\right|+\\left|co10\\right\\rangle \\left\\langle co11\\right|, \\\\ \\nonumber &&\\sigma _3=\\left|oc00\\right\\rangle \\left\\langle oc10\\right|+\\left|oc01\\right\\rangle \\left\\langle oc11\\right|, \\hspace{7.11317pt} \\sigma _4=\\left|oc10\\right\\rangle \\left\\langle oc11\\right|+\\left|oc00\\right\\rangle \\left\\langle oc01\\right|, \\\\ \\nonumber &&\\sigma _5=\\left|oo00\\right\\rangle \\left\\langle oo10\\right|+\\left|oo01\\right\\rangle \\left\\langle oo11\\right|, \\hspace{7.11317pt} \\sigma _6=\\left|oo00\\right\\rangle \\left\\langle oo01\\right|+\\left|oo10\\right\\rangle \\left\\langle oo11\\right|, \\\\ && \\gamma _1=\\gamma _2=\\gamma _3=\\gamma _4, \\hspace{7.11317pt} \\gamma _5=\\gamma _6=2\\gamma _1.$ Instead of solving the master equation, we have employed the quantum jump formalism, which allows one to work with Eq.",
"(REF ) instead of Eq.",
"(REF ), with the consequent simplicity.",
"The dissipation can be modeled with an additional term $H_D = \\frac{-i}{2} \\sum \\gamma _i \\sigma ^{\\dag }_{i}\\sigma _{i}$ in the first part of the time delayed equation in the absence of a decay event.",
"Therefore, in order to assure that the non-Hermitian Hamiltonian accounts for the real evolution of the system, one has to properly balance the relation between $\\kappa _1$ and $\\gamma _i$ .",
"$\\frac{d}{dt} \\left|\\psi (t)\\right\\rangle = -i \\left[ \\theta (t-t_i)\\theta (t_f -t)(\\kappa _1 H_1 + H_D)\\left|\\psi (t)\\right\\rangle + \\kappa _2 H_2 \\left( \\left|\\psi (t)\\right\\rangle - \\left|\\psi (t-\\delta )\\right\\rangle \\right) \\right].$ A non-dissipative alternative consists in the modification of the coupling constant associated with each of the control-target pairs in the unitary operation.",
"These two techniques allow us to find the shortest path between two nodes in a network once the natural basis of the unitary is known.",
"Another possible aspect to study is the extension of the algorithm to higher-dimensional building blocks.",
"We provide an example in which the optimal control state for a multitask controlled unitary operation acting on qutrits is obtained.",
"This operation $U$ is defined in terms of the control states $\\left|c_i\\right\\rangle $ as $U=\\left|c_1\\right\\rangle \\left\\langle c_1\\right|\\otimes \\mathbb {1} + \\left|c_2\\right\\rangle \\left\\langle c_2\\right|\\otimes \\left( \\begin{array}{ccc} 0&1&0\\\\0&0&1\\\\1&0&0 \\end{array} \\right) +\\left|c_3\\right\\rangle \\left\\langle c_3\\right|\\otimes \\left( \\begin{array}{ccc} 0&0&1\\\\1&0&0\\\\0&1&0 \\end{array} \\right).$ where the first qutrit belongs to the control subspace and the second one belongs to the target subspace.",
"Although no network is defined in this case, the goal of the algorithm is to find the control state that realizes the $\\left|\\text{in}\\right\\rangle $ -$\\left|\\text{out}\\right\\rangle $ transition in the target subspace.",
"In this problem, the system consists of a single control qutrit and a single target qutrit.",
"See Fig.",
"REF for a numerical simulation of the learning process in this particular case.",
"Figure: Learning curves for phase gates.",
"a) We plot the fidelity of the learning process as a function of the number of episodes for the problem of finding the appropriate phase gate.All examples discussed until this point consisted on $s$ gates whose effect can be understood as a permutation of the basis elements.",
"Let us consider now a different scenario in which the operations in the target subspace are phase gates, $s_i = \\mathbb {1} -2 |i\\rangle \\langle i |$ , therefore, the complete unitary operation reads $U=\\sum |i\\rangle \\langle i|\\otimes (\\mathbb {1} - 2 |i\\rangle \\langle i |)$ with $i \\in [1,n]$ .",
"If we choose the reference target states to be, $\\left|\\text{in}\\right\\rangle =\\frac{1}{\\sqrt{2}} (\\left|1\\right\\rangle +\\left|n\\right\\rangle ), \\hspace{7.11317pt} \\left|\\text{out}\\right\\rangle =\\frac{1}{\\sqrt{2}} (\\left|1\\right\\rangle -\\left|n\\right\\rangle ),$ we know a priori that the only solution is given by $s=\\mathbb {1}-2 |n\\rangle \\langle n|$ associated to control $\\left|c\\right\\rangle =|n\\rangle \\langle n|$ .",
"We perform a numerical experiment to analyze how the initial equally weighted control state, $\\frac{1}{\\sqrt{n}} \\sum \\left|i\\right\\rangle $ , converges to the solution under the action of $H_1=-2 \\sum {\\left|i\\right\\rangle \\left\\langle i\\right|\\otimes \\left|i\\right\\rangle \\left\\langle i\\right|}$ depending on the dimension of the system.",
"See Fig.",
"REF for the simulations.",
"The results show that our algorithm is particularly efficient for this selection of Hamiltonians, given that the solution is reached in $O(\\sqrt{n})$ for all the cases studied.",
"It is important to mention that the simulations and techniques we provide here constitute an analysis of our quantum machine learning algorithm, but our aim in this work is not to demonstrate scalability or quantum speedup.",
"It would be convenient to analytically solve Eq.",
"(REF ) in order to rigorously analyze the scope of the algorithm and be able to obtain information about its scalability for general problems.",
"Since we have not solved the dynamics analytically, we evaluate the performance by comparing our results with the ones obtained via different methods.",
"In particular, we follow two different strategies to determine the structure of the controlled unitary operation, measure it and analyze it by using machine learning techniques.",
"Here, the resources are quantified by the number of times the unitary operation has to be applied and the output measured in order to be able to determine its structure.",
"Here, we employ state-of-the-art classical machine learning algorithms to compare with our quantum protocol.",
"We show the results achieved for three different networks, the two-node line, and two different instances of the three-node line, all of them previously studied with our algorithm in Fig. 2.",
"The numerical experiment is designed for determining the optimal control state by evaluating the action of $U$ on the tensor product of a random control state and the fixed $\\left|\\text{in}\\right\\rangle $ .",
"The data consists of a recompilation of random control states, which cover the whole control subspace, with their correspondent fidelity for a fixed $\\lbrace \\left|\\text{in}\\right\\rangle , \\left|\\text{out}\\right\\rangle \\rbrace $ pair.",
"For each network, three data sets were used (small, medium, large) with a different number of instances.",
"It must be emphasized that all results are referred to test sets, i.e., obtained with data not used to train the models.",
"Therefore, they must be taken as a good estimation of the prediction capability of the models for new unseen data.",
"Cross-validation was implemented by means of a $k$ -fold approach [4], where $k$ =10 for all data sets, except for the small data set of the two-line network whose value was $k$ =5 due to the very limited number of instances.",
"All results were achieved by using Support Vector Regressors (SVRs) [22], whose characteristics make them especially adequate when dealing with sparse data sets (few instances and high dimension).",
"SVRs work by creating a transformed data space in which the problem is more easily solvable (ideally the problem is transformed into a linear one).",
"That transformation between spaces is carried out by the so-called kernels (Gaussian and polynomial kernels have been used in this experimentation).",
"The data used for training the models has been randomly selected from a set of multiple pairs of control state and fidelity.",
"Although other ML approaches, such as Reinforcement Learning (RL), might seem appropriate to solve this problem, note that the goal of the problem is actually a prediction of the efficiency of the solution rather than the optimal sequence of steps that link the input state with the output state, thus not matching the RL paradigm Tables REF , REF and REF report the results achieved by the SVR in the three analyzed networks.",
"These correspond to the two and three node lines analyzed in Fig.",
"2.",
"In the case of $n=3$ , the topology of networks A and B is the same, the one depicted in Fig.",
"REF ., but they are defined in a different control basis.",
"For each case, the state with the best fidelity is shown, together with the Mean Error (ME) and the Root Mean Square Error (RMSE).",
"ME is a measure of bias that represents the difference between the real and the predicted efficiencies, i.e., gives information about whether the model tends to make overestimations (negative values) or underestimations (positive values).",
"On the other hand, RMSE is a well-known robust measure of accuracy.",
"Table: Two-node line.",
"The optimal control state for this network is 1⊗1\\left|1\\right\\rangle \\otimes \\left|1\\right\\rangle , while the best result obtained with this analysis is (0.05350+0.99861)⊗(0.07860+0.99691)(0.0535\\left|0\\right\\rangle +0.9986\\left|1\\right\\rangle )\\otimes (0.0786\\left|0\\right\\rangle + 0.9969\\left|1\\right\\rangle ).Table: Three-node line A.",
"The optimal control state for this network is 1⊗1⊗1\\left|1\\right\\rangle \\otimes \\left|1\\right\\rangle \\otimes \\left|1\\right\\rangle , while the best solution that the machine learning protocol provides is (0.17850+0.98391)⊗(0.20630+0.97851)⊗(0.17540+0.98451)(0.1785\\left|0\\right\\rangle +0.9839\\left|1\\right\\rangle )\\otimes (0.2063\\left|0\\right\\rangle + 0.9785\\left|1\\right\\rangle )\\otimes (0.1754\\left|0\\right\\rangle + 0.9845\\left|1\\right\\rangle ).Table: Three-node line B.",
"The optimal control state for this network is 1 2[0+1]⊗1⊗[cos(π/3)0+sin(π/3)1]\\frac{1}{\\sqrt{2}}[\\left|0\\right\\rangle +\\left|1\\right\\rangle ]\\otimes \\left|1\\right\\rangle \\otimes [\\cos {(\\pi /3)}\\left|0\\right\\rangle +\\sin {(\\pi /3)}\\left|1\\right\\rangle ], while the result of the analysis is (0.75120+0.661)⊗(0.15990+0.98711)⊗(0.49360+0.86971)(0.7512\\left|0\\right\\rangle +0.66\\left|1\\right\\rangle )\\otimes (0.1599\\left|0\\right\\rangle + 0.9871\\left|1\\right\\rangle )\\otimes (0.4936\\left|0\\right\\rangle + 0.8697\\left|1\\right\\rangle ).An alternative method for solving the learning task would be to measure the input-output relation of the controlled unitary operation when strategically, and not randomly, exploring the control subspace.",
"Let us denote by $\\left|c_i\\right\\rangle $ the natural basis of the control subspace in $U$ , and by $\\left|b_i\\right\\rangle $ our guess for this basis in a Hilbert space of dimension $n$ .",
"The measurement protocol consists in applying the unitary operation to $\\left|b_i\\right\\rangle \\otimes \\left|\\text{in}\\right\\rangle $ , projecting this result on $\\left|\\text{out}\\right\\rangle \\left\\langle \\text{out}\\right|$ and tracing out the target subspace achieving $\\rho _i$ for each $b_i$ .",
"In the worst case, this operation has to be repeated for all $b_i$ to guarantee that the populations of the solutions, and not the internal phases, are found.",
"Afterwards, one has to find the appropriate basis $\\left|c_i\\right\\rangle $ as a linear combination of the proposed one $\\left|b_i\\right\\rangle $ .",
"Another approach is to determine each component of the unitary operation and change to a basis in which the unitary is expressed as a direct sum of the $s_{i}$ operations.",
"This particular strategy highlights the relation between our algorithm and the field of quantum process tomography.",
"In summary, the purely random approach analyzed with ML techniques requires in principle more resources than the quantum feedback algorithm with delayed equation.",
"Nevertheless, the fact that ML techniques are independent of the basis guarantees their success in any possible situation.",
"The comparison is made between the episodes, the number of times that the time delayed equation has to be repeated, and the instances, the amount of data employed in the ML algorithm.",
"Even if both methods are based on different training mechanisms, the information fed to both of them is the same, a figure of merit for each control state.",
"In the SVR the system is provided with pairs of control state and its correspondent fidelity, which requires the implicit knowledge of $\\lbrace \\left|\\text{in}\\right\\rangle , \\left|\\text{out}\\right\\rangle \\rbrace $ and the ideal $U$ operation.",
"The connection with the quantum algorithm is that the delay term in Eq.",
"2 provides a distance that works in an analogue way as the fidelity in the SVR.",
"Notice that in the quantum algorithm each episode only requires a pair of {$\\left|\\text{in}\\right\\rangle $ , $\\left|\\text{out}\\right\\rangle $ } states, therefore the number of episodes equals the number of instances.",
"A more realistic analysis would take into account the duration of each process, but for the moment we cannot make a precise estimation about the time for implementing a time delayed equation.",
"With respect to the complete measurement approach, recent studies bound its scalability in the order of $n^2$ or even $n$ , being the latter the dimension of the Hilbert space [23], [24], [25].",
"On the other hand, the measurement protocol does not provide the solution in a physical register, but it is the analysis of the unitary operation that provides the knowledge of it.",
"Moreover, each implementation of the controlled unitary operation is associated with a measurement, while in the quantum machine learning algorithm intermediate measurements are not required, because they are included as an intrinsic part of the dynamics, in contrast to the tomography approach.",
"Additionally, when measuring, one needs to perform a search for the convenient basis along the Hilbert space to retrieve the correct structure of $U$ .",
"Regarding the scalability of our algorithm, we have observed that the number of episodes for reaching the solution depends on the distance between both, the initial control state and the solution.",
"A direct consequence is that the protocol will not properly work when the initial control state is orthogonal to the solution.",
"This is important to consider because the way to notice the failure is to validate the result by measuring the outcome of the unitary operation.",
"In the simulations carried out here, we have employed $\\left|+\\right\\rangle ^{\\otimes n}$ as the initial control state, but this choice is not unique.",
"In some sense, our protocol can also be understood as a search algorithm.",
"Therefore, a comparison with Grover's result [26] may be in order.",
"Regarding the similarities, the conditional phase rotation in Grover's search algorithm requires the use of an oracle, whose role is played in our formalism by the combination of a controlled unitary operation and the time-delayed terms.",
"On the other hand, the main difference between both protocols is that on Grover's algorithm the basis in which the states to optimize are described is known, while in ours, the search is performed without previous knowledge of the basis, in a similar spirit to the analog algorithm by Farhi and Gutmann [27].",
"A positive property of our protocol, in contrast with the previously mentioned quantum search algorithms, is that the solution is reached asymptotically, i.e., the fidelity always increases with the number of episodes.",
"In conclusion, we have proposed a quantum machine learning algorithm in which the implementation of time-delayed dynamics allows one to avoid the intermediate measurements, and therefore provides a complementary strategy to conventional quantum machine learning algorithms [28], [29], [30], [31].",
"Moreover, we have shown how the framework of multitask controlled unitary operations is flexible enough to address different problems such as efficient excitation transport in networks.",
"This kind of protocol may be straightforwardly adapted to different quantum architectures, which is beyond the scope of this article.",
"We believe our study represents the first proposal for exploiting feedback-induced effects of delayed-equation dynamics without intermediate measurements in quantum machine learning algorithms.",
"The authors acknowledge support from Basque Government grants BFI-2012-322 and IT986-16, Spanish MINECO/FEDER FIS2015-69983-P, Ramón y Cajal Grant RYC-2012-11391, and UPV/EHU UFI 11/55.",
"U. Alvarez-Rodriguez, L. Lamata and E. Solano designed the time delayed equation, while P. Escandell-Montero and J. D. Martín-Guerrero analyzed the problem with SVR techniques.",
"The authors declare no competing financial interests."
]
] | 1612.05535 |
[
[
"Cavity-enhanced single photon source based on the silicon vacancy center\n in diamond"
],
[
"Abstract Single photon sources are an integral part of various quantum technologies, and solid state quantum emitters at room temperature appear as a promising implementation.",
"We couple the fluorescence of individual silicon vacancy centers in nanodiamonds to a tunable optical microcavity to demonstrate a single photon source with high efficiency, increased emission rate, and improved spectral purity compared to the intrinsic emitter properties.",
"We use a fiber-based microcavity with a mode volume as small as $3.4~\\lambda^3$ and a quality factor of $1.9\\times 10^4$ and observe an effective Purcell factor of up to 9.2.",
"We furthermore study modifications of the internal rate dynamics and propose a rate model that closely agrees with the measurements.",
"We observe lifetime changes of up to 31%, limited by the finite quantum efficiency of the emitters studied here.",
"With improved materials, our achieved parameters predict single photon rates beyond 1 GHz."
],
[
"Introduction",
"Single photon sources are a fundamental component of the toolbox for quantum information technologies that promise transformational advances in the communication and processing of information [1], [2].",
"There is thus large interest in developing scalable sources that fulfill the requirements of high purity (emitting exactly one photon at a time), high efficiency (obtaining a photon in a collectable mode), high brightness (large maximal excitation rate), and high spectral purity (a narrow, ideally Fourier-transform limited spectrum).",
"Solid-state-based quantum emitters [3], [4] have evidenced outstanding properties in this respect.",
"To achieve high collection efficiencies and large emission rates, coupling the emitter to photonic structures [5] such as optical resonators [6], [7], [8], [9], [10], [11], waveguides [12], [13], or antennas [14], [15] is desired.",
"This has been demonstrated for various systems like quantum dots [6], [7], [9], [16], molecules [17], [18], carbon nanotubes [19], [20], [21], nitrogen vacancy (NV) [22], [10], [23], [11], and silicon vacancy (SiV) [24], [25], [26], [27] centers in diamond.",
"While cryogenic experiments already come close to ideal single photon sources, it remains a challenge to achieve high efficiency and spectral purity under ambient conditions.",
"Here, we demonstrate an approach to achieve high efficiency, brightness, and spectral purity for a room-temperature source by coupling the emission of single SiV centers to a high quality factor microcavity.",
"Coupling quantum emitters to optical microcavities [28] increases the spontaneous emission rate by the Purcell factor $ C = \\frac{3(\\lambda /n)^3}{4\\pi ^2}\\frac{Q_\\mathrm {eff}}{V_m}\\zeta \\left|\\frac{\\vec{\\mu }\\vec{E}}{\\mu E_0}\\right|^2 $ , where $ \\lambda $ is the wavelength, $ n $ the refractive index, $ V_m $ the mode volume, $\\vec{\\mu }$ the dipole matrix element, $\\vec{E}$ the electric field vector at the location of the dipole, $E_0$ the maximal field in the cavity, and $ \\zeta $ the branching ratio into the zero phonon line (ZPL).",
"The effective quality factor $ Q_\\mathrm {eff}=(Q_c^{-1}+Q_\\mathrm {em}^{-1})^{-1} $ includes the cavity quality factor $ Q_c $ and the emitter quality factor $ Q_\\mathrm {em} $ obtained from the emitter linewidth [29], [30], motivating the choice of narrow emitters.",
"In this respect, the SiV [31] is particularly promising, due to its narrow ZPL that carries about 80% of the oscillator strength ($ \\zeta = 0.8$ ), and an excited state lifetime of $\\sim 1$ ns, favoring bright single photon emission [32], [33], [34], [35].",
"The emission is coupled to a well-collectable cavity mode with an efficiency $\\beta =C/(C+1)$ , which can be near-unity for large $C$ .",
"Furthermore, in the bad emitter regime, where the spectral width of the fluorescence is broader than the cavity mode, the spectral emission is determined primarily by the properties of the optical resonator rather than the electronic emitter.",
"This is attractive because it offers potential for exquisite control over photon emission even at ambient conditions."
],
[
"Experimental methods",
"We use a fiber-based Fabry-Perot microcavity [36] which combines a small mode volume and a high quality factor with full tunability and an open access design, allowing the investigation of different emitters with one and the same cavity.",
"The cavity consists of a macroscopic planar mirror and a micromirror at the endfacet of a single-mode optical fiber (Fig.",
"REF a).",
"The fiber is shaped by CO$_2$ laser machining [37] and has a conical tip with a remaining plateau of about $18~\\mu $ m diameter to allow for the shortest mirror separations [11].",
"In its center, we produce a concave profile with a radius of curvature of $26~\\mu $ m, (see Fig.",
"REF b).",
"The fiber and the planar mirror have different dielectric coatings to optimize excitation through the large mirror and up to 90% out-coupling of fluorescence to the fiber (see appendix for details).",
"The fiber is mounted on a shear piezo stack, which allows us to accurately tune the cavity length.",
"We measure the cavity finesse $\\mathcal {F}$ at 780 nm and obtain a value which is consistent with $\\mathcal {F}=3750$ at 740 nm according to the coating simulation.",
"To calibrate the optical cavity length, we record broad-band cavity transmission spectra with a supercontinuum laser and evaluate the separation of subsequent cavity resonances, see Fig.",
"REF c. We find that the smallest accessible effective cavity length is $d_\\mathrm {eff}=5\\lambda /2$ , corresponding to the longitudinal mode order $q = 5$ , limited by the profile depth (300 nm) and the field penetration into the coating (1160 nm at 740 nm).",
"At this separation, we obtain a cavity quality factor of $ Q_c = q \\mathcal {F}= 1.9 \\times 10^4$ .",
"From scanning-cavity microscopy measurements and calculations [38], we infer the mode waist to be $ w_0 = 1.0~\\mu $ m, resulting in a mode volume $ V_m = (\\pi w_0^2 d_\\mathrm {eff})/4 = 3.4 \\lambda ^3 $ .",
"Together, these cavity parameters yield an ideal Purcell factor of $C_0=3\\lambda ^3/(4\\pi ^2)Q_c/V_m=425$ .",
"We study nanodiamonds of about 100 nm in size produced by bead-assisted sonic disintegration of a polycrystalline chemical vapor deposition film [39].",
"The nanodiamonds are spin-coated onto the planar mirror and are first investigated by confocal microscopy with an excitation wavelength of 690 nm.",
"The crystals typically contain ensembles of SiV centers featuring a broad (7 nm) ZPL at the nominal wavelength of 738 nm.",
"In some crystals, we also observe narrow (down to 1 nm) emission lines that are spectrally shifted and which show pronounced photon antibunching.",
"We attribute these lines to emission from single SiV centers that are subject to local perturbations such as strain in the nanodiamonds [39].",
"It is desirable to study emitters with maximal radiative quantum yield, fluorescence stability, and optimal dipole orientation, which we favor by selecting emitters with high brightness.",
"The spectra of the single SiV centers studied here are shown in Fig.",
"REF a.",
"The central wavelengths range from 737 nm (the nominal emission wavelength) to 759 nm with spectral widths from 1.1 nm to 3.0 nm (see appendix ).",
"Figure: (a) Free space fluorescence spectra of single SiV centers studied for cavity coupling.",
"(b) Cavity fluorescence spectra of ND5 for different geometric cavity lengths (air gap), logarithmic color scale.",
"Top: Linear scale spectrum for d=0.875d=0.875~nm where maximal enhancement occurs.",
"Left: Linear scale plot of count rate for different lengths at emission wavelength λ=752.4\\lambda =752.4~nm.We have developed a setup which combines a confocal microscope and a tunable microcavity side-by-side sharing a single nanopositioner.",
"Calibrating the displacement between the confocal focus and the cavity enables the characterization of the same emitters both in free space and inside the cavity.",
"Nanodiamonds pre-characterized confocally can be easily found in the cavity via the Rayleigh scattering and absorption loss they introduce.",
"Therefore, we perform scanning cavity microscopy [40] and measure the cavity transmission of a supercontinuum laser filtered to a 33 nm spectral band around 747 nm.",
"On such transmission maps (see appendix ) the nanodiamonds appear as dark spots and can be directly related to the confocal fluorescence maps.",
"To achieve resonant coupling conditions with the SiV emission, we stepwise tune the cavity length to shift a cavity resonance across the emission spectrum and record the fluorescence spectra on a grating spectrometer.",
"On resonance, we observe enhanced emission into a single cavity resonance.",
"Figure REF b shows fluorescence spectra for varying mirror separation for the $q=5$ mode order in a logarithmic color scale to make the high signal-to-background level visible.",
"Emission away from the cavity resonance is suppressed, and we detect only dark counts (blue color).",
"The cavity resonance appears broadened due to the finite spectrometer resolution and some length jitter within the acquisition time of 1 s. The actual FWHM line width is 43 pm or $\\kappa =21$ GHz as inferred from the quality factor.",
"In addition to the fundamental mode, one can see the prominent second order transverse mode, and in between the odd first order mode, which couples weakly to the emitter."
],
[
"Results",
"The Purcell enhancement of the spontaneous emission leads to a reduction of the excited state lifetime, which we investigate by time-correlated single photon counting under pulsed excitation at 690 nm.",
"Exemplary traces are shown in Fig.",
"REF a.",
"We measure the instrument response function to be a Gaussian with $ \\sigma = 0.157 $ ns and convolute it with an exponential decay as a fit function to reproduce the lifetime data.",
"We perform such measurements both in free space and in the cavity at the longitudinal mode orders $q=5,6,7$ .",
"In the latter case, we stabilize the cavity on resonance by a piezo actuator and a software algorithm that maximizes the count rate.",
"The polarization of the excitation light is in all cases chosen to match the projection of the dipole orientation in the plane of the mirror.",
"Figure: (a) Lifetime measurement for ND3; data points and fit (solid line) with exponential function starting at t=0t=0 convoluted with system response function.",
"Blue: free space.",
"Orange: cavity (q=5 q = 5 ).",
"Green: cavity (q=6 q = 6 ) (b) Cavity saturation measurement of ND4 (Blue: data, red: fit).Table: Measured lifetimes in free space (τ 0 \\tau _0 ) and in the cavity (τ c \\tau _c : lowest reachable order, τ c,2 \\tau _{c,2} : following order).",
"The error is obtained from different fitting methods (see appendix ).We observe a lifetime reduction in the cavity, and the effect is larger for smaller mirror separation, i.e.",
"smaller mode volume.",
"Table REF summarizes the lifetime measurements taken for three different emitters and shows that lifetime reductions $ \\Delta \\tau / \\tau _0 $ between 17% and 31% are observed.",
"The lifetime of the narrow emitters is in many cases found to be shorter than the reported value of 1 to 1.7 ns [35], [41] in unstrained bulk diamond, which we also observe for SiV ensembles in this sample.",
"We find a large spread of lifetimes [33], which may originate from different strain in the nanodiamonds and a varying contribution of non-radiative decay [42].",
"The ratio of lifetimes in free space, $ \\tau _0 $ , and the cavity, $ \\tau _c $ , depends on the Purcell factor and the quantum efficiency $\\mathrm {QE}=\\gamma _r/(\\gamma _r+\\gamma _{nr})$ , where $ \\gamma _r $ is the radiative and $ \\gamma _r $ the non-radiative decay rate: $ \\tau _0/\\tau _c = C \\, \\mathrm {QE} + 1 $ .",
"For an unknown $ \\mathrm {QE} < 1 $ as expected for SiV centers [26], [34], we can thus not infer $C$ directly from the lifetime change.",
"Table: Results of saturation measurements.",
"I m,fs ∞ I_{m,fs}^{\\infty } (I m,c ∞ I_{m,c}^{\\infty }): Saturation count rate in free space (cavity).",
"I fs ∞ I_{fs}^{\\infty } (I c ∞ I_{c}^{\\infty }): Photon emission rate in free space (cavity) calculated from count rates.",
"All rates in MHz.",
"C th C_{th} : maximal theoretical Purcell factor.",
"C exp =I c ∞ /I fs ∞ C_{\\mathrm {exp}}=I_{c}^{\\infty } / I_{fs}^{\\infty }.To quantify the cavity enhancement of the emission and to obtain an estimate for the achieved Purcell factor, we compare the photon emission rate in free space and in the cavity under saturation conditions.",
"In free space, the emission rate is given by $ I^\\infty _{fs} = \\gamma _r n_2^\\infty $ , where $ n_2^\\infty $ is the equilibrium population of the excited state.",
"The emission rate into the cavity mode is $ I^\\infty _c = C \\gamma _r n_2^\\infty $ , and the ratio between the rates directly yields the Purcell factor, $C_{\\mathrm {exp}}= I^\\infty _c / I^\\infty _{fs}$ , independent of the quantum efficiency.",
"Experimentally, we measure the saturation count rate both in free space and in the cavity, and use the knowledge of collection and detection efficiencies to calculate back to the respective emission rates.",
"An example for the saturation in the cavity is shown in Fig.",
"REF b, where we have again optimized the polarization of the excitation light.",
"The measured rate $ I_m(P) $ as a function of the excitation power $ P $ can be described as $ I_m(P) = \\frac{P I^\\infty _m}{P+P_\\mathrm {sat}} + a_{bg} P $ with $ I^\\infty _m $ the count rate in the limit of large $ P $ , $ P_\\mathrm {sat} $ the saturation power, and $ a_{bg} P $ a linear term describing the contribution from background fluorescence.",
"We find $a_{bg}= 62~(40)\\times 10^3$ cts/(s mW) in free space (in the cavity) for ND4.",
"The obtained values for $ I^\\infty _m $ are given in table REF , where the errors are from the uncertainty of the fit.",
"We observe saturation count rates at the detector of up to $I_{m,c}^\\infty =1.78\\times 10^6$ cts/s, corresponding to a peak spectral density of $2I_{m,c}^\\infty /(\\pi \\kappa )= 54\\times 10^3$ cts/(s GHz).",
"The peak spectral density is more than 20-fold larger than in earlier room-temperature experiments [33], [8], [10], [26], [11].",
"To obtain the photon emission rates (emission rate into a solid angle of $ 4\\pi $ in free space, $I^\\infty _\\mathrm {fs}$ , and into the cavity mode, $ I^\\infty _c$ ) from measured count rates, we account for the collection and detection efficiency in both cases.",
"In free space, the light is collected with an NA 0.55 objective, and the emission is enhanced and directed due to the presence of the mirror.",
"In the cavity case, we consider the outcoupling efficiency through the fiber mirror $\\eta _c$ and the mode matching between the cavity and fiber modes $\\epsilon =0.47$ .",
"Values for $\\eta _c$ vary between 20% and 38% for the investigated emitters, limited by scattering and absorption of the nanodiamonds (see appendix for more details).",
"With an improved sample, this loss channel can be avoided.",
"The obtained photon emission rates are given in table REF .",
"In free space, we infer $I^\\infty _\\mathrm {fs}$ of up to 16 MHz for ND2, while in the cavity, we find a rate $ I^\\infty _c$ of more than 28 MHz.",
"From the ratio of the two rates, we obtain values for the Purcell factor of up to $ C_\\mathrm {exp} =9.2$ (ND5).",
"This corresponds to an efficiency to collect the emitted photons with the cavity mode of $ \\beta =90\\% $ .",
"The stated errors stem from the uncertainties of the fit of the saturation curves, and do not include further uncertainties.",
"We can also infer the enhancement of the spectral density compared to free space emission, $C_\\mathrm {exp} Q_c/Q_\\mathrm {em}$ , yielding a value of 237 for ND5.",
"We compare $C_\\mathrm {exp}$ to the expected maximal Purcell factor $C_\\mathrm {th}$ as calculated from $Q_c,Q_\\mathrm {em}$ , and $V_m$ for the respective emitters.",
"In the calculation, we obtain $Q_\\mathrm {em}$ from the linewidth of the measured emission spectra and calculate $Q_c$ for the respective emission wavelength.",
"Furthermore, we assume optimal coupling conditions, i.e.",
"$\\eta _E\\equiv \\left|\\frac{\\vec{\\mu }\\vec{E}}{\\mu E_0}\\right|^2=1$ as an upper bound.",
"The ideal value is almost reached for ND5, but the experimental results stay well below the ideal enhancement for the other three emitters (see table II).",
"This is explained by an unfavorable position of the emitter within the crystal, or a non-ideal dipole orientation, leading to $\\eta _E<1$ .",
"As those factors also enter the collection efficiencies, the values of $C_\\mathrm {exp}$ contain additional uncertainties.",
"From $C_\\mathrm {exp}$ and the lifetime change $ \\tau _0/\\tau _c $ , we can coarsely estimate the quantum efficiency for ND1 and 2, and find $\\mathrm {QE} \\approx 7\\%$ and 25%, respectively.",
"The former is comparable to previously published values [26], [34], the latter appears inconsistently high (see appendix ).",
"A low QE can also explain the low emission rate of ND1 despite its short lifetime.",
"Notably, the Purcell effect leads to an increased QE.",
"One finds that the QE in the cavity, $ \\mathrm {QE}_c $ , relates to the free-space QE via $ \\mathrm {QE}_c=(C+1)/(C+1/\\mathrm {QE}) $ , such that e.g.",
"for ND1, the QE increases from 7% to 30% in the cavity.",
"The overall device efficiency, which states the probability to obtain a photon in the fiber after excitation of the emitter, is given by $\\beta _\\mathrm {tot}= \\mathrm {QE}_c \\beta \\eta _c \\epsilon $ , and we obtain $\\beta _\\mathrm {tot}= \\mathrm {QE}_c\\times 16\\%$ for ND5.",
"Figure: (a) Free space g (2) g^{(2)} measurement of ND1 using pulsed excitation (50 ps pulse duration, 20 MHz repetition rate, wavelength 690 nm).",
"Red curve: data.",
"Blue histogram: Integrated over each peak, background subtracted.",
"(b) Free space g (2) g^{(2)} measurement (cw excitation at 690 nm) of ND1 for selected excitation powers.",
"Inset: Zoom into data for 0.17P sat 0.17 P_{sat} .",
"Black solid lines: Fit with g (2) g^{(2)} function convoluted with system response function.",
"Orange solid line: g (2) g^{(2)} function without convolution.",
"(c) Fit parameters as a function of power and fit with power-dependent deshelving model (solid lines).",
"Dashed: Fit with model from .To prove that the emitters show single photon emission, we measure the intensity correlation function $ g^{(2)}(\\tau ) $ with a Hanbury-Brown-Twiss (HBT) setup.",
"Figure REF a shows a pulsed $ g^{(2)} $ measurement yielding a value $ g^{(2)}(0) = 0.14 $ for zero time delay (see appendix for data of the other emitters).",
"For continuous wave excitation (Fig.",
"REF b), we observe a power-dependent photon bunching for intermediate time delays consistent with previous observations (see e.g.",
"[34]), which can be attributed to a meta-stable shelving state [31].",
"The dynamics of such a three level system is described by $g^{(2)}(\\tau ) = 1-(1+a) e^{-|\\tau |/\\tau _1} + a e^{-|\\tau |/\\tau _2},$ which fits the data well.",
"Since the shelving state might significantly reduce the achievable excited state population $n_2^\\infty $ and emission rate, it is important to understand the internal rate dynamics.",
"Therefore, we measure the $ g^{(2)} $ function for various powers both in the cavity and in free space and fit with Eq.",
"REF including uncorrelated background.",
"The fit parameters $ \\tau _1 $ , $ \\tau _2 $ , and $ a $ are given in Fig.",
"REF c as functions of excitation power.",
"We find that the antibunching time constant $ \\tau _1 $ is smaller in the cavity, as expected.",
"Note that $ \\tau _1(0) $ is equivalent to the spontaneous emission lifetime of the system being measured.",
"We observe a strong power dependence of the bunching time constant $ \\tau _2 $ [34], which is equally large in the cavity and in free space.",
"However, the proposed model for an intensity-dependent deshelving ([34], dashed line) does not accurately describe the data.",
"Rather than approaching a finite value, $ \\tau _2 $ converges to zero for large powers.",
"This can be explained by a revised model, which allows linearly power dependent excitation to a higher lying state both from the exited state and the shelving state (see appendix ).",
"An ionization process could explain these dynamics.",
"For the emitter studied, the model yields a rather large value $n_2^\\infty =0.34$ , and the short $ \\tau _2 $ at high power indicates the possibility of high repetition rates for pulsed excitation."
],
[
"Conclusion",
"We have shown significant Purcell enhancement of the single photon emission of SiV centers, achieving high efficiency ($\\beta =90\\%$ ), a high photon rate coupled into a single-mode fiber (4.1 MHz), and a narrow linewidth (21 GHz) at room temperature.",
"Several emitters show excited state lifetimes below 1 ns and bunching time constants that decay quickly with power, such that pulsed excitation at GHz rates is possible.",
"We have introduced a revised rate model to accurately describe the power dependent dynamics of the SiV center.",
"The reported experiments were limited by properties of the sample, such as excessive scattering and absorption loss, photobleaching of the emitters after excitation times ranging from minutes to weeks, and a moderate quantum efficiency.",
"On the cavity side, smaller mode volume and higher quality factor cavities have been fabricated and promise Purcell factors of about 40 for 1 nm emitter linewidth, and outcoupling efficiencies up to $\\eta _c=97\\%$ are achievable for small nanodiamonds.",
"For an improved SiV sample with $\\mathrm {QE}=30\\%$ [26] coupled to an optimized cavity, our approach has the prospect to achieve a device efficiency of $\\beta _\\mathrm {tot}=90\\%$ and yield single photon rates beyond 1 GHz.",
"Furthermore, using a high-$Q$ cavity, spectral purity can be improved to a level where indistinguishable single photons could be produced under ambient conditions [43], meeting the challenging requirements for all-optical quantum computation [44], [45]."
],
[
"Cavity properties",
"The fiber is coated with a dielectric mirror with a transmission of 1500 ppm at a center wavelength of 740 nm and 2500 ppm at the excitation wavelength of 690 nm, such that no excitation light enters the fiber to avoid fiber fluorescence.",
"The planar mirror has a coating centered at 780 nm with a transmission of 60 ppm (200 ppm at 740 nm) and is designed to yield an electric field maximum 30 nm above the mirror surface.",
"The coating is almost transparent at 690 nm, and we focus the excitation light into the cavity with an aspheric lens through the planar mirror.",
"The asymmetry in transmission leads to about 90% of the fluorescence light being emitted into the fiber, which is our collection channel.",
"Figure REF a shows a simulation of the fiber and mirror coating using the transfer matrix method.",
"Figure: (a) Computed mirror transmission of the fiber (yellow) and the planar mirror (blue).",
"(b) Computed finesse assuming the above mirror transmissions and 40 ppm absorption losses.Assuming a total absorption loss at the mirrors of 40 ppm, this yields the cavity finesse as shown in Fig.",
"REF b.",
"The measured finesse of 2000 at 780 nm coincides well with the computation such that it is a fair assumption that the computed value of 3750 at 740 nm is also reached.",
"The mirror is held by a gimbal mount for angular alignment and can be reproducibly moved in all three directions with a nanopositioning stage."
],
[
"Sample scans",
"The mirror can be shifted between an objective and the fiber using a single nanopositioning stage (attocube ECS3030), such that the same area can be investigated both with a confocal microscope and in the cavity.",
"Figure REF a shows a fluorescence map containing a narrow line single emitter (circle).",
"Then, a scanning cavity microscopy map of the same area is recorded (see Fig.",
"REF b), where nanodiamonds and other nanoparticles show up as dark spots.",
"A calibration of the offset between the confocal focus and the fiber enables us to quickly switch between the two observation methods and easily find a pre-characterized emitter in the cavity.",
"The extinction induced by a chosen emitter enables optimization of the spatial overlap with the cavity mode.",
"Figure: (a) Confocal fluorescence map showing the count rate in MHz.",
"The marked spot is a narrow line emitter.",
"(b) Cavity transmission scan of the same area (transmission normalized to 1) for a cavity length of about 10μ10~\\mu m. The marked spot is the same location as in a)."
],
[
"Extended data",
"In this paragraph, we show the complete set of emitter properties as extracted from the free space spectra and an exemplary comparison of saturation measurements in free space and in the cavity.",
"Table: Overview of considered nanodiamonds.",
"λ \\lambda : center wavelength of line.",
"δ \\delta : FWHM of emission spectrum.",
"Q em =λ/δ Q_\\mathrm {em} = \\lambda /\\delta .",
"The finesse at λ \\lambda was used to compute Q eff Q_\\mathrm {eff} and C eff C_\\mathrm {eff} .",
"g (2) (0) g^{(2)}(0) : Minimum of g (2) g^{(2)} function obtained from pulsed measurement.",
"Exception: Value for ND5 was obtained from cw measurement.Figure: (a) Free space saturation measurement of ND4.",
"Blue: data.",
"Red: fit.",
"(b) Cavity measurement of ND4.Table REF gives an overview of all emitters considered including their center wavelength and linewidth as obtained from a Lorentzian fit.",
"The cavity finesse was calculated for the wavelengths of all emission lines from the coating parameters.",
"Together with the width $ \\delta $ of the lines, $ Q_\\mathrm {em} = \\lambda /\\delta $ and a theoretical value for $ C_\\mathrm {eff} $ were computed and are also given in table REF .",
"It includes the branching factor $\\zeta =0.8$ of the ZPL.",
"The values are upper bounds for the case of optimal position and dipole orientation of the emitter, i.e.",
"$\\eta _E\\equiv \\left|\\frac{\\vec{\\mu }\\vec{E}}{\\mu E_0}\\right|^2=1$ .",
"The minimum values of the $ g^{(2)} $ function are obtained using pulsed excitation at 690 nm (pulse length 50 ps) strongly saturating the emitters.",
"They are thus to be seen as an upper bound as the background increases linearly with power and therefore contributes more when $ P > P_{sat} $ increases.",
"ND5 is an exception as it underwent photobleaching before pulsed measurements could be taken.",
"Here, we give the minimum of the fit to a continuous wave 690 nm $ g^{(2)} $ measurement.",
"Although being slightly larger than 0.5, the data is consistent with a single quantum emitter when taking into account the background fluorescence during this measurement (about 30%).",
"Figure REF shows a comparison of a saturation measurement in free space and the cavity together with a fit containing a linear background.",
"We find that the background is dominated by broadband fluorescence from the crystal, which greatly varies between crystals, and in the cavity, the background is suppressed.",
"The given count rate is the actually detected rate."
],
[
"Fitting of lifetime data",
"For obtaining the excited state lifetime, we apply different fitting methods to the data and give the half min-max deviation as a conservative error of the fit (see table REF ).",
"First, we fit the whole trace, i.e.",
"an exponential decay function starting at delay time zero and including a constant background, convoluted with the system response function.",
"For some emitters, a second exponential function has to be included to account for a fast decaying background.",
"This is mostly necessary for the free space time trace.",
"Next, only the decay is fitted to avoid systematic errors due to the positioning of the zero time delay.",
"Last, we fit the section of exponential decay only."
],
[
"Calculation of the photon emission rate from the count rate",
"To obtain the Purcell factor, one has to compare the photon emission rate into the cavity mode to the free space photon emission rate into $ 4\\pi $ .",
"In order to trace back to those rates from the count rates on the detector, the collection and detection efficiency have to be accounted for.",
"The detection efficiency of the APDs is $ \\eta _{det} = 70\\% $ for the emitter wavelengths.",
"From a measurement, we find that $ \\eta _{trans} = 64\\% $ of the fluorescence light is transmitted through the optics (including filters) from the first lens to the detector.",
"It is a fair assumption that this value is the same for both the free space and cavity situation as the light travels the same path.",
"As the nanodiamonds reside on the planar mirror even in the confocal measurement, the local density of states is different as compared to free space.",
"Here, we are not interested in the collected fraction of actually emitted light, but rather in the fraction of light collected as compared to free space emission into $ 4\\pi $ .",
"To obtain this value, we compute the complex reflectivity $ r_{s,p} $ of the dielectric mirror stack for both s- and p-polarization using the transfer matrix method [46], [47].",
"The power radiated by a dipole is given by $P_s(\\alpha ,\\theta ,\\phi ) = \\frac{3}{8\\pi } \\sin ^2 \\theta \\sin ^2 \\phi $ for s-polarization and $P_p(\\alpha ,\\theta ,\\phi ) = \\frac{3}{8\\pi } (\\cos \\theta \\sin \\alpha + \\sin \\theta \\cos \\alpha \\cos \\phi )^2$ for p-polarization, where $ \\alpha $ the azimuthal angle between the optical axis and the emission direction, $ \\phi $ the polar angle, and $ \\theta $ the angle between the dipole and the optical axis.",
"The total power emitted by the dipole on the mirror into a certain direction normalized to free space power is the interference of the directly emitted and reflected light in both polarizations [48]: $P_{em}(\\alpha ,\\theta ,\\phi ) = \\sum _{i\\in \\lbrace s,p\\rbrace } \\left| 1+|r_i|\\exp (2k\\cos \\alpha z_0 + \\arg r_i) \\right| ^2 \\\\ \\cdot P_i(\\alpha ,\\theta ,\\phi )$ The term $ 2k\\cos \\alpha z_0 $ with $ k $ the wave number, is an additional phase acquired if the dipole has a distance $ z_0 $ from the surface of the mirror.",
"Integrating this quantity over the solid angle given by the numerical aperture ($ \\mathrm {NA} = 0.55 $ ) yields the desired collection efficiency: $\\eta _{coll}(\\theta ) = \\int _{0}^{\\arcsin \\mathrm {NA}} \\int _{0}^{2\\pi } P_{em}(\\alpha ,\\theta ,\\phi ) \\sin \\alpha \\mathop {}\\!\\mathrm {d}\\phi \\mathop {}\\!\\mathrm {d}\\alpha $ For a dipole oriented parallel to the mirror surface, we obtain $ \\eta _{coll}(\\pi /2) = 45\\% $ , for perpendicular orientation $ \\eta _{coll}(0) = 8\\% $ .",
"As we choose very bright emitters for the experiment, there is a bias towards those with optimal, i.e.",
"parallel, dipole orientation.",
"We therefore use $ \\eta _{coll}(\\pi /2) $ as a good estimate for the collection efficiency.",
"This value carries an additional uncertainty, as we do not know the exact position of the dipole within the crystal.",
"Again, we assume that we preselect diamonds in which the dipole resides close to the field maximum as these will appear brighter.",
"The objective has a transmission of about $ \\eta _{obj} = 80\\% $ for the fluorescence wavelength.",
"In summary, the free space photon emission rate into $ 4\\pi $ can be calculated by $I_{fs}^\\infty = I_{m,fs}^\\infty /(\\eta _{det} \\eta _{trans} \\eta _{coll} \\eta _{obj}).$ Table: T/T 0 T/T_0 : extinction.",
"η c \\eta _c : outcoupling.",
"C exp C_{\\mathrm {exp}}: experimental Purcell factor.",
"β=C/(C+1) \\beta = C/(C+1): fraction of total emission into the cavity mode.",
"βη c \\beta \\eta _c: fraction of emission coupled out of the cavity.In the cavity case, we are interested in what part $ \\eta _c $ of the emitted light is coupled out through the fiber mirror, which is our collection channel.",
"It is given by the transmission of this mirror divided by all losses: $\\eta _c = T_f/(T_f+T_p+A+L),$ where $ T_f $ ($ T_p $ ) is the transmission of the fiber (planar) mirror, $ A $ the absorption and scattering losses of both mirrors, and $ L $ the extinction losses due to absorption and scattering of the nanocrystal.",
"$ L $ can be calculated from the extinction $ T/T_0 $ , which is given in table REF .",
"To obtain the extinction, we measure the cavity transmission and normalize it to the transmission of the empty cavity.",
"The cavity transmission is given by $T= \\frac{4T_s T_f}{(T_s+T_f+A+L)^2},$ so $ L $ can be determined by solving the following equation: $(T_0/T) (T_s+T_f+A)^2 = (T_s+T_f+A+L)^2$ As the light is collected through the fiber, one has to take into account the mode matching between the cavity mode and the mode guided by the fiber, whose mode field radius $ w_f $ is 2.5 µm.",
"For the situation that the modes are coaxial (for a well-centered fiber profile and good angular alignment), the power coupling efficiency can be computed as $\\epsilon = \\frac{4}{\\left(\\frac{w_f}{w_0}+\\frac{w_0}{w_f}\\right)^2+\\left(\\frac{s \\lambda }{\\pi w_0 w_f}\\right)^2},$ where $ w_0 $ is the mode waist and $ s = d + d_{mirror} $ is the optical distance between the two mode waists [49], [36].",
"The latter is composed of the geometric mirror separation $ d $ and the optical thickness of the fiber's dielectric mirror stack $ d_{mirror} $ .",
"The mode waists are calculated for all cavity lengths by optimizing the Gaussian mode for the given fiber profile (for details see [50], [38]).",
"For the lowest achievable cavity length $ d_\\mathrm {eff} = 5 \\lambda /2 $ , we obtain $ w_0 = 1.00 $ µm.",
"This leads to a mode matching $ \\epsilon = 46.6 \\% $ .",
"Note that this value is an upper boundary as already a slight misalignment of the fiber profile with respect to the fiber core can significantly reduce the mode matching.",
"Therefore, using $ \\epsilon $ for calculating the photon emission rate yields a conservative estimate.",
"At the glass-air-interface at the outcoupling port of the fiber, another 4% get lost, and we get an additional factor $ \\eta _{fiber} = 96\\% $ .",
"The photon emission rate into the cavity can then be calculated from the count rate as $I_{cav}^\\infty = I_{m,cav}^\\infty /(\\eta _{det} \\eta _{trans} \\eta _c \\eta _{fiber} \\epsilon ).$ Due to the large uncertainties of some of the factors, the obtained photon emission rates and Purcell factors are an estimate.",
"Table REF also gives the efficiency $ \\beta =C/(C+1) $ , which is the fraction of the total emission into the cavity mode.",
"We obtain values up to 90%.",
"The fraction of the light actually coupled out of the cavity $ \\beta \\eta _c $ is given in the last column.",
"Note that this could be significantly improved by choosing a sample with less extinction, i.e.",
"smaller crystal size and better crystal quality.",
"The photon emission rates given only include the emission into the cavity mode, i.e.",
"$ I^\\infty _{cav}=\\beta I^\\infty _\\mathrm {tot} $ where $ I^\\infty _\\mathrm {tot} $ would be the total rate into $ 4\\pi $ .",
"As $ I^\\infty _\\mathrm {tot} = (C+1) I^\\infty _{fs} $ , the ratio of cavity and free space emission rate yields just $ C $ : $I^\\infty _{cav}=\\beta I^\\infty _\\mathrm {tot}=\\beta (C+1) I^\\infty _{fs}=C I^\\infty _{fs}$"
],
[
"Theoretical Purcell factor calculation",
"The theoretical Purcell factors as stated in Table REF are calculated as $C_\\mathrm {eff} = \\frac{3(\\lambda /n)^3}{4\\pi ^2}\\frac{Q_\\mathrm {eff}}{V_m} \\zeta \\left|\\frac{\\vec{\\mu }\\vec{E}}{\\mu E_0}\\right|^2$ with $Q_\\mathrm {eff}=(Q_c^{-1}+Q_\\mathrm {em}^{-1})^{-1}$ being the effective quality factor, $ \\zeta = 80\\%$ , and assuming ideal dipole location and orientation, i.e.",
"$\\left|\\frac{\\vec{\\mu }\\vec{E}}{\\mu E_0}\\right|^2=1$ .",
"The mode volume $ V_m $ is calculated using the effective cavity length and the optimized mode waist.",
"$ Q_c $ is adjusted individually for all emitters as the additional extinction loss $ L $ alters the finesse on the emitter as compared to the bare cavity.",
"This is however not very critical, as $ Q_c \\gg Q_\\mathrm {em} $ still holds.",
"The most important quantity is $ Q_\\mathrm {em} = \\lambda /\\delta \\lambda $ and is obtained from the measured linewidth $ \\delta \\lambda $ .",
"The resulting effective Purcell factors set an upper limit for the achievable experimental values for the case the dipole is oriented parallel to the mirror surface and resides in the maximum of the electric field.",
"In reality, this does not have to be the case and the effects on the Purcell factor are discussed here.",
"Figure: Correction factor for Purcell factor due to the emitter's position relative to the electric field maximum.",
"Red line: for a color center residing in the middle of a 100 nm large diamond.",
"Shaded area: uncertainty of the correction factor due to the unknown position of the emitter.The planar mirror, on which the sample is placed, is a dielectric Bragg reflector at a central wavelength of 780 nm.",
"It is capped with a spacer layer of $ \\mathrm {SiO_2} $ such that the field maximum for 780 nm lies 30 nm above the surface.",
"Using the complex reflectivity $ r $ of the planar mirror, $ |1+r^2| $ gives the field intensity normalized to the value without a mirror.",
"Dividing this by the intensity maximum, we obtain a correction factor $|E/E_0|^2$ for the Purcell factor, which is plotted for different wavelengths in Fig.",
"REF .",
"The red line is computed assuming the color center resides in the middle of a 100 nm large diamond.",
"The shaded area gives the uncertainty of the correction factor due to the unknown position of the emitter.",
"As an example, at a wavelength of 754 nm, the correction factor can range from 56% to almost 100%.",
"So an unfortunate position of the color center within the crystal can decrease the Purcell factor to half its ideal value.",
"The second unknown quantity is the orientation of the dipole moment.",
"By maximizing the count rate, we align the polarization of the electric field with the in-plane component of the dipole moment.",
"Therefore, only the out of plane component remains unknown, which can be characterized by an angle $ \\phi = \\angle (\\vec{\\mu },\\vec{E}) $ .",
"Considering equation REF , we find that $ C \\propto \\sin ^2 \\phi $ .",
"Therefore, the Purcell factor goes down to zero for $ \\vec{\\mu } \\perp \\vec{E} $ .",
"But as the excitation goes likewise down, emitters with unfavorable orientation of $ \\vec{\\mu } $ are not bright and are likely not to be preselected."
],
[
"Rate dynamics",
"We assume a level scheme as depicted in Fig.",
"REF a, where 1 is the ground, 2 the excited, and 3 the shelving state.",
"The rate $ k_{12} $ is linearly dependent on the excitation power as we excite off-resonantly with subsequent fast relaxation into the phononic ground state.",
"A three level system with otherwise constant rates leads to a constant bunching time constant $ \\tau _2 $ [34], which is in contradiction with observation.",
"We attribute this deviation to excitation from levels 2 and 3 to some higher lying state or the valance band.",
"The excitation rates are also assumed to be linearly dependent on power.",
"All deexcitation rates are constant.",
"The cavity is resonant with transition $ 2\\rightarrow 1 $ .",
"For easier mathematical treatment, an equivalent level scheme is presented in Fig.",
"REF b leading to the same dynamics.",
"Here, green arrows indicate constant rates, red arrows rates that are linearly dependent on power and the blue rate $ k_{32} $ has both a linear and a constant term: Figure: (a) Proposed level scheme.",
"Green: constant rates.",
"Red: linearly power dependent rates.",
"(b) Equivalent three level system.",
"Blue: rate with linear and constant term.$k_{12} &= \\sigma P\\\\k_{32} &= d P\\\\k_{23} &= e P + k_{23}^0,$ where $ \\sigma $ , $ d $ , and $ e $ are proportionality constants.",
"The $ g^{(2)} $ function for this three level system is given by $g^{(2)}(\\tau ) = 1-(1+a) e^{-|\\tau |/\\tau _1} + a e^{-|\\tau |/\\tau _2},$ with $\\tau _{1,2} &= 2/(A \\pm \\sqrt{A^2-4B}) \\\\a &= \\frac{1-\\tau _2(k_{31}+k_{32})}{(k_{31}+k_{32})(\\tau _2-\\tau _1)}$ and $A &= k_{12} + k_{21} + k_{23} + k_{31} + k_{32} \\\\\\begin{split}B &= k_{23} k_{31} + k_{21} (k_{31} + k_{32})\\\\&+ k_{12} (k_{23} + k_{31} + k_{32}).\\end{split}$ Like in [34], we express the constant rates by values for the parameters at zero power: $k_{21} &= 1/\\tau _1^0 - k_{23}^0 \\equiv \\gamma _r + \\gamma _{nr}\\\\k_{31} &= 1/\\tau _2^0.$ In the limit of large powers, both time constants converge to zero.",
"The bunching parameter $ a $ is zero for zero power and reaches a finite value $ a^\\infty $ for large powers.",
"We use it for fitting instead of the proportionality constant $ e $ as the starting value is easier to choose: $e = \\frac{-a^\\infty d^2+ a^\\infty d \\sigma }{a^\\infty d + \\sigma }.$ In Section III, we compare the model with data taken from ND1.",
"We perform a global fit of six data sets for $g^{(2)}(\\tau )$ taken at different excitation power for $ a $ , $ \\tau _1 $ , and $ \\tau _2 $ both in free space and in the cavity.",
"The fit has eight free parameters all together ($ d $ , $ d_c $ , $ \\sigma $ , $ \\sigma _c $ , $ a^\\infty $ , $ a^\\infty _c $ , $ \\tau _2^0 $ , and $ k_{23}^0 $ ) as $ \\tau _2^0 $ and $ k_{23}^0 $ are the same in both cases.",
"From these parameters, one can calculate the equilibrium population of the excited state $ n_2^\\infty $ and finally the total deexcitation rate $ \\Gamma $ : $&n_2 = \\frac{k_{12}(k_{31}+k_{32})}{k_{23}k_{31}+k_{21}(k_{31}+k_{32})+k_{12}(k_{23}+k_{31}+k_{32})}\\nonumber \\\\&n_2^\\infty =\\lim _{P\\rightarrow \\infty }n_2 = \\frac{1}{1+e/d}\\\\&\\Gamma = n_2^\\infty k_{21}$ We obtain $ n_2^\\infty = 34\\% $ for ND1.",
"In free space (in the cavity), we get $ k_{21} = 1.7 $ GHz (2.2 GHz) and $ \\Gamma = 578 $ MHz (750 MHz), comparable to the values found in [34].",
"$ \\Gamma $ includes non-radiative deexcitation, such that it should be possible to extract the QE by a comparison with the measured photon emission rate $ I_{fs}^\\infty $ .",
"This yields a value for the QE of 0.1% to 0.5% depending on the orientation of dipole, significantly less than the 7% estimated from the experimental Purcell factor and the lifetime change.",
"The origin of this discrepancy remains unclear at this stage.",
"The emitter might show different levels of blinking at intermediate timescales, or have jumped into another state with a different emission rate between the measurements (both are phenomena we occasionally observe)."
],
[
"Sample properties",
"As previously observed [34], the used sample contains a small fraction of photostable emitters as well as emitters which feature blinking (see e.g.",
"Fig.",
"REF ) on different timescales (ranging from less than seconds to hours) and permanent photobleaching.",
"The latter can occur after illumination times from seconds to several weeks and is more probable for higher excitation powers.",
"This has prevented us from obtaining complete data sets for all emitters.",
"New samples [51], however, promise better emitter photostability.",
"Figure: Fluorescence timetrace (count rate on detector) of single emitter featuring blinking.The large scattering and absorption losses reduces the number of useful emitters for investigation.",
"It is therefore crucial to use smaller diamonds.",
"Absorption can be caused by sp2 hybridized carbon at surfaces, grain boundaries and lattice defects.",
"A higher crystal quality would thus reduce absorption and background fluorescence.",
"This could for example be obtained by annealing of the nanodiamonds under oxygen atmosphere.",
"We thank Roland Albrecht, Elke Neu, and Matthias Mader for contributions to the experiment.",
"Fruitful discussions with John Rarity are acknowledged.",
"The work has been funded by the European Union 7th framework Program under grant agreement no.",
"61807 (WASPS), the DFG Cluster of Excellence NIM, and the DFG project FOR1493.",
"T. W. Hänsch acknowledges funding from the Max-Planck Foundation."
]
] | 1612.05509 |
[
[
"3d steady Gradient Ricci Solitons with linear curvature decay"
],
[
"Abstract In this note, we prove that a 3-dimensional steady Ricci soliton is rotationally symmetric if its scalar curvature $R(x)$ satisfies $$\\frac{C_0^{-1}}{\\rho(x)}\\le R(x)\\le \\frac{C_0}{\\rho(x)}$$ for some constant $C_0>0$, where $\\rho(x)$ denotes the distance from a fixed point $x_0$.",
"Our result doesn't assume that the soliton is $\\kappa$-noncollapsed."
],
[
"Introduction",
"In his celebrated paper [13], Perelman conjectured that all 3-dimensional $\\kappa $ -noncollapsed steady (gradient) Ricci solitons must be rotationally symmetric.",
"The conjecture is solved by Brendle in 2012 [1].",
"For a general dimension $n\\ge 3$ , under an extra condition that the soliton is asymptotically cylindrical, Brendle also proves that any $\\kappa $ -noncollapsed steady Ricci soliton with positive sectional curvature must be rotationally symmetric [2].",
"In general, it is still open whether an $n$ -dimensional $\\kappa $ -noncollapsed steady Ricci soliton with positive curvature operator is rotationally symmetric for $n\\ge 4$.",
"For $\\kappa $ -noncollapsed steady Kähler-Ricci solitons with nonnegative bisectional curvature, the authors have recently proved that they must be flat [10], [11].",
"Recall from [2], Definition 1.1 An $n$ -dimensional steady Ricci soliton $(M,g,f)$ is called asymptotically cylindrical if the following holds: (i) Scalar curvature $R(x)$ of $g$ satisfies $\\frac{C_0^{-1}}{\\rho (x)} \\le R(x) \\le \\frac{C_0}{\\rho (x)},~\\forall ~\\rho (x)\\ge r_0, $ where $C_0>0$ is a constant and $\\rho (x)$ denotes the distance of $x$ from a fixed point $x_0$ .",
"(ii) Let $p_m$ be an arbitrary sequence of marked points going to infinity.",
"Consider rescaled metrics $ g_m(t) = r_m^{-1} \\phi ^*_{r_m t} g,$ where $r_m R(p_m) = \\frac{n-1}{2} + o(1)$ and $ \\phi _{ t}$ is a one-parameter subgroup generated by $X=-\\nabla f$ .",
"As $m \\rightarrow \\infty ,$ flows $(M, g_m(t), p_m)$ converge in the Cheeger-Gromov sense to a family of shrinking cylinders $( \\mathbb {R} \\times \\mathbb {S}^{n-1}(1), \\widetilde{g}(t)), t \\in (0, 1).$ The metric $\\widetilde{g}(t)$ is given by $ \\widetilde{g}(t) = dr^2+ (n - 2)(2 -2t) g_{\\mathbb {S}^{n-1}(1)},$ where $\\mathbb {S}^{n-1}(1)$ is the unit sphere of euclidean space.",
"In this note, we discuss 3-dimensional steady (gradient) Ricci solitons without assuming the $\\kappa $ -noncollapsed conditon.It is proved by Chen that any 3-dimensional ancient solution has nonnegative sectional curvature [7].",
"We prove Theorem 1.2 Let $(M,g,f)$ be a 3-dimensional steady Ricci soliton.",
"Then, it is rotationally symmetric if the scalar curvature $R(x)$ of $(M,g,f)$ satisfies $\\frac{C_0^{-1}}{\\rho (x)}\\le R(x)\\le \\frac{C_0}{\\rho (x)},$ for some constant $C >0$ , where $\\rho (x)$ denotes the distance from a fixed point $x_0$ .",
"Under the condition (REF ), we need to check the property (ii) in Definition REF to prove Theorem REF .",
"Actually, we show that for any sequence $p_{i}\\rightarrow \\infty $ , there exists a subsequence $p_{i_{k}}\\rightarrow \\infty $ such that $(M,g_{p_{i_{k}}}(t),p_{i_{k}})\\rightarrow (\\mathbb {R}\\times \\mathbb {S}^{2},\\widetilde{g}(t),p_{\\infty }),~for~t\\in (-\\infty ,1),$ where $g_{p_{i_{k}}}(t)=R(p_{i_{k}})g(R^{-1}(p_{i_{k}})t)$ and $(\\mathbb {R}\\times \\mathbb {S}^{2},\\widetilde{g}(t))$ is a shrinking cylinders flow, i.e.",
"$\\widetilde{g}(t)=dr^2+(2-2t)g_{\\mathbb {S}^{2}}.$ As in [10], we study the geometry of neighborhood $M_{p,k}=\\lbrace x\\in M| ~ f(p)-\\frac{k}{\\sqrt{R(p)}}\\le f(x)\\le f(p)+\\frac{k}{\\sqrt{R(p)}}\\rbrace $ around level set $\\Sigma _r=\\lbrace f(x)=f(p)=r\\rbrace $ for any $p\\in M$ .",
"We are able to give a uniform injective radius estimate for $(M,R_{p_i}g)$ at each sequence of $p_i$ .",
"Then we can still get a limit flow for rescaled flows $(M,g_{p_{i}}(t))$ , which will split off a line.",
"By using a classification result of Daskalopoulos-Hamilton-Sesum for ancient flows on a compact surface [8], we finish the proof of Theorem REF .",
"We remark that the curvature condition in Theorem REF cannot be removed, since there does exist a 3-dimensional non-flat steady Ricci soliton with exponential curvature decay.",
"For example, $(\\mathbb {R}^2\\times \\mathbb {S}^{1},g_{cigar}+ds^2)$ , where $(\\mathbb {R}^2,g_{cigar})$ is a cigar soliton.",
"Also, Theorem REF is not true for dimension $n\\ge 4$ by Cao's examples of steady Kähler-Ricci solitons with positive sectional curvature [3].",
"At last, we remark that it is still open wether there exists a 3-dimensional collapsed steady Ricci soliton with positive curvature.",
"Hamilton has conjectured that there should exist a family of collapsed 3-dimensional complete gradient steady Ricci solitons with positive curvature and $S^1$ -symmetry (cf.",
"[5]).",
"Our result shows that the curvature of Hamilton's examples could not have a linear decay.",
"Acknowledgements.",
"The work was done partially when the second author was visiting at the Mathematical Sciences Research Institute at Berkeley during the spring 2016 semester.",
"The author would like to thank her hospitality and the financial supports, National Sciences Foundation under Grant No.",
"DMS-1440140, and Simons Foundation."
],
[
"Positivity of Ricci curvature",
"$(M,g,f)$ is called a gradient steady Ricci soliton if a Riemannian metric $g$ on $M$ satisfies ${\\rm Ric}(g)=\\nabla ^2 f,$ for some smooth function $f$ .",
"We first show the positivity of Ricci curvature of $(M,g,f)$ under (REF ) of Theorem REF .",
"Lemma 2.1 Under (REF ), $(M,g,f)$ has positive sectional curvature.",
"We need to show that $(M,g)$ has positive Ricci curvature.",
"On the contrary, $(M,g)$ locally splits off a flat piece of line by Shi's splitting theorem [14].",
"Then, the universal cover $(\\widetilde{M},\\widetilde{g})$ of $(M,g)$ is isometric to a product Riemannian manifold of a real line and the cigar soliton.",
"Namely, $(\\widetilde{M},\\widetilde{g})=(\\mathbb {R}^2\\times \\mathbb {R},g_{cigar}+ds^2)$ .",
"Let $\\pi :\\widetilde{M}\\rightarrow M$ be a universal covering.",
"We fix $x_0\\in M$ and $\\widetilde{x}_0\\in \\widetilde{M}$ such that $\\pi (\\widetilde{x}_0)=x_0$ .",
"For any $x\\in M$ and $\\widetilde{x}\\in \\widetilde{M}$ such that $\\pi (\\widetilde{x})=x$ , one sees $\\rho (x,x_0)\\le \\widetilde{\\rho }(\\widetilde{x},\\widetilde{x}_0),$ where $\\rho $ and $\\widetilde{\\rho }$ are the distance functions w.r.t $g$ and $\\widetilde{g}$ respectively.",
"Let $\\lbrace \\widetilde{x}_i\\rbrace _{i\\ge 1}$ be a sequence of points so that $\\widetilde{x}_i=(p_i,0)\\in \\mathbb {R}^2\\times \\mathbb {R}$ tend to infinity.",
"Then, one may check that $\\widetilde{R}(\\widetilde{x}_i)\\rho (\\widetilde{x}_i,\\widetilde{x}_0)\\rightarrow 0,~{\\rm as}~i\\rightarrow \\infty .$ Since $R(x_i)=\\widetilde{R}(\\widetilde{x}_i)\\rightarrow 0$ , where $x_i=\\pi (\\widetilde{x}_i)$ , we see $d(x_i,x_0)\\rightarrow \\infty $ by (REF ).",
"Again by (REF ) and (REF ), we get $C_1\\le R(x_i)d(x_i,x_0)\\le \\widetilde{R}(\\widetilde{x}_i)d(\\widetilde{x}_i,\\widetilde{x}_0).$ This is a contradiction to (REF ).",
"Hence, the lemma is proved.",
"Corollary 2.2 $(M,g,f)$ in Theorem REF has a unique equilibrium point $o$ , i.e., $ \\nabla f(o)=0$ .",
"As a consequence, $\\Sigma _r=\\lbrace f(x)=r\\rbrace $ is diffeomorphic to $\\mathbb {S}^{2}$ , for any $r>f(o)$ .",
"Note that $|\\nabla f|^2+R=A.$ By taking covariant derivatives on both sides of (REF ), it follows $2{\\rm Ric}(\\nabla f,\\nabla f)=-\\langle \\nabla R,\\nabla f\\rangle .$ On the the hand, by (REF ), there exists a point $o$ such that $\\sup _M R(x)=R(o)=R_{{\\rm max}}.$ In particular, $\\nabla R(o)=0.$ Thus ${\\rm Ric}(\\nabla f,\\nabla f)=0.$ By Lemma REF , $\\nabla f(o)=0$ .",
"The uniqueness also follows from the positivity of Ricci curvature.",
"By the Morse theorem, $\\Sigma _r=\\lbrace f(x)=r>f(o)\\rbrace $ is diffeomorphic to $\\mathbb {S}^{2}$ (cf.",
"[10], Lemma 2.1)."
],
[
"Geometry of $M_{p,k}$",
"For any $p\\in M$ and number $k>0$ , we set $M_{p,k}=\\lbrace x\\in M| ~ f(p)-\\frac{k}{\\sqrt{R(p)}}\\le f(x)\\le f(p)+\\frac{k}{\\sqrt{R(p)}}\\rbrace .$ Let $g_{p}=R(p)g$ be a rescaled metric and denote $B(p,r; g_{p})$ a $r$ -geodesic ball centered at $p$ with respect to $g_{p}$ .",
"Then by Corollary REF , we have (cf.",
"[10], Lemma 3.3) Lemma 3.1 Under (REF ), for any $p\\in M$ and number $k>0$ with $f(p)-\\frac{k}{\\sqrt{R(p)}}>f(o)$ , it holds $B(p,\\frac{k}{\\sqrt{R_{max}}}; g_{p})\\subset M_{p,k}.$ By Lemma REF , we prove Lemma 3.2 Under (REF ), there exists a constant $C$ such that $\\frac{|\\Delta R|(p)}{R^2(p)}\\le C,~\\forall ~p\\in ~M.$ Fix any $p\\in M$ with $f(p)\\ge r_0>>1$ .",
"Then $|f(x)-f(p)|\\le \\frac{1}{\\sqrt{R(p)}},~ \\forall ~x\\in M_{p,1}.$ It is known by [4], $c_1\\rho (x)\\le f(x)\\le c_2 \\rho (x), ~\\forall ~\\rho (x)\\ge r_0.$ Thus by (REF ), (REF ) and (REF ), we get $c_2\\rho (x)\\ge f(p)-\\frac{1}{\\sqrt{R(p)}}\\ge c_1\\rho (p)-\\sqrt{C_0\\rho (p)}.$ It follows $\\frac{R(x)}{R(p)}\\le C_0^2\\frac{\\rho (p)}{\\rho (x)}\\le \\frac{2c_2C_0^2}{c_1},~\\forall x\\in M_{p,1}.$ On the other hand, by (REF ), we have $B(p,\\frac{1}{\\sqrt{R_{\\max }}};g_p)\\subseteq M_{p,1}.$ Hence $R(x)\\le C^{\\prime }R(p),~\\forall ~x\\in B(p,\\frac{1}{\\sqrt{R_{\\max }}};g_p).$ Let $\\phi _t$ be generated by $-\\nabla f$ .",
"Then $g(t)=\\phi _t^{\\ast }g$ satisfies the Ricci flow, $\\frac{\\partial g(t)}{\\partial t} &= -2{\\rm Ric}(g(t)).$ Also rescaled flow $g_p(t)=R(p) g(R^{-1}(p) t)$ satisfies (REF ).",
"Since the Ricci curvature is positive, $B(p,\\frac{1}{\\sqrt{R_{\\max }}};g_p(-t))\\subseteq B(p,\\frac{1}{\\sqrt{R_{\\max }}};g_p(0)),~t\\in ~[-1,0].$ Combining with (REF ), we get $R_{g_p(t)}(x)\\le C^{\\prime },~\\forall ~x\\in B(p,\\frac{1}{\\sqrt{R_{\\max }}};g_p(0)),~t\\in [-1,0].$ Thus, by Shi's higher order estimates, we obtain $|\\Delta _{g_p(t)} R_{g_p(t)}|(x)\\le C_{1}^{\\prime },~\\forall ~x\\in B(p,\\frac{1}{ 2\\sqrt{R_{\\max }}};g_p(-1)),~t\\in [-\\frac{1}{2},0].$ It follows $|\\Delta R|(x)\\le C_{1}^{\\prime }R^2(p),~\\forall ~x\\in B(p,\\frac{1}{2\\sqrt{R_{\\max }}};g_p(-1)).$ In particular, we have $|\\Delta R|(p)\\le C_{1}^{\\prime }R^2(p),~{\\rm as}~\\rho (p)\\ge r_0.$ The lemma is proved.",
"Remark 3.3 Under (REF ), by the same argument as in the proof of Lemma REF , for each $k\\in \\mathbb {N}$ , there exists a constant $C(k)$ such that $\\frac{|\\nabla ^k R|(p)}{R^{\\frac{k+2}{2}}(p)}\\le C(k),~\\forall ~p\\in ~M.$ Next, we want to show that $M_{p,k}$ is bounded by a finite ball $B(p, Ck ; g_p)$ , where $C$ is a uniform constant.",
"We need to use the Gauss formula, $R(X,Y,Z,W)=\\overline{R}(X,Y,Z,W)+\\langle B(X,Z),B(Y,W)\\rangle -\\langle B(X,W),B(Y,Z)\\rangle ,$ where $X,Y,Z,W\\in T\\Sigma _{r}$ and $B(X,Y)=(\\nabla _{X}Y)^{\\bot }$ .",
"Note that $B(X,Y)=&\\langle \\nabla _{X}Y,\\nabla f\\rangle \\cdot \\frac{\\nabla f}{|\\nabla f|^{2}}\\\\=&[\\nabla _{X}\\langle Y,\\nabla f\\rangle -\\langle Y,\\nabla _{X}\\nabla f\\rangle ]\\cdot \\frac{\\nabla f}{|\\nabla f|^{2}}\\\\=&-{\\rm Ric}(X,Y)\\cdot \\frac{\\nabla f}{|\\nabla f|^{2}}.$ We choose a normal basis $\\lbrace e_1,e_2\\rbrace $ on $(\\Sigma _r,\\bar{g})$ with the induced metric $\\bar{g}$ .",
"Then $\\lbrace e_1,e_2,\\frac{\\nabla f}{|\\nabla f|}\\rbrace $ spans a normal basis of $(M,g)$ .",
"Thus $R_{11}=&\\overline{R}_{11}+R(\\frac{\\nabla f}{|\\nabla f|},e_{1},e_{1},\\frac{\\nabla f}{|\\nabla f|})-\\frac{R_{11}R_{22}-R_{12}R_{21}}{|\\nabla f|^{2}},\\\\R_{22}=&\\overline{R}_{22}+R(\\frac{\\nabla f}{|\\nabla f|},e_{2},e_{2},\\frac{\\nabla f}{|\\nabla f|})-\\frac{R_{11}R_{22}-R_{12}R_{21}}{|\\nabla f|^{2}}.$ Since $(\\Sigma _r, \\bar{g})$ is a surface, $K=\\overline{R}_{11}=\\overline{R}_{22}$ .",
"Hence, we get Lemma 3.4 The Gauss curvature of $(\\Sigma _r, \\bar{g})$ is given by $K=\\frac{R}{2}-\\frac{{\\rm Ric}(\\nabla f,\\nabla f)}{|\\nabla f|^2}+\\frac{R_{11}R_{22}-R_{12}R_{21}}{|\\nabla f|^{2}}.$ Lemma 3.5 Under (REF ), there exists a uniform $B>0$ such that the following is true: for any $k\\in \\mathbb {N}$ , there exists $\\bar{r}_0=\\bar{r}_0(k)$ such that $M_{p,k}\\subset B(p,2\\pi \\sqrt{B}+\\frac{2k}{\\sqrt{R_{\\max }}} ; g_p), ~\\forall ~ \\rho (p)\\ge \\bar{r}_0.$ By (REF ) and (REF ), we have $\\frac{R_{\\max }}{2}\\le |\\nabla f|^2(x)\\le R_{\\max },~\\forall ~x\\in M_{p,k},$ as long as $\\rho (p)\\ge r_0>>1.$ Then by Lemma REF and Lemma REF , we get $|K-\\frac{R}{2}|=&|-\\frac{{\\rm Ric}(\\nabla f,\\nabla f)}{|\\nabla f|^2}+\\frac{R_{11}R_{22}-R_{12}R_{21}}{|\\nabla f|^{2}}|\\\\\\le &\\frac{|\\langle \\nabla R,\\nabla f\\rangle |}{2|\\nabla f|^2}+\\frac{R^2}{|\\nabla f|^{2}}\\\\\\le &\\frac{|\\Delta R+2|{\\rm Ric}|^2|}{2|\\nabla f|^2}+\\frac{R^2}{|\\nabla f|^{2}}\\\\\\le &\\frac{(C+4)R^2}{R_{\\max }}.$ It follows $\\frac{R(x)}{4}\\le K(x)\\le \\frac{3R(x)}{4},~\\forall ~x\\in M_{p,k}, ~\\rho (p)\\ge r_0.$ On the other hand, by (REF ), (REF ) and (REF ), we see $c_2^{-1}\\Big (c_1\\rho (p)-k\\sqrt{\\rho (p)C_0}\\Big )&\\le \\rho (x) \\\\&\\le c_1^{-1}(c_2\\rho (p)+k\\sqrt{\\rho (p)C_0}),~\\forall ~x\\in M_{p,k},$ as long as $\\rho (p)\\ge r_0.$ Then similar to (REF ), there exists a $\\bar{r}_0\\ge r_0$ such that $R(x)\\ge \\frac{c_1}{2c_2C_0^2} R(p), ~\\forall ~x\\in M_{p,k}.$ Thus by (REF ), we get $\\overline{R}_{ij}\\ge B^{-1} R(p)\\overline{g}_{ij}, ~\\forall ~x\\in \\Sigma _{f(p)}, ~\\rho (p)\\ge \\bar{r}_0,$ where $B>0$ is a uniform constant.",
"By the Myer's theorem, the diameter of $\\Sigma _{f(p)}$ is bounded by ${\\rm diam}(\\Sigma _{f(p)},g)\\le {\\rm diam}(\\Sigma _{f(p)},\\overline{g}_{f(p)})\\le 2\\pi \\sqrt{\\frac{B}{ R(p)}}.$ As a consequence, $\\Sigma _{f(p)}\\subset B(p,2\\pi \\sqrt{B}; R(p)g).$ For any $q\\in M_{p,k}$ , there exists $q^{\\prime }\\in \\Sigma _{f(p)}$ such that $\\phi _{s}(q)=q^{\\prime }$ for some $s\\in \\mathbb {R}$ .",
"Then by (REF ) and (REF ), we have $d(q,p)\\le & d(q^{\\prime },p) + d(q,q^{\\prime })\\\\\\le & {\\rm diam}(\\Sigma _{f(p)},g)+\\mathcal {L}(\\phi _{\\tau }|_{[0,s]})\\\\\\le & 2\\pi \\sqrt{\\frac{B}{R(p)}}+|\\int _{0}^{s}|\\frac{d\\phi _{\\tau }(q)}{d\\tau }|d\\tau |\\\\=& 2\\pi \\sqrt{\\frac{B}{ R(p)}}+\\int _{0}^{s}|\\nabla f(\\phi _{\\tau }(q))|d\\tau \\\\\\le & 2\\pi \\sqrt{\\frac{B}{R(p)}}+\\int _{0}^{s}|\\nabla f(\\phi _{\\tau }(q))|^{2}\\cdot \\frac{2}{\\sqrt{R_{\\max }}}d\\tau \\\\=& 2\\pi \\sqrt{\\frac{B}{ R(p)}}+|\\int _{0}^{s}\\frac{d(f(\\phi _{\\tau }(q)))}{d\\tau }\\cdot \\frac{2}{\\sqrt{R_{\\max }}}d\\tau |\\\\\\le & 2\\pi \\sqrt{\\frac{B}{R(p)}}+|f(q)-f(p)|\\cdot \\frac{2}{\\sqrt{R_{\\max }}}\\\\\\le & \\Big (2\\pi \\sqrt{B}+\\frac{2k}{\\sqrt{R_{\\max }}}\\Big )\\cdot \\frac{1}{\\sqrt{R(p)}}.$ Thus $M_{p,k}\\subset B(p,2\\pi \\sqrt{B}+\\frac{2k}{\\sqrt{R_{\\max }}} ; R(p)g).$ The lemma is proved.",
"By Lemma REF , we get the following volume estimate of $B(p,s;g_p)$ .",
"Proposition 3.6 Under (REF ) of Theorem REF , there exists $s_0$ and ${c}>0$ such that ${\\rm Vol}B(p,s;g_p)\\ge c s^3,~\\forall ~s\\le s_0~{\\rm and}~\\rho (p)\\ge r_0>>1.$ Moreover, the injective radius of $(M,g_p)$ at $p$ has a uniform lower bound $\\delta >0$ , i.e., ${\\rm inj}(p,g_p)\\ge \\delta ,~\\forall ~\\rho (p)\\ge r_0.$ By Lemma REF , we have $M_{p,1}\\subset B(p,2\\pi \\sqrt{B}+\\frac{2}{\\sqrt{R_{\\max }}} ; g_p).$ In the following, we give an estimate of ${\\rm Vol}(\\Sigma _l, \\bar{g})$ for any $l$ with $f(p)-\\frac{1}{\\sqrt{R(p)}}\\le l\\le f(p)+\\frac{1}{\\sqrt{R(p)}}$ .",
"By (REF ) and (REF ), we see ${C}_1^{-1}\\le \\frac{R(x)}{R(p)}\\le {C}_1, ~\\forall ~\\rho (p)\\ge r_0~{\\rm and}~x\\in M_{p,1}.$ By (REF ), it follows that the Gauss curvature $K_l$ of $(\\Sigma _l, g_p|_{\\Sigma _l})$ satisfies $\\frac{1}{4C_1}\\le K_l\\le \\frac{3C_1}{4}.$ Thus ${\\rm Vol}(\\Sigma _l, \\bar{g})=\\frac{1}{R(p)}{\\rm Vol}(\\Sigma _l,g_p|_{\\Sigma _l})\\ge \\frac{64\\pi {C}_1}{R(p)}.$ By the Co-Area formula, we get ${\\rm Vol}(M_{p,1},g)=&\\int _{f(p)-\\frac{1}{\\sqrt{R(p)}}}^{f(p)+\\frac{1}{\\sqrt{R(p)}}} \\frac{{\\rm Vol}(\\Sigma _l, \\bar{g} )}{|\\nabla f|} dl \\\\\\ge & 128\\pi {C}_1 R_{max}^{-\\frac{1}{2}} R^{-\\frac{3}{2}}(p).",
"$ Hence ${\\rm Vol}(B(p,2\\pi \\sqrt{B}+\\frac{2}{\\sqrt{R_{\\max }}} ; g_p))\\ge {\\rm Vol}(M_{p,1},g_p)\\ge 128\\pi {C}_1 R_{max}^{-\\frac{1}{2}}.$ By the volume comparison theorem, we derive from (REF ), $\\frac{{\\rm Vol}(B(p,s; g_p))}{s^3}\\ge & \\frac{{\\rm Vol}(B(p,2\\pi \\sqrt{B}+\\frac{2}{\\sqrt{R_{\\max }}} ; g_p))}{(2\\pi \\sqrt{B}+\\frac{2}{\\sqrt{R_{\\max }}})^3}\\\\\\ge &\\frac{128\\pi {C}_1R_{max}^{-\\frac{1}{2}} }{(2\\pi \\sqrt{B}+\\frac{2}{\\sqrt{R_{\\max }}})^3},$ for any $s\\le 2\\pi \\sqrt{B}+\\frac{2}{\\sqrt{R_{\\max }}}$ .",
"This proves (REF ).",
"By (REF ), we can apply a result of Cheeger-Gromov-Taylor for Riemannian manifolds with bounded curvature to get the injective radius estimate (REF ) immediately [6]."
],
[
"Proof of Theorem ",
"First we prove the following convergence of rescaled flows.",
"Lemma 4.1 Under (REF ), let $p_i\\rightarrow \\infty $ .",
"Then by taking a subsequence of $p_i$ if necessary, we have $(M,g_{p_{i}}(t),p_{i})\\rightarrow (\\mathbb {R}\\times N,\\widetilde{g}(t); p_{\\infty }),~for~t\\in (-\\infty ,0],$ where $g_{p_i}(t)=R(p_{i})g(R^{-1}(p_{i})t)$ , $\\widetilde{g}(t)=ds\\otimes ds+g_{ N}(t)$ and $( N, g_{N}(t))$ is an ancient solution of Ricci flow on $N$ .",
"For a fixed $\\overline{r}$ , as in (REF ), it is easy to see that there exists a uniform $C_1$ independent of $\\overline{r}$ such that $R(x)\\le C_1R(p_i),~\\forall ~x\\in M_{p_i,\\overline{r}\\sqrt{R_{\\max }}}$ as long as $i$ is large enough.",
"By Lemma REF , it follows $R_{g_{p_i}}(x)\\le C_1,~\\forall ~x\\in B(p_i,\\overline{r};g_{p_i}),$ where $g_{p_i}=g_{p_i}(0)$ .",
"Since the scalar curvature is increasing along the flow, we get $|{\\rm Rm}_{g_{p_i}(t)}(x)|_{g_{p_i}(t)}&\\le 3R_{g_{p_i}(t)}(x)\\\\&\\le 3R_{g_{p_i}}(x)\\le 3C_1,~\\forall ~x\\in B(p_i,\\overline{r};g_{p_i}),~t\\in (-\\infty ,0].$ Thus together with the injective radius estimate in Proposition REF , we can apply the Hamilton compactness theorem to see that $g_{p_i}(t)$ converges subsequently to a limit flow $(\\widetilde{M}, \\tilde{g}(t); p_\\infty )$ on $t\\in (-\\infty ,0]$ [12].",
"Moreover, the limit flow has uniformly bounded curvature.",
"It remains to prove the splitting property.",
"By Remark REF , we have $|{\\rm Ric}|(x)\\le CR(x),~\\forall ~x\\in B(p_{i},\\overline{r} ; {g_{p_i}}).$ It follows from (REF ), $|{\\rm Ric}|(x)\\le CR(p_i),~\\forall ~x\\in B(p_{i},\\overline{r} ; {g_{p_i}}).$ Let $X_{(i)}=R(p_{i})^{-\\frac{1}{2}}\\nabla f$ .",
"Then $\\sup _{ B(p_{i},r_{0} ; {g_{p_i}})}| \\nabla _{(g_{p_i})}X_{(i)}|_{g_{p_i}}&= \\sup _{ B(p_{i},r_{0} ; {g_{p_i}})}\\frac{|{\\rm Ric}|}{\\sqrt{R(p_{i})}}\\\\&\\le C\\sqrt{R(p_{i})} \\rightarrow 0.$ On the other hand, by Remark REF , we also have $\\sup _{ B(p_{i},r_{0} ; {g_{p_i}})}| \\nabla ^{m}_{(g_{p_i})}X_{(i)}|_{g_{p_i}}\\le C(n)\\sup _{ B(p_{i},r_{0} ; {g_{p_i}})}| \\nabla ^{m-1}_{(g_{p_i})}{\\rm Ric}({g_{p_i})}|_{g_{p_i}}\\le C_1.$ Thus $X_{(i)}$ converges subsequently to a parallel vector field $X_{(\\infty )}$ on $(\\widetilde{M}, \\tilde{g}(0))$ .",
"Moreover, $|X_{(i)}|_{g_{p_i}}( x)=|\\nabla f|(p_{i})=\\sqrt{R_{\\rm max}}+o(1)>0, ~\\forall ~ x\\in B(p_{i},r_{0} ; {g_{i}}),$ as long as $f(p_i)$ is large enough.",
"This implies that $X_{(\\infty )}$ is non-trivial.",
"Hence, $(\\widetilde{M},\\widetilde{g}(t))$ locally splits off a piece of line along $X_{(\\infty )}$ .",
"It remains to show that $X_{(\\infty )}$ generates a line through $p_\\infty $ .",
"By Lemma REF , $M_{p_i,k}\\subset B(p_i,2\\pi \\sqrt{B}+\\frac{2k}{\\sqrt{R_{\\max }}} ; g_{p_i}(0)), ~\\forall ~ p_i\\rightarrow \\infty .$ Let $\\gamma _{i,k}(s)$ , $s\\in (-D_{i,k},E_{i,k})$ be an integral curve generated by $X_{(i)}$ through $p_i$ , which restricted in $M_{p,k}$ .",
"Then $\\gamma _{i,k}(s)$ converges to a geodesic $\\gamma _\\infty (s)$ generated by $X_{(\\infty )}$ through $p_\\infty $ , which restricted in $B(p_\\infty ,2\\pi \\sqrt{B}+\\frac{2k}{\\sqrt{R_{\\max }}};\\widetilde{g}(0))$ .",
"If let $L_{i,k}$ be lengths of $\\gamma _{i,k}(s)$ and $L_{\\infty ,k}$ length of $\\gamma _\\infty (s)$ , $L_{i,k}=&\\int _{-D_{i,k}}^{E_{i,k}}|\\nabla f|_{g_{p_i}(0)} ds=\\int _{f(p_i)-\\frac{k}{\\sqrt{R(p_i)}}}^{f(p_i)+\\frac{k}{\\sqrt{R(p_i)}}} \\sqrt{R(p_i)} \\Vert \\nabla f\\Vert _{g} \\\\&\\ge R_{max}^{-\\frac{1}{2}} \\int _{f(p_i)-\\frac{k}{\\sqrt{R(p_i)}}}^{f(p_i)+\\frac{k}{\\sqrt{R(p_i)}}} \\sqrt{R(p_i)} \\Vert \\nabla f\\Vert _{g}^2 ds\\\\&= R_{max}^{-\\frac{1}{2}} \\int _{f(p_i)-\\frac{k}{\\sqrt{R(p_i)}}}^{f(p_i)+\\frac{k}{\\sqrt{R(p_i)}}} \\sqrt{R(p_i)} df =2 R_{max}^{-\\frac{1}{2}}k, $ and so, $L_{\\infty ,k}\\ge \\frac{1}{2}L_{i,k}\\ge R_{max}^{-\\frac{1}{2}} k. $ Thus $X_{(\\infty )}$ generates a line $\\gamma _\\infty (s)$ through $p_\\infty $ as $k\\rightarrow \\infty $ .",
"As a consequence, $(\\widetilde{M},\\widetilde{g}(0))$ splits off a line and so does the flow $(\\widetilde{M},\\widetilde{g}(t); p_{\\infty })$ .",
"The lemma is proved.",
"Next we estimate the curvature of $(N,g_{N}(t))$ .",
"Lemma 4.2 Under (REF ), there exists a constant $C$ independent of $t$ such that the scalar curvature $R_{N}(t)$ of $(N,g_{N}(t))$ satisfies $\\frac{R_{N}(x,t)}{R_{N}(y,t)}\\le C,~\\forall ~x,y\\in N,~t\\in (-\\infty ,0].$ Let $\\widetilde{R}(x, t)$ be the scalar curvature of $( \\mathbb {R}\\times N, \\widetilde{g}(t))$ .",
"It suffices to prove the following is true: $\\frac{\\widetilde{R}(x,t)}{\\widetilde{R}(y,t)}\\le C,~\\forall ~x,y\\in \\mathbb {R}\\times N,~t\\in (\\infty ,0],$ for some constant $C$ .",
"For any $x,y\\in \\mathbb {R}\\times N$ , we choose $\\overline{r}>0$ such that $x,y\\in B(p_{\\infty },\\overline{r};\\widetilde{g}(0))$ .",
"By the convergence of $g_{p_i}(t)$ , there are sequences $\\lbrace x_i\\rbrace $ and $\\lbrace y_i\\rbrace $ in $ B(p_{i},\\overline{r}; {g}_{p_i}(0))$ such that $x_i$ and $y_i$ converge to $x$ and $y$ in the Cheeger-Gromov sense, respectively.",
"By Lemma REF , we have $x_i,y_i\\subseteq B(p_{i},\\overline{r}; {g}_{p_i}(0))\\subseteq M_{p_i,\\overline{r}\\sqrt{R_{\\max }}}.$ Thus $f(x_i)= (1+o(1))f(p_i)~{\\rm and}~ f(y_i)=(1+o(1))f(p_i), ~{\\rm as}~p_i\\rightarrow \\infty .$ On the other hand, for a fixed $t<0$ , $\\frac{f(\\phi _{R^{-1}(p_i)t}(x_i))-f(x_i)}{|R^{-1}(p_i)t|}=\\frac{\\int _{R^{-1}(p_i)t}^0 |\\nabla f|^2ds}{|R^{-1}(p_i)t|}\\rightarrow R_{\\max },~{\\rm as}~p_i\\rightarrow \\infty $ and $ \\frac{f(\\phi _{R^{-1}(p_i)t}(y_i))-f(y_i)}{|R^{-1}(p_i)t|}=\\frac{\\int _{R^{-1}(p_i)t}^0 |\\nabla f|^2ds}{|R^{-1}(p_i)t|}\\rightarrow R_{\\max },~{\\rm as}~p_i\\rightarrow \\infty .$ By (REF ) and the fact $C_1\\le R(x)f(x)\\le C_2,~\\forall ~f(x)>>1, $ we get $\\frac{f(\\phi _{R^{-1}(p_i)t}(x_i))}{f(\\phi _{R^{-1}(p_i)t}(y_i))}\\rightarrow 1,~{\\rm as}~p_i\\rightarrow \\infty .$ It follows $\\frac{R(\\phi _{R^{-1}(p_i)t}(x_i))}{R(\\phi _{R^{-1}(p_i)t}(y_i))}\\le \\frac{C_2}{C_1}.",
"$ Hence we obtain $ \\frac{R_{N}(x,t)}{R_{N}(y,t)} &=\\lim _{i\\rightarrow \\infty } \\frac{ R^{-1}(p_i) R(x_i, R^{-1}(p_i)t)) }{ R^{-1}(p_i) R( y_i, R^{-1}(p_i)t )}\\\\&= \\lim _{i\\rightarrow \\infty } \\frac{ R^{-1}(p_i) R(\\phi _{R^{-1}(p_i)t}(x_i))}{ R^{-1}(p_i) R(\\phi _{R^{-1}(p_i)t}(y_i))}\\le \\frac{C_2}{C_1}.$ This proves (REF ).",
"The proof of Theorem REF is completed by the following lemma.",
"Lemma 4.3 $(N,g_{N}(t))$ in Lemma REF is a shrinking spheres flow.",
"Namely, $(N,g_{N}(t))=(\\mathbb {S}^2,(2-2t)g_{\\mathbb {S}^{2}}).$ By Lemma REF , the Gauss curvature of $(N,g_{N}(0))$ has a uniform positive lower bound.",
"Then $N$ is compact by Myer's Theorem.",
"On the other hand, by a classification theorem of Daskalopoulos-Hamilton-Sesum [8], an ancient solution on a compact surface $N$ is either a shrinking spheres flow or a Rosenau solution.",
"The Rosenau solution is obtained by compactifying $(\\mathbb {R}\\times \\mathbb {S}^1(2),h(x,\\theta ,t)=u(x,t)(dx^2+d\\theta ^2))$ by adding two points, where $u(x,t)=\\frac{\\sinh (-t)}{\\cosh (x)+\\cosh (t)}$ and $t\\in (-\\infty ,0)$ .",
"By a direct computation, $R_{h(t)}=\\frac{\\cosh (t)\\cosh (x)+1}{\\sinh (-t)(\\cosh (x)+\\cosh (t))}.$ It is easy to check that $R_{h(t)}$ doesn't satisfy (REF ) in Lemma REF as $t\\rightarrow -\\infty $ .",
"Hence, $(N,g_{N}(t))$ must be a shrinking spheres flow on $\\mathbb {S}^2$ .",
"Note that $\\widetilde{R}(p_{\\infty },0)=1$ .",
"Then it is easy to see that $g_{\\mathbb {S}^{2}}(t)=(2-2t)g_{\\mathbb {S}^{2}}$ ."
],
[
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] | 1612.05713 |
[
[
"The VIMOS Public Extragalactic Redshift Survey (VIPERS). Gravity test\n from the combination of redshift-space distortions and galaxy-galaxy lensing\n at $0.5 < z < 1.2$"
],
[
"Abstract We carry out a joint analysis of redshift-space distortions and galaxy-galaxy lensing, with the aim of measuring the growth rate of structure; this is a key quantity for understanding the nature of gravity on cosmological scales and late-time cosmic acceleration.",
"We make use of the final VIPERS redshift survey dataset, which maps a portion of the Universe at a redshift of $z \\simeq 0.8$, and the lensing data from the CFHTLenS survey over the same area of the sky.",
"We build a consistent theoretical model that combines non-linear galaxy biasing and redshift-space distortion models, and confront it with observations.",
"The two probes are combined in a Bayesian maximum likelihood analysis to determine the growth rate of structure at two redshifts $z=0.6$ and $z=0.86$.",
"We obtain measurements of $f\\sigma_8(0.6) = 0.48 \\pm 0.12$ and $f\\sigma_8(0.86) = 0.48 \\pm 0.10$.",
"The additional galaxy-galaxylensing constraint alleviates galaxy bias and $\\sigma_8$ degeneracies, providing direct measurements of $[f(0.6),\\sigma_8(0.6)] = [0.93 \\pm 0.22, 0.52 \\pm 0.06]$ and $f(0.86),\\sigma_8(0.86)] = [0.99 \\pm 0.19, 0.48 \\pm 0.04]$.",
"These measurements are statistically consistent with a Universe where the gravitational interactions can be described by General Relativity, although they are not yet accurate enough to rule out some commonly considered alternatives.",
"Finally, as a complementary test we measure the gravitational slip parameter, $E_G$ , for the first time at $z>0.6$.",
"We find values of $\\smash{\\overline{E}_G}(0.6) = 0.16 \\pm 0.09$ and $\\smash{\\overline{E}_G}(0.86) = 0.09 \\pm 0.07$, when $E_G$ is averaged over scales above $3 h^{-1} \\rm{Mpc}$.",
"We find that our $E_G$ measurements exhibit slightly lower values than expected for standard relativistic gravity in a {\\Lambda}CDM background, although the results are consistent within $1-2\\sigma$."
],
[
"Introduction",
"The origin of the late-time acceleration of the universal expansion is a major question in cosmology.",
"The source of this acceleration and its associated energy density are crucial in understanding the properties of the Universe and its evolution and fate.",
"In the standard cosmological model, this cosmic acceleration can be associated with the presence of a dark energy component, a cosmological fluid with negative pressure, which opposes the gravitational force on large scales.",
"However, this apparent acceleration can conversely be interpreted as a failure of the standard relativistic theory of gravity.",
"A key goal for cosmology is therefore to investigate the nature of gravity empirically.",
"To be clear, what can potentially be falsified is the validity of Einstein's field equations, rather than General Relativity itself; this sets a broader framework within which Einstein gravity or modified alternatives can operate.",
"The large-scale structure of the Universe has proved to be very powerful for testing the cosmological model through the use of various observables such as the two-point statistics of the galaxy distribution and its features [71], [20], [91], [29], [40], [72], [9], [12], [3], [1].",
"In this context, a unique probe of gravitational physics is the large-scale component of galaxy peculiar velocities affecting the observed galaxy distribution in redshift surveys [40], sensitive to the growth rate of structure $f$ defined as $\\mathrm {d}\\ln D/\\mathrm {d}\\ln a$ , where $D$ and $a$ are respectively the linear growth factor and scale factor.",
"In turn, the growth rate of structure tells us about the strength of gravity acting on cosmological scales and is a direct prediction of gravity theories.",
"The distortions induced by peculiar velocities in the apparent galaxy clustering, the so-called redshift-space distortions (RSD), are a very important cosmological probe of the nature of gravity.",
"In the last decade, they have been studied in large galaxy redshift surveys, showing a broad consistency with $\\Lambda \\rm {CDM}$ and General Relativity predictions [12], [10], [25], [82], [33], [18].",
"Although galaxy redshift surveys are powerful cosmological tools for understanding the geometry and the dynamics of the Universe, they are fundamentally limited by the inherent uncertainty related to the bias of galaxies, the fact that these are not faithful tracers of the underlying matter distribution.",
"Gravitational lensing represents a powerful probe that is complementary to galaxy redshift-space clustering.",
"In the weak regime in particular, the statistical shape deformations of background galaxies probe the relativistic gravitational deflection of light by the projected dark matter fluctuations due to foreground large-scale structure.",
"There are several techniques associated with weak gravitational lensing; one that is particularly useful for combining with galaxy clustering is galaxy-galaxy lensing.",
"This technique consists of studying the weak deformations of background galaxies around foreground galaxies, whose associated dark matter component acts as a gravitational lens.",
"This is particularly useful for probing the galaxy-matter cross-correlation, which in turn provides insights on the bias of foreground galaxies and the matter energy density $\\Omega _m$ , although the projected nature of the statistic makes it insensitive to redshift-space distortions.",
"The combination of galaxy-galaxy lensing with redshift-space galaxy correlations is therefore a very promising way to study gravitational physics, given that both lensing information on background sources and spectroscopic information on foreground galaxies are available on the same field.",
"Beyond the determination of the growth rate of structure, one can define consistency tests of gravity that are sensitive to both the Newtonian and curvature gravitational potentials, $\\Psi $ and $\\Phi $ respectively [86].",
"One is the gravitational slip, $E_G$ , which was originally proposed by [97] and implemented by [78] in terms of the ratio between the galaxy-galaxy lensing signal and the redshift-space distortions parameter $\\beta =f/b$ times the galaxy clustering signal of the lenses.",
"Here $b$ is the galaxy linear bias.",
"$E_G$ effectively tests whether the Laplacian of $\\Psi +\\Phi $ , to which gravitational lensing is sensitive, and that of $\\Psi $ , to which galaxy peculiar velocities are sensitive, are consistent with standard gravity predictions.",
"In the standard cosmological model, $E_G$ asymptotes to $\\Omega _m/f$ on large linear scales.",
"A failure of this test would either imply an incorrect matter energy density or a departure from standard gravity.",
"This test has been performed at low redshift in the SDSS survey by [78] and more recently at redshifts up to $z=0.57$ by [13] and [77].",
"The $E_G$ statistic is formally defined as $E_G=\\Upsilon _{gm} / (\\beta \\Upsilon _{gg})$ , where $\\Upsilon _{gm}$ and $\\Upsilon _{gg}$ are filtered versions of the real-space projected galaxy-matter and galaxy-galaxy correlation functions respectively, and $\\beta $ is the RSD parameter.",
"In practice, its implementation involves measuring $\\beta $ and the ratio $\\Upsilon _{gm} / \\Upsilon _{gg}$ separately, to finally combine them.",
"But since $\\beta $ and $\\Upsilon _{gg}$ are extracted from the same observable, namely the anisotropic two-point correlation function of lens galaxies, this is suboptimal and does not account for the covariance between them.",
"In this analysis, we follow a different approach.",
"We combine the galaxy-galaxy lensing quantity $\\Upsilon _{gm}$ and the redshift-space anisotropic correlation function monopole and quadrupole moments $\\xi _0$ and $\\xi _2$ (from which $\\beta $ can be estimated) in a joint likelihood analysis, to provide constraints on $f$ and gravity at redshifts above $z=0.6$ .",
"We note that we do not include $\\Upsilon _{gg}$ because of the redundant cosmological information shared with $\\xi _0$ and $\\xi _2$ .",
"The VIMOS Public Extragalactic Redshift Survey (VIPERS) is a large galaxy redshift survey probing the $z\\simeq 0.8$ Universe with an unprecedented density of spectroscopic galaxies of $5 \\times 10^{-3}\\,h^{3}~{\\rm Mpc^{-3}}$ and covering an overall area of $23.5~\\rm {deg}^2$ on the sky.",
"The prime goal of VIPERS is an accurate measurement of the growth rate of structure at redshift around unity.",
"A first measurement has been performed using the Public Data Release 1 (PDR-1), setting a reference measurement of $f\\sigma _8$ at $z=0.8$ [25].",
"The survey is now complete and several analyses including this one are using the final dataset to produce the VIPERS definitive growth rate of structure measurements, but following a variety of approaches.",
"The present analysis aims at maximizing the cosmological information available and takes advantage of the overlapping lensing information provided by CFHTLenS lensing survey, to provide a precise gravity test at redshifts $0.5<z<1.2$ by combining RSD and galaxy-galaxy lensing.",
"The paper is organized as follows.",
"The data are described in Sect.",
"2; Sect.",
"3 describes our methods for estimating galaxy clustering and galaxy-galaxy lensing; Sect.",
"4 describes the theoretical modelling that is tested in Sect.",
"5; Sect.",
"6 presents how the likelihood analysis is constructed; Sect.",
"7 describes the cosmological results, and Sect.",
"8 summarizes our findings and concludes.",
"Throughout this analysis and if not stated otherwise, we assume a flat $\\Lambda {\\rm CDM}$ ($\\Lambda $ -Cold Dark Matter) cosmological model with $(\\Omega _m,\\Omega _b,n_s)=(0.3,0.045,0.96)$ and a Hubble constant of $H_0=100~h~\\rm {km~s^{-1}~Mpc^{-1}}$ .",
"The VIPERS galaxy target sample was selected from the optical photometric catalogues of the Canada-France-Hawaii Telescope Legacy Survey Wide [37].",
"VIPERS covers $23.5$ deg$^2$ on the sky, divided over two areas within the W1 and W4 CFHTLS fields.",
"Galaxies are selected to a limit of $i_{\\rm AB}<22.5$ , applying a simple and robust $gri$ colour pre-selection to efficiently remove galaxies at $z<0.5$ .",
"Coupled with a highly optimized observing strategy [84], this allows us to double the galaxy sampling rate in the redshift range of interest, with respect to a pure magnitude-limited sample.",
"At the same time, the area and depth of the survey result in a relatively large volume, $5 \\times 10^{7}\\,h^{-3}~{\\rm Mpc^3}$ , analogous to that of the Two Degree Field Galaxy Redshift Survey (2dFGRS) at $z\\simeq 0.1$ [21], [22].",
"Such a combination of sampling rate and depth is unique amongst current redshift surveys at $z>0.5$ .",
"VIPERS spectra are collected with the VIMOS multi-object spectrograph [55] at moderate resolution ($R=220$ ) using the LR Red grism, providing a wavelength coverage of 5500-9500$\\rm {Å}$ and a redshift error corresponding to a galaxy peculiar velocity error at any redshift of $\\sigma _{\\rm vel}=163$$\\,{\\rm km\\, s^{-1}}$ .",
"The full VIPERS area of $23.5$ deg$^2$ is covered through a mosaic of 288 VIMOS pointings (192 in the W1 area, and 96 in the W4 area).",
"A discussion of the survey data reduction and management infrastructure is presented in [32].",
"A complete description of the survey construction, from the definition of the target sample to the actual spectra and redshift measurements, is given in the survey description paper [41].",
"The data used here correspond to the publicly released PDR-2 catalogue [85] that includes $86\\,775$ galaxy spectra, with the exception of a small sub-set of redshifts (340 galaxies missing in the range $0.6 < z< 1.1$ ), for which the redshift and quality flags were revised closer to the release date.",
"Concerning the analysis presented here, this has no effect.",
"A quality flag has been assigned to each object in the process of determining their redshift from the spectrum, which quantifies the reliability of the measured redshifts.",
"In this analysis (as with all statistical analyses presented in the parallel papers of the final science release), we use only galaxies with flags 2 to 9 inclusive, corresponding to objects with a redshift confidence level of $96.1\\%$ or larger.",
"This has been estimated from repeated spectroscopic observations in the VIPERS fields [85].",
"The catalogue used here, which we will refer to just as the VIPERS sample in the following, includes 76584 galaxies with reliable redshift measurements.",
"In addition to the VIPERS spectroscopic sample, we make use of the public lensing data from the Canada-France-Hawaii Lensing Survey [46], hereafter referred to as CFHTLenS.",
"The CFHTLenS survey analysis combined weak lensing data processing with theli [30], shear measurement with lensfit [63], and photometric redshift measurement with PSF-matched photometry [47].",
"A full systematic error analysis of the shear measurements in combination with the photometric redshifts is presented in [46], with additional error analyses of the photometric redshift measurements presented in [7]."
],
[
"Sample selection",
"For this analysis, we define two redshift intervals covering the full volume of the VIPERS survey: $0.5<z<0.7$ and $0.7<z<1.2$ .",
"The number density of galaxies in the combined W1 and W4 fields is presented in Fig.",
"REF , after correction with survey incompleteness weights $w^C$ (see Sect.",
"REF ).",
"It is worth emphasizing that after application of survey incompleteness corrections, the VIPERS spectroscopic sample represents a statistically unbiased subset of the parent $i_{\\rm AB}<22.5$ photometric catalogue [41], [32], [85].",
"The redshift distribution is modelled using the $V_{\\rm max}$ method [19], [25] and shown with the solid curve in the figure.",
"In this method, we randomly sample 500 times the $V_{\\rm max}$ of each galaxy, defined as the comoving volume between the minimum and maximum redshifts where the galaxy is observable given its apparent magnitude and the magnitude limit of VIPERS, $i_{\\rm AB}=22.5$ .",
"The redshift distribution thus obtained is regular and can be straightforwardly interpolated with a smooth function, showed with the solid curve in Fig.",
"REF .",
"In addition to VIPERS spectroscopic galaxies, photometric galaxies from the CFHTLenS survey on the overlapping areas with VIPERS survey, have been used for the galaxy-galaxy lensing.",
"The lens sample satisfies the VIPERS selection $i_{\\rm AB}<22.5$ and uses VIPERS spectroscopic redshifts when available (i.e.",
"for about $30\\%$ of objects) or CFHTLenS maximum likelihood photometric redshifts otherwise.",
"The sources have been selected to have $i_{\\rm AB}<24.1$ and thus have a higher surface density.",
"Sources inside the mask delimiting bad photometric areas in the CFHTLenS catalogue have been discarded.",
"We also make use of the individual source redshift probability distribution function estimates obtained from bpz [47] as described in Sect.",
"REF .",
"Source galaxies extend above $z_{\\rm phot}=1.4$ and their number density is represented with the unfilled histogram in Fig.",
"REF .",
"Figure: Number densities of VIPERS galaxies in the individual W1 andW4 fields and of CFHTLenS/VIPERS photometric redshift galaxies, as afunction of redshift.",
"The number densities of VIPERS galaxies arecorrected for the survey incompleteness by weighting each galaxy inthe counts by its associated inverse completeness weight w C w^C.",
"Thesolid curve corresponds to the model n(z)n(z) used in the analysis.",
"Itwas obtained by randomly sampling galaxy redshifts within theirV max V_{\\rm max} (see text for details)."
],
[
"Anisotropic galaxy clustering estimation",
"We estimate the redshift-space galaxy clustering by measuring the two-point statistics of the spatial distribution of galaxies in configuration space.",
"For this we infer the anisotropic two-point correlation function $\\xi (s,\\mu )$ using the [53] estimator: $\\xi (s,\\mu )=\\frac{GG(s,\\mu )-2GR(s,\\mu )+RR(s,\\mu )}{RR(s,\\mu )}, $ where $GG(s,\\mu )$ , $GR(s,\\mu )$ , and $RR(s,\\mu )$ are respectively the normalized galaxy-galaxy, galaxy-random, and random-random number of pairs with separation $(s,\\mu )$ .",
"Since we are interested in quantifying RSD effects, we have decomposed the three-dimensional galaxy separation vector $\\vec{s}$ into polar coordinates $(s,\\mu )$ , where $s$ is the norm of the separation vector and $\\mu $ is the cosine of the angle between the line-of-sight and separation vector directions.",
"This estimator minimizes the estimation variance and circumvents discreteness and finite volume effects [53], [42].",
"A random catalogue needs to be constructed, whose aim is to accurately estimate the number density of objects in the sample.",
"It must be an unclustered population of objects with the same radial and angular selection functions as the data.",
"In this analysis, we use random samples with 20 times more objects than in the data to minimize the shot noise contribution in the estimated correlation functions, and the redshifts of random points are drawn randomly from the model $n(z)$ presented in Fig.",
"REF .",
"In order to study redshift-space distortions, we further extract the multipole moments of the anisotropic correlation function $\\xi (s,\\mu )$ .",
"This approach has the main advantage of reducing the number of observables, compressing the cosmological information contained in the correlation function.",
"This eases the estimation of the covariance matrices associated with the data.",
"We adopt this methodology in this analysis and use the two first non-null moments $\\xi _0(s)$ and $\\xi _2(s)$ , where most of the relevant information is contained, and ignore the contributions of the more noisy subsequent orders.",
"The multipole moments are related to $\\xi (s,\\mu )$ as $\\xi _\\ell (s)=\\frac{2\\ell +1}{2}\\int _{-1}^{1}\\xi (s,\\mu )L_\\ell (\\mu )\\mathrm {d}\\mu , $ where $L_\\ell $ is the Legendre polynomial of order $\\ell $ .",
"In practice the integration of Eq.",
"REF is approximated by a Riemann sum over the binned $\\xi (s,\\mu )$ .",
"We use a logarithmic binning in $s$ with $\\Delta \\log (s)=0.1$ and a linear binning in $\\mu $ with $\\Delta \\mu =0.02$ .",
"VIPERS has a complex angular selection function which has to be taken into account carefully when estimating the correlation function.",
"This has been studied in detail for the VIPERS Public Data Release 1 (PDR-1) [41], [32] and particularly for the galaxy clustering estimation in [25] and [58].",
"We follow the same methodology to account for it in this analysis with only small improvements.",
"We summarize it in the following and refer the reader to the companion paper, [74], for further details and tests of the method when applied to the VIPERS final dataset.",
"The main source of incompleteness in the survey is introduced by the VIMOS slit positioner, SSPOC, and the VIPERS one-pass observational strategy.",
"This results in an incomplete and uneven spectroscopic sampling, described in detail in [41], [32].",
"In terms of galaxy clustering, the effect is to introduce an underestimation in the amplitude of the measured galaxy correlation function, which becomes scale-dependent on the smallest scales.",
"We demonstrate in [25] that this can be corrected by weighting each galaxy in the estimation of the correlation function.",
"For this we define a survey completeness weight, $w^C$ , which is defined for each spectroscopic galaxy as well as an angular pair weight, $w^A$ , which is applied only to GG pair counts.",
"The latter is obtained from the ratio of one plus the angular correlation functions of targeted and spectroscopic galaxies, as described in [25].",
"The improvements compared to the PDR-1 analysis only concern the estimation of survey completeness weights $w^C$ .",
"These in fact correspond to the inverse effective sampling rate, ${\\rm ESR}$ , and are defined for each galaxy as $w^C={\\rm ESR}^{-1}=({\\rm SSR}\\times {\\rm TSR})^{-1}, $ where ${\\rm SSR}$ , ${\\rm TSR}$ are respectively the spectroscopic and target sampling rates [41].",
"A significant effort has been invested in improving the estimation of the ${\\rm SSR}$ and ${\\rm TSR}$ .",
"In particular the ${\\rm SSR}$ , which characterizes our ability of measuring the redshifts from observed galaxy spectra, has been refined and now accounts for new galaxy property dependencies, as described in [85].",
"The ${\\rm TSR}$ , defined as the fraction of spectroscopically observed galaxies in the parent target catalogue, has been recomputed with better angular resolution, on rectangular apertures of 60 by 100 $\\rm {arcsec}^2$ around spectroscopic galaxies.",
"In order to mitigate the shot noise contribution in the galaxy counts in such small apertures, we use the Delaunay tesselation that naturally adapts to local density of points [74].",
"The accuracy of this new set of weights is tested in the next section and in [74].",
"By applying these weights we effectively up-weight galaxies in the pair counts.",
"It is important to note that the spatial distribution of the random objects is kept consistently uniform across the survey volume.",
"The final weights assigned to $GG$ , $GR$ , and $RR$ pairs combine the survey completeness and angular pair weights as GG(s,)=i=1NGj=i+1NGwCiwCjwA(ij)ij(s,) GR(s,)=i=1NGj=1NRwCiij(s,) RR(s,)=i=1NRj=i+1NRij(s,), where $\\Theta _{ij}(s,\\mu )$ is equal to unity for $\\log (s_{ij})$ in $[\\log (s)-\\Delta \\log (s)/2,\\log (s)+\\Delta \\log (s)/2]$ and $\\mu _{ij}$ in $[\\mu -\\Delta \\mu /2,\\mu +\\Delta \\mu /2]$ , and null otherwise.",
"We define the separation associated with each logarithmic bin as the median pair separation inside the bin.",
"This definition is more accurate than using the bin centre, particularly at large $s$ when the bin size is large.",
"One can also extract real-space clustering information from the anisotropic redshift-space correlation function.",
"This can be done by measuring the latter with the estimator of Eq.",
"REF , but where the redshift-space galaxy separation vector is decomposed in two components, $r_p$ and $\\pi $ , respectively perpendicular and parallel to the line-of-sight [31].",
"This decomposition allows the isolation of the effect of peculiar velocities as these modify only the component parallel to the line-of-sight.",
"This way, redshift-space distortions can then be mitigated by integrating $\\xi (r_p,\\pi )$ over $\\pi $ , thus defining the projected correlation function $w_p(r_p)=\\int ^{\\pi _{\\rm max}}_{-\\pi _{\\rm max}} \\xi (r_p,\\pi )\\mathrm {d}\\pi .$ We measure $w_p(r_p)$ using an optimal value of $\\pi _{\\rm max}=50\\,h^{-1}~{\\rm Mpc}$ , allowing us to reduce the underestimation of the amplitude of $w_p(r_p)$ on large scales and at the same time to avoid including noise from uncorrelated pairs with separations of $\\pi >50\\,h^{-1}~{\\rm Mpc}$ .",
"From the projected correlation function, one can derive the following quantity $\\Upsilon _{gg}(r_p,r_0) = \\rho _c \\left( \\frac{2}{r^2_p} \\int _{r_0}^{r_p} r w_p(r)\\,dr - w_p(r_p) + \\frac{r^2_0}{r^2_p} w_p(r_{0}) \\right), $ where $r_0$ is a cut-off radius, $\\rho _c=3H^2/(8\\pi G)$ is the critical density, $H(a)=\\dot{a}/a$ is the Hubble parameter, and $G$ is the gravitational constant.",
"This quantity is equivalent to $\\Upsilon _{gm}$ , which is measurable from galaxy-galaxy lensing (see next section), but for galaxy-galaxy correlations instead of galaxy-matter ones.",
"It enters the definition of the gravitational slip parameter $E_G$ .",
"In order to measure it in practice, since the logarithmic binning in $r_p$ is rather large in our analysis, we interpolate $w_p(r_p)$ using cubic spline interpolation before evaluating the integral in Eq.",
"REF numerically.",
"We find that $\\Upsilon _{gg}$ is more accurately measured with this technique than by modelling $w_p(r_p)$ as a power law to perform the integral, as is often done [57]."
],
[
"Galaxy-galaxy lensing estimation",
"We use in this analysis the weak lensing technique usually referred to as galaxy-galaxy lensing, in which one infers the tangential shear of background sources $\\gamma _t$ around foreground objects (lenses) induced by the projected matter distribution in between.",
"This quantity is sensitive to the projected cross-correlation between lens galaxies and the underlying matter distribution.",
"Since the shear signal is weak and the intrinsic ellipticity of galaxies is unknown, one has to average the former over a large number of foreground sources.",
"The quantity that is effectively measured is the differential excess surface density $\\Delta \\Sigma _{gm} (r_p)=\\Sigma _{\\rm crit} \\left< \\gamma _t(r_p)\\right>,$ where $\\Sigma _{\\rm crit}=\\frac{c^2}{4\\pi G}\\frac{D_{\\rm S}}{D_{\\rm LS}D_{\\rm L}}.$ In the above equations, $r_p$ is the comoving transverse distance between lens and source galaxies, $D_{\\rm S}$ , $D_{\\rm LS}$ , $D_{\\rm L}$ are the angular diameter observer-source, lens-source, and observer-lens distances, and $c$ is the speed of light in the vacuum.",
"We use the inverse variance-weighted estimator for the differential excess surface density [57]: $\\Delta \\Sigma _{gm} (r_p)= \\frac{\\sum ^{N_S}_{i=1} \\sum ^{N_L}_{j=1} w^S_i e_{t,i} \\Sigma ^{-1}_{{\\rm crit},~ij}\\Theta _{ij}\\left(r_p\\right)}{\\sum ^{N_{\\rm S}}_{i=1} \\sum ^{N_{\\rm L}}_{j=1} w^S_i \\Sigma ^{-2}_{{\\rm crit},~ij}\\Theta _{ij}\\left(r_p\\right)}, $ where the $i$ and $j$ indices run over source and lens galaxies respectively, $N_{\\rm S}$ and $N_{\\rm L}$ are respectively the number of source and lens galaxies, $e_{t,i}$ is the tangential ellipticity for each lens-source pair, $w^S$ are statistical weights accounting for biases in the determination of background source ellipticities, and $\\Theta _{ij}(r_p)$ is equal to unity for $r_{p,~ij}$ in $[r_p-\\Delta r_p/2,r+\\Delta r_p/2]$ and null otherwise.",
"The projected separation $r_p$ is calculated as $r_p=\\theta \\chi _{\\rm L}$ , where $\\theta $ and $\\chi _{\\rm L}$ are respectively the angular distance between the lens and the source, and the radial comoving distance of the lens.",
"This estimator includes an inverse-variance weight for each lens-source pair $\\smash{\\Sigma ^{-2}_{\\rm crit}}$ , which downweights the pairs at close redshifts that contribute little to the weak lensing signal [57].",
"This estimator is unbiased if the redshifts of the sources are perfectly known, but here we have only photometric redshift estimates: the maximum likelihood photometric redshift and the normalized redshift probability distribution function for each source $p_s(z)$ .",
"Using the maximum likelihood photometric redshift of sources in Eq.",
"REF and restricting the sum to pairs with $z_{\\rm S}>z_{\\rm L}$ can possibly lead to a dilution of the signal induced by the non-negligible probability that $z_{\\rm S}<z_{\\rm L}$ .",
"This effect can be mitigated by replacing $\\Sigma ^{-1}_{\\rm crit}$ in Eq.",
"REF by its average over the source redshift probability distribution function $p_s$ $\\left< \\Sigma ^{-1}_{\\rm crit} \\right> = \\int _{z_{\\rm L}}^\\infty dz_{\\rm S} p_s(z_{\\rm S})\\Sigma ^{-1}_{\\rm crit}(z_{\\rm L},z_{\\rm S}),$ which leads to the following estimator [64], [13]: $\\Delta \\Sigma _{gm} (r_p)= \\frac{\\sum ^{N_S}_{i=1} \\sum ^{N_L}_{j=1} w^S_i e_{t,i} \\left<\\Sigma ^{-1}_{{\\rm crit},~ij}\\right>\\Theta _{ij}\\left(r_p\\right)}{\\sum ^{N_S}_{i=1} \\sum ^{N_L}_{j=1} w^S_i \\left<\\Sigma ^{-1}_{{\\rm crit},~ij}\\right>^2\\Theta _{ij}\\left(r_p\\right)} .$ In principle, those estimators hold in the limit where the lens redshift distribution is narrow and lens redshifts accurate [66].",
"To better understand the importance of the effects introduced by an imperfect knowledge of the source and lens redshifts in the data, we perform a comparison of different estimates using Eq.",
"REF or Eq.",
"REF , and various assumptions on the source and lens redshifts.",
"This is presented in terms of the relative difference with respect to a fiducial estimate in Fig.",
"REF .",
"The fiducial estimate is that obtained by using Eq.",
"REF , which includes the individual redshift probability distribution function $p_s(z)$ of the sources, and for the lenses, the VIPERS spectroscopic redshift when available or the CFHTLenS maximum likelihood photometric redshift otherwise.",
"Figure: Relative difference between various estimates of ΔΣ gm \\Delta \\Sigma _{gm}, based on different assumptions for source and lensredshifts, and the fiducial estimate in the data at 0.5<z<0.70.5<z<0.7.",
"Thequantity shown in the figure is ΔΣ gm /ΔΣ gm fid -1\\Delta \\Sigma _{gm}/\\Delta \\Sigma ^{fid}_{gm}-1 as a function of the projected separationr p r_p.",
"The fiducial estimate ΔΣ gm fid \\Delta \\Sigma ^{fid}_{gm} is thatobtained by using Eq.",
", which includes theindividual redshift probability distribution function p s (z)p_s(z) ofthe sources, and for the lenses, the VIPERS spectroscopic redshift(z spec z_{spec}) when available or the CFHTLenS maximum likelihoodphotometric redshift (z phot z_{phot}) otherwise (see text).",
"Itcorresponds to the adopted estimate for the analysis.",
"The greyshaded area represents the relative statistical error expected inthe survey.We find that the estimate based on Eq.",
"REF , which only uses maximum likelihood photometric redshifts for both lenses and sources, underestimate the signal on all probed scales by about $15\\%$ with respect to the fiducial case.",
"Here, we impose $z_S > 0.1 + z_L$ , including the additive term of $0.1$ to account for typical photometric redshift errors [23].",
"Further including the source redshift probability distribution function through the estimator of Eq.",
"REF allows a slight improvement, reaching an underestimation of about $10\\%$ with respect to the fiducial case.",
"The two previous estimates are still affected by the uncertainty on the lens redshifts, which effectively tends to dilute the overall signal.",
"If we now use as lenses only VIPERS spectroscopic galaxies, which represents about $30\\%$ of all galaxies with $i_{AB}<22.5$ , we find a remarkably good agreement with the fiducial estimate.",
"In principle, this estimate may be considered as the reference unbiased estimate, however on the largest scales probed by the data, i.e.",
"at $r_p = 10-20\\,h^{-1}~{\\rm Mpc}$ , the signal drops significantly.",
"This can be imputed to the lack of source-lens pairs induced by the reduced number of lenses, directly affecting our ability to probe the largest scales signal.",
"However, we find that this effect can be mitigated by adding photometric lenses from the CFHTLenS catalogue, taking the maximum likelihood photometric redshifts: this corresponds to the fiducial estimate.",
"We note that the expected statistical uncertainty, which is shown in Fig.",
"REF with the grey shaded area, is not negligible particularly above $r_p =10\\,h^{-1}~{\\rm Mpc}$ , and higher than any residual systematic effect.",
"This test makes us confident that our fiducial estimate of $\\Delta \\Sigma _{gm}(r_p)$ is robust, given the expected level of statistical error in the data.",
"Similar results are found at $0.7<z<1.2$ , leading to the same conclusions.",
"A non-negligible source of systematics in weak lensing measurements is related to the measurement of background galaxy shapes.",
"This can lead to systematic biases in the lensing measurements.",
"The CFHTLenS collaboration has studied these extensively in [63] and [46], and we follow their method to correct our measurements.",
"We used the additive and multiplicative shear calibration corrections $c$ and $m$ , as well as the optimal weights $w^S$ provided by lensfit, which are available in the CFHTLenS catalogue.",
"In particular, to correct for the multiplicative bias we applied the correction factor $\\left(1+K(r_p)\\right)^{-1}$ to $\\Delta \\Sigma _{gm}(r_p)$ as described in [63] and [94].",
"We found this correction to boost the galaxy-galaxy lensing signal by about $5\\%$ independently of the scale.",
"For the purpose of constraining the cosmological model, it can be difficult to use $\\Delta \\Sigma _{gm}$ as its modelling is non-linear.",
"One of the difficulties is to model the non-linear scales and the intrinsic mixing of small-scale non-linear and large-scale linear information [4].",
"This is achievable but at the expense of introducing additional nuisance parameters in the model [15], [65].",
"An alternative approach, which we use in this analysis, consists of using a derived statistic that allows the mitigation of non-linearities: the annular differential surface density $\\Upsilon _{gm}$ , which is defined as [4] $\\Upsilon _{gm}(r_p, r_0) = \\Delta \\Sigma _{gm}(r_p) -\\frac{r^2_0}{r^2_p} \\Delta \\Sigma _{gm}(r_{0}).",
"$ This statistic removes the small-scale non-linear contribution of $\\Delta \\Sigma _{gm}$ below a cut-off radius $r_0$ .",
"We use this quantity in our analysis and study the impact of the choice of $r_0$ in Sect.",
"REF ."
],
[
"Galaxy biasing",
"Galaxies are not faithful tracers of the underlying matter distribution and this has to be taken into account in cosmological analyses, since cosmological models primarily predict matter observables.",
"The modelling of galaxy biasing is simplified when focusing on large scales, where bias can be considered as linear and simply be represented as a constant multiplicative factor in front of the matter power spectrum.",
"This is a common assumption in RSD analyses.",
"In our case, however, the relatively small survey volume means that much of our information lies below fully linear scales; for this reason, and because of the intrinsic non-linearities in the excess surface density $\\Delta \\Sigma _{gm}$ , additional care must be taken to model galaxy biasing.",
"We use a non-linear prescription for galaxy bias based on the cosmological perturbation theory that allows describing it more accurately down to translinear scales.",
"We adopt the non-linear non-local bias model of [60] that relates the galaxy overdensity $\\delta _{g}$ and matter overdensity $\\delta $ as: $ \\delta _{g}({\\bf x}) &=& b_1 \\delta ({\\bf x}) + \\frac{1}{2}b_2[\\delta ^2({\\bf x})-\\sigma ^2]+\\frac{1}{2}b_{s^2}[s^2({\\bf x})-\\langle s^2 \\rangle ] \\nonumber \\\\&& + O(s^3({\\bf x})),$ where $b_1$ and $b_2$ are the linear and second-order non-linear bias terms, $b_{s^2}$ the non-local bias term, $s$ is the tidal tensor term from which non-locality originates.",
"The $\\sigma ^2$ and $\\langle s^2\\rangle $ terms ensure the condition $\\langle \\delta _g\\rangle =0$ ."
],
[
"Annular differential excess surface density",
"The galaxy-galaxy lensing quantity that we observe is the differential excess surface density.",
"It is defined as $\\Delta \\Sigma _{gm} (r_p) = \\overline{\\Sigma }_{gm}(r_p) - \\Sigma _{gm}(r_p),$ where $\\overline{\\Sigma }_{gm}(r_p) = \\frac{2}{r^2_p} \\int ^{r_p}_0 \\Sigma _{gm} (r)\\,r\\,dr$ and $\\Sigma _{gm}(r_p)$ is the projected surface density defined as $\\Sigma _{gm}(r_p) = \\Omega _m\\rho _c\\int ^{\\infty }_{-\\infty } \\left(1+\\xi _{gm}(\\sqrt{r^2_p+\\chi ^2}\\right)d\\chi .$ In the above equation, $\\Omega _m$ is matter energy density and $\\chi $ is the radial comoving coordinate.",
"$\\Upsilon _{gm}$ can be predicted from $\\Delta \\Sigma _{gm}$ by using Eq.",
"REF or directly from the galaxy-matter cross-correlation function as [4] $\\Upsilon _{gm}(r_p) = \\int _0^\\infty \\xi _{gm}(x)W_\\Upsilon (x,r_p,r_0)dx,$ where $W_\\Upsilon (x,r_p,r_0)$ is the window function [4]: $W_\\Upsilon (x,r_p,r_0)&=&\\frac{4x}{r^2_p}\\left(\\sqrt{x^2-r^2_0}\\Theta (x-r_0) - \\sqrt{x^2-r_p^2}\\Theta (x-r_p) \\right) \\nonumber \\\\&& -\\frac{2x}{r^2_p}\\left( \\frac{r^2_p\\Theta (x-r_p)}{\\sqrt{x^2-r^2_p}} - \\frac{r_0^2\\Theta (x-r_0)}{\\sqrt{x^2-r^2_0}}\\right),$ where $\\Theta (x)$ is the Heaviside step function.",
"From these equations one can see explicitly that $\\Upsilon _{gm}$ is related to the galaxy-matter cross-correlation function $\\xi _{gm}$ or cross-power spectrum $P_{gm}$ .",
"If we assume the biasing model of Eq.",
"REF , $P_{gm}$ can be written as [60] $P_{gm}(k) &=& b_1 P_{\\delta \\delta }(k) + b_2 P_{b2,\\delta }(k) + b_{s^2} P_{bs2,\\delta }(k) \\nonumber \\\\&& + b_{3nl}\\sigma ^2_3(k)P_{\\rm lin}(k),$ where $P_{\\rm {\\delta \\delta }}$ is the non-linear matter density-density power spectrum, $b_{3nl}$ is a third-order non-local bias term, $P_{\\rm lin}$ is the linear matter power spectrum, and $P_{b2,\\delta }$ , $P_{bs2,\\delta }$ are 1-loop integrals given in Appendix .",
"In the local Lagrangian picture where one assumes no initial non-local bias, one can predict that the non-local bias terms at later time are related to $b_1$ such that [17], [80] $b_{s^2} &=& - \\frac{4}{7} (b_1-1) \\\\b_{3nl} &=& \\frac{32}{315} (b_1-1).$ We adopt these relations and our model has finally two galaxy biasing parameters: $b_1$ and $b_2$ , $b_1$ being the standard linear bias parameter."
],
[
"Redshift-space distortions",
"The most general formalism describing the redshift-space anisotropies in the power spectrum derives from writing the matter density conservation in real and redshift space [50].",
"In particular, in the plane-parallel approximation that is assumed in this analysis, the anisotropic power spectrum of matter has the general compact form [83] $P^s(k,\\nu )&=&\\int \\frac{\\mathrm {d}^3\\vec{r}}{(2\\pi )^3} e^{-i\\vec{k} \\cdot \\vec{r}}\\left<e^{-ikf\\nu \\Delta u_\\parallel } \\times \\right.",
"\\nonumber \\\\&& \\left.",
"[\\delta (\\vec{x})+f \\partial _{_\\parallel } u_{_\\parallel }(\\vec{x})][\\delta (\\vec{x}^\\prime )+f \\partial _{_\\parallel } u_{_\\parallel }(\\vec{x}^\\prime )]\\right> $ where $\\nu =k_\\parallel /k$ , $u_\\parallel (\\vec{r})=-v_\\parallel (\\vec{r})/(f aH(a))$ , $v_\\parallel (\\vec{r})$ is the line-of-sight component of the peculiar velocity, $\\delta $ is the matter density field, $\\Delta u_\\parallel =u_\\parallel (\\vec{x})-u_\\parallel (\\vec{x}^\\prime )$ and $\\vec{r}=\\vec{x}-\\vec{x}^\\prime $ .",
"It is worth noting that in Fourier space, for an irrotational velocity field, $\\smash{\\partial _{_\\parallel } u_{_\\parallel }}$ is related to the divergence of the velocity field $\\theta $ via $\\smash{\\partial _{_\\parallel }u_{_\\parallel }(\\vec{k})=\\nu ^2\\theta (\\vec{k})}$ .",
"Although exact, Eq.",
"REF is impractical and we use the approximation proposed by [89].",
"In the case of perfect matter tracers, the latter model takes the form $P^s(k,\\nu ) &=& D(k\\nu \\sigma _v)\\left[P_{\\rm {\\delta \\delta }}(k)+2\\nu ^2 f P_{\\rm {\\delta \\theta }}(k) + \\nu ^4 f^2 P_{\\rm {\\theta \\theta }}(k) \\right.",
"\\nonumber \\\\&& \\left.",
"+ C_A(k,\\nu ,f) + C_B(k,\\nu ,f) \\right],$ where $D(k\\nu \\sigma _v)$ is a damping function, $P_{\\rm {\\delta \\delta }}$ , $P_{\\rm {\\delta \\theta }}$ , $P_{\\rm {\\theta \\theta }}$ are respectively the non-linear matter density-density, density-velocity divergence, and velocity divergence-velocity divergence power spectra, and $\\sigma _v$ is an effective pairwise velocity dispersion that we can fit for and then treat as a nuisance parameter.",
"The expressions for $C_A(k,\\nu ,f)$ and $C_B(k,\\nu ,f)$ are given in [89], [24].",
"This phenomenological model can be seen in configuration space as a convolution of a pairwise velocity distribution, the damping function $D(k\\mu \\sigma _v)$ that we assume to be Lorentzian in Fourier space, i.e.",
"$D(k\\nu \\sigma _v)=(1+k^2\\nu ^2\\sigma ^2_v)^{-1},$ and a term involving the density and velocity divergence correlation functions and their spherical Bessel transforms.",
"This model can be generalized to the case of biased tracers by including a biasing model.",
"By introducing that of Eq.",
"REF , one obtains for the redshift-space galaxy power spectrum [11], [35] $P_g^s(k,\\nu ) &=& D(k\\nu \\sigma _v)\\left[P_{\\rm {gg}}(k)+2\\nu ^2 f P_{\\rm {g\\theta }}(k) + \\nu ^4 f^2 P_{\\rm {\\theta \\theta }}(k) \\right.",
"\\nonumber \\\\&& \\left.",
"+ C_A(k,\\nu ,f,b_1) + C_B(k,\\nu ,f,b_1) \\right] $ where $P_{gg}(k)&=&b_1^2 P_{\\delta \\delta }(k)+2b_2b_1P_{b2,\\delta }(k)+2b_{s^2}b_1P_{bs2,\\delta }(k) \\nonumber \\\\&& +b_2^2P_{b22}(k) +2b_2b_{s^2}P_{b2s2}(k)+b_{s^2}^2P_{bs22}(k) \\nonumber \\\\&& +2b_1b_{3\\rm nl}\\sigma _3^2(k)P_{\\rm lin}(k) + N, \\\\P_{g\\theta }(k)&=&b_1P_{\\delta \\theta }(k)+b_2P_{b2,\\theta }(k)+b_{s^2}P_{bs2,\\theta }(k) \\nonumber \\\\&& +b_{3\\rm nl}\\sigma _3^2(k)P_{\\rm lin}(k).$ In the above equations $P_{\\rm {\\delta \\theta }}$ is the non-linear matter density-velocity divergence power spectrum, $P_{\\rm lin}$ is the matter linear power spectrum, and $P_{b2,\\delta }$ , $P_{bs2,\\delta }$ , $P_{b2,\\theta }$ , $P_{bs2,\\theta }$ , $P_{b22}$ , $P_{b2s2}$ , $P_{bs22}$ , $\\sigma _3^2$ are 1-loop integrals given in Appendix .",
"The final model for $\\xi ^s_\\ell (s)$ is obtained from its Fourier counterpart $P^s_\\ell (k)$ defined as $ P^s_\\ell (k)=\\frac{2\\ell +1}{2} \\int _{-1}^1 P_g^s(k,\\nu ) L_\\ell (\\nu )\\,\\mathrm {d}\\nu ,$ where $ \\xi ^s_\\ell (s)=i^\\ell \\int \\frac{k^2}{2\\pi ^2} P^s_\\ell (k)j_\\ell (ks)\\,\\mathrm {d}k.$ In the above equation, $j_\\ell $ denotes the spherical Bessel functions.",
"The ingredients of the model are the non-linear power spectra of density and velocity divergence at the effective redshift of the sample.",
"These power spectra can be predicted from perturbation theory or simulations for different cosmological models.",
"The non-linear matter power spectrum can also be obtained to a great accuracy from semi-analytical prescriptions such as HALOFIT [87], for various cosmologies.",
"In particular, HALOFIT allows the prediction of $P_{\\rm {\\delta \\delta }}$ from the linear matter power spectrum and the knowledge of the scale of non-linearity at the redshift of interest, $k_{\\rm nl}(z)$ .",
"We note that at fixed linear matter power spectrum shape, variations of $\\sigma _8(z)$ can be straightforwardly mapped into variations of $k_{\\rm nl}(z)$ [87].",
"In this analysis, the linear matter power spectrum is predicted using the CLASS Boltzmann code [56], and we use the latest calibration of HALOFIT by [88] to obtain $P_{\\rm {\\delta \\delta }}$ .",
"To predict $P_{\\rm {\\theta \\theta }}$ and $P_{\\rm {\\delta \\theta }}$ , we use the nearly universal fitting functions of [6] that depend on the linear power spectrum and $\\sigma _8(z)$ as $P_{\\rm {\\theta \\theta }}(z)&=&P_{\\rm lin}(z) e^{-k m_1\\sigma ^{m_2}_8(z)}\\\\ P_{\\rm {\\delta \\theta }}(z)&=&\\left(P_{\\rm {\\delta \\delta }}(z) P_{\\rm lin}(z) e^{-kn_1\\sigma ^{n_2}_8(z)}\\right)^{1/2},$ where $P_{\\rm lin}$ is the linear power spectrum and $(m_1,m_2,n_1,n_2)$ are free parameters calibrated on simulations.",
"We adopt here the values $(m_1,m_2,n_1,n_2)=(1.906,2.163,2.972,2.034)$ .",
"These predictions for $P_{\\rm {\\theta \\theta }}$ and $P_{\\rm {\\delta \\theta }}$ are accurate at the few percent level up to $k\\simeq 0.7$ [6].",
"Therefore, the overall degree of non-linearity in $P_{\\rm {\\delta \\delta }}$ , $P_{\\rm {\\delta \\theta }}$ and $P_{\\rm {\\theta \\theta }}$ is solely controlled by $\\sigma _8(z)$ , which is left free when fitting the model to observations.",
"In the model, the linear bias and growth rate parameters, $b_1$ and $f$ , are degenerate with the normalization of the matter power spectrum parameter $\\sigma _8$ .",
"Generally with RSD, only the combination of $b_1\\sigma _8$ and $f\\sigma _8$ can be constrained if no assumption is made on the actual value of $\\sigma _8$ .",
"However in the [89] model, $b_1^2f\\sigma _8^4$ , $b_1f^2\\sigma _8^4$ , and $f^3\\sigma _8^4$ terms appear in the correction term $C_A$ [89], [24].",
"Accordingly, in the general case, $(f,b_1,b_2,\\sigma _8,\\sigma _v)$ are treated as separate parameters in the fit and we provide marginalized constraints on the derived $f\\sigma _8$ ."
],
[
"Redshift errors",
"Redshift errors can potentially affect the anisotropic RSD signal.",
"In the anisotropic correlation function they have a similar effect as galaxy random motions in virialized objects: they introduce a smearing of the correlation function along the line of sight at small transverse separations.",
"If the probability distribution function of redshift errors is known, their effect can be forward modelled by adding another multiplicative damping function in the redshift-space galaxy power spectrum of Eq.",
"19.",
"In that case, the damping function should be the Fourier transform of the error probability distribution function.",
"We follow this approach and the final model is obtained by multiplying Eq.",
"19 by a Gaussian with standard deviation set to the estimated pairwise redshift dispersion of VIPERS galaxies such that the final RSD model $\\smash{\\widehat{P_g^s}}$ reads $ \\widehat{P_g^s}(k,\\nu )=G(k\\nu \\sigma _z)P_g^s(k,\\nu ),$ where $P_g^s(k,\\nu )$ is taken from Eq.",
"REF , $G$ is the Fourier transform of the Gaussian kernel $G(k\\nu \\sigma _z)=\\exp \\left(-\\frac{k^2\\nu ^2\\sigma ^2_z}{2}\\right),$ and $\\sigma _z$ is the pairwise standard deviation associated with the redshift error probability distribution function.",
"Figure: Probability distribution function of redshift errors at0.5<z<0.70.5<z<0.7 and 0.7<z<1.20.7<z<1.2 in the VIPERS data.",
"This is obtainedfrom the redshift differences of reobserved galaxies, for whichthere are two independent redshift measurements.",
"The dotted anddashed curves are best-fitting Gaussians for the redshift intervals0.5<z<0.70.5<z<0.7 and 0.7<z<1.20.7<z<1.2 respectively.The Gaussian form is motivated by the data themselves as shown in Fig.",
"REF .",
"In this figure are shown the distributions of redshift differences at $0.5<z<0.7$ and $0.7<z<1.2$ in VIPERS reobservations (1061 at $0.5<z<0.7$ and 1086 at $0.7<z<1.2$ ), for which we have two independent redshift measurements for the same galaxies [85].",
"These distributions can be rather well modelled by Gaussians, and by doing so, we obtain values of $\\sigma _z=1.31 \\times 10^{-3}$ and $\\sigma _z=1.36 \\times 10^{-3}$ for the pairwise redshift standard deviations at $0.5<z<0.7$ and $0.7<z<1.2$ respectively.",
"These are further converted in comoving length assuming the fiducial cosmology to enter the model in Eq.",
"REF ."
],
[
"Alcock-Paczynski effect",
"Additional distortions can arise in galaxy clustering because of the need to assume a fiducial cosmology to convert redshift and angular positions into comoving distances, and the fact that this fiducial cosmology is not necessarily the true one.",
"This is the [2] effect (AP).",
"More specifically, since the line-of-sight separations require the knowledge of the Hubble parameter, $H(z)$ , and transverse separations that of the angular diameter distance, $D_A(z)$ , any difference in $H(z)$ and $D_A(z)$ between the fiducial and true cosmologies, translates into an anisotropic clustering, independently of RSD.",
"Although AP and RSD anisotropies are degenerate to some extent in the observables [5], [59], they have a fundamentally different origin: AP is sensitive to the geometry whereas RSD are sensitive to the growth of cosmological perturbations.",
"We follow [96] and model AP distortions using the $\\alpha $ and $\\epsilon $ parameters, which characterize respectively the isotropic and anisotropic distortion components associated with AP.",
"These are given by $\\alpha &=& \\left( \\frac{D^2_A}{D^{\\prime 2}_A} \\frac{H^{\\prime }}{H} \\right)^{1/3} \\\\\\epsilon &=& \\left( \\frac{D^{\\prime }_A}{D_A} \\frac{H^{\\prime }}{H} \\right)^{1/3} - 1,$ where quantities calculated in the fiducial cosmology are denoted with primes.",
"Those parameters modify the scales at which the correlation function is measured such that $r_{\\parallel }^{\\prime } &=& \\alpha (1+\\epsilon )^2 r_{\\parallel }\\\\r_{\\perp }^{\\prime } &=& \\alpha (1+\\epsilon )^{-1} r_{\\perp }.$ Therefore, for the model correlation function monopole and quadrupole in a tested cosmology, the corresponding quantities in the fiducial cosmology are obtained as [96] $\\xi ^{\\prime }_0(s^{\\prime }) &=& \\xi _0(\\alpha s) + \\frac{2}{5}\\epsilon \\left[ 3\\xi _2(\\alpha s) + \\frac{d\\xi _2(\\alpha s)}{d\\ln (s)} \\right] \\\\\\xi ^{\\prime }_2(s^{\\prime }) &=& 2\\epsilon \\frac{d\\xi _0(\\alpha s)}{d\\ln (s)}+\\bigg ( 1 + \\frac{6}{7}\\epsilon \\bigg )\\xi _2(\\alpha s)+\\frac{4}{7}\\epsilon \\frac{d\\xi _2(\\alpha s)}{d\\ln (s)} \\nonumber \\\\&& + \\frac{4}{7}\\epsilon \\bigg [ 5\\xi _4(\\alpha s) +\\frac{d\\xi _4(\\alpha s)}{d\\ln (s)} \\bigg ].$ In the case of the galaxy-galaxy lensing statistic that we are considering, since it is a function of the transverse separation $r_p$ , the corresponding $\\Upsilon _{gm}$ in the fiducial cosmology is simply given by $\\Upsilon ^\\prime _{gm}(r^\\prime _p)=\\Upsilon _{gm}\\left(\\alpha (1+\\epsilon )^{-1}r_p\\right).$"
],
[
"Cosmological insights from galaxy clustering and galaxy-galaxy lensing",
"Gravitational physics on cosmological scales can be tested from measurements of the growth rate of structure, which is well measured from RSD in the galaxy clustering pattern.",
"We have seen that in practice, the correlation function multipole moments depend not only on the growth rate of structure $f$ , but also on the shape and amplitude $\\sigma _8$ of the matter power spectrum, the galaxy bias parameters $b_1$ and $b_2$ , and the pairwise velocity dispersion $\\sigma _v$ .",
"To derive the growth rate of structure, one then needs to marginalise over those nuisances.",
"This is of course a source of uncertainty in the determination of the growth rate of structure.",
"Moreover, since there is a degeneracy between the amplitude of the matter power spectrum $\\sigma _8$ , the growth rate of structure $f$ , and the linear bias parameter $b_1$ , RSD alone are sensitive to the $f\\sigma _8$ and $b_1\\sigma _8$ parameter combinations.",
"On the other hand, galaxy-galaxy lensing probes the real-space galaxy-matter correlations that are described by the shape and amplitude $\\sigma _8$ of the matter power spectrum, the galaxy bias parameters $b_1$ and $b_2$ , and the matter density parameter $\\Omega _m$ .",
"Projected galaxy-galaxy correlations are also sensitive to $\\sigma _8$ , $b_1$ , and $b_2$ .",
"But by looking in detail at those dependencies, we can see that in the linear regime $\\Upsilon _{gm}\\propto \\Omega _m b_1 \\sigma ^2_8$ , while $\\Upsilon _{gg} \\propto b^2_1\\sigma ^2_8$ , such that by combining the two we can break the degeneracy between $b_1$ and $\\sigma _8$ .",
"We note that $\\xi (s,\\mu )$ , from which $\\xi _0$ and $\\xi _2$ are derived, has the same parameter dependences as $\\Upsilon _{gg}$ , except for the additional $f$ dependence.",
"Therefore, additional galaxy-galaxy lensing information brings an independent handle on the bias parameters $b_1$ and $b_2$ , and the power spectrum amplitude $\\sigma _8$ , reducing the uncertainties on the growth rate of structure induced by the lack of knowledge on the bias of galaxies, as well as a supplementary sensitivity to $\\Omega _m$ ."
],
[
"Simulated data",
"To test the robustness of redshift-space galaxy clustering, galaxy-galaxy lensing, and associated error estimates, we make use of a large number of mock galaxy samples, which are designed to be a realistic match to the VIPERS final dataset.",
"We used the mock lensing lightcones presented in [36].",
"These have been built upon the Big MultiDark dark matter N-body simulation [52], which assumes a flat $\\Lambda {\\rm CDM}$ cosmology with $(\\Omega _m,~\\Omega _\\Lambda ,~\\Omega _b,~h,~n,~\\sigma _8) = (0.307, 0.693,0.0482, 0.678, 0.960, 0.823)$ and covers a volume of $15.625\\,h^{-3}~{\\rm Gpc^3}$ .",
"These lightcones contain the shear information associated with simulated background galaxies distributed uniformly on the sky but following the redshift distribution of CFHTLenS galaxies.",
"More specifically, the lightcones have been built to match the effective number density and redshift distribution of the CFHTLenS lensing catalogue.",
"We added Gaussian random errors with standard deviation $\\smash{\\sigma _e=(\\sigma _{e_1}^2+\\sigma _{e_2}^2)^{1/2}=0.38}$ to the ellipticities to mimic those in the CFHTLenS data.",
"The size of the simulation allowed us to create 54 independent lightcones for W1 and W4, spanning the redshift range $0<z<2.3$ [36].",
"We populate these lightcones with foreground galaxies using the halo occupation distribution (HOD) technique and apply the detailed VIPERS selection function and observational strategy.",
"The haloes were identified in the simulation using a friends-of-friends algorithm with a relative linking length of $b=0.17$ times the inter-particle separation.",
"The mass limit to which the halo catalogues are complete is $10^{11.95}\\,h^{-1}~{\\rm M_{\\odot }}$ .",
"Because this limiting mass is too large to host the faintest galaxies observed with VIPERS, we use the method of [26] to reconstruct haloes below the resolution limit.",
"This method is based on stochastically resampling the halo number density field using constraints from the conditional halo mass function.",
"For this, one needs to assume the shapes of the halo bias factor and halo mass function at masses below the resolution limit and use the analytical formulae obtained by [92], [93].",
"With this method we are able to populate the simulation with low-mass haloes with a sufficient accuracy to have unbiased galaxy two-point statistics in the simulated catalogues [25].",
"The minimum reconstructed halo mass we consider for the purpose of creating VIPERS mocks is $10^{10}\\,h^{-1}~{\\rm M_{\\odot }}$ .",
"In this process, we populate each halo with galaxies according to its mass, the mean number of galaxies in a halo of a given mass being given by the HOD.",
"It is common usage to differentiate between central and satellite galaxies in haloes.",
"While the former are put at rest at halo centres, the latter are randomly distributed within each halo according to a NFW radial profile [67], [68].",
"The halo occupation function and its dependence on redshift and luminosity/stellar mass must be precisely chosen in order to obtain mock catalogues with realistic galaxy clustering properties.",
"We calibrated the halo occupation function directly on the VIPERS data, as presented in [25].",
"We add velocities to the galaxies and measure their redshift-space positions.",
"While the central galaxies are assigned the velocity of their host halo, satellite galaxies have an additional random component for which each Cartesian velocity component is drawn from a Gaussian distribution with a standard deviation that depends on the mass of the host halo.",
"Details about the galaxy mock catalogue construction technique are given in Appendix A of [25].",
"The final step in obtaining fully realistic VIPERS mocks is to add the detailed survey selection function.",
"We start by applying the magnitude cut $i_{\\rm AB}<22.5$ and the effect of the colour selection on the radial distribution of the mocks.",
"This is achieved by depleting the mocks at $z<0.6$ so as to reproduce the VIPERS colour sampling rate [41].",
"The mock catalogues that we obtain are then similar to the parent photometric sample in the data.",
"We next apply the slit-positioning algorithm with the same setting as for the data.",
"This allows us to reproduce the VIPERS footprint on the sky, the small-scale angular incompleteness and the variation of ${\\rm TSR}$ across the fields.",
"Finally, a random redshift error is added to the redshifts as in the data.",
"We are thus able to produce realistic mock galaxy catalogues that contain the detailed survey completeness function and observational biases of VIPERS, which we refer to as the `observed' mock catalogues in the following.",
"We note that another set of VIPERS mock catalogues spanning the redshift range of $0.4<z<1.2$ have been constructed.",
"This set, which comprises 306 and 549 lightcones of W1 and W4 fields respectively, has not been explicitly used in this analysis, but in accompanying VIPERS PDR-2 analyses [44], [74], [95], [79]."
],
[
"Systematics on the correlation function monopole and quadrupole",
"The mock samples are crucial for testing the redshift-space clustering estimation in VIPERS, which is not trivial given the complex selection function of the survey.",
"We first study the impact of the survey selection function on the measurement of the monopole and quadrupole correlation functions.",
"We measured these quantities in the observed mocks, applying the different weights defined in Sect.",
"REF , and compare them to the reference measurements obtained from the parent mocks, including VIPERS typical spectroscopic redshift errors.",
"The relative differences in $\\xi _0$ and $\\xi _2$ as a function of separation and averaged over the mocks are shown in Figs.",
"REF and REF , respectively for the two samples at $0.5<z<0.7$ and $0.7<z<1.2$ .",
"First of all, it is clear from these figures that without any correction the spectroscopic strategy introduces biases in the estimation of the galaxy clustering.",
"But when applying the survey completeness weights $w^C$ , one can recover within a few percent the correct amplitude of the correlation functions on scales above $5\\,h^{-1}~{\\rm Mpc}$ .",
"By further applying the angular weights $w^A$ , we obtain an almost unbiased estimate of the monopole and quadrupole down to a few $\\,h^{-1}~{\\rm Mpc}$ .",
"The statistical relative error induced by sample variance and estimated from the dispersion among the mock samples, is shown with the shaded area in these figures.",
"It is important to note that it is much larger than any residual systematics over the range of scales considered.",
"Finally, it is worth mentioning that in the quadrupole, the apparent higher level of systematics at around $s=10 \\,h^{-1}~{\\rm Mpc}$ is an artefact due to the zero crossing of the functions at slightly different separations."
],
[
"Systematics on the growth rate of structure",
"We further study our ability to determine $f\\sigma _8$ when combining RSD and galaxy-galaxy lensing measurements in a maximum likelihood analysis.",
"For this purpose we perform several analyses of the mean RSD and galaxy-galaxy lensing measurements in the observed mocks, for different minimum separations $s_{\\rm min}$ in the correlation functions and different cut-off scale $r_0$ in the annular differential excess surface density.",
"These analyses are performed on the mean quantities to reduce the impact of statistical errors and concentrate on systematics.",
"The precision matrix is estimated from the mocks as explained in Sect.",
", except that each element is further divided by the number of mocks to characterize the error on the mean.",
"As an illustration, we present in this section only the case of the sample at $0.5<z<0.7$ .",
"The sample at $0.7<z<1.2$ provides very similar systematic levels.",
"Fig.",
"REF presents the systematic errors on $f\\sigma _8$ , i.e.",
"the relative difference of recovered values with respect to the fiducial value of the mocks, as a function of $s_{\\rm min}$ and for $r_0=(1 h^{-1}{\\rm Mpc}, 1.5 h^{-1}{\\rm Mpc})$ .",
"We consider rather small minimum scales and cut-off radii to explore the extent to which our modelling is robust in the translinear regime.",
"We can see in this figure that our model allows the recovery of the fiducial value of $f\\sigma _8$ down to $s_{\\rm min}=6.3\\,h^{-1}~{\\rm Mpc}$ , with systematic errors below $5\\%$ , independently of the choice of $r_0$ .",
"In principle, values of $r_0$ smaller than the typical radius of haloes hosting these galaxies may not be optimal, since the non-linear contribution to correlations may dominate on those scales, which are more difficult to describe [4].",
"However, this also depends on the galaxy type and the redshift.",
"For VIPERS galaxies and the considered biasing model, we find that $r_0=1\\,h^{-1}~{\\rm Mpc}$ can be well described by our model [13].",
"This can be seen in Fig.",
"REF where is shown the comparison between the mean mock $\\Delta \\Sigma _{gm}$ and $\\Upsilon _{gm}$ obtained with $r_0=(1h^{-1}{\\rm Mpc},1.5h^{-1}{\\rm Mpc})$ and the predictions of our model, when $b_2$ is allowed to vary and $b_1$ is fixed to its fiducial value.",
"We can see that although the model fails to reproduce $\\Delta \\Sigma _{gm}$ on scales below about $3\\,h^{-1}~{\\rm Mpc}$ , it provides a good description of $\\Upsilon _{gm}$ for $b_2=-0.1$ .",
"Figure: Comparison of differential excess surface densityΔ gm \\Delta _{gm} and annular differential excess surface densityΥ gm \\Upsilon _{gm} predictions in the mocks (points and shaded regions)and by our theoretical model (curves) at 0.5<z<0.70.5<z<0.7.",
"The mockpredictions correspond to the mean signal among the mockrealizations (points) and its associated 1σ1\\sigma error (shadedregion).",
"The curves show the theoretical predictions for thefiducial parameters of the mocks, varying only the b 2 b_2 parameteras labeled.We finally test the impact of redshift errors on the recovery of $f\\sigma _8$ in Fig.",
"REF .",
"This figure shows the relative systematic error on $f\\sigma _8$ as a function $s_{\\rm min}$ in the case where $r_0$ is fixed to $1\\,h^{-1}~{\\rm Mpc}$ and typical VIPERS redshift errors are added randomly to mock galaxy redshifts.",
"By comparing it to the case without redshift errors, we can see that for the RSD model where a Gaussian damping term is added to account for redshift errors, the recovery of $f\\sigma _8$ is achieved without additional bias, with only a small relative bias of about $3\\%$ at $s_{\\rm min}=6.3\\,h^{-1}~{\\rm Mpc}$ and $-5\\%$ above.",
"We note that this is the ideal case where the redshift error probability distribution function is perfectly known.",
"Overall, these tests demonstrate that our model with $s_{\\rm min}=6.3\\,h^{-1}~{\\rm Mpc}$ and $r_0=1\\,h^{-1}~{\\rm Mpc}$ is robust enough to provide a precise measurement of $f\\sigma _8$ with VIPERS data, with residual systematics of the order of a few per cent only, but only representing about one fifth of expected statistical error on the measurement as shown with the grey shaded region in Fig.",
"REF .",
"Based on these tests, we adopt $(s_{\\rm min},r_0)=(6.3~h^{-1}{\\rm Mpc}, 1~h^{-1}{\\rm Mpc})$ values for the following analysis."
],
[
"Likelihood analysis and precision matrix",
"In order to derive cosmological parameters from the combination of RSD and galaxy-galaxy lensing measurements, we perform a maximum likelihood analysis in which we define the likelihood function $\\mathcal {L}$ such that $-2\\ln {\\mathcal {L}}=\\sum _{i=1}^{N_p}\\sum _{j=1}^{N_p}\\Delta _i \\hat{C}^{-1}_{ij} \\Delta _j,$ where $N_p$ is the number of data points in the fit, $\\bf \\Delta $ is the data-model difference vector, and $\\smash{{\\bf \\hat{C}^{-1}}}$ is the inverse data covariance matrix.",
"$\\bf \\Delta $ is defined such that each element is $\\Delta _i=d_i-m_i$ , where $\\bf d$ and $\\bf m$ are respectively the data and model prediction vectors.",
"In our case, $\\bf d$ is the concatenation of $\\xi _0$ , $\\xi _2$ , and $\\Upsilon _{gm}$ , for the set of considered separations.",
"The parameter space of the model is explored using a Monte Carlo Markov Chain (MCMC) method implementing the Metropolis-Hastings algorithm [61].",
"A robust estimation of the inverse data covariance matrix, or precision matrix, is crucial in order to achieve realistic posterior likelihood functions of the parameters.",
"The different bins in $\\xi _0$ , $\\xi _2$ , and $\\Upsilon _{gm}$ are correlated to some degree and this must be allowed for in the likelihood analysis.",
"We measure these three quantities in the 54 mocks and estimate the covariance matrix $\\bf C$ .",
"The generic elements of the matrix can be evaluated as $C_{ij}=\\frac{1}{N_{m}-1}\\sum _{k=1}^{N_m}\\left(d^k_i-\\bar{d_i}\\right)\\left(d^k_j-\\bar{d_j}\\right)\\, ,$ where $N_m$ is the number of mock realizations and the indices $i,j$ run over the data vector $\\bf d$ elements.",
"An unbiased estimate of the inverse covariance matrix, ${\\bf \\hat{C}^{-1}}$ , is obtained as [43] $ {\\bf \\hat{C}^{-1}}=\\frac{N_{m}-N_{p}-2}{N_{m}-1}{\\bf C}^{-1},$ for $N_m>N_p-2$ .",
"The resulting inverse covariance matrix obtained from mock realizations can be noisy, depending on how large $N_m$ is with respect to $N_p$ .",
"In our case, $N_m=54$ and $N_p=16$ , which suggests the presence of a non-negligible noise in the inverse covariance matrix.",
"In order to reduce the level of noise, we adopt the tapering technique of [51].",
"This technique has been introduced in the context of cosmological analysis by [70].",
"This technique relies on the assumption that large-scale covariances vanish, and consists of tapering the covariance matrix around the diagonal using a specific positive and compact taper function.",
"Contrary to other estimators such as shrinkage [76], the two-tapers estimator has the advantage of being unbiased.",
"The inverse tapered covariance matrix is obtained as ${\\bf \\hat{C_T}^{-1}}=\\left( \\frac{N_{m}-N_{p}-2}{N_{m}-1}\\right) ({\\bf C}\\circ {\\bf T})^{-1}\\circ {\\bf T},$ where `$\\circ $ ' denotes the element-wise matrix product and ${\\bf T}$ is the tapering matrix.",
"We follow [70] and use the tapering matrix defined as $T_{ij} = K(|{\\bf x}_i-{\\bf x}_j|),$ where $x_i$ is the $i^{th}$ measurement position in the data vector, and $K$ is the taper function that we take to be a Wendland function: K(x)={ lcl (1-xTp)4 (4xTp+1) if x<Tp 0 otherwise.",
".",
"This taper function has one free parameter, the tapering scale $T_p$ , which essentially represents the typical scale difference above which covariances are nullified.",
"Figure: Top panel: recovered errors on fσ 8 f\\sigma _8 normalized to thatobtained without tapering (equivalent to applying tapering withT p =∞T_p=\\infty ), as a function of the tapering scale T p T_p used in theestimation of the precision matrix.",
"This is obtained from the mocksat 0.5<z<0.70.5<z<0.7.",
"Bottom panel: recovered maximum likelihood valuesfor fσ 8 f\\sigma _8 and associated 1σ1\\sigma error as a function of thetapering scale T p T_p.Figure: Correlation matrix (left panel) and normalized precisionmatrix (right panel) for galaxy clustering and galaxy-galaxy lensingdata in the redshift interval 0.5<z<0.70.5<z<0.7.",
"These are defined asC ij /C ii C jj C_{ij}/\\sqrt{C_{ii}C_{jj}} andC ij -1 /C ii -1 C jj -1 C^{-1}_{ij}/\\sqrt{C^{-1}_{ii}C^{-1}_{jj}} respectively, whereC ij C_{ij} and C ij -1 C^{-1}_{ij} refer to covariance and precision matrixelements respectively.",
"In both panels, the upper triangular matrixrepresents the case without tapering, while the lower panel the casewith tapering.",
"The precision matrix is normalized such that diagonalelements are unity.In our case, the covariance matrix is associated with three different quantities as well as two different separation types, $s$ and $r_p$ .",
"One would then potentially need to use a combination of several taper functions, since one does not expect the large-scale covariance to vanish at the same scales for all quantities.",
"Although it may be sub-optimal to use a single taper function, we still expect to increase the signal-to-noise, and since the estimator is unbiased, one cannot introduce additional bias or error.",
"We therefore decided to use a single taper function for simplicity.",
"In the general case, it is not straightforward to define a priori the optimal tapering scale.",
"[70] introduced a simple empirical method, which consists of performing several maximum likelihood analyses of the data varying only the tapering scale, and taking as the optimal $T_p$ the one that minimizes the error on the parameter of interest.",
"We perform the same exercise on the mean mock predictions.",
"The marginalized $1\\sigma $ error on $f\\sigma _8$ as a function of $T_p$ is presented in the top panel of Fig.",
"REF .",
"We can see that the $T_p$ value that minimizes the error is around $15\\,h^{-1}~{\\rm Mpc}$ , and we adopt this value in our analysis.",
"We also verified that the maximum likelihood values for $f\\sigma _8$ remain unchanged for any value of $T_p$ as shown in the bottom panel of Fig.",
"REF .",
"To illustrate the method, we present in Fig.",
"REF the correlation matrix and normalized precision matrix, for the combined RSD and galaxy-galaxy lensing data in the redshift interval $0.5<z<0.7$ , when applying or not the tapering technique (lower and upper triangles respectively).",
"Those matrices are defined as $C_{ij}/\\sqrt{C_{ii}C_{jj}}$ and $C^{-1}_{ij}/\\sqrt{C^{-1}_{ii}C^{-1}_{jj}}$ respectively, where $C_{ij}$ and $C^{-1}_{ij}$ refer to covariance and precision matrix elements respectively.",
"We can see the reduction of noise, which is particularly clear in the normalized precision matrix for most off-diagonal terms.",
"The tapering technique allows a significant reduction of the noise level in the precision matrix, but cannot completely remove it.",
"The remaining noise can propagate through the likelihood analysis into derived parameter uncertainties.",
"In order to obtain realistic confidence limits on parameters one needs to account for the additional uncertainties coming from the precision matrix estimation [90].",
"[73] showed that this additional error can be described as a rescaling of the target parameter covariance, in the case when the precision matrix is estimated with the standard estimator of Eq.",
"REF .",
"But the appropriate degree of rescaling is unclear when the tapering estimator is used.",
"The improvement on the error that we find with the tapering estimator (i.e.",
"$26.5\\%$ ) is similar to or larger than what we would expect with the standard estimator using 300 mocks or more as predicted by [28], [73].",
"This gives us confidence that only a small correction, if any, would be necessary."
],
[
"Cosmological results",
"The comprehensive tests of the methodology described in previous sections make us confident that we can perform a robust combined analysis of RSD and galaxy-galaxy lensing with VIPERS and CFHTLenS dataset, and infer cosmology from it.",
"We present in this section the data measurements, growth rate of structure constraints, and derived gravitational slip parameters at $0.5<z<0.7$ and $0.7<z<1.2$ ."
],
[
"Galaxy clustering and galaxy-galaxy lensing measurements",
"The correlation function measurements are performed on the full VIPERS galaxy sample in the redshift intervals $0.5<z<0.7$ and $0.7<z<1.2$ .",
"We select all VIPERS galaxies above the limiting magnitude of the survey, and measure the monopole and quadrupole correlation functions in both W1 and W4 fields.",
"The combined W1+W4 measurements are obtained by summing up the pairs in the two fields, contributing to the anisotropic two-point correlation functions $\\xi (s,\\mu )$ , before deriving $\\xi _0$ and $\\xi _2$ from Eq.",
"REF .",
"The full anisotropic two-point correlation functions are presented in Fig.",
"REF , and the monopole and quadrupole moments in Fig.",
"REF .",
"In the latter figure, the individual mock measurements are superimposed, giving a visual appreciation of the error associated with these measurements in VIPERS.",
"We can see that the combined W1+W4 monopole and quadrupole correlation function measurements enable us to probe accurately the redshift-space galaxy clustering signal on scales below about $s=50\\,h^{-1}~{\\rm Mpc}$ .",
"Figure: Anisotropic correlation functions of VIPERS galaxies at0.5<z<0.70.5<z<0.7 (top panel) and 0.7<z<1.20.7<z<1.2 (bottom panel) as a functionparallel and transverse to the line-of-sight separations.Figure: Monopole (circles) and quadrupole (triangles) correlationfunctions of VIPERS galaxies at 0.5<z<0.70.5<z<0.7 (top panel) and0.7<z<1.20.7<z<1.2 (bottom panel).",
"Solid curves correspond to individualmock measurements.The differential excess surface density measurements are obtained by combining W1 and W4 individual field measurements in a similar fashion.",
"The lens galaxies are taken from the VIPERS catalogue or the CFHTLenS catalogue if no spectroscopic redshift is available.",
"They are selected to have $i_{\\rm AB}<22.5$ and a redshift in the intervals $0.5<z<0.7$ and $0.7<z<1.2$ .",
"The source galaxies are taken from the CFHTLenS catalogue and are selected to have $i_{\\rm AB}<24.1$ .",
"The differential excess surface density and annular differential excess surface density measurements for $r_0=1\\,h^{-1}~{\\rm Mpc}$ are presented in Fig.",
"REF .",
"As in Fig.",
"REF , the individual mock measurements are superimposed.",
"We can see that with the combined W1+W4 annular differential excess surface density measurements we can reach scales up to about $r_p=20\\,h^{-1}~{\\rm Mpc}$ .",
"Figure: Differential excess surface density (circles) and annulardifferential excess surface density (triangles) at 0.5<z<0.70.5<z<0.7 (toppanel) and 0.7<z<1.20.7<z<1.2 (bottom panel).",
"Solid curves correspond toindividual mock measurements.Our $\\Upsilon _{gm}$ measurements are more uncertain than the $\\xi _0$ and $\\xi _2$ ones.",
"This is essentially related to the way the former are estimated.",
"Weak lensing is fundamentally limited by the unknown intrinsic ellipticity of the sources, which dominates the error budget.",
"This can be mitigated by means of a larger number of sources.",
"Given the surface density of sources in our sample and its rather modest angular coverage of $23.5~\\rm {deg}^2$ , we obtain relative errors on $\\Upsilon _{gm}$ of about $25\\%$ , estimated from the mock samples.",
"In contrast, the typical relative error that we obtain on $\\xi _0$ is of $5\\%$ .",
"Therefore, in our combined analysis of the RSD and galaxy-galaxy lensing we expect $\\Upsilon _{gm}$ to have a much lower weight in the likelihood.",
"We finally remark that the observed $\\Upsilon _{gm}$ tend to exhibit lower amplitudes than expected in the mock samples, in particular in the highest redshift interval.",
"We discuss the cosmological implications of this in Sect.",
"REF ."
],
[
"Growth of structure constraints",
"We perform a combined maximum likelihood analysis of the monopole, quadrupole, and annular differential excess surface density to derive constraints on the growth rate of structure at $0.5<z<0.7$ and $0.7<z<1.2$ .",
"The effective redshifts associated with these intervals are $z=0.6$ and $z=0.86$ .",
"They correspond to the average redshift of pairs contributing the most to monopole and quadrupole correlation functions in these redshift intervals [82].",
"The theoretical model that we use is described in Sect.",
"; it depends on 11 parameters, ${\\bf p}=(f,b_1,b_2,\\sigma _v,\\sigma _8,\\epsilon ,\\alpha ,\\Omega _m,\\Omega _mh^2,\\Omega _bh^2,n_s)$ .",
"The last three describe the shape of the matter power spectrum and these are determined most accurately by CMB data.",
"Since our galaxy clustering and weak lensing measurements cannot provide such tight constraints on these parameters, we fix them to the best-fitting Planck 2015 TT+lowP+lensing parameters [75].",
"Consistently, $\\Omega _m$ is kept fixed to the Planck value in $\\Upsilon _{gm}$ .",
"Possible departures from those parameter values are only allowed through variations of the AP distortion parameters $\\epsilon $ and $\\alpha $ .",
"In the following, we first consider measurements of $f\\sigma _8$ , as a derived parameter, and later study the possibility of deriving independent measurements of $f$ and $\\sigma _8$ .",
"All those measurements are obtained by marginalizing over the nuisance parameters: ${\\bf p_n}=(b_1,b_2,\\sigma _v,\\epsilon ,\\alpha )$ .",
"The adopted uniform priors on the likelihood parameters are summarized in Table REF and the full posterior likelihood contours for the cases presented in the next section are given in Appendix .",
"Table: Adopted priors on the likelihood parameters."
],
[
"$f\\sigma _8$ measurements",
"In our standard configuration, the linear matter power spectrum shape is fixed to the best-fitting $\\Lambda \\rm {CDM}$ model from Planck 2015 TT+lowP+lensing data [75].",
"AP distortion parameters are set to $(\\epsilon ,\\alpha )=(0,1)$ and are not allowed to vary.",
"In this configuration we obtain $f\\sigma _8$ values of $f\\sigma _8(z=0.6)&=&0.48\\pm 0.12 \\\\f\\sigma _8(z=0.86)&=&0.48\\pm 0.10,$ after marginalizing over other parameters.",
"Associated reduced chi-squared values are $\\chi ^2_\\nu =1.52$ and $\\chi ^2_\\nu =1.62$ respectively.",
"These measurements use both RSD and galaxy-galaxy lensing information.",
"It is instructive to see the impact of adding the galaxy-galaxy lensing on the measurement of $f\\sigma _8$ .",
"Thus if we use the standard RSD approach without including galaxy-galaxy lensing information, we obtain $f\\sigma _8(z=0.6)&=&0.48\\pm 0.11 \\\\f\\sigma _8(z=0.86)&=&0.46\\pm 0.09,$ with a reduced chi-squared value of $\\chi ^2_\\nu =1.12$ for both redshifts.",
"In that case, we fixed $b_2=b_{s^2}=b_{3nl}=0$ in the RSD model, as bias non-linearities are negligible for VIPERS galaxies bias given the minimum scale used in the fit [74].",
"Moreover, the shape of non-linear power spectra in the model is fixed by setting $\\sigma _8$ to its fiducial value at the effective redshift of the sample, as is commonly done [25].",
"The recovered values and associated errors are very similar to the previous case.",
"We do not find an improvement on $f\\sigma _8$ accuracy when galaxy-galaxy lensing is included, in fact errors are marginally larger.",
"This can be explained by the lower number of degrees of freedom in the RSD-only case and the significant uncertainty associated with our galaxy-galaxy lensing measurements compared to the galaxy clustering ones in the VIPERS fields.",
"In fact the real gain is on contraining $f$ and $\\sigma _8$ separately as discussed in Sect.",
"REF ."
],
[
"Inclusion of Alcock-Paczynski distortions",
"As a robustness test, we relax the assumption on the shape of the linear matter power spectrum.",
"We allow the AP distortion parameters $(\\epsilon ,\\alpha )$ to vary, considering flat priors on $\\epsilon ,\\alpha $ parameters, extending by $\\pm 0.1$ around $(\\epsilon ,\\alpha )=(0,1)$ .",
"After marginalizing over those parameters as well, we obtain the following $f\\sigma _8$ measurements: $f\\sigma _8(z=0.6)&=&0.51\\pm 0.13, \\\\f\\sigma _8(z=0.86)&=&0.52\\pm 0.11,$ with reduced chi-squared values of $\\chi ^2_\\nu =1.58$ and $\\chi ^2_\\nu =1.3$ respectively.",
"As expected from the additional degrees of freedom introduced in the likelihood, the marginalized $68\\%$ errors on $f\\sigma _8$ are increased, although the constraints remain completely compatible with previous measurements when $\\epsilon $ and $\\alpha $ were fixed.",
"This test thus removes any potential concern that our measurements of $f\\sigma _8$ might lack robustness though being dependent on the assumption of a $\\Lambda \\rm {CDM}$ expansion history."
],
[
"Comparison with other measurements",
"In Fig.",
"REF we compare our $f\\sigma _8$ measurements with previous measurements from the literature, as well as predictions of the standard relativistic model for gravity.",
"Our measurements are consistent with previous measurements at lower or similar redshifts from VVDS [40], SDSS LRG [14], [81], WiggleZ [12], 6dFGS [10], VIPERS PDR-1 [25], MGS [49], FastSound [69], BOSS-LOWZ [33], and BOSS-CMASS [33], [18].",
"In particular, our measurement at $z=0.6$ is compatible within $1\\sigma $ with the WiggleZ $z=0.6$ [12] and BOSS-CMASS $z=0.57$ [33], [18] measurements.",
"Our results are also very close to the standard cosmological model predictions: they are consistent within $1\\sigma $ with General Relativity predictions in a $\\Lambda \\rm {CDM}$ model with cosmological parameters set to Planck CMB results [75].",
"Figure: fσ 8 f\\sigma _8 as a function of redshift, showing VIPERS resultscontrasted with a compilation of recent measurements.",
"The previousresults from VVDS , SDSS LRG, , WiggleZ , 6dFGS, VIPERS PDR-1 , MGS, FastSound , BOSS-LOWZ, BOSS-CMASS , , andVIPERS PDR-2 voids are shown with the differentsymbols (see labels).",
"The solid curve and associated shaded areacorrespond to the expectations and 68%68\\% uncertainty for GeneralRelativity in a Λ CDM \\Lambda \\rm {CDM} background model set toTT+lowP+lensing Planck 2015 predictions .These results are part of a combined effort of the VIPERS collaboration to estimate the growth rate of structure from the same data but using different complementary techniques.",
"Specifically, in [74] we provide a thorough investigation of the performances of different RSD models in configuration space, using a general consistent modelling of non-linear RSD; in [95] we use the clipping technique in Fourier space to minimise the impact of non-linearities; finally in [44] we use cosmic voids as RSD tracers.",
"In particular in [44], we make use of the void catalogue built from the VIPERS PDR-2 data and resulting from the earlier work by [62], to estimate the void-galaxy cross-correlation function in redshift space.",
"By modelling its anisotropy we obtain an estimate of $f\\sigma _8$ at $z = 0.73$ and derive a value of $f\\sigma _8(z=0.73) = 0.296^{+0.075}_{-0.078}$ , which is lower than those obtained here.",
"However, this technique is still in its infancy, with potential systematic errors not yet fully understood.",
"This and the other VIPERS measurements are all fully compatible within statistical errors.",
"More discussion is presented in the specific papers."
],
[
"$f, b_1, \\sigma _8$ degeneracy breaking",
"As discussed in Sect.",
", the use of RSD in the galaxy clustering pattern allows a measurement of the parameter combinations $f\\sigma _8$ , $b_1\\sigma _8$ , or $\\beta =f/b_1$ .",
"But with the additional constraint of galaxy-galaxy lensing, which exhibits different parameter dependencies, we expect to be able to break the $f-b_1-\\sigma _8$ degeneracy inherent to galaxy-galaxy correlations.",
"We investigate this by studying the posterior likelihood contours at $68\\%,95\\%,99\\%$ for the various pairs of $f$ , $b$ , $\\sigma _8$ parameters in our data.",
"This is done for the likelihood analyses presented in the previous sections, i.e.",
"when including or not galaxy-galaxy lensing.",
"The posterior likelihood contours are presented in Figs.",
"REF and REF for the two considered redshift intervals.",
"Figure: Same as Fig.",
"but for the redshiftinterval 0.7<z<1.20.7<z<1.2.These figures show strong degeneracies in the $f-b_1$ , $f-\\sigma _8$ , and $b_1-\\sigma _8$ planes when considering only RSD.",
"In particular, we can see in the $f-\\sigma _8$ plane the distribution of the likelihood contours along the regions with constant $f\\sigma _8$ , marked with solid and dashed curves in the figures.",
"Now with the inclusion of galaxy-galaxy lensing, we can see a shrinking of the contours, in particular along the $\\sigma _8$ direction, and to a lesser extent along the $b_1$ one.",
"Galaxy-galaxy lensing thus effectively provides a strong handle on the $\\sigma _8$ parameter.",
"This allows the $f-\\sigma _8$ degeneracy to be broken and therefore leads to the possibility of a direct measurement of the growth rate of structure, $f$ .",
"The $f-b_1$ degeneracy is also partially broken, even if the effect is milder.",
"We find that the $f-\\sigma _8$ degeneracy breaking is more efficient in the high-redshift interval, with measurements of $(f,\\sigma _8)=(0.93\\pm 0.22,0.52\\pm 0.06)$ and $(f,\\sigma _8)=(0.99\\pm 0.19,0.48\\pm 0.04)$ at $z=0.6$ and $z=0.86$ respectively.",
"These direct measurements of the growth rate of structure and $\\sigma _8$ are in agreement within $1\\sigma $ with Planck ${\\rm \\Lambda CDM+GR}$ predictions, which are $(f,\\sigma _8)=(0.79,0.60)$ and $(f,\\sigma _8)=(0.85,0.53)$ respectively at $z=0.6$ and $z=0.86$ .",
"Planck ${\\rm \\Lambda CDM+GR}$ predictions are represented with the stars in Figs.",
"REF and REF .",
"In Fig.",
"REF , we compare our $(f,\\sigma _8)$ constraints with those from [34], obtained by combining redshift-space galaxy power spectrum and bispectrum information in the BOSS survey at $z=0.57$ .",
"In [34], they use the galaxy bispectrum instead of galaxy-galaxy lensing to bring additional constraints on galaxy bias.",
"Although those measurements are quite uncertain, this parameter space and how it can be used as a cosmological model diagnostic, will be very interesting to explore for next-generation cosmological surveys, such as Euclid, which will allow a dramatical improvement on such measurement accuracy.",
"Figure: Joint (f,σ 8 (f,\\sigma _8) constraints at different redshifts.",
"Thecombined RSD and galaxy-galaxy lensing posterior likelihood contoursat 1σ1\\sigma and 2σ2\\sigma and those from ,obtained by combining redshift-space power spectrum and bispectruminformation in the BOSS survey, are presented.",
"The solid curve andassociated grey shaded area correspond to the expectations and68%68\\% uncertainty for General Relativity in a Λ CDM \\Lambda \\rm {CDM}background model set to TT+lowP+lensing Planck 2015 predictions, as a function of redshift from z=2z=2 to z=0z=0.Independent measurements of $\\sigma _8$ at different redshifts also carry information about the growth rate of structure.",
"Since $\\sigma _8$ grows with time proportionally to the growth factor, the growth rate can be written as $\\mathrm {d}\\ln \\sigma _8/\\mathrm {d}\\ln a$ .",
"In the case of two $\\sigma _8$ measurements at $a_1$ and $a_2$ , as in our analysis, this equation can be approximated through finite difference by $f\\simeq \\frac{\\ln \\left(\\sigma _8(a_1)/\\sigma _8(a_2)\\right)}{\\ln \\left(a_1/a_2\\right)}.$ By applying this to our $\\sigma _8$ measurements we obtain an additional, independent measurement of $f=0.57\\pm 0.96$ at the mean redshift of $z=0.73$ .",
"It is clear that this type of measurement is not compelling in our dataset, but can potentially be useful as an additional constraint to be combined with direct measurements in next-generation cosmological surveys.",
"Finally, we notice in Figs.",
"REF and REF that the addition of galaxy-galaxy lensing constraints significantly modifies the posterior probability distribution function of the linear bias parameter, $b_1$ , becoming more compact and skewed towards larger values.",
"This means that adding galaxy-galaxy lensing information reduces the uncertainties on $b_1$ , and pushes its maximum likelihood value towards values that are in excellent agreement with previous linear bias estimates that are not solely based on two-point statistics [27], [16], [39]."
],
[
"Gravitational slip",
"In addition to the growth rate of structure, we can measure the gravitational slip parameter $E_G$ .",
"This is done by taking the ratio of the measured $\\Upsilon _{gm}$ and $\\Upsilon _{gg}$ , and multiplying it by $\\beta ^{-1}$ .",
"The RSD distortion parameter $\\beta $ is estimated from the combined maximum likelihood analysis of the monopole and quadrupole correlation functions (the same as for the RSD-only case presented in Sect.",
"REF ).",
"After marginalizing over nuisance parameters we obtain $\\beta (z=0.6)&=&0.66\\pm 0.17 \\\\\\beta (z=0.86)&=&0.63\\pm 0.14.$ The $68\\%$ error on the $E_G$ measurements is obtained by adding in quadrature the fractional error on $\\Upsilon _{gm}/\\Upsilon _{gg}$ estimated from mock samples and the fractional error on $\\beta ^{-1}$ .",
"The $E_G(r_p)$ measurements are presented in Fig.",
"REF for the two redshift intervals under consideration, and compared with the linear predictions for ${\\rm \\Lambda CDM+GR}$ (horizontal line and associated $68\\%$ contour).",
"We find that our measurements at $z=0.6$ are compatible within $1\\sigma $ with the standard model, although the central values tend to be slightly lower.",
"We also report in this figure the averaged gravitational slip parameter over the range $3\\,h^{-1}~{\\rm Mpc}$ <$r_p$ <$50\\,h^{-1}~{\\rm Mpc}$ , $\\smash{\\overline{E}_G}$ , obtained by [13] in the similar redshift range $0.43<z<0.7$ .",
"It is represented with a stripe in the figure, with horizontal extent corresponding to the range of $r_p$ used to measure $\\smash{\\overline{E}_G}$ and vertical extent showing the $\\pm 1\\sigma $ error on the measurement.",
"By averaging our $E_G$ over $3\\,h^{-1}~{\\rm Mpc}$ <$r_p$ <$20\\,h^{-1}~{\\rm Mpc}$ we obtain $\\smash{\\overline{E}_G}(z=0.6)=0.16\\pm 0.09$ and $\\smash{\\overline{E}_G}(z=0.86)=0.09\\pm 0.07$ .",
"Our results are in good agreement with this measurement and also with that by [77] at much higher scales, which also exhibits a slightly lower value compared with ${\\rm \\Lambda CDM+GR}$ prediction.",
"The $E_G$ measurements are lower than ${\\rm \\Lambda CDM+GR}$ at $r_p>3\\,h^{-1}~{\\rm Mpc}$ but remain within $1-2\\sigma $ , depending on the scale.",
"At $z=0.86$ , the agreement with ${\\rm \\Lambda CDM+GR}$ is poorer than at lower redshift.",
"Figure: Gravitational slip parameter as a function of scale asmeasured at 0.5<z<0.70.5<z<0.7 (top panel) and 0.7<z<1.20.7<z<1.2 (bottompanel).",
"In both panels, the solid curves and associated shaded areascorrespond to the expectations and 68%68\\% uncertainties for GeneralRelativity in a Λ CDM \\Lambda \\rm {CDM} background model set toTT+lowP+lensing Planck 2015 predictions .",
"In the toppanel, the horizontal stripe shows the averagedE ¯ G \\smash{\\overline{E}_G} over the range 3h -1 Mpc 3\\,h^{-1}~{\\rm Mpc}<r p r_p<50h -1 Mpc 50\\,h^{-1}~{\\rm Mpc}obtained by at 0.43<z<0.70.43<z<0.7.",
"E G E_G asymptotes toΩ m /f\\Omega _m/f in the standard model, and the simplest way of erasingthe modest discrepancy with the model prediction would be to lowerthe density parameter.The origin of the tendency of our $E_G$ measurements to be smaller than expected at $r_p>3\\,h^{-1}~{\\rm Mpc}$ remains unclear.",
"Particularly given our statistical errors, we have to be cautious in interpreting this trend.",
"In any case, such a result could arise as a result of residual observational systematics or a misinterpretation of the observables, rather than any break-down of standard gravitational physics.",
"From the construction of $E_G$ , these low values of $E_G$ seem to be most probably caused by the rather low measured amplitude of $\\Upsilon _{gm}$ at $r_p>3\\,h^{-1}~{\\rm Mpc}$ , and seen in Fig.",
"REF .",
"If this discrepancy is upheld by further data, one possible interpretation is that weak lensing prefers a lower value of $\\Omega _m$ than that determined by CMB data.",
"It is worth noticing that a similar tension has already been identified in the CFHTLenS cosmic shear analysis of [45], as well as in the more recent analysis performed in the KiDS lensing survey [48].",
"It is clear that this point needs to be investigated in detail in the future, in particular in the preparation of next-generation very large surveys combining galaxy clustering and weak lensing observables."
],
[
"Conclusion",
"This paper has presented a combined analysis of redshift-space distortions and galaxy-galaxy lensing in the final VIPERS dataset, making use of complementary data from the CFHTLenS lensing survey over the same area.",
"We have built a consistent theoretical model of the two observables, which includes prescriptions for non-linear, non-local galaxy bias, as well as quasi-linear redshift-space distortions.",
"This model has been shown to enable robust measurements of the growth rate of structure.",
"The model robustness and adopted methodology have been tested by using a series of realistic mock surveys constructed for this purpose.",
"The main goal of VIPERS has been to provide an accurate measurement of the growth rate of structure using redshift-space distortions in a redshift regime where the growth is not well determined.",
"With the first data release we were able to provide an initial measurement of $f\\sigma _8$ at $z=0.8$ [25].",
"The final dataset increases the survey volume by a factor of $1.6$ , and by further adding galaxy-galaxy lensing information, we have been able to provide new accurate measurements of $f\\sigma _8$ at both $z=0.6$ and $z=0.86$ .",
"We have found values of $f\\sigma _8(z=0.6)=0.48\\pm 0.12$ and $f\\sigma _8(z=0.86)=0.48\\pm 0.10$ , which are consistent with previous measurements at lower or similar redshifts.",
"The additional galaxy-galaxy lensing constraint and the specific treatment of $\\sigma _8$ to describe the non-linearity level of the real-space power spectra entering the model alleviate the degeneracy between the galaxy bias parameter, $\\sigma _8$ , and $f$ , and has allowed direct measurements of these two parameters.",
"We have obtained values of $\\left(f,\\sigma _8\\right)=(0.93\\pm 0.22,0.52\\pm 0.06)$ and $\\left(f,\\sigma _8\\right)=(0.99\\pm 0.19,0.48\\pm 0.04)$ at $z=0.6$ and $z=0.86$ , respectively.",
"These measurements put new constraints on gravity at the epoch when the Universe was almost half its present age.",
"Our measurements are statistically consistent with a Universe where the gravitational interactions between structures on cosmological scales can be described by General Relativity, although they are not yet accurate enough to rule out some commonly considered alternatives to General Relativity.",
"In addition to measuring the growth rate of structure, we have been able to measure the gravitational slip parameter, $E_G$ , for the first time at $z>0.6$ .",
"This quantity, which can be directly constructed from galaxy clustering and galaxy-galaxy lensing observables, is sensitive to the growth rate of structure and mean matter density in the Universe.",
"We have obtained averaged values of the gravitational slip parameter of $\\smash{\\overline{E}_G}(z=0.6)=0.16\\pm 0.09$ and $\\smash{\\overline{E}_G}(z=0.86)=0.09\\pm 0.07$ .",
"Our $E_G$ measurements are consistent within $1-2\\sigma $ , although they exhibit slightly lower values than expected in the standard model for gravity in a $\\Lambda \\rm {CDM}$ background.",
"Overall, this analysis has demonstrated the importance of the combination of galaxy clustering in redshift space and galaxy-galaxy lensing in order to probe the origin of cosmic acceleration.",
"This combination can alleviate the inherent uncertainty related to galaxy bias in RSD analyses and provide new insights into the gravitational physics at work on cosmological scales.",
"This analysis and adopted methodology can be seen as a proof-of-concept in the context of the preparation of next-generation cosmological surveys such as Euclid [54], which will allow galaxy clustering and galaxy-galaxy lensing to be combined with exquisite precision.",
"We acknowledge the crucial contribution of the ESO staff for the management of service observations.",
"In particular, we are deeply grateful to M. Hilker for his constant help and support of this program.",
"Italian participation to VIPERS has been funded by INAF through PRIN 2008, 2010, 2014 and 2015 programs.",
"LG and BRG acknowledge support from the European Research Council through grant n. 291521.",
"OLF acknowledges support from the European Research Council through grant n. 268107.",
"JAP acknowledges support of the European Research Council through the COSFORM ERC Advanced Research Grant (# 670193).",
"GDL acknowledges financial support from the European Research Council through grant n. 202781.",
"RT acknowledges financial support from the European Research Council through grant n. 202686.",
"AP, KM, and JK have been supported by the National Science Centre (grants UMO-2012/07/B/ST9/04425 and UMO-2013/09/D/ST9/04030).",
"EB, FM and LM acknowledge the support from grants ASI-INAF I/023/12/0 and PRIN MIUR 2010-2011.",
"LM also acknowledges financial support from PRIN INAF 2012.",
"SDLT, EJ, and MP acknowledge the support of the OCEVU Labex (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the \"Investissements d'Avenir\" French government program managed by the ANR.",
"Research conducted within the scope of the HECOLS International Associated Laboratory, supported in part by the Polish NCN grant DEC-2013/08/M/ST9/00664.",
"TM and SA acknowledge financial support from the ANR Spin(e) through the French grant ANR-13-BS05-0005.",
"The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.",
"(www.gauss-centre.eu) and the Partnership for Advanced Supercomputing in Europe (PRACE, www.prace-ri.eu) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de)."
],
[
"Theoretical power spectra for biased tracers",
"This appendix presents the models describing the real-space galaxy-galaxy, galaxy-velocity divergence, and galaxy-matter power spectra, which enter the modelling of RSD and galaxy-galaxy lensing.",
"We adopt the non-linear non-local bias model of [60] that relates the galaxy overdensity $\\delta _{g}$ and matter overdensity $\\delta $ as: $ \\delta _{g}({\\bf x}) &=& b_1 \\delta ({\\bf x}) + \\frac{1}{2}b_2[\\delta ^2({\\bf x})-\\sigma ^2]+\\frac{1}{2}b_{s^2}[s^2({\\bf x})-\\langle s^2 \\rangle ] \\nonumber \\\\&& + O(s^3({\\bf x})).$ In this equation, $b_1$ and $b_2$ are the linear and second-order non-linear bias terms, $b_{s^2}$ the non-local bias term, $s$ is the tidal tensor term from which non-locality originates.",
"The $\\sigma ^2$ and $\\langle s^2\\rangle $ terms ensure the condition $\\langle \\delta _g\\rangle =0$ .",
"From the bias model of Eq.",
"REF one can derive the following power spectra for galaxy-galaxy, galaxy-velocity divergence ($\\theta $ ), and galaxy-matter correlations: $P_{gg}(k)&=&b_1^2 P_{\\delta \\delta }(k)+2b_2b_1P_{b2,\\delta }(k)+2b_{s^2}b_1P_{bs2,\\delta }(k) \\nonumber \\\\&& +b_2^2P_{b22}(k) +2b_2b_{s^2}P_{b2s2}(k)+b_{s^2}^2P_{bs22}(k) \\nonumber \\\\&& +2b_1b_{3\\rm nl}\\sigma _3^2(k)P_{\\rm lin}(k) + N, \\\\P_{g\\theta }(k)&=&b_1P_{\\delta \\theta }(k)+b_2P_{b2,\\theta }(k)+b_{s^2}P_{bs2,\\theta }(k) \\nonumber \\\\&& +b_{3\\rm nl}\\sigma _3^2(k)P_{\\rm lin}(k), \\\\P_{gm}(k) &=& b_1 P_{\\delta \\delta }(k) + b_2 P_{b2,\\delta }(k) + b_{s^2} P_{bs2,\\delta }(k) \\nonumber \\\\&& + b_{3nl}\\sigma ^2_3(k)P_{\\rm lin}(k),$ where [11], [35] $P_{b2,\\delta }(k)&=&\\int \\frac{d^3q}{(2\\pi )^3}\\, P_{\\rm lin}(q)P_{\\rm lin}(|{\\bf k}-{\\bf q}|) F_2({\\bf q},{\\bf k-q}),\\\\P_{bs2,\\delta }(k)&=& \\int \\frac{d^3q}{(2\\pi )^3}\\, P_{\\rm lin}(q)P_{\\rm lin}(|{\\bf k}-{\\bf q}|) F_2({\\bf q},{\\bf k-q}) \\nonumber \\\\&& S_2({\\bf q},{\\bf k-q}), \\\\P_{b2,\\theta }(k)&=&\\int \\frac{d^3q}{(2\\pi )^3}\\,P_{\\rm lin}(q)P_{\\rm lin}(|{\\bf k}-{\\bf q}|)G_2({\\bf q},{\\bf k}-{\\bf q}),\\\\P_{bs2,\\theta }(k)&=&\\int \\frac{d^3q}{(2\\pi )^3}\\,P_{\\rm lin}(q)P_{\\rm lin}(|{\\bf k}-{\\bf q}|)G_2({\\bf q},{\\bf k}-{\\bf q}) \\nonumber \\\\&& S_2({\\bf q},{\\bf k}-{\\bf q}), \\\\P_{b2s2}(k)&=& -\\frac{1}{2}\\int \\frac{d^3q}{(2\\pi )^3}\\, P_{\\rm lin}(q)\\left[ \\frac{2}{3}P_{\\rm lin}(q) - P_{\\rm lin}(|{\\bf k}-{\\bf q}|) \\right.",
"\\nonumber \\\\&& \\left.",
"S_2({\\bf q},{\\bf k}-{\\bf q}) \\right], \\\\P_{bs22}(k)&=& -\\frac{1}{2} \\int \\frac{d^3q}{(2\\pi )^3}\\, P_{\\rm lin}(q)\\left[ \\frac{4}{9}P_{\\rm lin}(q) - P_{\\rm lin}(|{\\bf k}-{\\bf q}|) \\right.",
"\\nonumber \\\\&& \\left.",
"S_2({\\bf q},{\\bf k}-{\\bf q})^2\\right], \\\\P_{b22}(k)&=& -\\frac{1}{2} \\int \\frac{d^3q}{(2\\pi )^3}\\, P_{\\rm lin}(q)\\left[ P_{\\rm lin}(q)-P_{\\rm lin}({\\bf k}-{\\bf q}|)\\right],\\\\\\sigma ^2_3(k)&=&\\int \\frac{d^3{\\bf q}}{(2\\pi )^3}\\,P_{\\rm lin}(q)\\left[ \\frac{5}{6}+\\frac{15}{8}S_2({\\bf q},{\\bf k}-{\\bf q}) \\right.",
"\\nonumber \\\\&& \\left.",
"S_2(-{\\bf q},{\\bf k}) -\\frac{5}{4}S_2({\\bf q},{\\bf k}-{\\bf q}) \\right].$ In the above equations, $S_2$ , $F_2$ , $G_2$ perturbation theory kernels are defined by [38], [8] $S_2({\\bf k}_i,{\\bf k}_j)&=&\\frac{({\\bf k}_i\\cdot {\\bf k}_j)^2}{(k_ik_j)^2}-\\frac{1}{3}, \\\\F_2({\\bf k}_i,{\\bf k}_j)&=& \\frac{5}{7}+\\frac{1}{2}\\frac{{\\bf k}_i\\cdot {\\bf k}_j}{k_ik_j}\\left(\\frac{k_i}{k_j}+\\frac{k_j}{k_i}\\right)+\\frac{2}{7}\\left[ \\frac{{\\bf k}_i\\cdot {\\bf k}_j}{k_ik_j} \\right]^2, \\\\G_2({\\bf k}_i,{\\bf k}_j)&=& \\frac{3}{7}+\\frac{1}{2}\\frac{{\\bf k}_i\\cdot {\\bf k}_j}{k_ik_j}\\left(\\frac{k_i}{k_j}+\\frac{k_j}{k_i}\\right)+\\frac{4}{7}\\left[ \\frac{{\\bf k}_i\\cdot {\\bf k}_j}{k_ik_j} \\right]^2.$"
],
[
"Posterior likelihood contours",
"In this appendix are provided the posterior likelihood contours of all pairs of parameters appearing in the likelihood analyses presented in Sect.",
"REF .",
"Fig.",
"REF shows the posterior likelihood contours in the case of the RSD-only analysis, while Fig.",
"REF in the case where RSD and galaxy-galaxy lensing are combined.",
"In both figures, the three types of shaded regions in each subpanel correspond to the posterior likelihood contours at $68\\%$ , $95\\%$ , and $99\\%$ .",
"Figure: Posterior likelihood contours for ff, σ 8 \\sigma _8, b 1 b_1, andσ v \\sigma _v parameters at z=0.6z=0.6 (left panel) and z=0.86z=0.86 (rightpanel) in the case where RSD are considered alone (seeSect.",
").Figure: Posterior likelihood contours for ff, σ 8 \\sigma _8, b 1 b_1,b 2 b_2, and σ v \\sigma _v parameters at z=0.6z=0.6 (left panel) andz=0.86z=0.86 (right panel) in the case where RSD and galaxy-galaxylensing are combined (see Sect.",
")."
]
] | 1612.05647 |
[
[
"Temporal ghost imaging with twin photons"
],
[
"Abstract We use twin photons generated by spontaneous parametric down conversion (SPDC) to perform temporal ghost imaging of a single time signal.",
"The retrieval of a binary signal containing eight bits is performed with an error rate below 1%."
],
[
"Introduction",
"For the two last decades, ghost imaging has emerged as a kind of magical way to form images of a spatial object, typically a spatially varying transparency, with a Single Point Detector (SPD) that does not have spatial resolution.",
"The initial works used the quantum nature of entanglement of a two-photons state, where photons of a pair are spatially and temporally correlated, to detect temporal coincidences.",
"While one of the photons passing through the object was detected by a photon counter with no spatial resolution, its twin photon was detected with spatial resolution by scanning the transverse plane with a single detector [1], or recently by an intensified charge-coupled device (ICCD) [2].",
"Later, ghost imaging exploiting the temporal correlations of the intensity fluctuations of classical [3] or pseudothermal light [4] was proposed.",
"The ability to retrieve the object with unity contrast seems the only property that belongs to quantum experiments on their own [5].",
"The extension of ghost imaging to a time object, i.e.",
"a temporally varying transparency, has been recently demonstrated experimentally [6], [7], [8].",
"In [6], the light was transmitted through a \"time object\" and detected with a slow SPD which cannot resolve the time object, while, in the reference arm, the light that did not interact with the temporal object was detected with a fast SPD.",
"Measurements over several thousands copies of the temporal signal were necessary to retrieve a binary signal with a good signal-to-noise ratio.",
"To retrieve a non-reproducible time object using a single shot acquisition, we proposed in [7] the exact space-time transposition of computational ghost imaging [9], [10]: a single shot acquisition of the time object was performed by multiplying it with computer-generated random images, ensuring spatial multiplexing of temporal intensity correlations before detecting the sum image with no temporal resolution.",
"While very simple and costless, this method is slow.",
"To increase the speed to a $kHz$ rate, we reported the use of speckle patterns [8] i.e.",
"the temporal transposition of spatial ghost imaging with pseudothermal light [4].",
"In the present paper, we demonstrate temporal ghost imaging with twin photons generated by spontaneous parametric down conversion (SPDC), i.e the temporal transposition of the first ghost imaging experiments [1]: while the photons passing through the temporal object are detected by a photon-counting camera with no temporal resolution, their twins do not interact with the object but are detected with temporal and spatial resolution by a second camera.",
"Note that the use of biphotons for temporal imaging has been studied theoretically in [11]."
],
[
"Experimental overview",
"In the setup represented in Figure REF , a type 2 oriented Beta Barium Borate (BBO) nonlinear crystal, with a diameter of 5 mm and a thickness of 0.8 mm, is enlightened over its entire surface with a 354.65 nm UV pulsed laser.",
"From their interaction with the BBO, pump photons are annihilated and generate twin photons that form the signal and idler SPDC beams at 708.5$\\pm $ 8 nm.",
"The filters used have a quasi-rectangular spectrum transmission profile, centered at 708.5$\\pm $ 8 nm, i.e.",
"not exactly at the degeneracy wavelength of the SPDC.",
"A variable density made with a liquid crystal variable retarder controlled by a function generator, and followed by a polarizer, is placed on the way of the signal beam.",
"Hence, the signal images are first weighted one by one by the variable density whose values are given by the temporal signal to be retrieved, then summed together, without any temporal resolution, on an electron multiplying charge coupled device (EMCCD1).",
"At exactly the same time, the unweighed idler images are acquired one by one by a second camera (EMCCD2) as reference patterns, with for each image $I_n$ a time exposure synchronized with the step $n$ of the time signal.",
"The signal and idler images are recorded in the image plane of the BBO crystal, in order to ensure a good match between the positions of the signal and idler photons in twin images, even if their wavelengths are slightly different due to the 16 nm width filters.",
"The $512\\times 512$ pixels EMCCDs (Andor iXon3 897) sensors are cooled at $-100^\\circ $ C, and ensure a quantum efficiency over 90% at 708 nm.",
"The photon localizations are recorded by applying a thresholding procedure, (as shown in Figure REF ).",
"The mean flux is set between $0.10$ and $0.20$ photon per pixel (ph/px) on the sum image, i.e.",
"less on the reference images, in order to minimize the whole number of false detections [12].",
"The equivalent quantum efficiency $\\eta $ of the setup is given for twin images by the number of detected signal (or idler) photons corresponding to a true pair divided by the total number of detected photons.",
"This parameter takes into account the overall quantum efficiency of the setup, affected by the random absorption of photons by the optical components or no detection by the cameras, but also parasitic fluorescence of the optical components and false decisions during the thresholding procedure.",
"We have shown in [13] that sources of single photons, like parasitic fluorescence or photons transmitted at the edge of the filters with no transmission at the twin wavelength, have an effect similar to a decrease of the quantum efficiency.",
"Here, the equivalent quantum efficiency of the filters is estimated at 92% and the combined maximum transmission of the retarder and the polarizer is estimated at $\\eta _L=81\\%$ .",
"Noises from the detector, like readout noise or clock induced charges (CIC) result also in single photoelectrons: either a non genuine photon, due for example to CIC, is detected (false positive error) or a genuine photon is not detected, because the associated level at the output of the multiplication register remains below the threshold (false negative error).",
"This latter case is directly equivalent to a decrease of the quantum efficiency.",
"The former, false positive error, cannot be considered equivalent as a variation of the quantum efficiency since its occurrence does not depend of the light flux [13].",
"However, it does result in the creation of single detected photons and its effect is completely similar to a decrease of the quantum efficiency in integral measurements like that performed in this experiment.",
"The background noises were estimated by recording images with the pump beam off.",
"They are estimated at $0.018~ph/px$ for the signal image and $0.0064~ph/px$ for one of the idler images.",
"Figure: (a)(a): mean on 900 realizations of the normalized cross-correlation coefficient of two twin images, without binning.",
"(b)(b): the same for two independent signal and idler images.",
"(c)(c): the cross-correlation coefficient of figure (a)(a) after a binning of 16×516\\times 5 pixels.The experimental value of $\\eta $ is directly given by the normalized cross-correlation coefficient of the signal and idler pictures.",
"However, in our case, the size of the spatial coherence cells of the SPDC beams does not correspond to the pixels and scales as the inverse of the phase matching angular range [14].",
"The cross-correlation then displays a peak with a Gaussian like shape that spreads on several pixels.",
"To obtain the full cross-correlation peak on one pixel, a grouping (binning) of B pixels of the cross-correlation figure must be performed.",
"To estimate $\\eta $ , we recorded one series of 900 twin images, with no time modulation (transmission set to one) and the same integration time on each side.",
"The mean filling ratio $m_s$ was set at $0.04 ph/px$ by adjusting the acquisition time of the cameras.",
"The number of pixels $D=506\\times 506$ in each picture used for the cross-correlation is then given by the effective area of the cameras sensors.",
"The average normalized cross-correlation is given in figure REF .",
"The size of the spatial coherence cell is estimated at $B=16\\times 5$ pixels.",
"However, the binning of the cross-correlation figure is not used here since the average on 900 cross-correlations allows the distinction of the pixels of the peak from those of the cross-correlation background.",
"The integration of the cross-correlation peak gives an overall equivalent quantum efficiency $\\eta $ of $30.2\\%$ ."
],
[
"Retrieval of a ghost time signal",
"After acquisition of the weighted and of the reference images, the reconstruction of the time signal is performed by computing the cross-correlation of the signal image and each reference idler image.",
"Here, since we use only one pair of images to determine each cross-correlation coefficient, the cross-correlation peak must be binned in order to be distinguished from the cross-correlation background.",
"The successive cross-correlation coefficients are then plotted over the time to retrieve the shape of the time signal.",
"In an ideal experiment with unity quantum efficiency, a photoelectron detected in the integrated signal image corresponds always to a photon at the same position in one of the reference idler images.",
"However, the random distribution of SPDC photons provides, for different time steps for the signal and the idler or different temporal modes in a single time step, statistically independent photon repartitions.",
"Consequently two signal and idler photons that are not twin can be situated in a coherence cell at the same position.",
"Those photons create accidental matches when cross-correlating the associated signal and idler images.",
"We thus need to consider in the computation of the cross-correlation coefficient a contribution related to accidental coincidences of independent events, but also a fluctuation of the twin and accidental coincidences due to the random nature of the events.",
"The reconstruction of the time signal can be performed properly in a single operation only if we can distinguish at least two levels in the signal (case of a binary signal).",
"For a Gaussian distribution, this distinction is feasible in 99.3% of cases if the signal to noise ratio $SNR_n$ associated to the step $n$ verifies : $SNR_n = \\frac{C_n}{\\sigma _n}\\ge 2.45~L~T_n$ where $\\sigma _n$ is the overall standard deviation of the number of coincidences, $L=2$ is the number of levels in the signal, $T_n$ is the binary transmission ($T_n=0$ or 1) induced by the variable attenuation for the step $n$ of the time signal, and $C_n$ is the mean total number of twin coincidences between the sum picture $S$ and the idler image $I_n$ .",
"The value $2.45\\times \\sigma $ represents the abscissa at which the cumulative density function of a Gaussian distribution takes a value 99.3%.",
"The average number of accidental coincidences is shifted to zero by removing in each picture the deterministic shape of the SPDC beams.",
"This step ensures the statistical independence of two non twin images.",
"Two effective methods can be applied here.",
"The first one consists in assimilating the shape of the SPDC beam as a Gaussian, since the intensity of the SPDC beams is proportional to the intensity of the pump beam.",
"Each signal and idler picture is fitted by a Gaussian profile that is then removed from the image.",
"This method is efficient if the experiment is limited to the retrieval of a unique time signal and not repeated [15].",
"However, if the shape of the beams is not perfectly Gaussian, this method could leave some residual deterministic correlations.",
"The second method consists in recording a large number of images to determine the average deterministic shape of the signal and idler beams, as a calibration of the system.",
"The average shape is then removed from each picture before performing the cross-correlation [14].",
"If feasible, this last method is slightly more efficient and much more rapid.",
"It will be used in the following.",
"$C_n$ can be determined from our experimental parameters, after this subtraction, as [13]: $C_n \\simeq T_n D (\\eta m_i-m_i^2)$ where $m_i$ is the mean number of photons (events) per pixel of the idler images.",
"The approximation is valid if the incident photon flux per time step $m_i/\\eta $ is much smaller than one, which allows the probability of two photons incident on the same pixel to be neglected.",
"$\\sigma _n$ takes into account the fluctuations of both twin and accidental coincidences : $\\sigma _n = {(V_{c,n} + V_{a,n})}^{1/2}$ where $V_{c,n}$ and $V_{a,n}$ are respectively the variances of the total number of twin and accidental coincidences between $S$ and $I_n$ , at the location of the peak.",
"Because of the poissonian distribution of the SPDC pattern, we have directly $V_{c,n} = C_n $ .",
"Since $V_{a,n}$ is not related to the number of twin coincidences between the signal integrated image $S$ and one of the idler reference image $I_n$ , it does not depend on the step $n$ , consequently $V_{a,n}=V_{a}$ .",
"Its value can be assessed as follows, in a similar manner as in ref [13].",
"We want to assess the fluctuations of $\\widehat{cov}(N_s,N_{i_k})$ , the estimator of the covariance between one pixel $s$ of the signal picture $S$ and the same pixel $i_k=s$ of the idler picture $I_k$ , for independent events (no twin coincidences).",
"$N_s$ and $N_{i_k}$ are the intensities of the $s$ and ${i_k}$ pixels, that are in our case either 1 for one photon or 0 for no photon.",
"$\\widehat{cov}(N_s,N_{i_k})$ is given for each couple $s=i_k$ of the area D by : $\\begin{split}\\ \\widehat{cov}(N_s,N_{i_k}) = \\frac{1}{D} \\sum \\limits _{s=i_k=1}^{D} (N_{s}-\\overline{N}_{s}) (N_{i_k}-\\overline{N}_{i_k})\\\\=\\overline{N_{s}N_{i_k}}-\\overline{N}_{s}\\overline{N}_{i_k}\\end{split}$ Because of the independence of the events, $<\\widehat{cov}(N_s,N_{i_k})>=0$ , where $< >$ stands for the true mean (mathematical expectation).",
"In the second term of Eq.REF , the variance of $\\overline{N}_{s}\\overline{N}_{i_k}$ is negligible with respect to the variance of $\\overline{N_{s}N_{i_k}}$ .",
"The only possible values of $N$ are 0 and 1.",
"Hence we have : $\\begin{split}\\ var(\\overline{N_{s}N_{i_k}})=\\frac{1}{D}(<(N_{s}N_{i_k})^{2}>-<N_{s}N_{i_k}>^{2})\\\\=\\frac{1}{D}m_s m_i-(m_s m_i)^{2}\\simeq \\frac{1}{D}m_s m_i\\end{split}$ where $m_s$ and $m_i$ are respectively the true means of the signal and idler images.",
"In the last approximative equality, we assume that $m_s$ and $m_i$ are both $<<1$ .",
"The total number of coincidences is given also by Eq.REF , but without the division by the number of pixels.",
"Hence, we have, if no binning: $\\begin{split}V_{a}=var\\left(\\sum \\limits _{s=i_k=1}^{D} N_{s} N_{i_k}\\right)=D~ m_s m_i\\end{split}$ The last step consists in calculating the variance $V_{a}$ of the total number of coincidences between two areas obtained by summing $B$ adjacent pixel values of the correlation image: $\\begin{split}V_{a}=var\\left(\\sum \\limits _{b=1}^{B}\\sum \\limits _{s=i_k=1}^{D} N_{s} N_{i_k}\\right)=D~ B~ m_s m_i\\end{split}$ The signal to noise ratio hence becomes : $SNR_n = \\frac{T_n D (\\eta m_i-m_i^2)}{(T_n D \\eta m_i + D B~ m_s m_i)^{1/2}}$ Because of the binning B, the second term of the denominator, due to accidental coincidences, is much greater than the first term, due to the fluctuations of the number of twin coincidences.",
"By neglecting this first term and the second term of the numerator, and by assuming a binary signal with M bits at one, we obtain an approximation of $SNR_n$ as: $SNR_n \\simeq T_n \\eta \\left(\\frac{D}{B~ M}\\right)^{1/2}$ The approximation of Eq.REF , though not very precise (the second term of the numerator is not completely negligible) gives us a practical clue.",
"We have to find an optimal compromise for the binning: increasing $B$ allows the surface of the cross-correlation peak to be entirely covered, resulting in an increase of $\\eta $ , but at the expense of a decreasing of the number of resolution cells $D/B$ .",
"Experimentally, the lowest error rate has been attained for a binning of 5 pixels on the y axis and 16 pixels on the x axis.",
"This reduced binning unfortunately brings us to ignore the fourth of the coincidences, that are situated on pixels outside the position of the binned peak.",
"The equivalent quantum efficiency thus becomes $\\eta =23\\%$ .",
"By taking into account these values, we have chosen to perform the reconstruction of a time signal of 2 levels (binary) and 8 steps, that should result in a $SNR$ around 6."
],
[
"Experimental application",
"Although the purpose of the experiment is to benefit of a safe reconstruction on a single operation, the process is repeated 990 times.",
"This allows the estimation of the average experimental $SNR$ on the reconstructed steps.",
"The ghost signal reconstructed here is made of 4 bits at \"1\" and 4 bits at \"0\".",
"The reconstruction shown in figure REF displays the average cross-correlation coefficient of each step of the time signal, given as a number of coincidences.",
"Let us recall that the average images have been subtracted, meaning that the mean numbers of accidental coincidences have been set to 0, even if these accidental coincidences are the main source of noise, as shown above.",
"The error bars represent the experimental standard deviation of the computed numbers of coincidences associated to each step.",
"The $SNR$ of the steps \"1\" is here equal to $4.9$ , while a direct application of Eq.REF gives a $SNR$ of 6.3.",
"The most important factors that explain this difference between are: - the gaussian shape of the beams result in an effective number of pixels which is smaller than the number $D$ of physical pixels.",
"- In the low light level parts of the image, the detector noises are more important than taken into account by the effective quantum efficiency.",
"- The fluctuations of the pixels in the correlation image are not completely independent, probably because of some smearing.",
"Experimentally, for B=1 the standard deviation of the correlation image has a value outside the twin peak equal to $37.3$ , in rather good agreement with its theoretical value $D (m_s m_i)^{1/2}=35.8$ .",
"On the other hand, its value of 373 for B=80 is greater than the expected 320.",
"The error rate of $0.7\\%$ is in full agreement with the experimentally measured $SNR$ .",
"It has been obtained by a simple method of thresholding, where a threshold (red full line in figure REF ) is placed at the middle between the mean 1 level and the 0 one."
],
[
"Conclusion",
"We showed in this experiment of temporal ghost imaging that it is possible to reconstruct single sequences of time signals, using quantum correlated photons of SPDC.",
"The number of coherence cells contained in the images determines the available whole length and the error rate in the reconstruction process.",
"The relatively low value of this number, around 3000 here, can be increased in two ways : - a thinner crystal allows the decrease of the size of a coherence cell in the image plane by increasing the phase matching range in the Fourier plane, - a wider crystal allows increasing the number of coherence cells in a transverse section.",
"However, the conservation of the SPDC gain would require a more powerfull pump beam, as the surface illuminated is larger and the interaction time between the pump pulses and the crystal is smaller.",
"Likewise, more efficient detectors and a reduction of the parasitic fluorescence could result in an increase of the equivalent quantum efficiency.",
"The phase matching constraints explain that performances remain below that obtained with classical means [7], [8].",
"Nevertheless, this experiment shows that temporal ghost imaging can be performed by using either twin photons or classical correlations, just as for spatial ghost imaging."
],
[
"Funding",
"This work was supported by the Labex ACTION program (ANR-11-LABX-0001-01)."
]
] | 1612.05723 |
[
[
"Building Efficient Query Engines in a High-Level Language"
],
[
"Abstract Abstraction without regret refers to the vision of using high-level programming languages for systems development without experiencing a negative impact on performance.",
"A database system designed according to this vision offers both increased productivity and high performance, instead of sacrificing the former for the latter as is the case with existing, monolithic implementations that are hard to maintain and extend.",
"In this article, we realize this vision in the domain of analytical query processing.",
"We present LegoBase, a query engine written in the high-level language Scala.",
"The key technique to regain efficiency is to apply generative programming: LegoBase performs source-to-source compilation and optimizes the entire query engine by converting the high-level Scala code to specialized, low-level C code.",
"We show how generative programming allows to easily implement a wide spectrum of optimizations, such as introducing data partitioning or switching from a row to a column data layout, which are difficult to achieve with existing low-level query compilers that handle only queries.",
"We demonstrate that sufficiently powerful abstractions are essential for dealing with the complexity of the optimization effort, shielding developers from compiler internals and decoupling individual optimizations from each other.",
"We evaluate our approach with the TPC-H benchmark and show that: (a) With all optimizations enabled, LegoBase significantly outperforms a commercial database and an existing query compiler.",
"(b) Programmers need to provide just a few hundred lines of high-level code for implementing the optimizations, instead of complicated low-level code that is required by existing query compilation approaches.",
"(c) The compilation overhead is low compared to the overall execution time, thus making our approach usable in practice for compiling query engines."
],
[
"Introduction",
"During the last decade, we have witnessed a shift towards the use of high-level programming languages for systems development.",
"Examples include the Singularity Operating System , the Spark and DryadLINQ frameworks for efficient, distributed data processing, the FiST platform for specifying stackable file systems and GPUs programming .",
"All these approaches collide with the traditional wisdom which calls for using low-level languages like C for building high-performance systems.",
"This shift is necessary as the productivity of developers is severely diminished in the presence of complicated, monolithic, low-level code bases, making their debugging and maintenance very costly.",
"High-level programming languages can remedy this situation in two ways.",
"First, by offering advanced software features (modules, interfaces, object orientation, etc.",
"), they allow the same functionality to be implemented with significantly less code (compared to low-level languages).",
"Second, by providing powerful type systems and well-defined design patterns, they allow programmers not only to create abstractions and protect them from leaking but also to quickly define system modules that are reusable (even in contexts very different from the one these were created for) and easily composable .",
"All these properties can reduce the number of software errors of the systems and facilitate their verification.",
"Yet, despite these benefits, database systems are still written using low-level languages.",
"The reason is that increased productivity comes at a cost: high-level languages increase indirection, which in turn has a pronounced negative impact on performance.",
"For example, abstraction generally necessitates the need of containers, leading to costly object creation and destruction operations at runtime.",
"Encapsulation is provided through object copying rather than object referencing, thus similarly introducing a number of expensive memory allocations on the critical path.",
"Even primitive types such as integers are often converted to their object counterparts for use with general-purposes libraries.",
"As a result of these overheads, the use of high-level languages for developing high-performance databases seems (deceptively) prohibited.",
"The abstraction without regret vision , argues that it is indeed possible to use high-level languages for building database systems that allow for both productivity and high performance, instead of trading off the former for the latter.",
"By programming databases in a high-level style and still being able to get good performance, the time saved can be spent implementing more database features and optimizations.",
"In addition, the language features of high-level languages can grant flexibility to developers so that they can easily experiment with various design choices.",
"Figure: Comparison of the performance/productivity trade-off for all approachespresented in this article.In this article, we realize the abstraction without regret vision on the domain of ad-hoc, analytical query processing.",
"We make the following contributions: We present LegoBase, an in-memory query execution engine written in the high-level programming language, Scala, being the first step towards providing a full DBMS written in a high-level language.",
"To avoid the overheads of a high-level language (e.g.",
"complicated memory management) while maintaining well-defined abstractions, we opt for using generative programming , a technique that allows for programmatic removal of abstraction overhead through source-to-source compilation.",
"This is a key benefit as, in contrast to traditional, general-purpose compilers – which need to perform complicated and sometimes brittle analyses before maybe optimizing programs – generative programming in Scala takes advantage of the type system of the language to provide programmers with strong guarantees about the structure of the generated code.",
"For example, developers can specify optimizations that are applied during compilation in order to ensure that certain abstractions (e.g.",
"generic data structures and function calls) are definitely optimized away during compilation.",
"Generative programming can be used to optimize any piece of Scala code.",
"This allows LegoBase to perform whole-system specialization and compile all components, data structures and auxiliary functions used inside the query engine to efficient C code.",
"This design significantly contrasts our approach with existing query compilation approaches (e.g.",
"the one proposed in ) for three reasons.",
"First, a compiler that handles only queries cannot optimize and inline their code with the remaining code of the database system (which is typically precompiled), thus missing a number of optimization opportunities.",
"Second, in their purest form, query compilation approaches simply optimize or inline the code of individual operators in the physical query plan, thus making cross-operator code optimization inside the query compiler impossible.",
"Finally, existing approaches perform compilation using low-level code generation templates.",
"These essentially come in stringified form, making their development and automatic type checking very difficultFor example, templates can be used to convert the code of individual query operators – typically written today in C/C++ – to optimized LLVM code.",
"In that case, developers must handle a number of low-level concerns themselves, like register allocation..",
"The LegoBase query engine uses a new optimizing compiler called SC .",
"When performing whole-system compilation, an optimizing compiler effectively needs to specialize high-level systems code which will naturally employ a hierarchy of components and libraries from relatively high to very low level of abstraction.",
"To scale to such complex code bases, an optimizing compiler must guarantee two properties, not offered by existing compiler frameworks for applying generative programming.",
"First, to achieve maximum efficiency, developers must have tight control on the compiler's phases – admitting custom optimization phases and phase orderings.",
"This is necessary as code transformers with different optimization objectives may have to be combined in every possible ordering, depending on architectural, data, or query characteristics.",
"However, existing generative programming frameworks do not offer much control over the compilation processFor instance, Lightweight Modular Staging (LMS) applies all user-specified, domain-specific optimizations in a single optimization step.",
"It does so to avoid the well-known phase-ordering problem in compilers, where applying two (or more) optimizations in an improper order can lead not only to suboptimal performance but also to programs that are semantically incorrect .",
"We analyze how the design of the new optimizing compiler, SC, differs from that of LMS in Section of this article..",
"This absence of control effectively forces developers to provision for all possible optimization orderings.",
"This pollutes the code base of individual optimizations, making some of them dependent on other, possibly semantically independent, optimizations.",
"In general, the code complexity grows exponentially with the number of supported transformationsAs an example, consider the case of a compiler that is to support only two optimizations: 1) data-layout optimizations (i.e.",
"converting a row layout to a column or PAX-like layout ) and 2) data-structure specialization (i.e.",
"adapting the definition of a data structure to the particular context in which it is used).",
"This means that if the second optimization handles three different types of specialization, one has to provision for $2 \\times 3 = 6$ cases to handle all possible combinations of these optimizations.. Second, existing optimizing compilers expose a large number of low-level, compiler internals such as nodes of an intermediate representation (IR), dependency information encoded in IR nodes, and code generation templates to their users.",
"This interaction with low-level semantics when coding optimizations, but also the introduction of the IR as an additional level of abstraction, both significantly increase the difficulty of debugging as developers cannot easily track the relationship between the source code, the optimization for it – expressed using IR constructs – and the final, generated code , .",
"Instead, the SC compiler was designed from the beginning so that it allows developers to have full control over the optimization process without exporting compiler internals such as code generation templates.",
"It does so by delivering sufficiently powerful programming abstractions to developers like those afforded by modern high-level programming languages.",
"The SC compiler along with all optimizations are both written in plain Scala, thus allowing developers to be highly productive when optimizing all components of the query engine.",
"We demonstrate the ease of use of the new SC compiler for optimizing system components that differ significantly in structure and granularity of operations.",
"We do so by providing (i) an in-depth presentation of the optimizations applied to the LegoBase query engine and (b) a description of the high-level compiler interfaces that database developers need to interact with when coding optimizations.",
"We show that the design and interfaces of our optimizing compiler provide a number of nice properties for the LegoBase optimizations.",
"These are expressed as library components, providing a clean separation from the base code of LegoBase (e.g.",
"that of query operators), but also from each other.",
"This is achieved, (as explained later in more detail in Section ) by applying them in multiple, distinct optimization phases.",
"Optimizations are (a) adjustable to the characteristics of workloads and architectures, (b) configurable, so that they can be turned on and off on demand and (c) composable, so that they can be easily chained but also so that higher-level optimizations can be built from lower-level ones.",
"For each such optimization, we present: (a) the domain-specific conditions that need to be satisfied in order to apply it (if any) and (b) possible trade-offs (e.g.",
"improved execution time versus increased memory consumption).",
"Finally, we examine which categories of database systems can benefit from applying each of our optimizations by providing a classification of the LegoBase optimizations.",
"We perform an experimental evaluation in the domain of analytical query processing using the TPC-H benchmark .",
"We show how our optimizations can lead to a system that has performance competitive to that of a standard, commercial in-memory database called DBX (that does not employ compilation) and the code generated by the query compiler of the HyPer database .",
"In addition, we illustrate that these performance improvements do not require significant programming effort as even complicated optimizations can be coded in LegoBase with only a few hundred lines of code.",
"We also provide insights on the performance characteristics and trade-offs of individual optimizations.",
"We do so by comparing major architectural decisions as fairly as possible, using a shared codebase that only differs by the effect of a single optimization.",
"Finally, we conclude our analysis by demonstrating that our whole-system compilation approach incurs negligible overhead to query execution.",
"Figure: Motivating example showing missed optimizations opportunities byexisting query compilers that use template expansion.Motivating Example.",
"To better understand the differences of our work with previous approaches, consider the SQL query shown in Figure REF .",
"This query first calculates some aggregations from relation S in the group by operator $\\Gamma $ .",
"Then, it joins these aggregations with relation R, the tuples of which are filtered by the value of column Q.",
"The results are then returned to the user.",
"Careful examination of the execution plan of this query, shown in the same figure, reveals the following three basic optimization opportunities missed by existing query compilers that use template expansion: First, the limited scope of existing approaches usually results in performing the evaluation of aggregations in precompiled DBMS code.",
"Thus, each aggregation is evaluated consecutively and, as a result, common sub-expression elimination cannot be performed in this case (e.g.",
"in the calculation of expressions [ language=Scala, breaklines=true, showspaces=false, showtabs=false, showstringspaces=false, breakatwhitespace=true, numbers=none, numberstyle=, basicstyle=, keywordstyle=,columns=fullflexible, escapeinside=(*@@*) ]|1-S.B| and [ language=Scala, breaklines=true, showspaces=false, showtabs=false, showstringspaces=false, breakatwhitespace=true, numbers=none, numberstyle=, basicstyle=, keywordstyle=,columns=fullflexible, escapeinside=(*@@*) ]|S.A*(1-S.B)|).",
"This shows that, if we include the evaluation of all aggregations in the compiled final code, we can get an additional performance improvement.",
"This motivates us to extend the scope of compilation in this work.",
"Second, template-based approaches may result in unnecessary computation.",
"This is because operators are not aware of each other.",
"In this example, the generated code includes two materialization points: (a) at the group by and (b) when materializing the left side of the join.",
"However, there is no need to materialize the tuples of the aggregation in two different data structures as the aggregations can be immediately materialized in the data structure of the join.",
"Such inter-operator optimizations are hard to express using template-based compilers.",
"By high-level programming, we can instead easily pattern match on the operators, as we show in Section REF .",
"Finally, the data structures have to be generic enough for all queries.",
"As such, they incur significant abstraction overhead, especially when these structures are accessed millions of times during query evaluation.",
"Current query compilers cannot optimize the data structures since these belong to the precompiled part of the DBMS.",
"Our approach eliminates these overheads as it performs whole-program optimization and compiles, along with the operators, the data structures employed by a query.",
"This significantly contrasts our approach with previous work.",
"The rest of this article is organized as follows.",
"Section presents the overall design of LegoBase, along with a detailed description of the APIs provided by the new SC optimizing compiler.",
"Section gives an in-depth presentation of all supported compiler optimizations of our system in multiple domains.",
"Section presents our evaluation, where we experimentally show that our approach using the SC optimizing compiler can lead to significant benefits compared to (i) a commercial DBMS that does not employ compilation and (ii) a database system that uses low-level, code-generation templates during query compilation.",
"We also give insights about the memory footprint, data loading time and programming effort required when working with the LegoBase system.",
"Section presents related work in the area of compilation and compares our approach with existing query compilers and engines.",
"Finally, Section concludes."
],
[
"System Design",
"In this section, we present the design of the LegoBase system.",
"First, we describe the overall system architecture of our approach (Subsection REF ).",
"Then, we describe in detail the SC compiler that is the core of our proposal (Subsection REF ) as well as how we efficiently convert the entire high-level Scala code of the query engine (not just that of individual operators) to optimized C code for each incoming query (Subsection REF ).",
"While doing so, we give concrete code examples of how (a) physical query operators, (b) physical query plans, and, (c) compiler interfaces look like in our system."
],
[
"Overall System Architecture",
"LegoBase implements the typical query plan operators found in traditional database systems, including equi, semi, anti, and outer joins, all on a high level.",
"In addition, LegoBase supports both a classical Volcano-style query engine as well as a push-style query interface In a push engine, the meaning of child and parent operators is reversed compared to the usual query plan terminology: Data flows from the leaves (the ancestors, usually being scan operators) to the root (the final descendant, which computes the final query results that are returned to the user)..",
"The overall system architecture of LegoBase is shown in Figure REF .",
"First, for each incoming SQL query, we must get a query plan which describes the physical query operators needed to process this query.",
"For this work, we consider traditional query optimization (e.g.",
"determining join ordering) as an orthogonal problem and we instead focus more on experimenting with the different optimizations that can be applied after traditional query optimization.",
"Thus, to obtain a physical query plan, we pass the incoming query through any existing query optimizer.",
"For example, for our evaluation, we choose the query optimizer of a commercial, in-memory database system.",
"Figure: Overall system architecture.",
"The domain-specific optimizations ofLegoBase are applied during the SC compiler optimization phase.Then, we pass the generated physical plan to LegoBase.",
"Our system, in turn, parses this plan and instantiates the corresponding Scala implementation of the operators.",
"Figure REF presents an example of how query plans and operators are written in LegoBase, respectively.",
"That is, the Scala code example shown in Figure REF loads the data, builds a functional tree from operator objects and then starts executing the query by calling [ language=Scala, breaklines=true, showspaces=false, showtabs=false, showstringspaces=false, breakatwhitespace=true, numbers=none, numberstyle=, basicstyle=, keywordstyle=,columns=fullflexible, escapeinside=(*@@*) ]|next| for the root operator.",
"It is important to note that operator implementations like the one presented in Figure REF are exactly what one would write for a simple query engine that does not involve compilation at all.",
"However, without further optimizations, this engine cannot match the performance of existing databases: it consists of generic data structures (e.g.",
"the one declared in line of Figure REF ) and involves expensive memory allocations on the critical pathNote that such memory allocations are not always explicit (i.e.",
"at object definition time through the [ language=Scala, breaklines=true, showspaces=false, showtabs=false, showstringspaces=false, breakatwhitespace=true, numbers=none, numberstyle=, basicstyle=, keywordstyle=,columns=fullflexible, escapeinside=(*@@*) ]|new| keyword in object-oriented languages like Java and Scala).",
"For instance, in line of Figure REF , the HashMap data structure may have to expand (in terms of allocated memory footprint) and be reorganized by the Scala runtime in order to more efficiently store data for future lookup operations.",
"We talk more about this issue and its consequences to performance later in this article., both properties that can significantly affect performance.",
"Figure: Code Snippet for the Partitioning Transformer"
]
] | 1612.05566 |
[
[
"Survey Paper on Rising Threats of Subverting Privacy Infrastructure"
],
[
"Abstract One of the main challenges faced by a user today is protecting their privacy, especially during widespread surveil- lance.",
"This led to the development of privacy infrastructures whose main purpose is to guarantee users privacy.",
"However, they are being misused by attackers who consider them to be an exploitable resource to perform illegitimate activities.",
"This paper proposes to analyze and assess the threats that occur due to subverting privacy infrastructures.",
"It begins with an outline of critical privacy infrastructures that were developed.",
"An overview of rising threats of subverting the most prominent privacy infrastructure, Tor is presented.",
"Additionally, a brief assessment of the severity of these threats is discussed and the paper concludes by recommending the scope for further research to mitigate the risks due to such threats."
],
[
"INTRODUCTION",
"Due to the rapid usage and development of Internet, privacy has become a major concern for individuals, particularly in case of widespread surveillance.",
"With the flexibility and advancement of communication technologies, the open nature of Internet also encouraged illegitimate activities such as prying on user’s personal information, selling information to advertising companies, exploiting information by government agencies thereby compromising the privacy of user information.",
"Hence, privacy infrastructures emerged as a solution.",
"(ALSABAH and GOLDBERG 2016) Tor, acronym for the Onion Router is an example of such a system.",
"VPN, Proxy servers, Remailers, JAP, I2P are few other examples.",
"These systems are responsible for hiding the identities of communicating parties from their peers, as well as from the adversaries who intend to eavesdrop the traffic flowing through the network.",
"Among them, VPN, proxy servers, TOR and I2P are actively used.",
"(Chakravarty 2014)(Bingdong Li 2013).",
"The success of privacy infrastructures further encouraged the attackers to perform illegal activities by subverting these infrastructures through sophisticated botnets and ransomware.",
"Botnets are generally used by attackers to perform malicious activities such as DDoS, personal data theft, spam, bitcoin mining and cyber-espionage.",
"Botnets are centralized overlay networks in which Command-and-Control (C&C) servers are a single entity responsible for control.",
"Bots connect to these servers to be reachable and a botmaster exists that manages the bots and is aware of the overlay network structure.",
"Major disadvantage in this architecture is that the C&C servers act as a single point of failure.",
"Hence, if the C&C servers are compromised, the complete botnet is defeated.",
"Attackers have resorted to this problem by adopting a more resilient unstructured P2P network that has a distributed architecture, or using improved techniques to reduce the detection capability of the C&C servers.",
"The latter was achieved by taking advantage of the anonymity features of Tor network.",
"Through Tor, a botmaster can access the C&C servers anonymously and an encrypted routing system is created to avoid detection through traffic analysis.",
"Moreover, Tor provides hidden services in which the client does not need to know the actual address or location of the service, and botmasters can configure the C&C servers as hidden services.",
"(Casenove and Miraglia 2014).In this paper, we will briefly describe an overview of privacy infrastructures, rising threats of subverting these infrastructures and assess the criticality of such threats based on its capabilities and previous attacks.",
"A proxy server is used for surfing the web, wherein it acts as an intermediary between the client and server, and client requests are sent with the proxy server’s identity rather than that of the real user thereby hiding the user’s identity.",
"Due to this, the destination server does not log the real IP address and other device information of user.",
"It is not very prominent for anonymous communication because it can be easily attacked or traced.",
"Before the traffic is sent to the proxy, it has to travel through the ISP’s physical network, making it susceptible to man-in-the-middle attacks.",
"An attacker can monitor the traffic between user and proxy server and obtain sensitive information.",
"(Towards an Autonomous System Monitor for Mitigating Correlation Attacks in the Tor Network 2016) (Erdin 2012) (Bingdong Li 2013)"
],
[
"Remailers",
"Remailers are servers which forwards to destination the messages it receives with embedded instructions, without revealing the sender’s information.",
"In particular, remailers manipulate the addresses in the e-mail headers of the transmitting node with fake addresses.",
"Hence anonymity in e-mails are achieved.",
"Remailers are of three types, Type I (Cypherpunk) that forwards messages to several servers before sending it to the destination and provides multiple layer encryption through PGP public keys, Type II (Mixmaster) which includes padding, message pools and is more resistant to traffic analysis, Type III (MixMinion) which is most secure due to free routing algorithm.",
"(Towards an Autonomous System Monitor for Mitigating Correlation Attacks in the Tor Network 2016) (Erdin 2012) (Bingdong Li 2013)"
],
[
"Virtual Private Network (VPN)",
"Unlike proxy server, VPN server provides secured communication to the destination by encrypting the traffic between the user and server.",
"The traffic is encrypted irrespective of the type of application being used, thus mitigating the risks of attacks such as man-in-the-middle.",
"Even though it offers secured communication when compared to proxy server, user’s privacy is not guaranteed.",
"There are many cases where VPN providers share user’s information for business profits.",
"(Erdin 2012)"
],
[
"Onion Routing",
"It is the most prevalent design for low latency communications.",
"The protocol consists of a structure similar to that of an onion, where message is encapsulated with layer by layer encryption.",
"In this mechanism, there is a set of servers called onion routers that is responsible to relay traffic to the clients.",
"Every node has a public and private key.",
"The public key is known to the client in order to set the path of communication.",
"Initially, the client constructs an encrypted tunnel called circuit using public-key cryptography.",
"Once it is set, symmetric key cryptography is used to transfer the data.",
"This protocol has different variations such as Onion Router(TOR) and Invisible Internet Project(I2P) (Erdin 2012)"
],
[
"Invisible Internet Project (I2P)",
"It is an alternative to TOR that supports all regular internet activities such as e-mail, web browsing.",
"Unlike TOR, I2P is more focused on accessing closed darknet rather than the regular Internet websites.",
"I2P offers anonymity services to identity-sensitive applications by building an overlay network of volunteer systems.",
"It is strictly based on UDP, but security can be achieved by including the libraries that allow reliable stream communication on top of the I2P network.",
"It is a closed system in which traffic is routed through other peers and by announcing its peers, it enables new users to join the network.",
"Many applications such as email, peer-to-peer, IRC interact with I2P but it is usually not preferred for low latency applications due to the lack of focus in end-to-end delay.",
"(Bingdong Li 2013) (Tails 2016)"
],
[
"Onion Router (TOR) ",
"Tor is a low latency anonymity overlay network that is known to be the most robust privacy tool.",
"It is helpful in preventing user discovery to any entities that are monitoring the network.",
"When packets are transmitted between the user and a destination host, a random path with three nodes are used so that no single node is aware of the complete transmission process thereby providing anonymity.",
"Furthermore, TOR connections are encrypted using TLS protocol.",
"However, connection between exit node and destination is not encrypted and hence exit nodes can observe the content of messages.",
"(Bingdong Li 2013)"
],
[
"Functioning of TOR",
"The directory authorities in TOR are centralized, trusted servers that track the complete TOR network.",
"Nodes or routers are voluntary computers that are distributed and categorized based on their respective functionality and positions.",
"Before transmitting the data packets to the destination node, it is encrypted several times using public key cryptography.",
"At every relay, one encryption layer is decrypted, which reveals the IP address of the next relay that the packet needs to be forwarded to.",
"This is done until the packet reaches the destination and the reverse process is followed for messages sent to the client.",
"The client’s privacy is maintained,since without the ability to analyze traffic, none of the relays can detect the corresponding message's sender and destination.",
"Through the hidden services feature, services are accessible to anyone with a Tor client without revealing any knowledge about the IP address or location of the server.For additional security, the Tor client does not select same router or relays in the same /16 subnet to be in the same circuit.",
"Tor’s telescopic approach to circuit establishment has two major benefits, one of which is that perfect forward secrecy is achieved due to discarding the session keys when circuit is closed.",
"The other benefit is that, routers need not store the hashes of previously processed onions to prevent replay attacks since replaying one side of a Diffie- Hellman handshake results in a different key which is not of any use to the attacker.",
"(Bingdong Li 2013)"
],
[
"Java Anonymous Proxy (JAP)",
"JAP is a low latency mix cascades that uses the server provided by volunteers to access the Internet.",
"Several mixes are used to encrypt the packet and maintain the rate of traffic constant in order to avoid rate-based traffic analysis.",
"The program can display all the active mixes from which user can choose the JAP cascades.",
"(Erdin 2012) JAP software is available from many years and is second in popularity after Tor.",
"Commercially, it is called JonDonym."
],
[
"Rising threats of subverting privacy infrastructure",
"Among the privacy infrastructures mentioned, Tor provides highest anonymity.",
"The stealthiness and untraceability features of Tor motivated the attackers to take advantage of this capability and develop Tor-based botnets.",
"By placing the botnet infrastructure in Tor, the Tor hidden services provides anonymous C&C servers which is difficult to detect or destroy.",
"Furthermore, attacking the server with DDoS seems unfeasible as it would result in the attack of complete Tor network."
],
[
"Tor-based Botnets [Past Work]",
"The idea of hiding botnets in Tor was discussed in 2010, in particular at the DefCon18 (D.Brown 2010) where a C&C server anonymity implementation using Tor was shown by DannisBrown.",
"Later in 2012, Guarnieri in (Guarnieri 2012) detected and analyzed the first Tor-based botnet.",
"The botnet was a modified version of Zeus consisting of DDoS, bitcoin mining and credential theft capabilities.",
"The malware comprised of Zeus bot, Tor client, GMinerbitcoin mining tool, and few libraries for GPU hash cracking.",
"The bots ran inside hidden services and all C&C communication was within the Tor network.",
"The botmaster tried to reduce the traceability by avoiding the use of exit nodes, and used IRC protocol to communicate and issue commands to the bots.",
"The botmaster also made the bots act as relays thereby exploiting and enhancing the Tor network simultaneously.",
"In 2013, due to a post on the Tor mailing list, Tor's network usage and the number of users accessing grew rapidly.",
"Researchers couldn’t explain the reason at first but on analysis came to conclusion that it was due to a large botnet that suddenly switched to Tor.",
"The botnet used centralized structure with HTTP protocol and a preconfigured earlier version of Tor to connect to the network.",
"The significant increase in the amount of Tor communication that was being established through relays resulted in the reduction of Tor system’s responsiveness.",
"(Casenove and Miraglia 2014) Similarly,in the last few years, various types of botnets were found in (Constantin 2012) (Dunn 2013) (Gottesman 2014)that made use of Tor infrastructure.",
"It provided a hideout for malware by deploying the command and control server as a hidden service with specific onion address that the other bots are configured with.",
"Such botnets and referencing the ones mentioned earlier as well, result in significant degradation in the performance of Tor.",
"One other example is the spike caused by Mevade/Sefnit botnet in 2013.",
"The spike was close to 600% increase in the number of clients, causing a network overload through C&C descriptors and creating circuit requests.",
"(ALSABAH and GOLDBERG 2016)"
],
[
"OnionBots [Future Threat]",
"According to (Sanatinia and Noubir 2015), onion bots are believed to be the next generation of resilient and stealthy botnets.",
"It uses privacy infrastructures such as Tor to stay undetected and decouple itself from the infected hosts.",
"Since onion bots are different from the peer-to-peer botnets, the existing solutions that are used for the peer-to-peer bot are not applicable to onion bots.",
"The design is also resilient to the current mitigation and analysis techniques such as botnet mapping, hijacking and assessing the botnet size.",
"Moreover, it is also proven to achieve low diameter, degree and high resiliency and repair if any event of a take-down of a fraction of botnet node occurs.",
"Anonymity is achieved through the periodically changing address of the bot during waiting stage.",
"Unlike current botnets, secure communications is achieved by encryption in OnionBot through Tor and SSL.",
"The threat environment will continue to grow due to its capability to offer new services such as botnet-for-rent and distribution computation platform for rent.",
"By taking advantage of payment through Bitcoins, Silk Road 2.0 in Tor business operations can be carried out and botnets can be instructed to perform CPU intensive operations such as bitcoin mining or password cracking.",
"Furthermore, the estimated threats due to OnionBots could be higher since they are robust to partitioning even when large fraction of these bots are taken down simultaneously.",
"(Sanatinia and Noubir 2015) (Chaabane, Manils and Kaafar 2010)"
],
[
"Threats due to ransomware",
"According to (Cygnus Business Media Inc. 2015) ,threats due to ransomware are one of the biggest threats.",
"In ransomware, the user machine is infected and files and applications are held hostage until a fee is paid.",
"The threat is very high because ransomware is an automatic customer service-type model where the malware will install and then victim can pay through bitcoins.",
"It doesn’t involve any human interaction for payment process.Cryptolocker and Cryptowall are two of the ransomwares mentioned in (Cygnus Business Media Inc. 2015).Furthermore, the report also states that malware economy is ever growing and ransomware and cryptocurrency such as bitcoins are helping attacker monetize their actions."
],
[
"Threat Assessment",
"According to the survey of anonymity technologies in (Bingdong Li 2013), newer systems such as Tor, I2P are gaining popularity.",
"Among the current anonymity systems, Tor networks have more number of volunteers than I2P.",
"From previous attacks it is evident that, Botnet over Tor is growing and among all the privacy infrastructures threats due to subverting Tor will be highest.",
"According to (Casenove and Miraglia 2014), even though botnets over Tor is good solution it is still not perfect.",
"They do not provide ultimate resilience and even with Tor, a centralized botnet can have vulnerability such as single point of failure.",
"When a botnet is integrated with the Tor infrastructure, a lot of attention is raised as it creates instability in a stable network such as Tor.",
"Hence when a botnet comprising of millions of nodes joins the Tor network in a short span, it can be easily detected.",
"Even from a client point of view, a botnet using Tor can leave traces.",
"A malware runs Tor client as external process and if the client was not installed previously, exposure of malware activity would be trivial.",
"By cross verifying the running processes, malware can be detected by identifying the Tor client process.",
"By using such detection techniques, the estimated threat due to the tor-based-botnets can be greatly reduced.",
"In case of OnionBot, the estimated level of threat is very high due to its robust nature and anonymity.",
"By using the suggested mitigation technique Sybil Onion Attack Protocol (SOAP), the botnets can be neutralized.",
"This might reduce the risk due to such bots but due to the continuous development of various variants of such botnets, threats due to these corresponding models also increases.",
"Successful removal of such threats may not be guaranteed but measures to reduce the threat’s effectiveness and mitigation of risks due to such threats need to be addressed."
],
[
"Conclusion",
"In this paper, various kinds of privacy infrastructures are discussed and the rising threats of subverting the privacy infrastructure are analyzed.Since Tor is the most prominent privacy infrastructure, it is attracting more threats.Major threat is due to botnet over Tor infrastructure which has been the basis for prominent botnet attacks since 2010.Severity of threats with respect to new variations of botnets such as OnionBots are high due to its robustness and resiliency.Since research with respect to Onionbots propose effective mitigation techniques such as SOAP,risk due to the exploitation of these threats can be reduced but not completely avoided.",
"Moreover, the severity of threats is still high.",
"With the rapid development of various robust botnet variants, design mitigation strategies need to be implemented rapidly and effectively.",
"Further research is recommended to achieve effectiveness in detection and mitigation of threats due to these infrastructures."
]
] | 1612.05806 |
[
[
"Reverses and Refinements of Jensen's Inequality for Positive Linear\n Functionals on Hermitian Unital Banach *-Algebras"
],
[
"Abstract We establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach *-algebra.",
"Reverses and refinements of Jensen's and Slater's type inequalities are provided.",
"Some examples for particular convex functions of interest are given as well."
],
[
"Introduction",
"We need some preliminary concepts and facts about Banach $\\ast $ -algebras.",
"Let $A$ be a unital Banach $\\ast $ -algebra with unit 1.",
"An element $a\\in A$ is called selfadjoint if $a^{\\ast }=a.$ $A$ is called Hermitian if every selfadjoint element $a$ in $A$ has real spectrum $\\sigma \\left( a\\right) ,$ namely $\\sigma \\left( a\\right) \\subset \\mathbb {R}$ .",
"We say that an element $a$ is nonnegative and write this as $a\\ge 0 $ if $a^{\\ast }=a$ and $\\sigma \\left( a\\right) \\subset \\left[ 0,\\infty \\right) .$ We say that $a$ is positive and write $a>0$ if $a\\ge 0$ and $0\\notin \\sigma \\left( a\\right) .$ Thus $a>0$ implies that its inverse $a^{-1}$ exists.",
"Denote the set of all invertible elements of $A$ by ${Inv}\\left( A\\right) .$ If $a,$ $b\\in {Inv}\\left( A\\right) ,$ then $ab\\in {Inv}\\left( A\\right) $ and $\\left( ab\\right)^{-1}=b^{-1}a^{-1}.$ Also, saying that $a\\ge b$ means that $a-b\\ge 0$ and, similarly $a>b$ means that $a-b>0.$ The Shirali-Ford theorem asserts that if $A$ is a unital Banach $\\ast $ -algebra [14] (see also [2]), then $\\left|a\\right|^{2}:=a^{\\ast }a\\ge 0\\text{ for every }a\\in A.",
"\\qquad \\mathrm {(SF)}$ Based on this fact, Okayasu [13], Tanahashi and Uchiyama [15] proved the following fundamental properties (see also [9]): If $a,$ $b\\in A,$ then $a\\ge 0,$ $b\\ge 0$ imply $a+b\\ge 0$ and $\\alpha \\ge 0$ implies $\\alpha a\\ge 0;$ If $a,$ $b\\in A,$ then $a>0,$ $b\\ge 0$ imply $a+b>0;$ If $a,$ $b\\in A,$ then either $a\\ge b>0$ or $a>b\\ge 0$ imply $a>0;$ If $a>0,$ then $a^{-1}>0;$ If $c>0,$ then $0<b<a$ if and only if $cbc<cac,$ also $0<b\\le a$ if and only if $cbc\\le cac;$ If $0<a<1,$ then $1<a^{-1};$ If $0<b<a,$ then $0<a^{-1}<b^{-1},$ also if $0<b\\le a,$ then $0<a^{-1}\\le b^{-1}.$ In order to introduce the real power of a positive element, we need the following facts [2].",
"Let $a\\in A$ and $a>0,$ then $0\\notin \\sigma \\left( a\\right) $ and the fact that $\\sigma \\left( a\\right) $ is a compact subset of $\\mathbb {C}$ implies that $\\inf \\lbrace z:z\\in \\sigma \\left( a\\right) \\rbrace >0$ and $\\sup \\lbrace z:z\\in \\sigma \\left( a\\right) \\rbrace <\\infty .$ Choose $\\gamma $ to be close rectifiable curve in $\\lbrace {Re}z>0\\rbrace ,$ the right half open plane of the complex plane, such that $\\sigma \\left( a\\right) \\subset {ins}\\left( \\gamma \\right) ,$ the inside of $\\gamma .$ Let $G$ be an open subset of $\\mathbb {C}$ with $\\sigma \\left( a\\right) \\subset G.$ If $f:G\\rightarrow \\mathbb {C}$ is analytic, we define an element $f\\left( a\\right) $ in $A$ by $f\\left( a\\right) :=\\frac{1}{2\\pi i}\\int _{\\gamma }f\\left( z\\right) \\left(z-a\\right) ^{-1}dz.$ It is well known (see for instance [3]) that $f\\left(a\\right) $ does not depend on the choice of $\\gamma $ and the Spectral Mapping Theorem (SMT) $\\sigma \\left( f\\left( a\\right) \\right) =f\\left( \\sigma \\left( a\\right)\\right)$ holds.",
"For any $\\alpha \\in \\mathbb {R}$ we define for $a\\in A$ and $a>0,$ the real power $a^{\\alpha }:=\\frac{1}{2\\pi i}\\int _{\\gamma }z^{\\alpha }\\left( z-a\\right)^{-1}dz,$ where $z^{\\alpha }$ is the principal $\\alpha $ -power of $z.$ Since $A$ is a Banach $\\ast $ -algebra, then $a^{\\alpha }\\in A.$ Moreover, since $z^{\\alpha } $ is analytic in $\\lbrace {Re}z>0\\rbrace ,$ then by (SMT) we have $\\sigma \\left( a^{\\alpha }\\right) =\\left( \\sigma \\left( a\\right) \\right)^{\\alpha }=\\lbrace z^{\\alpha }:z\\in \\sigma \\left( a\\right) \\rbrace \\subset \\left(0,\\infty \\right) .$ Following [9], we list below some important properties of real powers: If $0<a\\in A$ and $\\alpha \\in \\mathbb {R}$ , then $a^{\\alpha }\\in A$ with $a^{\\alpha }>0$ and $\\left( a^{2}\\right) ^{1/2}=a,$ [15]; If $0<a\\in A$ and $\\alpha ,$ $\\beta \\in \\mathbb {R}$ , then $a^{\\alpha }a^{\\beta }=a^{\\alpha +\\beta };$ If $0<a\\in A$ and $\\alpha \\in \\mathbb {R}$ , then $\\left( a^{\\alpha }\\right) ^{-1}=\\left( a^{-1}\\right) ^{\\alpha }=a^{-\\alpha };$ If $0<a,$ $b\\in A$ , $\\alpha ,$ $\\beta \\in \\mathbb {R}$ and $ab=ba, $ then $a^{\\alpha }b^{\\beta }=b^{\\beta }a^{\\alpha }.$ Okayasu [13] showed that the Löwner-Heinz inequality remains valid in a Hermitian unital Banach $\\ast $ -algebra with continuous involution, namely if $a,$ $b\\in A$ and $p\\in \\left[ 0,1\\right] $ then $a>b$ $\\left( a\\ge b\\right) $ implies that $a^{p}>b^{p}$ $\\left( a^{p}\\ge b^{p}\\right) .$ Now, assume that $f\\left( \\cdot \\right) $ is analytic in $G$ , an open subset of $\\mathbb {C}$ and for the real interval $I\\subset G$ assume that $f\\left(z\\right) \\ge 0$ for any $z\\in I.$ If $u\\in A$ such that $\\sigma \\left(u\\right) \\subset I,$ then by (SMT) we have $\\sigma \\left( f\\left( u\\right) \\right) =f\\left( \\sigma \\left( u\\right)\\right) \\subset f\\left( I\\right) \\subset \\left[ 0,\\infty \\right)$ meaning that $f\\left( u\\right) \\ge 0$ in the order of $A.$ Therefore, we can state the following fact that will be used to establish various inequalities in $A,$ see also [5].",
"Lemma 1 Let $f\\left( z\\right) $ and $g\\left( z\\right) $ be analytic in $G$ , an open subset of $\\mathbb {C}$ and for the real interval $I\\subset G,$ assume that $f\\left( z\\right) \\ge g\\left( z\\right) $ for any $z\\in I.$ Then for any $u\\in A$ with $\\sigma \\left( u\\right) \\subset I$ we have $f\\left(u\\right) \\ge g\\left( u\\right) $ in the order of $A.$ Definition 1 Assume that $A$ is a Hermitian unital Banach $\\ast $ -algebra.",
"A linear functional $\\psi :A\\rightarrow \\mathbb {C}$ is positive if for $a\\ge 0 $ we have $\\psi \\left( a\\right) \\ge 0.$ We say that it is normalized if $\\psi \\left( 1\\right) =1.$ We observe that the positive linear functional $\\psi $ preserves the order relation, namely if $a\\ge b$ then $\\psi \\left( a\\right) \\ge \\psi \\left(b\\right) $ and if $\\beta \\ge a\\ge \\alpha $ with $\\alpha ,$ $\\beta $ real numbers, then $\\beta \\ge \\psi \\left( a\\right) \\ge \\alpha .$ In the recent paper [6] we established the following McCarthy type inequality: Theorem 1 Assume that $A$ is a Hermitian unital Banach $\\ast $ -algebra and $\\psi :A\\rightarrow \\mathbb {C}$ a positive normalized linear functional on $A.",
"$ (i) If $p\\in \\left( 0,1\\right) $ and $a\\ge 0,$ then $\\psi ^{p}\\left( a\\right) \\ge \\psi \\left( a^{p}\\right) \\ge 0; $ (ii) If $q\\ge 1$ and $b\\ge 0,$ then $\\psi \\left( b^{q}\\right) \\ge \\psi ^{q}\\left( b\\right) \\ge 0; $ (iii) If $r<0,$ $c>0$ with $\\psi \\left( c\\right) >0,$ then $\\psi \\left( c^{r}\\right) \\ge \\psi ^{r}\\left( c\\right) >0.",
"$ In [7] and [8] we obtained the following result for analytic convex functions: Theorem 2 Let $f\\left( z\\right) $ be analytic in $G$ , an open subset of $\\mathbb {C}$ and the real interval $I\\subset G.$ If $f$ is convex (in the usual sense) on the interval $I$ and $\\psi :A\\rightarrow \\mathbb {C}$ is a positive normalized linear functional on $A,$ then for any selfadjoint element $c\\in A$ with with $\\sigma \\left( c\\right) \\subseteq \\left[ m,M\\right] \\subset I$ for some real numbers $m<M,$ $0& \\le \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right)\\right) \\le \\psi \\left( cf^{\\prime }\\left( c\\right) \\right) -\\psi \\left(c\\right) \\psi \\left( f^{\\prime }\\left( c\\right) \\right) \\\\& \\\\& \\le \\left\\lbrace \\begin{array}{l}\\frac{1}{2}\\left( M-m\\right) \\left[ \\psi \\left( \\left[ f^{\\prime }\\left(c\\right) \\right] ^{2}\\right) -\\psi ^{2}\\left( f^{\\prime }\\left( c\\right)\\right) \\right] ^{1/2} \\\\\\\\\\frac{1}{2}\\left[ f^{\\prime }\\left( M\\right) -f^{\\prime }\\left( m\\right) \\right] \\left( \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right)^{1/2}\\end{array}\\right.",
"\\\\& \\\\& \\le \\frac{1}{4}\\left( M-m\\right) \\left[ f^{\\prime }\\left( M\\right)-f^{\\prime }\\left( m\\right) \\right] .",
"$ Motivated by these results we establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach $\\ast $ -algebra.",
"Reverses and refinements of Jensen's and Slater's type inequalities are provided.",
"Some examples for particular convex functions of interest are given as well."
],
[
"Some Reverses",
"We have: Theorem 3 Let $f\\left( z\\right) $ be analytic in $G$ , an open subset of $\\mathbb {C}$ and the real interval $I\\subset G.$ If $f$ is convex on the interval $I$ and $\\psi :A\\rightarrow \\mathbb {C}$ is a positive normalized linear functional on $A,$ then for any selfadjoint element $c\\in A$ with $\\sigma \\left( c\\right) \\subseteq \\left[ m,M\\right] \\subset I$ for some real numbers $m<M,$ $0& \\le \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right)\\right) \\\\& \\\\& \\le \\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left(c\\right) -m\\right) }{M-m}\\sup _{t\\in \\left( m,M\\right) }\\Theta _{f}\\left(t;m,M\\right) \\\\& \\\\& \\le \\left\\lbrace \\begin{array}{c}\\frac{1}{4}\\left( M-m\\right) \\sup _{t\\in \\left( m,M\\right) }\\Theta _{f}\\left(t;m,M\\right) \\\\\\\\\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right) -m\\right)\\frac{f^{\\prime }\\left( M\\right) -f^{\\prime }\\left( m\\right) }{M-m}\\end{array}\\right.",
"\\\\& \\\\& \\le \\frac{1}{4}\\left( M-m\\right) \\left[ f^{\\prime }\\left( M\\right)-f^{\\prime }\\left( m\\right) \\right] $ provided $\\psi \\left( c\\right) \\in \\left( m,M\\right) ,$ where $\\Theta _{f}\\left( \\cdot ;m,M\\right) :\\left( m,M\\right) \\rightarrow \\mathbb {R}$ is defined by $\\Theta _{f}\\left( t;m,M\\right) =\\frac{f\\left( M\\right) -f\\left( t\\right) }{M-t}-\\frac{f\\left( t\\right) -f\\left( m\\right) }{t-m}.$ We also have $0& \\le \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right)\\right) \\le \\frac{1}{4}\\left( M-m\\right) \\Theta _{f}\\left( \\psi \\left(c\\right) ;m,M\\right) \\\\& \\\\& \\le \\frac{1}{4}\\left( M-m\\right) \\sup _{t\\in \\left( m,M\\right) }\\Theta _{f}\\left( t;m,M\\right) \\le \\frac{1}{4}\\left( M-m\\right) \\left[ f^{\\prime }\\left( M\\right) -f^{\\prime }\\left( m\\right) \\right] , $ provided $\\psi \\left( c\\right) \\in \\left( m,M\\right) .$ By the convexity of $f$ on $\\left[ m,M\\right] $ we have for any $z\\in \\left[m,M\\right] $ that $f\\left( z\\right) \\le \\frac{z-m}{M-m}f\\left( M\\right) +\\frac{M-z}{M-m}f\\left( m\\right) .",
"$ Using Lemma REF we have by (REF ) for any selfadjoint element $c\\in A$ with $\\sigma \\left( c\\right) \\subseteq \\left[ m,M\\right] $ that $f\\left( c\\right) \\le f\\left( M\\right) \\frac{c-m}{M-m}+f\\left( m\\right)\\frac{M-c}{M-m} $ in the order of $A.$ If we take in this inequality the functional $\\psi $ we get the following reverse of Jensen's inequality $\\psi \\left( f\\left( c\\right) \\right) \\le f\\left( M\\right) \\frac{\\psi \\left(c\\right) -m}{M-m}+f\\left( m\\right) \\frac{M-\\psi \\left( c\\right) }{M-m}.$ This generalizes the scalar Lah-Ribarić inequality for convex functions that is well known in the literature, see for instance [10] for an extension to selfadjoint operators in Hilbert spaces.",
"Define $\\Delta _{f}\\left( t;m,M\\right) :=\\frac{\\left( t-m\\right) f\\left( M\\right)+\\left( M-t\\right) f\\left( m\\right) }{M-m}-f\\left( t\\right) ,\\quad t\\in \\left[ m,M\\right] ,$ then we have $\\Delta _{f}\\left( t;m,M\\right) & =\\frac{\\left( t-m\\right) f\\left( M\\right)+\\left( M-t\\right) f\\left( m\\right) -\\left( M-m\\right) f\\left( t\\right) }{M-m} \\\\& =\\frac{\\left( t-m\\right) f\\left( M\\right) +\\left( M-t\\right) f\\left(m\\right) -\\left( M-t+t-m\\right) f\\left( t\\right) }{M-m} \\\\& =\\frac{\\left( t-m\\right) \\left[ f\\left( M\\right) -f\\left( t\\right) \\right]-\\left( M-t\\right) \\left[ f\\left( t\\right) -f\\left( m\\right) \\right] }{M-m}\\\\& =\\frac{\\left( M-t\\right) \\left( t-m\\right) }{M-m}\\Theta _{f}\\left(t;m,M\\right) $ for any $t\\in \\left( m,M\\right) .$ From (REF ) we have for $\\psi \\left( c\\right) \\in \\left( m,M\\right) $ that $& \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right) \\right) \\\\& \\le \\frac{\\left( \\psi \\left( c\\right) -m\\right) f\\left( M\\right) +\\left(M-\\psi \\left( c\\right) \\right) f\\left( m\\right) }{M-m}-f\\left( \\psi \\left(c\\right) \\right) \\\\& =\\Delta _{f}\\left( \\psi \\left( c\\right) ;m,M\\right) =\\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right) -m\\right) }{M-m}\\Theta _{f}\\left( \\psi \\left( c\\right) ;m,M\\right) \\\\& \\le \\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left(c\\right) -m\\right) }{M-m}\\sup _{t\\in \\left( m,M\\right) }\\Theta _{f}\\left(t;m,M\\right) .",
"$ We also have $\\sup _{t\\in \\left( m,M\\right) }\\Theta _{f}\\left( t;m,M\\right) & =\\sup _{t\\in \\left( m,M\\right) }\\left[ \\frac{f\\left( M\\right) -f\\left( t\\right) }{M-t}-\\frac{f\\left( t\\right) -f\\left( m\\right) }{t-m}\\right] \\\\& \\le \\sup _{t\\in \\left( m,M\\right) }\\left[ \\frac{f\\left( M\\right) -f\\left(t\\right) }{M-t}\\right] +\\sup _{t\\in \\left( m,M\\right) }\\left[ -\\frac{f\\left(t\\right) -f\\left( m\\right) }{t-m}\\right] \\\\& =\\sup _{t\\in \\left( m,M\\right) }\\left[ \\frac{f\\left( M\\right) -f\\left(t\\right) }{M-t}\\right] -\\inf _{t\\in \\left( m,M\\right) }\\left[ \\frac{\\Phi \\left( t\\right) -\\Phi \\left( m\\right) }{t-m}\\right] \\\\& =f^{\\prime }\\left( M\\right) -f^{\\prime }\\left( m\\right)$ and since, obviously $\\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right)-m\\right) }{M-m}\\le \\frac{1}{4}\\left( M-m\\right)$ we have the desired result (REF ).",
"From (REF ) we have $& \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right) \\right)\\le \\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right)-m\\right) }{M-m}\\Theta _{f}\\left( \\psi \\left( c\\right) ;m,M\\right) \\\\& \\le \\frac{1}{4}\\left( M-m\\right) \\Theta _{f}\\left( \\psi \\left( c\\right);m,M\\right) \\le \\frac{1}{4}\\left( M-m\\right) \\sup _{t\\in \\left( m,M\\right)}\\Theta _{f}\\left( t;m,M\\right) \\\\& \\le \\frac{1}{4}\\left( M-m\\right) \\left[ f^{\\prime }\\left( M\\right)-f^{\\prime }\\left( m\\right) \\right]$ that proves (REF ).",
"We also have: Theorem 4 With the assumptions of Theorem REF we have $0& \\le \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right)\\right) \\\\& \\\\& \\le \\left( 1+2\\frac{\\left|\\psi \\left( c\\right) -\\frac{m+M}{2}\\right|}{M-m}\\right) \\left[ \\frac{f\\left( m\\right) +f\\left( M\\right) }{2}-f\\left( \\frac{m+M}{2}\\right) \\right] \\\\& \\\\& \\le f\\left( m\\right) +f\\left( M\\right) -2f\\left( \\frac{m+M}{2}\\right) .$ First of all, we recall the following result obtained by the author in [4] that provides a refinement and a reverse for the weighted Jensen's discrete inequality: $& n\\min _{i\\in \\left\\lbrace 1,...,n\\right\\rbrace }\\left\\lbrace p_{i}\\right\\rbrace \\left[ \\frac{1}{n}\\sum _{i=1}^{n}\\Phi \\left( x_{i}\\right) -\\Phi \\left( \\frac{1}{n}\\sum _{i=1}^{n}x_{i}\\right) \\right] \\\\& \\le \\frac{1}{P_{n}}\\sum _{i=1}^{n}p_{i}\\Phi \\left( x_{i}\\right) -\\Phi \\left( \\frac{1}{P_{n}}\\sum _{i=1}^{n}p_{i}x_{i}\\right) \\\\& n\\max _{i\\in \\left\\lbrace 1,...,n\\right\\rbrace }\\left\\lbrace p_{i}\\right\\rbrace \\left[ \\frac{1}{n}\\sum _{i=1}^{n}\\Phi \\left( x_{i}\\right) -\\Phi \\left( \\frac{1}{n}\\sum _{i=1}^{n}x_{i}\\right) \\right] , $ where $\\Phi :C\\rightarrow \\mathbb {R}$ is a convex function defined on the convex subset $C$ of the linear space $X,$ $\\left\\lbrace x_{i}\\right\\rbrace _{i\\in \\left\\lbrace 1,...,n\\right\\rbrace }\\subset C$ are vectors and $\\left\\lbrace p_{i}\\right\\rbrace _{i\\in \\left\\lbrace 1,...,n\\right\\rbrace }$ are nonnegative numbers with $P_{n}:=\\sum _{i=1}^{n}p_{i}>0.$ For $n=2$ we deduce from (REF ) that $& 2\\min \\left\\lbrace t,1-t\\right\\rbrace \\left[ \\frac{\\Phi \\left( x\\right) +\\Phi \\left(y\\right) }{2}-\\Phi \\left( \\frac{x+y}{2}\\right) \\right] \\\\& \\le t\\Phi \\left( x\\right) +\\left( 1-t\\right) \\Phi \\left( y\\right) -\\Phi \\left( tx+\\left( 1-t\\right) y\\right) \\\\& \\le 2\\max \\left\\lbrace t,1-t\\right\\rbrace \\left[ \\frac{\\Phi \\left( x\\right) +\\Phi \\left( y\\right) }{2}-\\Phi \\left( \\frac{x+y}{2}\\right) \\right] $ for any $x,y\\in C$ and $t\\in \\left[ 0,1\\right] .$ If we use the second inequality in (REF ) for the convex function $f:I\\rightarrow \\mathbb {R}$ and $m,$ $M\\in \\mathbb {R}$ , $m<M$ with $\\left[ m,M\\right] \\subset I,$ we have for $t=\\frac{M-\\psi \\left( c\\right) }{M-m}$ that $& \\frac{\\left( M-\\psi \\left( c\\right) \\right) f\\left( m\\right) +\\left( \\psi \\left( c\\right) -m\\right) f\\left( M\\right) }{M-m} \\\\& -f\\left( \\frac{m\\left( M-\\psi \\left( c\\right) \\right) +M\\left( \\psi \\left(c\\right) -m\\right) }{M-m}\\right) \\\\& \\le 2\\max \\left\\lbrace \\frac{M-\\psi \\left( c\\right) }{M-m},\\frac{\\psi \\left(c\\right) -m}{M-m}\\right\\rbrace \\left[ \\frac{f\\left( m\\right) +f\\left( M\\right) }{2}-f\\left( \\frac{m+M}{2}\\right) \\right] , $ namely $& \\frac{\\left( M-\\psi \\left( c\\right) \\right) f\\left( m\\right) +\\left( \\psi \\left( c\\right) -m\\right) f\\left( M\\right) }{M-m}-f\\left( \\psi \\left(c\\right) \\right) \\\\& \\le \\left( 1+2\\frac{\\left|\\psi \\left( c\\right) -\\frac{m+M}{2}\\right|}{M-m}\\right) \\left[ \\frac{f\\left( m\\right) +f\\left( M\\right) }{2}-f\\left( \\frac{m+M}{2}\\right) \\right] \\\\& \\times \\left[ \\frac{f\\left( m\\right) +f\\left( M\\right) }{2}-f\\left( \\frac{m+M}{2}\\right) \\right] .",
"$ On making use of the first inequality in (REF ) and (REF ) we get the first part of (REF ).",
"The last part follows by the fact that $m\\le \\psi \\left( c\\right) \\le M.$"
],
[
"Refinements and Reverses",
"We start with the following result: Theorem 5 Let $f\\left( z\\right) $ be analytic in $G$ , an open subset of $\\mathbb {C}$ and the real interval $I\\subset G,$ $\\left[ m,M\\right] \\subset I$ for some real numbers $m<M,$ and $\\psi :A\\rightarrow \\mathbb {C}$ is a positive normalized linear functional on $A.$ If there exists the constants $K>k\\ge 0$ such that $K\\ge f^{\\prime \\prime }\\left( z\\right) \\ge k\\text{ for any }z\\in \\left[ m,M\\right] , $ then for any selfadjoint element $c\\in A$ with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset I,$ $\\frac{1}{2}K\\psi \\left[ \\left( c-t\\right) ^{2}\\right] \\ge \\psi \\left(f\\left( c\\right) \\right) -f^{\\prime }\\left( t\\right) \\left( \\psi \\left(c\\right) -t\\right) -f\\left( t\\right) \\ge \\frac{1}{2}k\\psi \\left[ \\left(c-t\\right) ^{2}\\right] $ and $\\frac{1}{2}K\\psi \\left[ \\left( c-t\\right) ^{2}\\right] \\ge \\psi \\left(cf^{\\prime }\\left( c\\right) \\right) -t\\psi \\left( f^{\\prime }\\left( c\\right)\\right) +f\\left( t\\right) -\\psi \\left( f\\left( c\\right) \\right) \\ge \\frac{1}{2}k\\psi \\left[ \\left( c-t\\right) ^{2}\\right] , $ for any $t\\in \\left[ m,M\\right] .$ Using Taylor's representation with the integral remainder we can write the following identity $f\\left( z\\right) =\\sum _{k=0}^{n}\\frac{1}{k!",
"}f^{\\left( k\\right) }\\left(t\\right) \\left( z-t\\right) ^{k}+\\frac{1}{n!",
"}\\int _{t}^{z}f^{\\left( n+1\\right)}\\left( s\\right) \\left( z-s\\right) ^{n}ds $ for any $z,$ $t\\in \\mathring{I},$ the interior of $I.$ For any integrable function $h$ on an interval and any distinct numbers $c,$ $d$ in that interval, we have, by the change of variable $s=\\left(1-s\\right) c+sd,$ $s\\in \\left[ 0,1\\right] $ that $\\int _{c}^{d}h\\left( s\\right) ds=\\left( d-c\\right) \\int _{0}^{1}h\\left( \\left(1-s\\right) c+sd\\right) ds.$ Therefore, $& \\int _{t}^{z}f^{\\left( n+1\\right) }\\left( s\\right) \\left( z-s\\right) ^{n}ds\\\\& =\\left( z-t\\right) \\int _{0}^{1}f^{\\left( n+1\\right) }\\left( \\left(1-s\\right) t+sz\\right) \\left( z-\\left( 1-s\\right) t-sz\\right) ^{n}ds \\\\& =\\left( z-t\\right) ^{n+1}\\int _{0}^{1}f^{\\left( n+1\\right) }\\left( \\left(1-s\\right) t+sz\\right) \\left( 1-s\\right) ^{n}ds.$ The identity (REF ) can then be written as $f\\left( z\\right) & =\\sum _{k=0}^{n}\\frac{1}{k!",
"}f^{\\left( k\\right) }\\left(t\\right) \\left( z-t\\right) ^{k} \\\\& +\\frac{1}{n!",
"}\\left( z-t\\right) ^{n+1}\\int _{0}^{1}f^{\\left( n+1\\right)}\\left( \\left( 1-s\\right) t+sz\\right) \\left( 1-s\\right) ^{n}ds.",
"$ For $n=1$ we get $f\\left( z\\right) =f\\left( t\\right) +\\left( z-t\\right) f^{\\prime }\\left(t\\right) +\\left( z-t\\right) ^{2}\\int _{0}^{1}f^{\\prime \\prime }\\left( \\left(1-s\\right) t+sz\\right) \\left( 1-s\\right) ds $ for any $z,$ $t\\in \\mathring{I}.$ By the condition (REF ) we have $K\\int _{0}^{1}\\left( 1-s\\right) ds\\ge \\int _{0}^{1}f^{\\prime \\prime }\\left(\\left( 1-s\\right) t+sz\\right) \\left( 1-s\\right) ds\\ge k\\int _{0}^{1}\\left(1-s\\right) ds,$ namely $\\frac{1}{2}K\\ge \\int _{0}^{1}f^{\\prime \\prime }\\left( \\left( 1-s\\right)t+sz\\right) \\left( 1-s\\right) ds\\ge \\frac{1}{2}k,$ and by (REF ) we get the double inequality $\\frac{1}{2}K\\left( z-t\\right) ^{2}\\ge f\\left( z\\right) -f\\left( t\\right)-\\left( z-t\\right) f^{\\prime }\\left( t\\right) \\ge \\frac{1}{2}k\\left(z-t\\right) ^{2} $ for any $z,$ $t\\in \\mathring{I}.$ Fix $t\\in \\left[ m,M\\right] $ .",
"Using Lemma REF and the inequality (REF ) we obtain for the element $c\\in A$ with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset I$ the following inequality in the order of $A$ $\\frac{1}{2}K\\left( c-t\\right) ^{2}\\ge f\\left( c\\right) -f\\left( t\\right)-\\left( c-t\\right) f^{\\prime }\\left( t\\right) \\ge \\frac{1}{2}k\\left(c-t\\right) ^{2}.$ If we take in this inequality the functional $\\psi $ we get (REF ).",
"Fix $z\\in \\left[ m,M\\right] .$ Using Lemma REF and the inequality (REF ) we obtain for the element $c\\in A$ with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset I$ the following inequality in the order of $A$ $\\frac{1}{2}K\\left( c-z\\right) ^{2}\\ge f\\left( z\\right) -f\\left( c\\right)-zf^{\\prime }\\left( c\\right) +cf^{\\prime }\\left( c\\right) \\ge \\frac{1}{2}k\\left( c-z\\right) ^{2}.",
"$ If we take in this inequality the functional $\\psi $ we get $\\frac{1}{2}K\\psi \\left[ \\left( c-z\\right) ^{2}\\right] & \\ge \\psi \\left(cf^{\\prime }\\left( c\\right) \\right) -z\\psi \\left( f^{\\prime }\\left( c\\right)\\right) -\\psi \\left( f\\left( c\\right) \\right) +f\\left( z\\right) \\\\& \\ge \\frac{1}{2}k\\psi \\left[ \\left( c-z\\right) ^{2}\\right] ,$ for any $z\\in \\left[ m,M\\right] .$ If we replace $z$ with $t$ we get the desired result (REF ).",
"Corollary 1 With the assumptions of Theorem REF we have the Jensen's type inequalities $\\frac{1}{2}K\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right]\\ge \\psi \\left( f\\left( c\\right) \\right) -f\\left( \\psi \\left( c\\right)\\right) \\ge \\frac{1}{2}k\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left(c\\right) \\right] $ and $\\frac{1}{2}K\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right]& \\ge \\psi \\left( cf^{\\prime }\\left( c\\right) \\right) -\\psi \\left( c\\right)\\psi \\left( f^{\\prime }\\left( c\\right) \\right) +f\\left( \\psi \\left( c\\right)\\right) -\\psi \\left( f\\left( c\\right) \\right) \\\\& \\ge \\frac{1}{2}k\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] .",
"$ Follows by Theorem REF on choosing $t=\\psi \\left( c\\right) \\in \\left[m,M\\right] .$ Corollary 2 With the assumptions of Theorem REF we have $& \\frac{1}{2}K\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& \\ge \\psi \\left( f\\left( c\\right) \\right) -f^{\\prime }\\left( \\frac{m+M}{2}\\right) \\left( \\psi \\left( c\\right) -\\frac{m+M}{2}\\right) -f\\left( \\frac{m+M}{2}\\right) \\\\& \\ge \\frac{1}{2}k\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right]$ and $& \\frac{1}{2}K\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& \\ge \\psi \\left( cf^{\\prime }\\left( c\\right) \\right) -\\frac{m+M}{2}\\psi \\left( f^{\\prime }\\left( c\\right) \\right) +f\\left( \\frac{m+M}{2}\\right)-\\psi \\left( f\\left( c\\right) \\right) \\\\& \\ge \\frac{1}{2}k\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] .$ Follows by Theorem REF on choosing $t=\\frac{m+M}{2}.$ Corollary 3 With the assumptions of Theorem REF and, if, in addition, $t=\\frac{\\psi \\left( cf^{\\prime }\\left( c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\in \\left[ m,M\\right] $ with $\\psi \\left( f^{\\prime }\\left( c\\right) \\right) \\ne 0,$ then we have the Slater's type inequalities $\\frac{1}{2}K\\psi \\left[ \\left( c-\\frac{\\psi \\left( cf^{\\prime }\\left(c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right)^{2}\\right] & \\ge f\\left( \\frac{\\psi \\left( cf^{\\prime }\\left( c\\right)\\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right) -\\psi \\left( f\\left( c\\right) \\right) \\\\& \\ge \\frac{1}{2}k\\psi \\left[ \\left( c-\\frac{\\psi \\left( cf^{\\prime }\\left(c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right)^{2}\\right] , $ and $& \\frac{1}{2}K\\psi \\left[ \\left( c-\\frac{\\psi \\left( cf^{\\prime }\\left(c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right)^{2}\\right] \\\\& \\ge f^{\\prime }\\left( \\frac{\\psi \\left( cf^{\\prime }\\left( c\\right)\\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right) \\frac{\\psi \\left( cf^{\\prime }\\left( c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left(c\\right) \\right) }-\\psi \\left( c\\right) f^{\\prime }\\left( \\frac{\\psi \\left(cf^{\\prime }\\left( c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right)\\right) }\\right) \\\\& -f\\left( \\frac{\\psi \\left( cf^{\\prime }\\left( c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right) +\\psi \\left( f\\left(c\\right) \\right) \\\\& \\ge \\frac{1}{2}k\\psi \\left[ \\left( c-\\frac{\\psi \\left( cf^{\\prime }\\left(c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right) \\right) }\\right)^{2}\\right] .",
"$ Follows by Follows by Theorem REF on choosing $t=\\frac{\\psi \\left(cf^{\\prime }\\left( c\\right) \\right) }{\\psi \\left( f^{\\prime }\\left( c\\right)\\right) }\\in \\left[ m,M\\right] .$ We observe that a sufficient condition for this to happen is that $f^{\\prime }\\left( c\\right) >0$ and $\\psi \\left(f^{\\prime }\\left( c\\right) \\right) >0.$ Corollary 4 With the assumptions of Theorem REF we have $& \\frac{1}{4}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] \\\\& \\ge \\frac{1}{2}\\left[ \\psi \\left( f\\left( c\\right) \\right) +\\frac{\\left(M-\\psi \\left( c\\right) \\right) f\\left( M\\right) +\\left( \\psi \\left( c\\right)-m\\right) f\\left( m\\right) }{M-m}\\right] \\\\& -\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt \\\\& \\ge \\frac{1}{4}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] $ and $& \\frac{1}{2}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] \\\\& \\ge \\frac{1}{M-m}\\int _{m}^{M}f\\left( z\\right) dz-\\psi \\left( f\\left(c\\right) \\right) -\\frac{m+M}{2}\\psi \\left( f^{\\prime }\\left( c\\right)\\right) -\\psi \\left( cf^{\\prime }\\left( c\\right) \\right) \\\\& \\ge \\frac{1}{2}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] .",
"$ If we take the integral mean over $t$ on $\\left[ m,M\\right] $ in the inequality (REF ) we get $& \\frac{1}{2}K\\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) ^{2}dt \\\\& \\ge f\\left( z\\right) -\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt-\\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) f^{\\prime }\\left( t\\right) dt \\\\& \\ge \\frac{1}{2}\\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) ^{2}dt $ for any $z\\in \\left[ m,M\\right] .$ Observe that $\\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) ^{2}& =\\frac{\\left( M-z\\right)^{3}+\\left( z-m\\right) ^{3}}{3\\left( M-m\\right) } \\\\& =\\frac{1}{3}\\left[ \\left( z-m\\right) ^{2}+\\left( M-z\\right) ^{2}-\\left(z-m\\right) \\left( M-z\\right) \\right] \\\\& =\\frac{1}{3}\\left[ \\frac{1}{4}\\left( M-m\\right) ^{2}+3\\left( z-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& =\\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( z-\\frac{m+M}{2}\\right) ^{2}$ and $& \\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) f^{\\prime }\\left( t\\right) dt \\\\& =\\frac{1}{M-m}\\left[ \\left.",
"\\left( z-t\\right) f\\left( t\\right) \\right|_{m}^{M}+\\int _{m}^{M}f\\left( t\\right) dt\\right] \\\\& =\\frac{1}{M-m}\\left[ \\int _{m}^{M}f\\left( t\\right) dt-\\left( M-z\\right)f\\left( M\\right) -\\left( z-m\\right) f\\left( m\\right) \\right] \\\\& =\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt-\\frac{\\left( M-z\\right)f\\left( M\\right) +\\left( z-m\\right) f\\left( m\\right) }{M-m}.$ Then by (REF ) we get $& \\frac{1}{2}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( z-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& \\ge f\\left( z\\right) -\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt-\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt \\\\& +\\frac{\\left( M-z\\right) f\\left( M\\right) +\\left( z-m\\right) f\\left(m\\right) }{M-m} \\\\& \\ge \\frac{1}{2}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( z-\\frac{m+M}{2}\\right) ^{2}\\right]$ namely $& \\frac{1}{4}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( z-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& \\ge \\frac{1}{2}\\left[ f\\left( z\\right) +\\frac{\\left( M-z\\right) f\\left(M\\right) +\\left( z-m\\right) f\\left( m\\right) }{M-m}\\right] -\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt \\\\& \\ge \\frac{1}{4}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( z-\\frac{m+M}{2}\\right) ^{2}\\right] $ for any $z\\in \\left[ m,M\\right] .$ Using Lemma REF and the inequality (REF ) we obtain for the element $c\\in A$ with $\\sigma \\left( c\\right) \\subseteq \\left[ m,M\\right]\\subset I$ the following inequality in the order of $A$ $& \\frac{1}{4}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& \\ge \\frac{1}{2}\\left[ f\\left( c\\right) +\\frac{\\left( M-c\\right) f\\left(M\\right) +\\left( c-m\\right) f\\left( m\\right) }{M-m}\\right] -\\frac{1}{M-m}\\int _{m}^{M}f\\left( t\\right) dt \\\\& \\ge \\frac{1}{4}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] .$ If we apply to this inequality the functional $\\psi $ we get (REF ).",
"If we take the integral mean over $z$ on $\\left[ m,M\\right] $ in the inequality (REF ) we get $& \\frac{1}{2}K\\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) ^{2}dz \\\\& \\ge \\frac{1}{M-m}\\int _{m}^{M}f\\left( z\\right) dz-f\\left( t\\right) -\\left(\\frac{m+M}{2}-t\\right) f^{\\prime }\\left( t\\right) \\\\& \\ge \\frac{1}{2}k\\frac{1}{M-m}\\int _{m}^{M}\\left( z-t\\right) ^{2}dz,$ namely $& \\frac{1}{2}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( t-\\frac{m+M}{2}\\right) ^{2}\\right] \\\\& \\ge \\frac{1}{M-m}\\int _{m}^{M}f\\left( z\\right) dz-f\\left( t\\right) -\\left(\\frac{m+M}{2}-t\\right) f^{\\prime }\\left( t\\right) \\\\& \\ge \\frac{1}{2}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\left( t-\\frac{m+M}{2}\\right) ^{2}\\right] $ for any $t\\in \\left[ m,M\\right] .$ Using (REF ) and a similar argument as above, we get the desired result (REF )."
],
[
"Some Examples",
"Assume that $A$ is a Hermitian unital Banach $\\ast $ -algebra and $\\psi :A\\rightarrow \\mathbb {C}$ a positive normalized linear functional on $A.$ Let $c\\in A$ be a selfadjoint element with $\\sigma \\left( c\\right) \\subseteq \\left[ m,M\\right] $ for some real numbers $m<M.$ If we take $f\\left(t\\right) =t^{2}$ and calculate $\\Theta _{f}\\left( t;m,M\\right) =\\frac{M^{2}-t^{2}}{M-t}-\\frac{t^{2}-m^{2}}{t-m}=M-m$ then by (REF ) we get $0\\le \\psi \\left( c^{2}\\right) -\\left( \\psi \\left( c\\right) \\right) ^{2}\\le \\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right) -m\\right)\\le \\frac{1}{4}\\left( M-m\\right) ^{2}.",
"$ Consider the convex function $f:\\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) \\rightarrow \\left( 0,\\infty \\right) ,$ $f\\left( t\\right) =t^{p},$ $p>1.$ Using the inequality (REF ) we have $0& \\le \\psi \\left( c^{p}\\right) -\\left( \\psi \\left( c\\right) \\right)^{p}\\le p\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right)-m\\right) \\frac{M^{p-1}-m^{p-1}}{M-m} \\\\& \\le \\frac{1}{4}p\\left( M-m\\right) \\left( M^{p-1}-m^{p-1}\\right) $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ If we use the inequality (REF ) we also get $0& \\le \\psi \\left( c^{p}\\right) -\\left( \\psi \\left( c\\right) \\right) ^{p} \\\\& \\le \\left( 1+2\\frac{\\left|\\psi \\left( c\\right) -\\frac{m+M}{2}\\right|}{M-m}\\right) \\left[ \\frac{m^{p}+M^{p}}{2}-\\left( \\frac{m+M}{2}\\right) ^{p}\\right] \\\\& \\le m^{p}+M^{p}-2^{1-p}\\left( m+M\\right) ^{p} $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Since $f^{\\prime \\prime }\\left( t\\right) =p\\left( p-1\\right) t^{p-2},$ $t>0$ then $k_{p}& :=p\\left( p-1\\right) \\left\\lbrace \\begin{array}{c}M^{p-2}\\text{ for }p\\in \\left( 1,2\\right) \\\\\\\\m^{p-2}\\text{ for }p\\in [2,\\infty )\\end{array}\\right.",
"\\\\& \\\\& \\le f^{\\prime \\prime }\\left( t\\right) \\le K_{p}:=p\\left( p-1\\right)\\left\\lbrace \\begin{array}{c}m^{p-2}\\text{ for }p\\in \\left( 1,2\\right) \\\\\\\\M^{p-2}\\text{ for }p\\in [2,\\infty )\\end{array}\\right.",
"$ for any $t\\in \\left[ m,M\\right] .$ Using (REF ) and (REF ) we get $\\frac{1}{2}K_{p}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] \\ge \\psi \\left( c^{p}\\right) -\\left( \\psi \\left( c\\right) \\right)^{p}\\ge \\frac{1}{2}k_{p}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left(c\\right) \\right] $ and $\\frac{1}{2}K_{p}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] & \\ge \\left( p-1\\right) \\psi \\left( c^{p}\\right) +\\left( \\psi \\left( c\\right) \\right) ^{p}-p\\psi \\left( c\\right) \\psi \\left( c^{p-1}\\right) \\\\& \\ge \\frac{1}{2}k_{p}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left(c\\right) \\right] , $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Using (REF ) and (REF ) we get $\\frac{1}{2}K_{p}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{2}\\right] & \\ge \\left( \\frac{\\psi \\left(c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{p}-\\psi \\left(c^{p}\\right) \\\\& \\ge \\frac{1}{2}k_{p}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{2}\\right] , $ and $& \\frac{1}{2}K_{p}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{2}\\right] \\\\& \\ge p\\left( \\frac{\\psi \\left( c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{p-1}\\left( \\frac{\\psi \\left( c^{p}\\right) }{\\psi \\left(c^{p-1}\\right) }-\\psi \\left( c\\right) \\right) -\\left( \\frac{\\psi \\left(c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{p}+\\psi \\left(c^{p}\\right) \\\\& \\ge \\frac{1}{2}k_{p}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{p}\\right) }{\\psi \\left( c^{p-1}\\right) }\\right) ^{2}\\right] $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Using (REF ) and (REF ) we also have $& \\frac{1}{4}K\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[ \\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] \\\\& \\ge \\frac{1}{2}\\left[ \\psi \\left( c^{p}\\right) +\\frac{\\left( M-\\psi \\left( c\\right) \\right) M^{p}+\\left( \\psi \\left( c\\right) -m\\right) m^{p}}{M-m}\\right] \\\\& -\\frac{M^{p+1}-m^{p+1}}{\\left( p+1\\right) \\left( M-m\\right) } \\\\& \\ge \\frac{1}{4}k\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] $ and $& \\frac{1}{2}K_{p}\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] \\\\& \\ge \\frac{M^{p+1}-m^{p+1}}{\\left( p+1\\right) \\left( M-m\\right) }-p\\frac{m+M}{2}\\psi \\left( c^{p-1}\\right) -\\left( p+1\\right) \\psi \\left(c^{p}\\right) \\\\& \\ge \\frac{1}{2}k_{p}\\left[ \\frac{1}{12}\\left( M-m\\right) ^{2}+\\psi \\left[\\left( c-\\frac{m+M}{2}\\right) ^{2}\\right] \\right] $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Consider the convex function $f:\\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) \\rightarrow \\left( 0,\\infty \\right) $ , $f\\left( t\\right) =\\frac{1}{t}$ .",
"We have $\\Theta _{f}\\left( t;m,M\\right) =\\frac{\\frac{1}{M}-\\frac{1}{t}}{M-t}-\\frac{\\frac{1}{t}-\\frac{1}{m}}{t-m}=\\frac{M-m}{tmM},$ which implies that $\\sup _{t\\in \\left( m,M\\right) }\\Theta _{f}\\left( t;m,M\\right) =\\frac{M-m}{m^{2}M}.$ From (REF ) we get $0& \\le \\psi \\left( c^{-1}\\right) -\\psi ^{-1}\\left( c\\right) \\le \\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right) -m\\right)}{m^{2}M} \\\\& \\le \\left\\lbrace \\begin{array}{l}\\frac{1}{4m^{2}M}\\left( M-m\\right) ^{2} \\\\\\\\\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right) -m\\right)\\frac{M+m}{m^{2}M^{2}}\\end{array}\\right.",
"\\le \\frac{1}{4}\\left( M-m\\right) ^{2}\\frac{M+m}{M^{2}m^{2}} $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ From (REF ) we have $0\\le \\psi \\left( c^{-1}\\right) -\\psi ^{-1}\\left( c\\right) \\le \\frac{1}{4}\\frac{\\left( M-m\\right) ^{2}}{mM}\\psi ^{-1}\\left( c\\right) \\le \\frac{1}{4m^{2}M}\\left( M-m\\right) ^{2} $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ From (REF ) we also have $0& \\le \\psi \\left( c^{-1}\\right) -\\psi ^{-1}\\left( c\\right) \\le \\frac{\\left( M-m\\right) ^{2}}{2mM\\left( m+M\\right) }\\left( 1+2\\frac{\\left|\\psi \\left( c\\right) -\\frac{m+M}{2}\\right|}{M-m}\\right) \\\\& \\le \\frac{\\left( M-m\\right) ^{2}}{mM\\left( m+M\\right) } $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Since $f^{\\prime \\prime }\\left( t\\right) =\\frac{2}{t^{3}},$ $t>0,$ then $\\frac{2}{m^{3}}\\ge f^{\\prime \\prime }\\left( t\\right) \\ge \\frac{2}{M^{3}}$ and by (REF ) and (REF ) we get $\\frac{1}{m^{3}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] \\ge \\psi \\left( c^{-1}\\right) -\\psi ^{-1}\\left( c\\right) \\ge \\frac{1}{M^{3}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right]$ and $\\frac{1}{m^{3}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] & \\ge \\frac{1}{2}\\left[ \\psi \\left( c\\right) \\psi \\left(c^{-2}\\right) +\\psi ^{-1}\\left( c\\right) \\right] -\\psi \\left( c^{-1}\\right) \\\\& \\ge \\frac{1}{M^{3}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left(c\\right) \\right] , $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ From (REF ) and (REF ) we also have $\\frac{1}{m^{3}}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{-1}\\right) }{\\psi \\left( c^{-2}\\right) }\\right) ^{2}\\right] & \\ge \\frac{\\psi \\left(c^{-2}\\right) }{\\psi \\left( c^{-1}\\right) }-\\psi \\left( c^{-1}\\right) \\\\& \\ge \\frac{1}{M^{3}}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{-1}\\right) }{\\psi \\left( c^{-2}\\right) }\\right) ^{2}\\right] , $ and $& \\frac{1}{m^{3}}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{-1}\\right) }{\\psi \\left( c^{-2}\\right) }\\right) ^{2}\\right] \\\\& \\ge \\psi \\left( c^{-1}\\right) -\\frac{\\psi \\left( c^{-1}\\right) }{\\psi \\left( c^{-2}\\right) }+\\psi \\left( c\\right) \\frac{\\psi ^{2}\\left(c^{-2}\\right) }{\\psi ^{2}\\left( c^{-1}\\right) }-\\frac{\\psi \\left(c^{-2}\\right) }{\\psi \\left( c^{-1}\\right) } \\\\& \\ge \\frac{1}{M^{3}}\\psi \\left[ \\left( c-\\frac{\\psi \\left( c^{-1}\\right) }{\\psi \\left( c^{-2}\\right) }\\right) ^{2}\\right] $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Similar results may be stated for the convex functions $f\\left( t\\right)=t^{r},$ $r<0$ and $f\\left( t\\right) =-t^{q}$ , $q\\in \\left( 0,1\\right) .$ The case of logarithmic function is also of interest.",
"If we take the function $f\\left( t\\right) =-\\ln t$ in (REF ), then we get $0\\le \\ln \\left( \\psi \\left( c\\right) \\right) -\\psi \\left( \\ln c\\right) \\le \\frac{\\left( M-\\psi \\left( c\\right) \\right) \\left( \\psi \\left( c\\right)-m\\right) }{mM}\\le \\frac{1}{4}\\frac{\\left( M-m\\right) ^{2}}{mM}$ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ From (REF ) we have $0& \\le \\ln \\left( \\psi \\left( c\\right) \\right) -\\psi \\left( \\ln c\\right)\\le \\ln \\left( \\frac{m+M}{2\\sqrt{mM}}\\right) \\left( 1+2\\frac{\\left|\\psi \\left( c\\right) -\\frac{m+M}{2}\\right|}{M-m}\\right) \\\\& \\le \\ln \\left( \\frac{m+M}{2\\sqrt{mM}}\\right) ^{2} $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Since $f^{\\prime \\prime }\\left( t\\right) =\\frac{1}{t^{2}}$ and $\\frac{1}{m^{2}}\\ge f^{\\prime \\prime }\\left( t\\right) \\ge \\frac{1}{M^{2}}$ for any $t\\in \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) ,$ then by (REF ) and (REF ) we have $\\frac{1}{2m^{2}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] \\ge \\ln \\left( \\psi \\left( c\\right) \\right) -\\psi \\left( \\ln c\\right) \\ge \\frac{1}{2M^{2}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] $ and $\\frac{1}{2m^{2}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left( c\\right) \\right] & \\ge \\psi \\left( \\ln c\\right) -\\ln \\left( \\psi \\left( c\\right)\\right) +\\psi \\left( c\\right) \\psi \\left( c^{-1}\\right) -1 \\\\& \\ge \\frac{1}{2M^{2}}\\left[ \\psi \\left( c^{2}\\right) -\\psi ^{2}\\left(c\\right) \\right] , $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ Finally, by making use of (REF ) and (REF ) we have $\\frac{1}{2m^{2}}\\psi \\left[ \\left( c-\\psi ^{-1}\\left( c^{-1}\\right) \\right)^{2}\\right] & \\ge \\psi \\left( \\ln c\\right) -\\ln \\left( \\psi ^{-1}\\left(c^{-1}\\right) \\right) \\\\& \\ge \\frac{1}{2M^{2}}\\psi \\left[ \\left( c-\\psi ^{-1}\\left( c^{-1}\\right)\\right) ^{2}\\right] , $ and $\\frac{1}{2m^{2}}\\psi \\left[ \\left( c-\\psi ^{-1}\\left( c^{-1}\\right) \\right)^{2}\\right] & \\ge \\psi \\left( c\\right) \\psi \\left( c^{-1}\\right) -1-\\psi \\left( \\ln c\\right) +\\ln \\left( \\psi ^{-1}\\left( c^{-1}\\right) \\right) \\\\& \\ge \\frac{1}{2M^{2}}\\psi \\left[ \\left( c-\\psi ^{-1}\\left( c^{-1}\\right)\\right) ^{2}\\right] $ for any $c\\in A$ a selfadjoint element with $\\sigma \\left( c\\right)\\subseteq \\left[ m,M\\right] \\subset \\left( 0,\\infty \\right) .$ The interested reader may obtain other similar inequalities by using the convex functions $f\\left( t\\right) =t\\ln t,$ $t>0$ and $f\\left( t\\right)=\\exp \\left( \\alpha t\\right) ,$ $t,$ $\\alpha \\in \\mathbb {R}$ and $\\alpha \\ne 0.$"
]
] | 1612.05685 |
[
[
"Surface enhanced nonlinear Cherenkov radiation in one-dimensional\n nonlinear photonic crystal"
],
[
"Abstract We study the configuration of efficient nonlinear Cerenkov diffraction generated from a one-dimensional nonlinear photonic crystal surface, which underlies the incorporation of both quasi-phase-matching and total internal reflection by the crystal surface.",
"Multidirectional radiation spots with different Cerenkov angles are demonstrated experimentally, which results from different orders of reciprocal vectors.",
"At specific angles, the incident light and total internal reflect light associating with quasi-phase-matching format completely phase-matching scheme, leading to great enhancement of harmonic efficiency."
],
[
"Introduction",
"When the phase velocity of nonlinear polarization wave ($\\nu _p$ ) exceeds that of the harmonic waves ($\\nu ^\\prime $ ) in the nonlinear medium, it will emit coherent electromagnetic waves called nonlinear Cerenkov radiation (NCR) [1].",
"The Cerenkov angle is defined as $\\theta =arccos(\\nu _p/\\nu ^\\prime )$ , which implies the automatically longitudinal phase-matching condition between the fundamental and the radiated harmonic waves.",
"With $\\chi ^{(2)}$ photonic crystals [2], waveguides [3], [4] or other micro-structures assistance are introduced, Cerenkov radiation could demonstrate possibilities of a wide variety of phase-matching types and diverse patterns of spatial modulation.",
"For instance, the phase-tuned Cerenkov-type interaction in two dimensional nonlinear photonic crystals [5], the quasi-phase-matching associated NCR generated in waveguides or nonlinear $\\chi ^{(2)}$ crystals [6], domain wall enhanced high-order NCR [7], [8], [9], and so on.",
"Such modulation mechanism has greatly expanded the NCR radiation characteristics, providing potential applications of short wavelength lasers [10], broadband frequency doubling [11] and optical imaging [12], [13].",
"In addition, the efficiency of NCR is mainly effected by the abrupt change of the second-order nonlinearity $\\chi ^{(2)}$ , which contains not only the -1 to 1 $\\chi ^{(2)}$ modulation corresponding to the domain wall but also the modulation 0 to 1 corresponding to the crystal surface [14], [15].",
"By using sum frequency polarization wave generated by incident and internal total reflected waves [16], previous studies have achieved enhanced NCR on the crystal surface [17].",
"Such NCR can provide good light quality and relatively high efficiency, which allows further practical applications, such as nondestructive diagnostics, harmonic conversion and ultrashort pulse characterization.",
"In this work, we study the behavior of NCR generated from the crystal surface and modulated by the $\\chi ^{(2)}$ microstructure on the surface.",
"Using coupled wave equation, we also demonstrate the effect of reciprocal vectors of photonic crystals to the radiation angles of NCR.",
"By utilizing the internal reflection inside the crystal boundary, the sum-frequency polarization of the incidences associated with different orders of reciprocal vectors can emit multiple NCR which exhibits $\\chi ^{(2)}$ spatial modulated pattern.",
"Particularly, at specific incident angles, one can achieve degenerated NCR which leads to remarkable enhancement on the efficiency."
],
[
"Phenomenon and Analysis",
"For simplicity, here we choose a one-dimensional (1D) periodically poled $\\mathrm {LiNbO_3}$ crystal (PPLN) which provides uniform collinear reciprocal vectors along x-axis as shown in Fig.",
"1(a).",
"The poling period of the sample $\\Lambda =6.92 ~\\mathrm {\\mu m}$ and it was put on a rotation stage which can be adjusted in the y-z plane.",
"Regarding to the calculation of the Sellmeier equation of the sample [18], one can find the refractive index of ordinary-polarized fundamental wave is larger than that of the extraordinary-polarized second harmonic wave when the wavelength of pump is longer than 1023 nm.",
"Consequently, it provides an anomalous-dispersion-like medium by utilizing type I (oo-e) SHG phase-matching interaction scheme [19].",
"The light source we used was an optical parametric amplifier (TOPAS, Coherent Inc.) producing 80 femtosecond pulses (1000 Hz rep. rate) at the variable wavelengths from 280 nm to 2600 nm.",
"A quarter-wave plate and a Glan-prism were set to adjust the polarization of incidences.",
"The laser beam was loosely focused into the x-z plane of the sample and a screen located 10 cm behind the sample to receive the emitted patterns.",
"The operating temperature was kept at 20 $^\\circ $ C. Figure: (a) Structure of the PPLN sample; (b) Multiple Cerenkov diffraction pattern.When the ordinary polarized fundamental wave (FW) with a wavelength of 1250 nm injecting into the y-z plane of the sample, there was no NCR emerge owing to the phase-matching condition being not satisfied in such anomalous dispersion.",
"As rotating the sample along x-axis, the fundamental beam would reach the x-y plane and be reflected on this surface while the incidence angle exceed the requirement of total reflection inside the sample.",
"By adjusting the incident angle of FW, and combining the total reflection wave and reciprocal vectors, the sum-frequency polarization along the crystal surface would give birth to NCR.",
"The far-filed image on the screen is shown in Fig.",
"1(b).",
"The straight line of dots in middle is the nonlinear Raman-Nath diffraction generated from the sum-frequency of incidences and total reflection, while the right arc line belongs to the conical scattering second harmonic (SH).",
"The fascinating phenomenon happened on the left of the sum-frequency Raman-Nath diffraction, where appeared multiple Cerenkov diffraction pattern with transverse angular dispersion lying in an arc array.",
"Distinguished from the phenomena in bulk material, the pattern exhibits periodically spatial distribution, which denotes to the reciprocal-involved NCR by total reflection on the PPLN surface.",
"To analyse the distribution of SH, the coupled wave equation under paraxial and small-signal approximation was solved using the Fourier transform, and the intensity of SH $I_2$ was expressed as [15], [20]: $I_2(k_x,k_z)=\\Big [\\frac{k_2}{2n_2^2}\\chi ^{(2)}\\Big ]^2I_1^2L^2\\mathrm {sinc}^2\\Big [\\Big (k_2-2k_1\\mathrm {cos}\\alpha -\\frac{k_x^2+k_z^2}{2k_2}\\Big )\\frac{L}{2}\\Big ]|F(k_x)|^2|G(k_z)|^2,$ where $n_2$ is the refractive index of the SH, $I_1$ denotes the complex amplitudes of the Gaussian FW with width $a$ , $L$ is the interaction distance of the nonlinear process, $\\alpha $ is the incident angle of FW to $y$ axis, $k_1$ and $k_2$ are the wave vectors of the FW and SH, respectively.",
"In the expression, $2k_1\\mathrm {cos}\\alpha =|\\vec{k_1}+\\vec{k_1^\\prime }|$ , where $\\vec{k_1^\\prime }$ is the wave vector of reflected FW.",
"$k_x$ and $k_z$ are the components of $k_2$ in $x$ and $z$ directions.",
"$F(k_x)=\\sqrt{\\frac{\\pi }{2}}a\\sum _{n}g_ne^{-a^2(nG_0-k_x)^2/8}$ and $G(k_z)=\\sqrt{\\frac{\\pi }{8}}ae^{-a^2k_z^2/8}+i\\frac{\\sqrt{2}}{2}aD\\big (\\frac{ak_z}{8}\\big )$ are the Fourier transform of $\\chi ^{(2)}$ modulated structure which are introduced by periodically reversed domains and the crystal boundary, respectively.",
"And $g_n$ are the fourier coefficients which can expressed as: $ g_n={\\left\\lbrace \\begin{array}{ll}~2\\mathrm {sin}(n\\pi d)/(n\\pi ) & n\\ne 0\\\\~2d-1 & n=0,\\end{array}\\right.",
"}$ where $n$ are integers.",
"And $G_0=2\\pi /\\Lambda $ denotes the 0-order reciprocal vector of PPLN, $d$ is the duty ratio of domain reversal.",
"$D\\big (\\frac{ak_z}{8}\\big )$ denotes the Dawson function.",
"Figure: (a) Simulation result; (b) Phase-matching geometry.The simulation result is shown in Fig.",
"2(a) under the same condition of experiment and the incident angle of FW $\\alpha =20^\\circ $ .",
"The distribution of SH has the similar pattern with Fig.",
"1(b).",
"But the right half was total reflected in experiment.",
"From Eq.",
"(1), we can find that, when $k_z=0$ , the SH intensity $I_2$ gathers in the direction defined by $k_x=nG_0$ which represent the nonlinear Raman-Nath diffraction of sum-frequency.",
"When $k_z\\ne 0$ , SH will radiate at angles satisfied the conditions $k_x=nG_0$ and $k_2-2k_1\\mathrm {cos}\\alpha -\\frac{k_x^2+k_z^2}{2k_2}=0$ .",
"Under the paraxial approximation, the solution of the latter is $k_x^2+k_y^2+(2k_1\\mathrm {cos}\\alpha )^2=k_2^2$ , which is exactly the longitudinal phase-matching condition of NCR.",
"The corresponding phase-matching geometry is shown in Fig.",
"2(b).",
"The emission of 0-order NCR in the experimental pattern is in accordance with the situation in bulk material [16].",
"For high-order NCR, one should take the reciprocal vectors into consideration, which is associated with Fourier components of the $\\chi ^{(2)}$ modulation in terms of quasi-phase-matching (QPM) along the x-axis.",
"The nonlinear sum-frequency polarization wave along the reflection interface (x-y plane) associating with the reciprocal vectors along the y-axis simulates an effective polarization wave which could emit the high order Cerenkov radiation.",
"The wave vector of nonlinear polarization wave of $n$ -order NCR has the form as: $\\vec{k_p^n}=|\\vec{k_1}+\\vec{k_1^\\prime }+n\\vec{G_0}|$ .",
"And we can deduce the radiation angle of $n$ -order NCR along $x$ axis: $\\varphi _n=\\mathrm {arctan}\\frac{nG_0}{2k_1\\mathrm {cos}\\alpha },$ and along $z$ axis: $\\theta _n=\\mathrm {arccos}\\frac{\\sqrt{(nG_0)^2)+(2k_1\\mathrm {cos}\\alpha )^2}}{k_2}.$"
],
[
"Experiment Results",
"To experimentally demonstrate the calculated relationship of n-order NCR angles, we investigate the external angles of different order Cerenkov radiations varying with the incident wavelength, with fixed external incident angle of FW $i=30^\\circ $ .",
"And with fixed wavelength of FW $\\lambda =1250$ nm, we measured the relationship between the external emergence angles and the external incident angles.",
"The experimental results, as shown in Fig.",
"3(a) and Fig.",
"3(b), respectively, demonstrate good agreement with theoretical predications.",
"According to the phase-matching condition and the experimental results, we verify that the polarization wave is always confined along the crystal surface.",
"Figure: The external angles along z axis of different order NCR versus the incident wavelength of FW (a) and external incident angle (b).",
"The relationship of external angles along x axis of 1-order NCR varying with the incidentwavelength (c) and the poling period of PPLN (d).",
"Theoretical prediction (solid curves) and experimental results (signs) are in well agreement with each other.Furthermore, we investigate the transverse distribution of the nonlinear diffraction along x axis.",
"With external incident angle fixed as $30^\\circ $ , the external angles of the 1st Cerenkov radiations varied with the incident wavelength [Fig.",
"3(c)].",
"When the wavelength of FW was fixed as 1250 nm, we draw the relationship between diffraction angles and the periods by using several samples with different poling periods, as shown in Fig.",
"3(d).",
"The angular position of nonlinear Cerenkov diffraction patterns are varying associating with the variance of the samples, which implies the periodical structure are modulating the surface reflecting Cerenkov radiation.",
"The transversely spatial distribution (along x-axis) of nonlinear Cerenkov patterns coincided with nonlinear Raman-Nath diffraction in 1D nonlinear crystal.",
"Figure: (a) Recorded pattern of enhanced 0-order NCR.",
"(b) The external FW angles of enhanced 0-order NCR versus the incident wavelength of FW.In Fig.",
"3(b), we note that each order of NCR has a cutoff angle respectively.",
"At these incident angles, the emission angle of corresponding order NCR equals to 0.",
"The phase-matching condition can be written as $\\vec{k_p^n}=|\\vec{k_1}+\\vec{k_1^\\prime }+n\\vec{G_0}|=\\vec{k_2}$ , where the polarization wave is collinear with the Cerenkov harmonic wave and the phase-mismatch is minimized, so that each order of NCR would be greatly enhanced at these specific angles.",
"In experiment, it's clearly that such enhancement occurred at the beginning of the NCR emergence.",
"We have experimentally recorded the enhanced 0-order NCR with wavelength at 1250 nm and sample period of 6.92 $\\mathrm {\\mu m}$ , as shown in Fig.4 (a).",
"In Fig.",
"4(b) we verify the dependence of the incident angle of the enhanced 0-order NCR as a function of the incident wavelength.",
"The incident angle increased proportionally with the incident wavelength, which is consistent with theoretical analysis.",
"So far, we have realized modulating the both diffraction pattern and the emit efficiency of Cerenkov diffraction on the crystal surface, which provide more plentiful radiation patterns compared with NCR on bulk crystal surface."
],
[
"Conclusion",
"In summary, we theoretically and experimentally demonstrated the $\\chi ^{(2)}$ modulated nonlinear Cerenkov diffraction on the crystal surface.",
"The sum-frequency polarization wave generated by incident and reflected waves is confined on the crystal surface and modulated by the periodical $\\chi ^{(2)}$ structure.",
"The diffraction angle and transverse distribution of the QPM-NCR are investigated experimentally, which shows a good agreement with the theoretical calculation.",
"In addition, with proper incident angles, the multiple diffraction would be greatly enhanced, which features the NCR emergence.",
"It's can present more plentiful radiation patterns could be expected in other various $\\chi ^{(2)}$ structures, such as aperiodic, quasi-periodic, random, chirp, two-dimensional or other desirable patterns.",
"Further, making appropriate artificial structures on the crystal surface would allow us to control the behavior of harmonic generation more efficiently."
],
[
"Funding",
"The National Basic Research Program 973 of China under Grant (2011CB808101); the National Natural Science Foundation of China under Grant (61125503, 61235009, 61205110, 61505189 and 11604318); the Innovative Foundation of Laser Fusion Research Center; the Presidential Foundation of the China Academy of Engineering Physics (201501023)."
]
] | 1612.05378 |
[
[
"Exploiting sparsity to build efficient kernel based collaborative\n filtering for top-N item recommendation"
],
[
"Abstract The increasing availability of implicit feedback datasets has raised the interest in developing effective collaborative filtering techniques able to deal asymmetrically with unambiguous positive feedback and ambiguous negative feedback.",
"In this paper, we propose a principled kernel-based collaborative filtering method for top-N item recommendation with implicit feedback.",
"We present an efficient implementation using the linear kernel, and we show how to generalize it to kernels of the dot product family preserving the efficiency.",
"We also investigate on the elements which influence the sparsity of a standard cosine kernel.",
"This analysis shows that the sparsity of the kernel strongly depends on the properties of the dataset, in particular on the long tail distribution.",
"We compare our method with state-of-the-art algorithms achieving good results both in terms of efficiency and effectiveness."
],
[
"Introduction",
"Collaborative filtering (CF) techniques can make recommendation to a user exploiting information provided by similar users.",
"The typical CF setting consists of a set $\\mathcal {U}$ of $n$ users, a set $\\mathcal {I}$ of $m$ items, and the so-called rating matrix $\\mathbf {R} = \\lbrace r_{ui}\\rbrace \\in \\mathbb {R}^{n \\times m}$ .",
"Ratings represent, numerically, the interactions between users and items.",
"These interactions can be of two different types: explicit feedbacks and implicit feedbacks.",
"The former are interactions performed by users in a direct way, such as, by giving a one-to-five star rate, thumbs-up versus thumbs-down, and so on.",
"The latter, instead, are actions performed by users without the awareness of giving some kind of feedback to the system, e.g., clicks, views, elapsed time on a web page and so on.",
"The explicit setting have got most of the attention from the CF research community and a lot of rating prediction algorithms have been proposed.",
"More recently however, the focus is constantly shifting towards the implicit feedback scenario since users are not always willing to give their opinion explicitly.",
"Furthermore, implicit interactions are easier to collect: for an already existing system, it is sufficient to store the operations done by a user in a session without changing the front-end and consequently there are no additional burden for the user.",
"One problem arising in implicit feedback is that typically the feedback is asymmetric, that is, while there can be evidence that a certain user interacted and hence showed some interest towards a product (unambiguous feedback), the opposite is not true, that is the missing evidence of interaction does not imply that a given user dislikes that item.",
"In this paper we focus on the implicit feedback scenario, and so we assume binary ratings, $r_{ui} \\in \\lbrace 0,1\\rbrace $ , where $r_{ui} = 1$ means that user $u$ interacted with the item $i$ (unambiguous feedback) and $r_{ui} = 0$ means there is no evidence that user $u$ interacted with the item $i$ (ambiguous feedback).",
"Specifically, the main contribution of this paper is a CF framework for top-N item recommendation based on the seminal work described in [1].",
"Starting from the original convex optimization problem, we propose an efficient variation of that formulation which makes the algorithm able to manage very large scale datasets, by preserving the effectiveness of the original algorithm.",
"This new formulation is then extended to be used with general kernels thus augmenting the expressiveness of the representation in collaborative filtering.",
"It is well known that kernels are not suited for large datasets since they have a prohibitive computational complexity.",
"Datasets used in CF domains are usually large: rating matrices $\\mathbf {R}$ have tens of thousands of rows and columns and so the application of kernels seems to be difficult.",
"Nevertheless, we can observe that the complexity of the computation of a kernel strictly depends on its sparsity and hence being able to control the sparsity makes the application of kernel methods to very large datasets feasible.",
"Unfortunately, kernels are usually dense unless we are working on very large and sparse feature spaces and this is in contrast with what we would like to have.",
"Exploiting a well known result from harmonic theory [2], we are able to demonstrate how using dot product kernels, a generalization of normalized homogeneous polynomial kernels, kernels as sparse as the standard cosine kernel can be obtained without changing the solution of the problem.",
"However, it should be noted that there is no guarantees about the sparsity of a cosine kernel because it is closely related to the distribution of the non-zero values in the matrix $\\mathbf {R}$ .",
"CF datasets are known to be very sparse, and also they are usually extracted from e-services which makes them subject to the long tail phenomenon that is often associated with the power law distribution.",
"Even if not every long tail is a power law [3], this tailed distribution over the item ratings in $\\mathbf {R}$ poses a strong bias on the density of the resulting kernel.",
"In fact, assuming the long tail, in $\\mathbf {R}$ there will be few dense columns and many sparse ones, which intuitively leads to a rather sparse kernel.",
"On the other side, a long tailed distribution over the user ratings can lead to dense kernels matrices since different items tend to be rated by mostly the same users.",
"We provide a deep analysis of this phenomenon showing theoretically and empirically which are the conditions for which a dataset is likely to produce a sparse dot product kernel.",
"It is worth to notice that these results are not only applicable to CF contexts, but they apply on every other context where the data distribution is long tailed.",
"Finally, in the experimental section we compare our framework with two state-of-the-art methods in top-N recommendation.",
"The empirical work shows how our method can achieve good results in terms of AUC (Area Under the ROC Curve) and also in terms of efficiency.",
"Summarizing, the contribution of this paper is 3-fold: Starting from the seminal work described in [1] we propose an optimized CF framework for top-N recommendation.",
"Specifically, our proposed method results far more efficient of the original enabling the application of the method to large and very large datasets (e.g.",
"in the order of 1 million of users/items and 50 million of rates) while preserving the state-of-the-art effectiveness of the original algorithm.",
"The formulation depicted above is generalized to be used with non-linear kernels as dot-product kernels, that is kernel functions in the form $k(\\mathbf {x},\\mathbf {y}) = f(\\mathbf {x} \\cdot \\mathbf {y})$ where $f$ is a non-linear function admitting a Maclaurin expansion with non-negative coefficients.",
"A sparsification method for dot-product kernels is also provided making the sparsity of any dot-product kernel equal to the sparsity of the linear or cosine kernel.",
"A theoretical and empirical analysis concerning the sparsity of the standard cosine kernel is proposed.",
"This analysis shows that the sparsity of the kernel produced depends on the properties of the user activity and item popularity long-tails.",
"In particular, long-tails observed on item popularity improve the sparsity of the kernels while long-tails observed on user activity is detrimental for the sparsity of the kernels.",
"This paper is an extended version of a preliminary paper presented at ESANN 2016 [4].",
"In particular, preliminary work about the contribution 1 and 2 above was already presented in that work while contribution 3 is novel.",
"Moreover, the extensive experimental work of this paper was not present in the preliminary version.",
"The rest of the paper is organized as follows.",
"In Section we will introduce the notation used throughout the paper.",
"Section presents related works on top-N recommendation for implicit feedback.",
"Sections and describes our framework with a particular focus on the applicability of our proposed kernel method.",
"Finally, Sections and show the experimental results and which directions our research can follow in the future."
],
[
"Notation",
"In this section we provide some useful notation used throughout the paper.",
"Recommender algorithms are thought to give suggestions to users about items.",
"We call the set of users as $\\mathcal {U}$ such that $|\\mathcal {U}|=n$ , the set of items as $\\mathcal {I}$ such that $|\\mathcal {I}|=m$ and the set of ratings $\\mathcal {R} = \\lbrace (u,i)\\rbrace $ .",
"We refer to the binary rating matrix with $\\mathbf {R} = \\lbrace r_{ui}\\rbrace \\in \\mathbb {R}^{n \\times m}$ , where users are on the rows and items on the columns.",
"We add a subscription to both user and item sets to indicate, respectively, the set of items rated by a user $u$ ($\\mathcal {I}_u$ ) and the set of users who rated the item $i$ ($\\mathcal {U}_i$ )."
],
[
"Related works",
"Top-N recommendation finds application in many different domains such as TV and movies [5], books [1], music [6], [7], social media [8] and so on.",
"Top-N recommendation methods can be divided into two macro categories.",
"The first is the neighbourhood-based CF algorithms [9], in which the recommendation for a target user is made by using the ratings of the most similar users.",
"This category comprises the so called memory-based methods that do not need the construction of a model, but they directly use the data inside the rating matrix.",
"Despite, in general, these methods suffer from low accuracy, in 2013 the winner of the remarkable challenge organized by Kaggle, the Million Songs Dataset challenge [10], was an extension of the well known item-based nearest-neighbors (NN) algorithm [11].",
"This extension [6] (here called MSDW) introduced an asymmetric similarity measure, called asymmetric cosine.",
"In a classic item-based CF method, the scoring function for a user-item pair $(u,i)$ is computed by a weighted sum over the items liked by $u$ in the past, that is: $\\hat{r}_{ui} = \\sum \\limits _{j \\in \\mathcal {I}} w_{ij} r_{uj} = \\sum \\limits _{j \\in \\mathcal {I}_u} w_{ij},$ where $w_{ij}$ expresses the similarity between item $i$ and item $j$ .",
"One of the main contribution, presented in [6], is the asymmetric cosine (asymC) similarity.",
"The intuition behind asymC comes from the observation that the cosine similarity can be expressed, over a pair of items $(a,b)$ , as the square root of the product of the reciprocal conditional probabilities.",
"Let $\\mathbf {a}, \\mathbf {b} \\in \\lbrace 0,1\\rbrace ^n$ be respectively the binary vector representations of items $a$ and $b$ .",
"The idea of asymmetric cosine similarity is to give different weights to the conditional probabilities, that is $S_{\\alpha }(a, b) = \\frac{\\mathbf {a}^\\top \\mathbf {b}}{\\Vert \\mathbf {a}\\Vert ^{2\\alpha } \\Vert \\mathbf {b}\\Vert ^{2(1-\\alpha )}} = P(a|b)^{\\alpha } P(b|a)^{1-\\alpha },$ with $0 \\le \\alpha \\le 1$ .",
"In case of binary rates this asymmetric similarity can be computed as in the following.",
"Let $\\mathcal {U}_i$ represents the set of users who rated the item $i$ , then the asymC between item $i$ and item $j$ is defined by: $w_{ij} = S_{\\alpha }(i,j) = \\frac{|\\mathcal {U}_i \\cap \\mathcal {U}_j|}{|\\mathcal {U}_i|^{\\alpha }|\\mathcal {U}_j|^{1-\\alpha }}.$ Besides its outstanding performance in terms of mAP@500, the MSD winning solution is also easily scalable to very large datasets.",
"However, one drawback of this solution is that it is not theoretically well founded.",
"The second category is the model-based CF techniques, which construct a model of the information contained in the rating matrix $\\mathbf {R}$ .",
"In the last two decades many different model-based approaches have been proposed.",
"A particular attention has been devoted to latent factor models which try to factorize the rating matrix into two low-rank matrices, $\\mathbf {R}=\\mathbf {W}\\mathbf {X}$ , which represent user-factors ($\\mathbf {W}$ ) and item-factors ($\\mathbf {X}$ ).",
"These factors are “meta-features” that define the user tastes and how much of these features represent an item.",
"Usually these methods are referred to as matrix factorization methods.",
"The prediction for a user-item pair is simply done by a dot product of the corresponding row and column in the factorized matrices.",
"One of the most used matrix factorization approach for implicit feedback is presented in [5] (WRMF: Weighted Regularized Matrix Factorization).",
"In this work Hu et al.",
"propose an adaptation of the classic SVD (Singular Value Decomposition) method in which they minimize the square-loss using two regularization terms in order to avoid overfitting.",
"Their optimization criterion is defined as: $\\sum \\limits _{u\\in \\mathcal {U}}\\sum \\limits _{i\\in \\mathcal {I}} c_{ui} (\\mathbf {w}_u^\\top \\mathbf {x}_i - 1)^2 + \\lambda \\Vert \\mathbf {W}\\Vert ^2 + \\lambda \\Vert \\mathbf {X}\\Vert ^2,$ where $c_{ui}$ are a-priori weights for each pair $(u,i)$ such that positive feedbacks have higher weights.",
"Actually, this method uses information about the rating values (not binary) to give more importance to user-item interactions with high rating values, in fact, $c_{ui}$ is calculated by: $c_{ui} = 1 + \\alpha r_{ui}$ .",
"In their experiments the best performances has been achieved with $\\alpha = 40$ , but with binary rating matrices this parameter losses a lot of its importance.",
"As we will see in the experimental section, changing the value of $\\alpha $ does not change the performance of the algorithm.",
"A similar approach has been used by Wang et al.",
"[12] where they proposed a framework for broadcast email prioritization based on a novel active learning method.",
"In [13] the top-N recommendation problem has been formulated as a ranking problem.",
"Rendle et al.",
"proposed a Bayesian Personalized Ranking (BPR) criterion that is the maximum posterior estimator derived from a Bayesian analysis.",
"They also provide a learning method based on stochastic gradient descent with bootstrap sampling.",
"They finally show how to adopt this criterion for kNN (BPRkNN) and MF methods (BPRMF).",
"In [14] Ning et al.",
"presented SLIM (Sparse LInear Method) which generates top-N recommendation by aggregating from user rating profiles.",
"SLIM learns an aggregation coefficient matrix by solving a regularized optimization problem.",
"More recently, a new principled algorithm for CF, which explicitly optimizes the AUC, has been proposed with very nice performances on the MovieLens dataset [1].",
"Since this algorithm represents the seminal work for our framework, we will discuss about it in details in the next section.",
"A more direct approach in order to build a good ranking over the items is the so called learning to rank[15].",
"Learning to rank methods exploit supervised machine learning to solve ranking problems.",
"These techniques can be divided into three categories: pointwise approaches [8], [16], [17] in which for each query-document (i.e., user-item) pair a score is predicted and then used to build the ranking; pairwise approaches [18], [19] face the ranking problem as a binary classification one (positive document versus negative ones) in which they try to minimize the number of inversions in the ranking; listwise approaches [20], [21] try to directly optimize one of the ranking evaluation measures.",
"The big challenge here is the fact that most of the measures are not continuous function w.r.t.",
"the model's parameters and for this reason approximations have to be used."
],
[
"CF-OMD",
"In this section we present the seminal CF algorithm, called CF-OMD (Optimization of the Margin Distribution) [1], for top-N recommendation inspired by preference learning [22][23], and designed to explicitly maximize the AUC (Area Under the ROC Curve).",
"Consider the matrix $\\mathbf {W} \\in \\mathbb {R}^{n \\times k}$ be the embeddings of users in a latent factor space and $\\mathbf {X} \\in \\mathbb {R}^{k \\times m}$ be the embeddings of items in the space.",
"Given a user, a ranking over items can be induced by the factorization $\\hat{\\mathbf {R}} = \\mathbf {W}\\mathbf {X}$ , where $\\hat{r}_{ui} = \\mathbf {w}_u^\\top \\mathbf {x}_i$ with the constraint $\\Vert \\mathbf {w}_u\\Vert = \\Vert \\mathbf {x}_i\\Vert = 1$ .",
"The model parameters (i.e., $\\mathbf {W}, \\mathbf {X}$ ) are generally computed minimizing a regularized loss function over the ground truth matrix $\\mathbf {R}$ .",
"Let now fix the item representation as $\\mathbf {x}_i = \\mathbf {r}_i/\\Vert \\mathbf {r}_i\\Vert $ , and let $\\rho (i \\prec _u j)=(\\hat{r}_{ui}-\\hat{r}_{uj})/2 = \\mathbf {w}_u^\\top (\\mathbf {x}_i - \\mathbf {x}_j)/2$ be the margin for an item pair $(i,j)$ for user $u$ .",
"Let also define the probability distribution over the positive and negative items for $u$ , $\\mathbf {A}_u = \\lbrace \\alpha _u \\in \\mathbb {R}_+^m | \\sum _{i \\in \\mathcal {I}_u} \\alpha _{ui} = 1, \\sum _{i \\notin \\mathcal {I}_u} \\alpha _{ui} = 1\\rbrace .$ In [1] it is proposed an approach to maximize the minimum margin inspired by preference learning where the ranking task is posed as a two-player zero-sum game.",
"Let $P_{\\textit {max}}$ and $P_{\\textit {min}}$ be the players: on each round of the game, $P_{\\textit {min}}$ picks a preference $i \\prec _u j$ and, simultaneously, $P_{\\textit {max}}$ picks an hypothesis $\\mathbf {w}_u$ with the aim of maximizing the margin $\\rho (i \\prec _u j)$ .",
"The value of the game, i.e., the expected margin, is computed by: $\\mathbb {E}_\\alpha [\\rho ] = \\frac{1}{2} \\mathbf {w}_u^\\top \\mathbf {X} \\mathbf {Y}_u \\alpha _u,$ where $\\mathbf {Y}_u$ is a diagonal matrix, $\\mathbf {Y}_u = \\textit {diag}(\\mathbf {y}_u)$ , such that $y_{ui} = 1$ if $i \\in \\mathcal {I}_u$ , $-1$ otherwise.",
"It can be demonstrated that the $\\mathbf {w}^*_u$ maximizing the expected margin is equal to $\\mathbf {w}_u^* = \\mathbf {X}\\mathbf {Y}_u\\mathbf {\\alpha }_u$ normalized.",
"Finally, the best strategy for $P_{\\textit {min}}$ can be expressed as a convex quadratic optimization problem: $\\alpha _u^* = \\operatornamewithlimits{argmin}\\limits _{\\alpha _u \\in \\mathbf {A}_u} \\;\\;\\alpha _u^\\top \\left( \\mathbf {Y}_u \\mathbf {X}^\\top \\mathbf {X} \\mathbf {Y}_u + \\Lambda \\right) \\alpha _u,$ in which $\\Lambda $ is a diagonal matrix such that $\\Lambda _{ii} = \\lambda _p$ if $i \\in \\mathcal {I}_u$ , otherwise $\\Lambda _{ii} = \\lambda _n$ , where $\\lambda _p$ and $\\lambda _n$ are regularization parameters $(\\lambda _p, \\lambda _n \\ge 0)$ .",
"Although this algorithm has shown state-of-the-art results in terms of AUC, it is not suitable to deal with large datasets.",
"In fact, let assume that each optimization problem can be solved by an algorithm with a complexity quadratic on the number of parameters.",
"Then the global complexity would be $O(n_{\\textit {ts}} m^2)$ , where $n_{\\textit {ts}}$ is the number of users in the test set, and for the MSD it would be $O(10^{19})$ ."
],
[
"Efficient CF-OMD",
"The main issue of CF-OMD, in terms of efficiency, is the number of parameters which is equal to the cardinality of the item set.",
"Analyzing the results reported in [1], we noticed that high values of $\\lambda _n$ did not particularly affect the results, because it tends to flatten the contribution of the ambiguous negative feedbacks toward the average, mitigating the relevance of noisy information.",
"In CF contexts the data sparsity is particularly high, this means, on average, that the number of ambiguous negative feedbacks is orders of magnitude greater than the number of positive feedbacks.",
"Formally, given a user $u$ , let $m_u^+ = |\\mathcal {I}_u|$ and $m_u^- = |\\mathcal {I} \\setminus \\mathcal {I}_u|$ then $m = m_u^- + m_u^+$ , where $m_u^+ \\ll m_u^-$ , and generally $O(m) = O(m_u^-)$ .",
"On the basis of this observation, we can simplify the optimization problem (REF ), by fixing $\\lambda _n = +\\infty $ , which means that $\\forall i \\notin \\mathcal {I}_u, \\alpha _{ui} = 1/m_u^-$ : $\\alpha _u^* &= \\operatornamewithlimits{argmin}\\limits _{\\alpha _u} \\: \\Vert \\alpha _{u^+}^\\top \\mathbf {X}_{u^+} - \\mu _u^-\\Vert ^2 + \\lambda _p \\Vert \\alpha _{u^+}\\Vert ^2 \\\\&= \\operatornamewithlimits{argmin}\\limits _{\\alpha _u} \\: \\Vert \\alpha _{u^+}^\\top \\mathbf {X}_{u^+} \\Vert ^2 - \\Vert \\mu _u^-\\Vert ^2 -2\\alpha ^\\top _{u^+}\\mathbf {X}_{u^+}^\\top \\mu _{u}^- + \\lambda _p \\Vert \\alpha _{u^+}\\Vert ^2 \\\\&= \\operatornamewithlimits{argmin}\\limits _{\\alpha _{u^+} \\in \\mathbf {A}_u} \\;\\;\\alpha _{u^+}^\\top \\mathbf {X}_{u^+}^\\top \\mathbf {X}_{u^+} \\alpha _{u^+} + \\lambda _p\\Vert \\alpha _{u^+}\\Vert ^2 - 2\\alpha ^\\top _{u^+}\\mathbf {X}_{u^+}^\\top \\mu _{u}^-, $ where $\\mu _{u}^- = \\frac{1}{m_u^-} \\sum _{i \\notin \\mathcal {I}_u} \\mathbf {x}_i$ is the centroid of the convex hull spanned by the negative items and $\\alpha _{u^+}$ are the probabilities associated with the positive items, $\\mathbf {X}_{u^+}$ is the sub-matrix of $\\mathbf {X}$ containing only the columns corresponding to the positive items.",
"The number of parameters in (REF ) is $m_u^+$ and hence the complexity from $O(n_{\\textit {ts}} m^2)$ drops to $O(n_\\textit {ts} {\\overline{m}_u^+}^2)$ , where $\\overline{m}_u^+ = \\mathbb {E}[|\\mathcal {I}_u|]$ is the expected cardinality of the positive item set.",
"In MSD $\\overline{m}_u^+ \\approx 47.46$ which leads to a complexity $O(10^8)$ ."
],
[
"Implementation trick",
"Notwithstanding the huge improvement in terms of complexity, a naïve implementation would have an additional cost due to the calculation of $\\mu _{u}^-$ .",
"For all users in the test set the cost would be $O(n_{\\textit {ts}} n \\overline{m}_u^-)$ , where $\\overline{m}_u^- = \\mathbb {E}[|\\mathcal {I} \\setminus \\mathcal {I}_u|]$ , and it can be approximated with $O(n_{\\textit {ts}}nm)$ .",
"To overcome this bottleneck, we propose an efficient incremental way of calculating $\\mu _{u}^-$ .",
"Consider the mean over all items $\\mu = \\frac{1}{m} \\sum _{i \\in \\mathcal {I}} \\mathbf {x}_i,$ then, for a given user $u$ , we can express $\\mu _{u}^- = \\frac{1}{m_u^-} \\left( m \\cdot \\mu - \\sum _{i \\in \\mathcal {I}_u} \\mathbf {x}_i \\right).$ From a computational point of view, it is sufficient to compute the sum $\\sum _{i \\in \\mathcal {I}} \\mathbf {x}_i$ once (i.e., $m \\cdot \\mu $ ) and then, for every $\\mu _u^-$ , subtract the sum of the positive items.",
"Using this simple trick, the overall complexity drops to $O(nm) + O(n_{\\textit {ts}}^2\\overline{m}_u^+)$ .",
"In the experimental section we successfully applied this algorithm to the MSD achieving competitive results against the state-of-the-art method but with higher efficiency."
],
[
"Kernelized CF-OMD",
"The method proposed in Section REF , can be seen as a particular case of a kernel method.",
"In fact, $\\mathbf {X}_{u^+}^\\top \\mathbf {X}_{u^+}$ is a kernel matrix, let call it $\\mathbf {K}_{u^+}$ with the corresponding (linear) kernel function $K: \\mathbb {R}^{m_u^+} \\times \\mathbb {R}^{m_u^+} \\rightarrow \\mathbb {R}$ .",
"Given $K$ we can reformulate (REF ) as: $\\alpha _{u^+}^* = \\operatornamewithlimits{argmin}\\limits _{\\alpha _{u^+} \\in \\mathbf {A}_u} \\;\\;\\alpha _{u^+}^\\top \\mathbf {K}_{u^+} \\alpha _{u^+} + \\lambda _p\\Vert \\alpha _{u^+}\\Vert ^2 - 2\\alpha _{u^+}^\\top \\mathbf {q}_u,$ where elements of the vector $\\mathbf {q}_u \\in \\mathbb {R}^{{m_u^+}}$ are defined as $\\quad q_{ui} = \\frac{1}{m_u^-} \\sum _{j \\notin \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j).$ Actually, inside the optimization problem (REF ) we can plug any kernel function.",
"Throughout the paper we will refer to this method as CF-KOMD.",
"Generally speaking, the application of kernel methods on a huge dataset have an intractable computational complexity.",
"Without any shrewdness the proposed method would not be applicable because of the computational cost of the kernel matrix and $\\mathbf {q}_u$ .",
"An important observation is that the complexity is strictly connected with the sparsity of the kernel matrix which is, unfortunately, commonly dense.",
"However, we can leverage on a well known result from harmonic theory [2] to keep the kernel as sparse as possible without changing the solution of CF-KOMD.",
"Theorem 1 A function $f:\\mathbb {R}\\rightarrow \\mathbb {R}$ defines a positive definite kernel $k : \\mathbf {B}(0, 1) \\times \\mathbf {B}(0,1)$ as $k: (\\mathbf {x}, \\mathbf {y}) \\mapsto f(\\mathbf {x}\\cdot \\mathbf {y})$ iff $f$ is an analytic function admitting a Maclaurin expansion with non-negative coefficients, $f(x) = \\sum _{s=0}^\\infty a_s x^s, a_s \\ge 0$ .",
"As emphasized in [2], [24], many kernels used in practice [25] satisfy the above-mentioned condition.",
"These kernels are called dot product kernels because they are defined as a function of the dot product of the input vectors.",
"Table REF gives some example of these kind of kernels.",
"Table: Some examples of dot product kernels.We can observe that the kernel matrices induced by these kernels are, in general, dense due to the zero degree term (i.e., $s = 0$ ) which is a constant added to all the entries.",
"Adding a constant to a whole matrix means a space translation and we can demonstrate that this operation does not affect the margin in CF-KOMD (this is valid also in the generic formulation of CF-OMD).",
"Let $\\mathbf {K} = \\mathbf {K}_0 + \\hat{\\mathbf {K}}$ be a dot product kernel matrix where $\\mathbf {K}_0$ is the constant matrix induced by the 0 degree term of the MacLaurin expansion (i.e.",
"$s = 0$ ).",
"Let also $\\mathbf {q}_u$ be consequently defined as: $\\quad q_{ui} &= \\frac{1}{m_u^-} \\sum _{j \\notin \\mathcal {I}_u} \\left( K_0(\\mathbf {x}_i, \\mathbf {x}_j) + \\hat{K}(\\mathbf {x}_i, \\mathbf {x}_j) \\right) \\\\&= \\frac{1}{m_u^-} \\left[ m_u^- \\cdot k_0 + \\sum _{j \\notin \\mathcal {I}_u} \\hat{K}(\\mathbf {x}_i, \\mathbf {x}_j) \\right] \\\\&= k_0 + \\hat{q}_{ui} \\Rightarrow \\mathbf {q}_u = \\mathbf {k}_0 + \\hat{\\mathbf {q}}_u.$ We can rewrite the optimization problem (REF ) as (we omit the $u^+$ subscription for brevity): $\\alpha ^* &= \\operatornamewithlimits{argmin}\\limits _{\\alpha \\in \\mathbf {A}_u} \\;\\;\\alpha ^\\top (\\mathbf {K}_0 + \\hat{\\mathbf {K}}) \\alpha + \\lambda _p\\Vert \\alpha \\Vert ^2 - 2\\alpha ^\\top (\\mathbf {k}_0 + \\hat{\\mathbf {q}}_u) \\\\&= \\operatornamewithlimits{argmin}\\limits _{\\alpha \\in \\mathbf {A}_u} \\;\\;\\alpha ^\\top \\mathbf {K}_0 \\alpha + \\alpha ^\\top \\hat{\\mathbf {K}} \\alpha + \\lambda _p\\Vert \\alpha \\Vert ^2 - 2\\alpha ^\\top \\mathbf {k}_0 - 2\\alpha ^\\top \\hat{\\mathbf {q}}_u$ where both $\\alpha ^\\top \\mathbf {K}_0 \\alpha $ and $-2\\alpha ^\\top \\mathbf {k}_0$ are constant values independent from $\\alpha $ : $\\alpha ^\\top \\mathbf {K}_0 \\alpha = k_0\\sum \\limits _{i\\in \\mathcal {I}_u} \\sum \\limits _{j\\in \\mathcal {I}_u} \\alpha _i \\alpha _j = k_0\\sum \\limits _{i\\in \\mathcal {I}_u} \\alpha _i \\sum \\limits _{j\\in \\mathcal {I}_u} \\alpha _j = k_0;$ $-2\\alpha ^\\top \\mathbf {k}_0 = -2\\sum \\limits _{i \\in \\mathcal {I}_u} k_0 \\alpha _i = -2k_0 \\sum \\limits _{i\\in \\mathcal {I}_u} \\alpha _i = -2k_0;$ and hence the solution of the optimization problem does not depend on $\\mathbf {K}_0$ : $\\alpha ^* = \\operatornamewithlimits{argmin}\\limits _{\\alpha \\in \\mathbf {A}_u} \\;\\;\\alpha ^\\top \\hat{\\mathbf {K}} \\alpha + \\lambda _p\\Vert \\alpha \\Vert ^2 - 2\\alpha ^\\top \\hat{\\mathbf {q}}_u.$ and it is the same as in (REF ).",
"For this reason we can “sparsify” these kernels (we will call theme RDP Kernels: Reduced Dot Product Kernels) by removing the zero degree factor obtaining kernel matrices whose sparsities depend only on the distribution of the input data since they are defined as linear combination of powers of dot products.",
"This also implies that the sparsity of a RDP kernel is exactly the same as in the simple linear kernel (i.e., $\\mathbf {K} = \\mathbf {X}^\\top \\mathbf {X}$ )."
],
[
"Sparsity and long tail distribution",
"As mentioned in Section REF , CF datasets are, in most of the cases, very sparse and in general the distribution of the ratings has a long tail form from the items perspective [26].",
"This means that a small set of items, the most popular ones, receive great part of the whole set of ratings.",
"What we need to understand are the conditions under which a RDP kernel remains sufficiently sparse.",
"To study this phenomenon we use the linear kernel, but results also apply for every other RDP kernel as well because they contain exactly the same zero entries as the linear one (which is in fact a special case of RDP kernel).",
"Let $\\mathbf {K} = \\mathbf {X}^\\top \\mathbf {X}$ ($\\mathbf {X} \\in \\mathbb {R}^{n\\times m}$ ) be a kernel matrix and let $\\mathbb {P}(K_{ij} \\ne 0)$ be the probability that the entry $K_{ij}$ is not zero.",
"Given an a-priori probability distribution over the ratings, and assuming the independence of the ratings, we can estimate the probability of having a value different from zero in the kernel matrix with: $\\mathbb {P}(K_{ij} \\ne 0) &= 1 - \\mathbb {P}(K_{ij} = 0) \\\\&= 1 - \\prod \\limits _h \\mathbb {P}(x_{ih}\\cdot x_{jh} = 0) \\\\&= 1 - \\prod \\limits _h (1 - \\mathbb {P}(x_{ih}\\cdot x_{jh} \\ne 0)) \\\\&= 1 - \\prod \\limits _h (1 - \\mathbb {P}(x_{ih} \\ne 0)\\mathbb {P}(x_{jh} \\ne 0)) \\\\&= 1 - (1 - \\mathbb {P}(x_{ih}\\ne 0) \\cdot \\mathbb {P}(x_{jh}\\ne 0))^n$ where $\\mathbb {P}(x_{ih}\\ne 0)$ and $\\mathbb {P}(x_{jh}\\ne 0)$ are the probability of having a non zero entry in the rating matrix.",
"It is worth to notice that this probability, in the uniform case, is actually the density of the matrix.",
"However, it does not take into account the fact that all the elements in the diagonal of the kernel are for sure non zero.",
"With this consideration in mind we can define an estimate of the kernel density $d(\\mathbf {K})$ , with $\\mathbf {K} \\in \\mathbb {R}^{m\\times m}$ , as follows: $d(\\mathbf {K}) = \\frac{1}{m^2} \\left[m + (m^2-m)\\mathbb {P}(K_{ij} \\ne 0) \\right].$ Intuitively, we can argue that anytime both $\\mathbf {x}_i$ and $\\mathbf {x}_j$ are popular items, i.e., $\\mathbb {P}(x_{ih}\\ne 0)$ and $\\mathbb {P}(x_{jh}\\ne 0)$ are close to 1, then $\\mathbb {P}(K_{ij} \\ne 0)$ tends to be high and hence $\\mathbf {K}$ is likely to be dense.",
"On the contrary, when one of the two vectors represents an unpopular item, then the probability $\\mathbb {P}(K_{ij} \\ne 0)$ is likely close to zero.",
"Since we are assuming a long tail distribution over the items, most of the kernel entries result from a dot product of two unpopular items with very few users that rated them and so the kernel tends to be sparse.",
"However, up to now, we are assuming a uniform distribution over the users and in real datasets this is often not the case.",
"Any other probability distribution would highly affect our estimation $d(\\mathbf {K})$ .",
"This is because, if we assume a long tail distribution over the users, the probability of having at least one user in common between two items would be generally high, since the ratings for an item are likely to be concentrated to the (few) most active users.",
"Considering that a mathematical proof of this intuition is quite complicated, in the next section we provide an empirical analysis on real CF datasets.",
"In order to validate our thesis, we empirically analyze a set of famous CF datasets comparing the theoretical sparsity with uniform ratings distribution with the sparsity of the linear kernel.",
"We expect that the long tail distribution of the ratings will tend to lower the likelihood of having a non zero value in the kernel matrix, with respect to the $d(\\mathbf {K})$ estimate.",
"The empirical analysis has been performed as follows: for each dataset, we build the corresponding rating matrix $\\mathbf {R}$ , we calculate the expected density using (REF ) by fixing $\\mathbb {P}(x_{ih} \\ne 0)$ equals to the density of $\\mathbf {R}$ ; we calculate the linear kernel $\\mathbf {K}=\\mathbf {R}^\\top \\mathbf {R}$ and finally we compare the sparsity of the kernel with $d(\\mathbf {K})$ .",
"Table REF summarizes the results.",
"Table: Analysis of the sparsity of the linear kernel.Although our intuition seems to work with most of the datasets, we can notice that with the Ciao and Book Crossing datasets the kernels are more dense than the estimation $d(\\mathbf {K})$ .",
"The reason why these datasets behave differently is clearly depicted in the plots REF - REF .",
"Every pair of plots show the distribution of the item popularity and the user activity.",
"The plots have a loglog scale and the blue line represents the best fitting power low function.",
"The fitting has been made using the least square method.",
"Figure: Book Crossing.",
"The plots are in loglog scale.Figure: Ciao.",
"The plots are in loglog scale.Figure: Delicious.",
"The plots are in loglog scale.Figure: Film Trust.",
"The plots are in loglog scale.Figure: LastFM.",
"The plots are in loglog scale.Figure: Movielens.",
"The plots are in loglog scale.Figure: Netflix.",
"The plots are in loglog scale.From the plots we can observe that: none of the item distributions follows exactly a power low, especially in the head of the distribution; datasets with a very dense kernel tend to have shorter tails: the right part of the plot exceed the power low line; generally items are long tailed while users tend to be more uniform; users distributions are in general not well fitted by a power law, with the exception of Ciao and Book Crossing datasets.",
"The observations listed above point out that Ciao and Book Crossing are the only two datasets with a well defined long tail distribution over both users and items.",
"This confirm our intuition about the likelihood of having a denser kernel with both long tailed distributions.",
"In conclusion, the long tail distribution over the items keeps the RDP kernels sparse while a long tail distribution over the users increase the density."
],
[
"Approximation of $\\mathbf {q}_u$",
"Using the RDP kernels, we can further optimize the complexity by providing a good approximation of $\\mathbf {q}_u$ that can be computed only once, instead of $n_{\\textit {ts}}$ times.",
"The idea consists in replacing every $q_{ui}$ with an estimate of $\\mathbb {E}[K(\\mathbf {x}_i, \\mathbf {x})]$ which is the expected value of the kernel between the item $i$ and every other items.",
"Formally, consider, without any loss of generality, a normalized kernel function $K$ and let the approximation of $\\mathbf {q}_u$ be $\\tilde{\\mathbf {q}}$ such that: $\\tilde{q}_i = \\frac{1}{m} \\sum _{j \\in \\mathcal {I}} K(\\mathbf {x}_i, \\mathbf {x}_j).$ At each component of $\\hat{\\mathbf {q}}$ , the approximation error is bounded by $\\frac{2m_u^+}{m}$ , which is linear on the sparsity of the dataset.",
"$|\\hat{q}_i - q_{ui}| &= \\left| \\frac{1}{m} \\sum \\limits _{j \\in \\mathcal {I}} K(\\mathbf {x}_i, \\mathbf {x}_j) - \\frac{1}{m_u^-} \\sum \\limits _{j \\notin \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) \\right| \\\\&= \\left| \\frac{1}{m} \\left[ \\sum \\limits _{j \\in \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) + \\sum \\limits _{j \\notin \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) \\right] - \\frac{1}{m_u^-} \\sum \\limits _{j \\notin \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) \\right| \\\\&= \\left| \\frac{1}{m} \\sum \\limits _{j \\in \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) - \\frac{m - m_u^-}{m \\cdot m_u^-} \\sum \\limits _{j \\notin \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) \\right| \\\\&\\le \\left| \\frac{1}{m} \\sum \\limits _{j \\in \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) \\right| + \\left| \\frac{m - m_u^-}{m \\cdot m_u^-} \\sum \\limits _{j \\notin \\mathcal {I}_u} K(\\mathbf {x}_i, \\mathbf {x}_j) \\right| \\\\&\\le \\left| \\frac{m_u^+}{m} \\right| + \\left| \\frac{m-m_u^-}{m \\cdot m_u^-}m_u^- \\right| \\le \\frac{m_u^+ + m - m_u^-}{m} = 2\\frac{m_u^+}{m}.$"
],
[
"Experiments and Results",
"Experiments have been performed comparing the proposed methods against the state-of-the-art method on MSD (MSDW) with respect to the ranking quality and computational performance.",
"We also compared our framework, in terms of AUC, against other state-of-the-art methods on top-N recommendation with implicit feedback, namely WRMF and BPR.",
"Our framework and MSDW are both implemented in PythonWe used CVXOPT package to solve the optimization problemThe MSDW implementation is available at http://www.math.unipd.it/ aiolli/CODE/MSD/The framework implementation is available at https://github.com/makgyver/pyros, while for WRMF and BPR we used the java implementation provided by the open source LibRec library http://www.librec.net/."
],
[
"Datasets",
"In this section we introduce the datasets used in the experiments.",
"Table REF shows a brief description of the datasets.",
"Table: Brief description of the used datasets.The MSD dataset is used only to demonstrate the applicability of our kernel method to huge datasets.",
"All the other datasets are used to compare our framework with the state-of-the-art method in top-N recommendation with implicit feedback."
],
[
"Experimental setting",
"Experiments have been performed 5 times for each dataset.",
"Datasets have been pre-processed as described in the following: we split randomly the users in 5 sets of the same dimension; for each user in a set we further split its ratings in two halves; at each round test, we use all the ratings in 4 sets of users plus the first half of ratings of the remaining set as training set, and the rest as test set.",
"This setting avoids situations of cold start for users, because in the training phase we have at least a rating for every user.",
"We also force users with less than 5 ratings to be in the training set.",
"The results reported below are the averages (with its standard deviations) over the 5 folds."
],
[
"Evaluation measures",
"The rankings' evaluation metric used to compare the performances of the methods is the AUC (Area Under the receiver operating characteristic Curve) defined as in the following: $\\textit {AUC} = \\frac{1}{|\\mathcal {U}|} \\sum \\limits _{u\\in \\mathcal {U}} \\frac{1}{|\\mathcal {I}_u| \\cdot |\\mathcal {I}\\setminus \\mathcal {I}_u|} \\sum \\limits _{i \\in \\mathcal {I}_u} \\sum \\limits _{j \\notin \\mathcal {I}_u} \\mathbb {I}[\\hat{r}_{ui} > \\hat{r}_{uj}]$ where $\\mathbb {I} : \\textit {Bool} \\rightarrow \\lbrace 0,1\\rbrace $ is the indicator function which returns 1 if the predicates is true 0 otherwise.",
"In the experiments which use the MSD dataset we also compared the algorithms using the same metric as in the challenge, that is the Mean Average Precision at 500 (mAP@500), which is the mean over all users of the average precision at $N$ ($N=500$ ).",
"Formally is defined as: $AP(\\pi _u)@N = \\frac{1}{\\min (|\\mathcal {I}_u|, N)} \\sum \\limits _{k=1}^{N} P(\\pi _u)@k \\cdot \\hat{r}_{u\\pi _u(k)},$ where $\\mathcal {I}_u$ is the set of positive associated items with the user $u$ , $\\pi _u$ is the items ranking for user $u$ , such that $\\pi _u(k) = i$ means that item $i$ is ranked at position $k$ , and $P(\\pi _u)@k$ is the precision at $k$ : $P(\\pi _u)@k = \\frac{1}{k} \\sum \\limits _{p=1}^k \\hat{r}_{u\\pi _u(p)}.$"
],
[
"MSD",
"We used MSD as described in the Kaggle challengehttps://www.kaggle.com/c/msdchallenge: the training set is composed by 1M users (plus 10K users as validation set) with all their listening history and for the rest (i.e., 100K users) only the first half of the history is provided, while the other half constitutes the test set.",
"In these experiments we fixed the $\\lambda _p$ parameter to 0.01.",
"Results are presented in Table REF .",
"In this case MSDW maintains its record performance in terms of mAP@500, while for the AUC all methods have very good results.",
"This underline the fact that both ECF-OMD and CF-KOMD (polynomial kernel with $c=1$ ) try to optimize the AUC rather than the mAP.",
"Table: Ranking accuracy on MSD using AUC and [email protected] computational costs on this dataset are reported in Figure REF .",
"Figure: Average computational time in hours for 1K users.The results are the average computing time over 1K test users.",
"All methods run on a machine with 150Gb of RAM and 2 x Eight-Core Intel(R) Xeon(R) CPU E5-2680 0 @ 2.70GHz.",
"Actually the times in Figure REF have a constant overhead due to read operations.",
"Results show that ECF-OMD and CF-K (abbreviation for CF-KOMD) are almost 5 time faster than MSDW even though they require more RAM to store the kernel matrix.",
"It is worth to notice that CF-K has a computational time very close to ECF-OMD, and this highlights the positive effects of the complexity optimization presented in this paper."
],
[
"Other datasets",
"This section shows the performance for the top-N recommendation task, in terms of AUC, achieved by our framework.",
"We compared our methods with some state-of-the-art techniques.",
"In particular, we used the following settings for each of the competing algorithm: ECF-OMD : we fixed the regularization parameter $\\lambda _p = 0.01$ ; CF-K$_P$ : it is the CF-KOMD method with the polynomial kernel.",
"We tested different values for $c \\in \\lbrace 0.5, 1, 2, 4\\rbrace $ and we fixed $\\lambda _p = 0.01$ ; CF-K$_T$ : it is the CF-KOMD method with the tanimoto kernel.",
"We fixed the regularization parameter $\\lambda _p = 0.01$ ; MSDW : tests have been performed varying the value of the $\\alpha $ parameter in the real range [0,1] with a step of 0.25; WRMF : we reported only the performance achieved with $\\alpha =1$ since the results obtained with different $\\alpha $ were substantially the same.",
"We fixed the maximum number of iteration to 30, the learning rate $\\rho =0.001$ , the regularization term $\\lambda =0.001$ and the number of factor $k=100$ ; BPR : we do not have free parameter.",
"We fixed, as in WRMF, the maximum number of iteration to 30, the learning rate $\\rho =0.001$ and the regularization term $\\lambda =0.001$ .",
"Table REF summarizes the obtained results.",
"Table: AUC results of our framework against state-of-the-art methods.On the MovieLens dataset, methods of our framework have significant better performance against all the other, achieving an AUC of 0.896 with CF-KOMD with the polynomial kernel ($c=4$ ).",
"CF-KOMD has the higher AUC (0.964) also in the FilmTrust dataset, but this time with the Tanimoto kernel.",
"In this dataset, anyway, the polynomial achieved result comparable with ECF-OMD.",
"Good results are also achieved by the MSDW.",
"On the Netflix dataset, the best AUC is 0.943 by ECF-OMD.",
"We got similar result with the Tanimoto kernel.",
"Surprisingly, with the Ciao dataset, MSDW obtained very good result while all the other approaches are quite behind,n particular, WRMF and BPR have a very poor performances.",
"We also compared the execution time of the algorithm.",
"All these experiments has been made on a MacBook Pro late 2012 with 16GB of RAM and CPU Intel(R) Core i7 @ 2.70GHz.",
"The results are shown in Figure REF .",
"Figure: Execution time took by the tested methods.The first thing we can notice is that the WRMF algorithm is always the most time consuming one.",
"To keep the plot readable, we cut the bars longer than 100 minutes.",
"Actually, in both Netflix and Ciao datasets WRMF took more than 7 hours (i.e., 420 minutes).",
"The two kernel based methods, CF-K$_P$ and CF-K$_T$ took almost the same time and they are a little bit slower than the linear (ECF-OMD) method which is often the fastest one.",
"MSDW is in general quite fast, but it seems to suffer when the number of items increase (e.g., Ciao dataset).",
"These results show how our framework have, both in efficacy and efficiency, performances at the state-of-the-art."
],
[
"Conclusions",
"In this paper we have proposed a collaborative filtering kernel-based method for the top-N recommendation.",
"The method belongs to a more general framework, inspired by preference learning and designed to explicitly maximize the AUC.",
"We have also proposed a strategy for the “sparsification” of the dot product kernels and in which conditions this strategy works.",
"Our analysis, conducted over CF datasets, have shown the effect of the long tail distribution on the sparsity of the kernel.",
"Since this kind of distribution is very common, our results can apply in many domains other than CF.",
"Finally, the experiments we reported have shown that the proposed kernel-based method achieve good result in terms of AUC and it is also efficient even with large scale datasets."
],
[
"Acknowledge",
"This work was supported by the University of Padova under the strategic project BIOINFOGEN."
]
] | 1612.05729 |
[
[
"Density-functional calculations of transport properties in the\n non-degenerate limit and the role of electron-electron scattering"
],
[
"Abstract We compute electrical and thermal conductivities of hydrogen plasmas in the non-degenerate regime using Kohn-Sham Density Functional Theory (DFT) and an application of the Kubo-Greenwood response formula, and demonstrate that for thermal conductivity, the mean-field treatment of the electron-electron (e-e) interaction therein is insufficient to reproduce the weak-coupling limit obtained by plasma kinetic theories.",
"An explicit e-e scattering correction to the DFT is posited by appealing to Matthiessen's Rule and the results of our computations of conductivities with the quantum Lenard-Balescu (QLB) equation.",
"Further motivation of our correction is provided by an argument arising from the Zubarev quantum kinetic theory approach.",
"Significant emphasis is placed on our efforts to produce properly converged results for plasma transport using Kohn-Sham DFT, so that an accurate assessment of the importance and efficacy of our e-e scattering corrections to the thermal conductivity can be made."
],
[
"Introduction",
"There has been a rapid increase of publications over the past fifteen years on the computation of electrical and thermal conductivities for warm dense matter (i.e., from warm liquids to hot dense plasmas) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] using Kohn-Sham DFT [11], [12].",
"In these studies, molecular dynamics (MD) simulations are first performed for classical ions moving in the force fields provided by the self-consistently determined electron density within the Born-Oppenheimer approximation.",
"The resulting thermally occupied Kohn-Sham states from individual ionic snapshots are then inserted into Kubo-Greenwood [13], [14] formulas to calculate the appropriate current-current correlation functions.",
"Finally, the results from different uncorrelated snapshots are averaged together and electrical ($\\sigma $ ) and thermal ($\\kappa $ ) conductivities are obtained.",
"Because the temperatures are high enough so that many electrons are free to conduct, and thermal electrons move so much faster than thermal ions, $\\sigma $ and $\\kappa $ for such systems are governed entirely by the behavior of the electron currents: the charge current ${\\bf j}_{e}$ , for $\\sigma $ , and the heat current ${\\bf j}_{Q}$ , for $\\kappa $ .",
"The calculations then amount to a determination of the degradation of these currents resulting from the interactions of the current-carrying electrons with the rest of the plasma, leading to resistance.",
"The advantage of using a DFT-based approach for dense plasmas is that it is unnecessary to decide a priori which electrons are “bound\" and which are “free\", as the degree of localization of a given single-electron state is determined in the course of solving the effective mean-field Schrödinger-like equation.",
"However, there is also a disadvantage: The electron-electron interaction is treated in a manner in which the electrons are considered as an aggregate, through their total charge density, rather than individually.",
"This is in sharp contrast to kinetic theory approaches such as the Boltzmann equation, in which explicit encounters between individual particles are considered in the collision terms.",
"With the exception of DFT's use of an exchange-correlation potential (which itself depends only on the total electron charge density), the treatment of the e-e interaction is essentially equivalent to that in the Vlasov equation; explicit e-e collisions are absent.",
"The classic plasma kinetic theory for $\\sigma $ and $\\kappa $ is that of Spitzer and Härm [15], in which a Fokker-Planck equation is solved to determine the steady-state electron velocity distribution resulting from the application of ${\\bf E}$ or ${\\nabla } T$ , from which ${\\bf j}_{e}$ and ${\\bf j}_{Q}$ are calculated.",
"The collisions are treated with Coulomb logarithms [16], $\\log \\lambda _{ei}$ and $\\log \\lambda _{ee}$ , which account for screening of the two-body interactions in a manner suited to the limit of weak plasma coupling (e.g., small-angle scattering).",
"Yet a more sophisticated kinetic theory approach is that of the Lenard-Balescu (LB) equation [17], in which the bare Coulomb collisions are dressed by the multicomponent wave vector and frequency dependent dielectric function; while resulting in answers identical to that of a Fokker-Planck equation with appropriately chosen Coulomb logarithms for sufficiently weak coupling, LB constitutes a predictive theory in which arbitrary distributions and particle species can be considered.",
"Conductivities of quantum plasmas as predicted by QLB are available [18], [19], [20], and comparisons between LB predictions for classical plasmas and the results of classical MD have proven very favorable for comparable regimes of plasma coupling [21], [22].",
"However unlike in the DFT treatment, only free electrons are considered, and therefore the bound versus free distinction must be made at the outset when studying real plasmas.",
"A method which attempts to combine some of the positive features of the DFT-MD and kinetic theory approaches (though predating the former) is the average-atom prescription, exhibited generally in the Ziman resistivity formula [23].",
"Here, various means (including DFT) are used to compute the interaction between an electron and a representative ion, together with its surrounding screening cloud of other electrons.",
"This interaction is then used in scattering theories of varying sophistication to produce the electronic contributions to $\\sigma $ and $\\kappa $ [24], [25], [26], [27], [28], once a statistical distribution of ionic positions is assumed.",
"If the treatment of the representative ion is sufficiently detailed, bound and free electrons can be treated on similar footing.",
"However the method only treats the electron-ion scattering; as in the Kohn-Sham DFT, e-e scattering is not included.",
"This is not a serious restriction for low-$T$ liquid metals (for which the original Ziman work was intended [23]), since the imposition of electron degeneracy, and the associated Pauli blocking, suppresses the effects of e-e scattering on the electron distribution function [29].",
"But it produces results which are in significant disagreement with, for instance, the Spitzer-Härm theory [15], particularly for high-$T$ and for low-$Z$ ions, since the effects of the e-e interaction are not outweighed by those of e-i [30].",
"As such, it is customary to employ separate multiplicative “Lorentz gas\" [31] corrections to $\\sigma $ and $\\kappa $ as determined from average-atom theories, which ensure that the final results agree with the weak-coupling limits of plasma kinetic theory [32], [25], [26].",
"These corrections reduce $\\sigma $ and $\\kappa $ at high-$T$ , relative to their values as predicted by theories in which an e-e collision term is absent, such as in average-atom descriptions and in specialized Lenard-Balescu treatments focused primarily on degenerate electrons [33].",
"Ziman resistivity formula results which are fit to expressions employing Coulomb logarithms, together with $T$ -dependent corrections accounting for e-e interaction, form the basis for many wide-ranged models for plasma conduction [34], [35], [36] used in continuum simulations of astrophysical objects [37], [38], [39], inertial confinement fusion (ICF) [40], and pulsed power applications [41].",
"Indeed, the importance of these applications for dense plasmas has fueled several of the DFT-based investigations mentioned at the outset [4], [5], [6], [7], [9], [10].",
"As such, these DFT works featured comparisons to some of these models for $\\sigma $ and $\\kappa $ .",
"In Ref.",
"[6] for instance, comparisons were made to the high-$T$ limit of the Lorenz number (${\\kappa }/{\\sigma T}$ ) for hydrogen plasmas, as predicted by Spitzer-Härm [15].",
"Though reasonable agreement was found, it has since been established that this agreement was spurious, resulting from an incomplete convergence of the DFT calculation of $\\kappa $ with respect to the number of Kohn-Sham states included in the computation [42].",
"While it is certainly reasonable to use DFT-based approaches.",
"[5], [6], [7], [9], [10] to attempt to go beyond the many approximations inherent in more conventional plasma descriptions.",
"[15], [18], [19], [34], [35], [36], it is equally important to uncover potential weaknesses in the assumptions underlying current implementations of Kohn-Sham DFT for plasma conduction.",
"This in turn requires that these DFT-based predictions of $\\sigma $ and $\\kappa $ are well-converged.",
"In this work we use the Kohn-Sham DFT prescription, complete with DFT-based MD and the Kubo-Greenwood approach mentioned above, to predict $\\sigma $ and $\\kappa $ for hydrogen plasmas at sufficiently high-$T$ to make a meaningful comparison to the predictions of the quantum LB equation.",
"Plasma conditions are chosen to be $\\rho = $ 40 g/cm$^{3}$ and $T$ between 500 eV and 900 eV, to coincide with a previous study [22] of hydrogen using classical MD and statistical potentials [43], [44], where it was demonstrated that the weak-coupling assumptions underlying LB are valid.",
"These conditions offer the added advantage that the hydrogen atoms are fully ionized, removing a potential discrepancy between the two approaches.",
"We show that our DFT prediction for $\\sigma $ is in excellent agreement with that of LB, while our prediction of $\\kappa $ is far too high in this regime.",
"From this, we posit that there are two distinct contributions of the e-e interaction to plasma conduction: 1.",
"A mean-field reshaping of the electron distribution function which is present in the DFT (as well as in any theory containing a Vlasov or Hartree term), and 2.",
"A binary e-e scattering piece which is missing in our current implementation of DFT, but which is present in various plasma kinetic theories (Fokker-Planck, LB, etc.).",
"We argue that while the first contribution affects both $\\sigma $ and $\\kappa $ , the second contribution only plays a role for $\\kappa $ , due to the inability of binary e-e scattering to degrade the electron charge current in a system in which the conservation of total electron momentum is mandated.",
"By alternately turning off the e-e and e-i collision terms in the quantum LB equation, we demonstrate that an e-e scattering correction to the DFT thermal conductivity can be written in the form: $1/\\kappa = 1/\\kappa _{\\rm DFT} + 1/\\kappa ^{\\rm OCP}_{ee}$ , where $\\kappa ^{\\rm OCP}_{ee}$ is the thermal conductivity of the electron one-component plasma (OCP) as predicted by QLB.",
"We also find that the reshaping contributions to $\\sigma $ and $\\kappa $ are practically identical.",
"This general framework is further justified by appealing to arguments derived from a quantum Boltzmann theory using the Zubarev approach [45], [46], [47], [48].",
"Our conclusions extend and amplify those made in a recent work investigating the effect of e-e scattering on the electrical conductivity of plasmas [49].",
"The remainder of this paper is organized as follows: In Section II, we outline the specific methods we use to produce converged results for the conductivities of hydrogen plasmas at high-$T$ using Kohn-Sham DFT.",
"In Section III, we discuss the comparison of these DFT results to those of QLB, and construct our e-e scattering correction for $\\kappa $ ; additional motivation for this correction using a quantum Boltzmann approach is deferred to the Appendix.",
"We conclude in Section IV."
],
[
"The DFT methods",
"Density functional molecular dynamics (DFT-MD) simulations, performed with VASP [50], are used to generate atomic configurations for hydrogen with $\\rho = $ 40 g/cm$^{3}$ and $T$ of 500, 700, and 900 eV.",
"The electronic temperature is established through a Fermi occupation of the electronic states.",
"All calculations are performed in the Local Density Approximation [12], [51].",
"Given the very high densities being explored ($r_s = 0.41$ bohr), we employ a bare proton $1/r$ potential for the hydrogen atom.",
"The plane wave cutoff energy is set to 3800 eV, and the electronic density and single-particle wave functions are sampled at a single $\\bf k$ -point at the $\\Gamma $ -point (${\\bf k}= 0$ ) in the Brillouin zone corresponding to the supercell (see below).",
"The practical limit on our calculations proved to be 256 hydrogen atoms in a periodically-repeated cubic supercell.",
"Simulations with more atoms were intractable in combination with the very high temperatures and corresponding need for a very large number of bands for the transport properties and the very high plane wave cutoff energy.",
"In each of these three cases, the electrons are fully ionized from the hydrogen nuclei.",
"Correspondingly, we calculate the fundamental dimensionless plasma parameters for these three cases, namely the ion-ion coupling factor $\\Gamma _{ii} \\equiv e^2/(k_{B}TR_i) $ , where $R_i = (3/(4 \\pi n_i))^{1/3}$ is the Wigner-Seitz radius for the protons, and the electron degeneracy $\\theta \\equiv k_{B}T/E_{\\rm Fermi}$ as shown in Table REF .",
"Note that even for the highest temperature of 900 eV, where $\\theta \\sim 3$ , we expect some residual consequences of electron degeneracy.",
"Table: Dimensionless plasma parameters Γ ii \\Gamma _{ii} and θ\\theta for a fully ionized 40 g/cm 3 ^3 hydrogen plasma.Following the DFT-MD simulations, a set of atomic configurations, well separated in time, are selected for subsequent calculation of the transport properties following the treatment in Ref. [7].",
"Achieving convergence on the electrical and thermal conductivities for these high density, high temperature systems requires a very large number of bands, from several thousand to well in excess of ten thousand.",
"A simple one shot calculation with such high band numbers is unfeasible owing to poor convergence during the self-consistent determination of the electronic density and Kohn-Sham wave functions.",
"We resort to a stepwise approach building up successively more bands by doing a sequence of calculations with increasing band numbers and using prior runs to initialize the simulation.",
"A consequence of the high band numbers is the need to continue increasing the plane wave cutoff energy as there must be plane waves of sufficient energy to represent the highest bands.",
"The combination of these two requirements leads to very poor scaling as the temperature is increased.",
"Using the Kubo relation [13], [14] for the current-current correlation functions ($\\langle {\\bf j}_{e}{\\bf j}_{e}\\rangle $ for $L_{11}$ , $\\langle {\\bf j}_{Q}{\\bf j}_{Q}\\rangle $ for $L_{22}$ , etc.",
"), one obtains [7] $L_{ij}(\\omega ) & = & \\frac{2\\pi (-1)^{i+j}}{3Vm^{2}\\omega }\\sum _{{\\bf k}\\nu \\mu } \\left( f_{{\\bf k}\\nu } - f_{{\\bf k}\\mu } \\right)\\left|\\langle {\\bf k}\\mu |\\hat{\\bf p}|{\\bf k}\\nu \\rangle \\right|^2 \\cr & \\times & \\epsilon _{{\\bf k} \\nu \\mu }^{i + j -2} \\delta (E_{{\\bf k}\\mu } - E_{{\\bf k}\\nu } - \\hbar \\omega ),$ where $i$ and $j$ are labeled by 1 and 2 for charge and heat currents, respectively.",
"$V$ is the system volume, ${\\bf k}$ is the electron wave vector, $\\nu $ and $\\mu $ are electron band indices, and $f_{{\\bf k}\\nu ,\\mu }$ are the corresponding Fermi occupations.",
"The $E_{{\\bf k}\\nu }$ are the Kohn-Sham band energies and $\\langle {\\bf k}\\mu |\\hat{\\bf p}|{\\bf k}\\nu \\rangle $ are the dipole matrix elements.",
"The Onsager weights are given by $\\epsilon _{{\\bf k}\\nu \\mu } \\equiv \\frac{1}{2}(E_{{\\bf k}\\nu } + E_{{\\bf k}\\mu }) - h$ where $h$ is the enthalpy per electron.",
"The appearance of the wave vector, ${\\bf k}$ , assumes that we are dealing with a periodic system (supercell, in our case).",
"Though we are ultimately interested in DC ($\\omega $ = 0) conductivities in this work, we perform computations of the $L_{ij}(\\omega )$ for small values of $\\omega $ and take the limit $\\omega \\rightarrow 0$ , from which we compute $\\sigma $ and $\\kappa $ .",
"The optical conductivity is given by $\\sigma (\\omega )=e^2{L}_{11},$ with the DC conductivity obtained in the limit $\\omega \\rightarrow 0$ .",
"The convergence of the optical conductivity with respect to the number of bands in the system is readily checked through the sum rule [52] $S = \\frac{2m V}{\\pi e^2 N_e}\\int _0^\\infty \\sigma (\\omega )d\\omega = 1.$ Likewise, the thermal conductivity is obtained from $\\kappa =\\frac{1}{T} \\biggl ( {L}_{22} - \\frac{{L}_{12}\\times {L}_{21}}{{L}_{11}} \\biggr ).$ There are two fundamental challenges in evaluating $\\sigma $ and $\\kappa $ using the expressions above: 1.",
"The $\\omega \\rightarrow 0$ limit can be problematic, because a finite-sized cell of electrons always possesses a nonzero minimum energy gap (and hence a nonzero minimum value of $E_{{\\bf k}\\mu } - E_{{\\bf k}\\nu }$ ) even though an infinite collection of electrons at sufficient density generally does not.",
"Thus, it is necessary to determine the DC limit by fitting the $\\sigma (\\omega )$ and $\\kappa (\\omega )$ results to $\\omega $ -dependent forms which have the correct behavior for an infinite system while extrapolating the simulations to infinite size.",
"For the systems considered here, where the electronic density of states is free-electron like, and the optical conductivity is well described with the Drude formula, this extrapolation of $\\omega \\rightarrow 0$ is straightforward.",
"2.",
"For the high temperature plasmas of our interest here, it is necessary to use a very large number of bands, $(\\mu ,\\nu )$ , in order to saturate the values of the $L_{ij}$ .",
"This is especially true for $L_{22}$ needed for $\\kappa $ , since the larger power of factors involving the single-particle energies more heavily weighs high-energy states.",
"For this, we find it necessary to extrapolate our conductivities to an infinite number of bands (or, equivalently, an infinite maximum eigenvalue) by performing a series of calculations using an increasing number of bands for each density and temperature condition we study.",
"Figure: (Color online) ϵ max →∞\\epsilon _{\\rm max} \\rightarrow \\infty scaling fit to the thermal conductivity for T=T = 500, 700, and 900 eV and ρ=\\rho = 40 g/cm 3 ^3.We find that the assumption of a simple power law behavior for the dipole matrix elements describes the asymptotic scaling of the thermal conductivity calculations with the maximum eigenvalue very well.",
"Noting that $L_{22} \\sim E^2$ and representing the dipole matrix elements in the limit $\\omega \\rightarrow 0$ by $E^\\gamma $ we fit a series of calculations with increasing maximum eigenvalue $\\epsilon _{\\rm max}$ to the following functional form $\\kappa (\\epsilon _{\\rm max})= \\kappa _\\infty \\frac{\\int _{-\\infty }^{\\epsilon _{\\rm max}} E^2 E^\\gamma \\frac{\\partial f}{\\partial E} dE}{\\int _{-\\infty }^{\\infty }E^2 E^\\gamma \\frac{\\partial f}{\\partial E}dE},$ where $f$ gives the Fermi occupations for the temperature and Fermi energy of the system in question.",
"The values of $\\gamma $ and $\\kappa _\\infty $ are then chosen for best fit to the series of calculations for each $\\epsilon _{\\rm max}$ at a given temperature.",
"The results of these fits for the thermal conductivity are displayed in Fig.",
"REF .",
"The assumed functional form captures the behavior of the calculated thermal conductivity very well in the limit of high $\\epsilon _{\\rm max}$ , giving us high confidence in the resulting value of $\\kappa _\\infty $ .",
"The best fit values of $\\gamma $ varied little between the three cases, ranging from 3.3 at 900 eV to 3.4 at 500 eV.",
"We show the results of the same procedure (as in Eq.",
"REF , but with $E^2 \\rightarrow 1$ ) applied to the electrical conductivity in Fig.",
"REF .",
"Note the significantly more rapid convergence of the electrical conductivity with increasing $\\epsilon _{\\rm max}$ .",
"It is important to note that even under conditions in which the sum rule (REF ) on $\\sigma (\\omega )$ is satisfied to a high degree, the calculation of $\\kappa $ could still be substantially in error.",
"For example, the sum rules for the 900 eV case range from 93.9% at the lowest $\\epsilon _{\\rm max}$ to 98.5% at the highest $\\epsilon _{\\rm max}$ .",
"Figure: (Color online) ϵ max →∞\\epsilon _{\\rm max} \\rightarrow \\infty scaling fit to the electrical conductivity for T=T = 500, 700, and900 eV and ρ=\\rho = 40 g/cm 3 ^3.Given our added confidence in these extrapolated predictions of $\\sigma $ and (especially) $\\kappa $ in these conditions, relative to earlier predictions [6], we are now in a position to compare them to the results of other approaches."
],
[
"Comparisons between Kohn-Sham DFT and Lenard-Balescu",
"In the moderate-to-weak plasma coupling regime of our interest in this work, we know of no highly constraining experimental results for $\\sigma $ or $\\kappa $ for hydrogen plasmas.",
"Therefore, we can only compare our extrapolated Kohn-Sham DFT results to the predictions from other theories.",
"Fortunately, there is an ab initio plasma kinetic theory which should provide very accurate estimates in this particular regime: Lenard-Balescu theory [17].",
"A recent work by some of us [22] demonstrated that classical LB theory reproduces MD computations of $\\sigma $ and $\\kappa $ for a semiclassical model of hydrogen (in which statistical two-body interaction potentials were used [43], [44]) for the very same conditions we study here.",
"The primary approximations in LB theory pertain to the neglect of large-angle scattering and a specific treatment of density fluctuations as modeled by the Random Phase Approximation (RPA) [53].",
"Since these approximations play very similar roles in both classical and quantum variants of the theory, we take the excellent agreement displayed in Ref.",
"[22] as a strong indication of the validity of quantum-LB here [54].",
"The mathematical and numerical prescription we use to generate quantum-LB predictions of $\\sigma $ and $\\kappa $ is outlined briefly in Section 2.2 of Ref.",
"[22], and the results we show here are in fact identical to those plotted as the thick dark blue lines in Figs.",
"1 and 2 of that work.",
"We note that these predictions are extremely close to those of Williams and DeWitt [18] for $\\sigma $ and $\\kappa $ derived from the quantum-LB equation, though with the minor caveat we mention in Ref. [55].",
"For the purposes of the discussion which follows, it is important to understand that the LB kinetic equation for hydrogen possesses two collision terms, $C_{ei}$ and $C_{ee}$ , each of which involve [17], [18], [19], [22]: 1.",
"The Fourier transforms of the bare two-body interactions; 2.",
"Occupation factors, $f$ , evaluated at the momenta of the colliding particles; and 3.",
"The two-species dynamical RPA dielectric function, $\\epsilon (q,\\omega )$ , evaluated at frequencies involving the center-of-mass energies of the colliding particles.",
"In practice, the effects of quantum diffraction manifest through the occupation factors, while the effects of screening arise through the dielectric function.",
"Figure: (Color online) (a) Electrical conductivity of hydrogen at ρ=\\rho = 40 g/cm 3 ^{3} as computed by Kohn-Sham DFT, extrapolated to an infinite number ofsingle-particle states, σ DFT \\sigma _{\\rm DFT} (solid green circles); as computed by the QLB equation using the prescription outlined in Ref.",
"(blue curve); the electrical conductivity in the absence of e-e collisions from the QLB calculations, σ ei \\sigma _{ei} (open blue circles).",
"(b) Thermal conductivityof hydrogen at ρ=\\rho = 40 g/cm 3 ^{3} extrapolated to an infinite number of single-particle states, κ DFT \\kappa _{\\rm DFT} (solid green circles); and as computedby the QLB equation using the prescription outlined in Ref.",
"(blue curve).Fig.",
"REF a shows our extrapolated DFT results (solid green circles) for the electrical conductivities of hydrogen plasmas along the $\\rho = $ 40 g/cm$^{3}$ isochore, as a function of temperature.",
"The general increase of $\\sigma $ with $T$ is expected from all theories [15], [18], [33], [25], provided that $T > T_{\\rm Fermi}$ , as is the case here.",
"Furthermore, the precise magnitude is very much in line with our calculation of $\\sigma $ using QLB theory [17], [22], shown as the blue curve [55].",
"The slightly lower $\\sigma $ values from QLB can be attributed to our neglect of electron degeneracy in the QLB calculation, given that $k_{\\rm B}T_{\\rm Fermi}$ is as high as 303 eV at this density [56].",
"Though quantum diffraction is accounted for in our implementation of QLB, Pauli blocking is not, as the collision terms we use do not possess the proper $1 - f$ factors needed to account for Pauli exclusion [22].",
"Nevertheless, the good agreement shown here establishes that upon extrapolation, our Kohn-Sham Kubo-Greenwood calculation of $\\sigma $ accounts for the bulk of the physics also included in the Lenard-Balescu treatment.",
"This physics involves not only scattering of the conducting electrons off the spatially distributed ions dressed by their individual dynamic screening clouds, but also the contribution of the e-e interaction in determining the precise shape of the steady-state electron distribution, $f(v)$ [57].",
"The tendency of the e-e collision term within kinetic theory to reshape the distribution at high-$T$ is well-known in the literature; if the simple assumption of the shifted equilibrium distribution [29] is made, $\\sigma $ is too low by a factor of 1.97 [33].",
"Indeed, this fact necessitates the application of correction factors when theories which make this assumption are used [25], [26], [32].",
"Though explicit e-e collisions are not included in the DFT, it is clear from this comparison that the mean-field Hartree (or Vlasov) term is allowing for the proper reshaping of the distribution upon the application of a weak uniform ${\\bf E}$ -field, since the precise magnitude of $\\sigma $ is very sensitive to the shape of $f(v)$ [15], [33].",
"The electrical conductivity in the absence of this proper reshaping contribution of electron-electron collisions, $\\sigma _{ei}$ , as calculated with the QLB equations, is shown with open circles in Fig.",
"REF a for comparison (see the discussion below for our precise definition of $\\sigma _{ei}$ ).",
"Figure REF b shows the corresponding comparison for thermal conductivity.",
"Here, the extrapolated DFT values are higher than those of QLB by around a factor of two in this regime.",
"Prior to the realization that this extrapolation was necessary here, the (under-converged) DFT predictions of $\\kappa $ would have been in better agreement with the QLB results [6].",
"As for $\\sigma $ , $\\kappa $ is also known to be affected by the e-e interaction within a plasma kinetic theory framework [15], [18], [33].",
"The correction factor needed to account for its effects, relative to a theory in which the low-$T$ shifted equilibrium distribution [29] is assumed, is distinct from that needed for $\\sigma $ .",
"This difference is the combined result of the different forcing terms on the left-hand side of the kinetic equation (corresponding to $\\nabla T$ rather than ${\\bf E}$ [58]), and the marked differences between the dependences of ${\\bf j}_{e}$ and ${\\bf j}_{Q}$ on the electron velocities.",
"In particular, within a semiclassical framework (see Ref.",
"[7] and the previous section for more precise expressions using Kohn-Sham states), ${\\bf j}_{e} \\propto \\sum _{i} e {\\bf v}_{i}$ , where $i$ indexes individual electronic states.",
"Note that this is proportional to the total electron momentum, $\\sum _{i} m{\\bf v}_{i}$ .",
"We therefore expect individual two-body e-e scatterings to do nothing to degrade ${\\bf j}_{e}$ , for the same reason that these intra-species collisions must leave the total electron momentum unchanged.",
"Because of this, the electron one- component plasma (OCP) has infinite static electrical conductivity; the application of a constant electric field to a uniform electron gas results in resistance-less current.",
"In contrast, ${\\bf j}_{Q} \\propto \\sum _{i}(\\frac{1}{2}m v_{i}^{2}){\\bf v}_{i} \\sim \\sum _{i}\\frac{1}{2}mv_{i}^{3}$ , assuming that the potential energy contributions to the heat current are negligible in comparison to the kinetic ones, as is the case for the weak coupling conditions studied here [22].",
"Two-body e-e scatterings can therefore change ${\\bf j}_{Q}$ (due to the fact that ${\\bf j}_{Q}$ is no longer proportional to a conserved quantity) and this results in a finite thermal conductivity for an electron OCP [58].",
"It is then reasonable to expect that an extra contribution to $\\kappa $ from the e-e interaction may result which, in contrast to $\\sigma $ , depends explicitly on e-e collisions.",
"This beyond-Vlasov/Hartree effect would indeed be absent from the Kohn-Sham DFT prescription we employ here [49].",
"With these observations in mind, we posit the following relations inspired by Matthiessen's rule [59], in which these two distinct manifestations of e-e interaction, 1. mean-field reshaping of $f(v)$ , and 2. binary scattering degradation of ${\\bf j}$ , are added “in series\": $\\frac{1}{\\sigma }= \\frac{1}{S_{\\sigma }\\sigma _{ei}} + \\frac{1}{\\sigma ^{\\rm OCP}_{ee}}= \\frac{1}{S_{\\sigma }\\sigma _{ei}},$ $\\frac{1}{\\kappa }= \\frac{1}{S_{\\kappa }\\kappa _{ei}} + \\frac{1}{\\kappa ^{\\rm OCP}_{ee}}.$ $\\sigma ^{\\rm OCP}_{ee}$ and $\\kappa ^{\\rm OCP}_{ee}$ are the electrical and thermal conductivities of the electron OCP, which can in principle be obtained by scaling down the $C_{ei}$ collision term in a kinetic equation otherwise possessing both $C_{ee}$ and $C_{ei}$ pieces.",
"$\\sigma _{ei}$ and $\\kappa _{ei}$ are the conductivities obtained by turning off the e-e interaction in the precise manner discussed in the following paragraph.",
"The second equality in Eq.",
"REF arises from the fact that $\\sigma ^{\\rm OCP}_{ee}= \\infty $ , as mentioned above.",
"The factors $S_{\\sigma }$ and $S_{\\kappa }$ are the reshaping corrections which result from the mean-field part of the e-e interaction.",
"Our contention is that within our Kohn-Sham Kubo-Greenwood prescription: $\\sigma _{\\rm DFT}= S_{\\sigma }\\sigma _{ei} = \\sigma ,$ $\\kappa _{\\rm DFT}= S_{\\kappa }\\kappa _{ei} = \\frac{\\kappa }{1 -{\\kappa }/{\\kappa ^{\\rm OCP}_{ee}}},$ where $\\sigma $ and $\\kappa $ are the true conductivities for the hydrogen plasma, i.e., as predicted by quantum-LB if we assume it to be perfectly valid in the conditions of interest.",
"Before we motivate Eqs.",
"REF , REF , REF , and REF further with direct numerical comparisons, we must clarify what we mean by $\\sigma _{ei}$ and $\\kappa _{ei}$ here.",
"Consider the QLB calculations of $\\sigma $ and $\\kappa $ for hydrogen.",
"As mentioned above, the kinetic equation for the electron distribution function has two collision terms, $C_{ei}$ and $C_{ee}$ , each of which involves the 2-component dielectric function, $\\epsilon $ .",
"This dielectric function depends on all three fundamental interactions, $\\phi _{ee}$ , $\\phi _{ii}$ , $\\phi _{ei}$ [53].",
"The collision terms, $C_{ei}$ and $C_{ee}$ , involve the screened interactions $(\\phi _{ei}/\\epsilon )$ and $(\\phi _{ee}/\\epsilon )$ , respectively.",
"The conductivities $\\sigma $ and $\\kappa $ are obtained by including both collision terms, while $\\sigma _{ei}$ and $\\kappa _{ei}$ are obtained by including only $C_{ei}$ .",
"However, it is important to note that $\\sigma _{ei}$ and $\\kappa _{ei}$ still include the effects of the e-e interaction within the dielectric function which screens $\\phi _{ei}$ , causing it to be reduced relative to its bare value.",
"This inclusion is crucial, and is taken into account in many theories less sophisticated than LB, such as Spitzer-Härm (embedded in their assumption, $b_{\\rm max}=$ Debye screening length) [15], and the various Ziman formula approaches in which the effective electron-ion scattering potential is taken to be $\\phi _{ei}/\\epsilon _{\\rm electron}$ [23], [25], [26], [33].",
"It is also accounted for in the rather sophisticated quantum-Boltzmann approach of Refs.",
"[48], [49] where the fundamental interaction within their $C_{ei}$ is $\\phi _{ei}$ statically screened by the electrons.",
"The construction of the various terms in Eqs.",
"REF , REF , REF , and REF from LB is then straightforward: Quantum-LB calculations including both $C_{ee}$ and $C_{ei}$ produce $\\sigma $ and $\\kappa $ ; calculations including only $C_{ei}$ produce $\\sigma _{ei}$ and $\\kappa _{ei}$ .",
"Calculations in which we apply a multiplier to $C_{ei}$ to force it to zero give us the electron OCP results, $\\sigma ^{\\rm OCP}_{ee}$ and $\\kappa ^{\\rm OCP}_{ee}$ .",
"As discussed, we recover $\\sigma ^{\\rm OCP}_{ee} \\rightarrow \\infty $ for all densities and temperatures, while we obtain finite values for $\\kappa ^{\\rm OCP}_{ee}$ as expected [58].",
"The need for the reshaping correction factors, $S_{\\sigma }$ and $S_{\\kappa }$ , is obviated within LB by first comparing $\\sigma $ with $\\sigma _{ei}$ .",
"Table REF shows our quantum-LB results for hydrogen along the $\\rho =$ 40 g/cm$^{3}$ isochore.",
"The ratio $\\sigma /\\sigma _{ei}\\equiv S_\\sigma $ varies between 0.64 and 0.72 within this temperature range.",
"Even though the inclusion of e-e interactions does nothing to degrade the electrical current of an OCP, the inclusion of $C_{ee}$ here reduces $\\sigma $ by an appreciable amount, and this occurs even as the contributions of the e-e interaction within the screening function, $\\epsilon $ , are left unchanged.",
"This reduction is due to the tendency for the e-e interaction to make the electron distribution function more isotropic in velocity space, which is seen clearly when the full solution is obtained by expanding $f$ in polynomials [18], [19], [22] using the standard Chapman-Enskog procedure [58].",
"The fact that our Kohn-Sham DFT electrical conductivities agree quite well with the quantum-LB $\\sigma $ , and far worse for $\\sigma _{ei}$ (see Fig.",
"REF ), indicates that this reshaping effect is within the purview of a self-consistent mean-field Hartree/Vlasov approach [60].",
"Our assertion appearing in Eq.",
"REF is therefore justified.",
"Turning to $\\kappa $ , if we now assume the relation of Eq.",
"REF , our LB computations of $\\kappa $ , $\\kappa _{ei}$ , and $\\kappa ^{\\rm OCP}_{ee}$ allow us to solve for $S_{\\kappa }$ .",
"Fig.",
"REF shows the product, $S_{\\kappa }\\cdot \\kappa _{ei}$ , vs. $T$ as the solid black diamonds.",
"These are extremely close to our results for $\\kappa _{\\rm DFT}$ , shown as the solid green circles.",
"This justifies our supposition of Eq.",
"REF , and points to a way to correct our Kohn-Sham DFT results for the thermal conductivity of hydrogen plasmas: $\\frac{1}{\\kappa }= \\frac{1}{\\kappa _{\\rm DFT}} + \\frac{1}{\\kappa ^{\\rm OCP}_{\\rm ee}}.$ The nearly coincident blue curve and solid blue squares at the bottom of Fig.",
"REF shows the comparison of the $\\kappa _{\\rm QLB}$ with that of $\\kappa _{\\rm DFT}$ when corrected in this manner.",
"In passing, we note that our quantum-LB results for hydrogen show that the reshaping correction factors for $\\sigma $ and $\\kappa $ are quite similar: $S_{\\sigma } \\sim S_{\\kappa }$ .",
"This is illustrated by the relative closeness of $S_\\sigma \\kappa _{ei}$ (open diamonds) to $S_\\kappa \\kappa _{ei}$ (solid black diamonds) in Fig.",
"REF .",
"Likewise, Table REF displays $\\kappa $ as computed by Eq.",
"REF but with $S_{\\sigma }$ used instead of $S_{\\kappa }$ .",
"Differences are less than $10\\%$ and are decreasing as $T$ is increased.",
"As shown in the Appendix, we have also used the Zubarev quantum-Boltzmann prescription of Refs.",
"[46], [47], [48] to affect the decompositions in Eqs.",
"REF and REF , and within this approach $S_{\\sigma }$ and $S_{\\kappa }$ are the same to within $5\\%$ in the Spitzer limit for these conditions.",
"Alternatively we show that under the assumption $S_{\\sigma } = S_{\\kappa }$ , the ansatz of Eq.",
"REF is satisfied to within 3% at the level of 4 moments, and within 2.2% in the Spitzer limit.",
"We emphasize that we are able to assert the efficacy of the correction in Eq.",
"REF only because we are operating in a regime where we expect Lenard-Balescu to be accurate.",
"One might then ask: Why use DFT at all, if LB is assumed to be better?",
"The answer is that for stronger-coupling and/or for plasmas and conditions for which (unlike in the present cases) ionization is incomplete, DFT is sure to provide much benefit over LB, since LB as such is only able to describe weakly-coupled plasmas with no bound states, etc.",
"Nevertheless, our primary aim in this work is to point out that an uncorrected thermal conductivity from Kohn-Sham DFT [5], [6], [7], [9], [10] is very possibly incomplete in its description, and that some relation like that of Eq.",
"REF which accounts for the effects of explicit e-e collisions may be more appropriate.",
"It is also possible that for higher-Z plasmas, such as those studied in Ref.",
"[10], the larger Z may cause the e-i interactions to outweigh the e-e interactions to the point where such effects are significantly less important [30].",
"This will very likely depend on density and temperature, and additionally so because highly degenerate electrons will feel minimal effects from e-e scattering.",
"More work must be done to further investigate these issues.",
"Table: σ\\sigma and σ ei \\sigma _{ei} as determined from quantum Lenard-Balescu for hydrogen at ρ=\\rho = 40 g/cm 3 ^{3};S σ =σ/σ ei S_{\\sigma } = \\sigma /\\sigma _{ei}.",
"See text for details.Table: Various quantities for hydrogen as computed with quantum Lenard-Balescu at ρ=40\\rho = 40 g/cm 3 ^{3} :κ\\kappa ; κ S σ \\kappa _{S_{\\sigma }} (κ\\kappa as computed fromEq.",
", but with S σ S_{\\sigma } instead of S κ S_{\\kappa }); the percent difference between κ S σ \\kappa _{S_{\\sigma }}and κ\\kappa .",
"See text for details.Figure: (Color online) Thermal conductivity of hydrogen at ρ=40\\rho = 40 g/cm 3 ^{3} as computed within a number of approximations.",
"QLB resultfor κ\\kappa (blue curve); κ DFT \\kappa _{\\rm DFT} extrapolated to an infinite number of single-particle states (solid green circles); S κ κ ei S_{\\kappa }\\kappa _{ei} as determined from QLB (solid black diamonds); S σ κ ei S_\\sigma \\kappa _{ei} as determined from QLB (open diamonds);κ DFT \\kappa _{\\rm DFT} corrected with κ ee \\kappa _{ee} from QLB using Eq.",
"(solid blue squares)."
],
[
"Conclusion",
"We have presented an investigation of the electrical and thermal conductivities of hydrogen plasmas for $\\rho = 40$ g/cm$^{3}$ and $T$ between 500 eV and 900 eV using Kohn-Sham DFT together with a Kubo-Greenwood response framework to compute the relevant current-current correlation functions.",
"In order to obtain converged results especially for the thermal conductivity, it was necessary to conduct a detailed extrapolation of transition dipole matrix elements to arrive at the results corresponding to an infinite number of high-lying Kohn-Sham states.",
"The resulting electrical conductivities are in excellent agreement with the predictions of quantum Lenard-Balescu theory, while the thermal conductivities are roughly a factor of two larger than the Lenard-Balescu values.",
"By conducting separate Lenard-Balescu studies in which electron-ion and electron-electron collision terms are independently switched off, we argue that the discrepancy in the thermal conductivity results from the neglect of explicit two-body electron-electron collisions in the (effectively mean-field) DFT prescription.",
"In contrast, the electrical conductivity is well- predicted by the DFT, suggesting that the well-known effect of the reshaping of the electron distribution function for that quantity is appropriately handled at the Hartree or Vlasov level.",
"We propose the following correction to the thermal conductivity as predicted by Kohn-Sham DFT, at least for hydrogen plasmas: $1/\\kappa = 1/\\kappa _{\\rm DFT} + 1/\\kappa ^{\\rm OCP}_{ee}$ , where $\\kappa ^{\\rm OCP}_{ee}$ is the thermal conductivity of the electron one-component plasma at the same $(\\rho ,T)$ .",
"It remains to be seen if such a correction is sensible for plasmas other than hydrogen.",
"In particular, it is not clear as to what should replace $1/\\kappa ^{\\rm OCP}_{ee}$ for matter in which the “free\" electron density is less approximately represented by an electron OCP.",
"Recent work on the electrical conductivity of warm, dense iron [61] has used a correction supplied by Dynamical Mean Field Theory, and there are other works in which corrections to mean-field electronic structure approaches have been proposed along similar lines [62].",
"More fundamentally, it is likely of great interest to know if a more consistent formulation within the rubric of Time-Dependent DFT [63] and/or Current-DFT [64] might admit a framework in which explicit electron-electron scattering can appear naturally in linear transport.",
"These important questions we leave for future studies."
],
[
"Acknowledgments",
"The authors gratefully thank John Castor, Frank Graziani, Gerd Röpke, Heidi Reinholz, and Martin French for helpful discussions.",
"RR thanks the Deutsche Forschungsgemeinschaft (DFG) for support within the SFB 652.",
"Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.",
"Portions of this work were performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344, and were funded by the Laboratory Directed Research and Development Program at LLNL under tracking code No.",
"12-SI-005 as part of the Cimarron Project.",
"*"
],
[
"The electrical and thermal conductivity are well known in the classical non-degenerate limit using kinetic theory, see e.g., Spitzer [15] and Chapman-Enskog [58], or linear response theory as outlined by Zubarev which will be employed here; for details, see [45], [46], [47], [48], [49], [65].",
"In the quantum-Boltzmann approach of Zubarev, which contains both the Ziman theory and the Spitzer theory as limiting cases, the conductivities are determined in linear response theory to arbitrary order in generalized momenta, while permitting arbitrary electron degeneracy and strong scattering.",
"For the analysis here, the collision integrals are regularized through the assumption of statically screened Coulomb potentials and the corresponding introduction of a Coulomb logarithm.",
"The electrons are assumed to be non-degenerate.",
"In particular, the influence of electron-electron collisions can be studied in order to validate the ansatz (REF ) and the relation for the prefactors $S_\\sigma \\sim S_\\kappa $ .",
"We start from the definition of the conductivities, $\\sigma = e^2 L_{11} \\;,\\; \\kappa = \\frac{1}{T} \\left(L_{22} - \\frac{L_{12}L_{21}}{L_{11}} \\right) ,$ where the Onsager coefficients $L_{ik}$ are defined as $L_{ik} = - \\frac{h^{(i+k-2)}}{\\Omega _0 \\mid d \\mid } \\;\\begin{array}{|cc|}0 & \\frac{k-1}{\\beta h} \\hat{N}_1 - \\hat{N}_0 \\\\\\frac{i-1}{\\beta h} N_1 - N_0 & (D)\\end{array} \\;;$ $h$ denotes the enthalpy per particle, $\\beta =1/k_BT$ , and $\\Omega _0$ is the system volume.",
"The vectors $\\hat{N}_n$ , $N_n$ and the matrix $(D)$ in Eq.",
"(REF ) contain correlation functions which can be calculated for arbitrary densities and temperatures, see [49], [65].",
"Using a finite set of $P$ moments to calculate the conductivities (i.e., the nonequilibrium distribution function) we have $\\hat{N}_m &=& ( \\hat{N}_{m0}, \\hat{N}_{m1}, \\ldots , \\hat{N}_{mP} )\\;,\\nonumber \\\\N_m &=& \\left( \\begin{array}{c}N_{om} \\\\ N_{1m} \\\\ \\vdots \\\\ N_{Pm}\\end{array} \\right)\\!,(D) = \\left( \\begin{array}{ccc}D_{00} & \\ldots & D_{0P} \\\\\\vdots & \\ddots & \\vdots \\\\D_{P0} & \\ldots & D_{PP}\\end{array} \\right)\\!.$ Generalized moments of the electron system are used to calculate the correlation functions, ${\\bf P}_n = \\sum _{\\bf k} \\hbar {\\bf k} [\\beta E_e(k)]^n a_e^{\\dagger }(k) a_e(k) ,$ and the time derivatives $\\dot{\\bf P}_n = \\frac{i}{\\hbar }[H_s,{\\bf P}_n]$ .",
"$H_s$ is the Hamilton operator of the system, the kinetic energy of the electrons is $E_e(k) = \\hbar ^2 k^2/(2m_e)$ , and $a_e^{\\dagger }(k)$ and $a_e(k)$ are creation and annihilation operators for electronic states $k$ , respectively.",
"The correlation functions are given as Kubo scalar products and its Laplace transforms: $N_{nm} &=& \\frac{1}{m_e}({\\bf P}_n,{\\bf P}_m) ,\\nonumber \\\\\\hat{N}_{nm} &=& N_{nm} + \\frac{1}{m_e} \\langle {\\bf P}_n(\\varepsilon );\\dot{\\bf P}_m \\rangle ,\\\\D_{nm} &=& \\langle \\dot{\\bf P}_n(\\varepsilon );\\dot{\\bf P}_m \\rangle .\\nonumber $ In the nondegenerate limit, the terms $\\langle {\\bf P}_n(\\varepsilon );\\dot{\\bf P}_m \\rangle $ can be neglected since they are related to the Debye-Onsager relaxation effect and we have $N_{nm}=\\hat{N}_{nm}$ .",
"According to the Hamilton operator $H_s=T+V_{ei}+V_{ee}$ , the force-force correlation functions $D_{nm}$ in Eq.",
"(REF ) can be separated with respect to electron-electron and electron-ion scattering, i.e.",
"$D_{nm} = D_{nm}^{\\rm ee} + D_{nm}^{\\rm ei}$ , for which analytical expressions can be given for hydrogen plasma ($N_i=N_e$ ) in the nondegenerate limit, see [48], [49], [66]: $N_{nm} &=& N_e \\frac{\\Gamma (n+m+5/2)}{\\Gamma (5/2)} ,\\\\D_{nm} &=& d \\, \\left\\lbrace \\left( \\frac{n+m}{2} \\right)!",
"+ c_{nm}^{ee} \\sqrt{2} \\right\\rbrace ,\\\\d &=& \\frac{4}{3} \\sqrt{2\\pi } \\frac{e^4}{(4\\pi \\varepsilon _0)^2}\\frac{\\sqrt{m_e}}{(k_BT)^{3/2}} n_e N_i \\Phi (\\Lambda ) ,$ with the Coulomb logarithm $\\Phi (\\Lambda )$ .",
"The weighting factors for the e-e correlation functions are given by $c_{0m}^{ee}=c_{m0}^{ee}=0$ , $c_{11}^{ee}=1$ , $c_{12}^{ee}=c_{21}^{ee}=11/2$ , $c_{22}^{ee}=157/4$ , $\\ldots $ The conductivities can be represented as $\\sigma &=& f \\sigma ^\\ast ,\\;\\sigma ^\\ast = \\frac{(4\\pi \\varepsilon _0)^2 (k_BT)^{3/2}}{\\sqrt{m_e} e^2 \\Phi (\\Lambda )} ,\\\\\\kappa &=& L \\left( \\frac{k_B}{e} \\right)^2 T \\sigma ,$ where $L$ is a Lorenz number.",
"The Spitzer theory [15] gives the correct values in this limit with $f_{\\rm Sp}^{ei+ee}=0.5908$ and $L_{\\rm Sp}^{ei+ee}=1.6220$ if e-i and e-e interactions are considered.",
"In the case of a Lorentz gas, i.e., neglecting e-e scattering, we get the values $f_{\\rm Sp}^{ei}=1.0159$ and $L_{\\rm Sp}^{ei}=4.0$ .",
"The prefactors for solutions up to 4th order within the Zubarev approach are given in Table REF .",
"They demonstrate a rapid convergence against the Spitzer values for the Lorentz gas and the fully interacting electron-ion system; see [66], [67].",
"Table: Prefactors for the electrical (ff) and thermal conductivity (LL) according to Eqs.",
"()and () in the nondegenerate limit.",
"The Zubarev approach using an increasing number of momentsP n P_n () is compared with the correct Spitzer values, see , .",
"Furthermore, theLorenz number ℓ ee \\ell ^{ee} of an electron OCP defined by Eq.",
"() is given.",
"Finally, the thermal conductivityaccording to the ansatz () can be expressed by the Lorenz number ℓ ei+ee \\ell ^{ei+ee} defined inEqs.",
"() - ().We now calculate the conductivities for an electron OCP model with only e-e interactions.",
"This can be done with the Hamilton operator $H_s = T + \\epsilon V_{ei} + V_{ee}$ by taking the limit $\\epsilon \\rightarrow 0$ after calculating the $L_{ik}$ .",
"Otherwise, the Onsager coefficients are divergent ($L_{11}$ ) or indefinite ($L_{12}$ , $L_{22}$ ) in the nondegenerate limit.",
"We have treated the electron OCP model by using up to four moments $P_n$ .",
"The result for the thermal conductivity can be represented as: $\\kappa ^{ee} &=&\\kappa ^{\\rm OCP}_{ee} = \\left( \\frac{k_B}{e} \\right)^2 T \\sigma ^\\ast \\cdot \\ell ^{ee} .$ The values for the factor $\\ell ^{ee}$ are given in Table REF .",
"We now explore the ansatz (REF ) in the nondegenerate limit $\\frac{1}{\\kappa } &\\stackrel{?",
"}{=}& \\frac{1}{S_{\\kappa }\\kappa _{ei}} + \\frac{1}{\\kappa ^{\\rm OCP}_{ee}} ,$ within the approximation $S_{\\kappa } = S_{\\sigma } $ .",
"We begin by writing $S_{\\kappa }\\kappa _{ei} = L^{ei} S_{\\kappa } \\sigma _{ei} \\left( \\frac{k_B}{e} \\right)^2 T \\approx L^{ei} S_{\\sigma } \\sigma _{ei}\\left( \\frac{k_B}{e} \\right)^2 T.$ Noting that $ \\sigma ^* = \\sigma _{ei}/{f^{ei}} $ and $ S_{\\sigma } = f^{ei + ee}/f^{ei}$ we can rewrite the ansatz as ${\\kappa } = {\\sigma \\left( \\frac{k_B}{e} \\right)^2 T} \\ell ^{ei+ee}.$ where $\\ell ^{ei+ee}= \\frac{L^{ei}\\ell ^{ee}}{\\ell ^{ee} + f^{ei+ee}L^{ei}}.$ The factor $\\ell ^{ei+ee}$ , to be compared to the exact value $L^{ei+ee}$ , is listed in the last column of Table REF .",
"We observe a fast convergence as before to 1.6605 at the level of 4 moments.",
"The deviation from the correct value in 4th order (1.6114) is just 3.0%, i.e., similar to the deviations of the numerical data from the quantum LB equation; see Figs.",
"REF and REF .",
"If instead we use the Spitzer values for $f^{ei+ee}$ and $L^{ei + ee}$ agreement with the Spitzer Lorenz number is within 2.2%.",
"Note that the coincidence of the values for $L^{ei+ee}$ and $\\ell ^{ei+ee}$ , i.e., of the direct and sum of the inverse thermal conductivities representing electron-ion and electron-electron scattering contributions, is only obtained if we use the prefactor $S_\\sigma =f^{ei+ee}/f^{ei}$ , which contains the influence of both electron-ion and electron-electron scattering.",
"Alternatively, we can take the ansatz (REF ) as an equality and solve for $S_\\kappa /S_\\sigma $ $\\frac{S_{\\kappa }}{S_{\\sigma }}= \\frac{1}{L^{ei}} \\left(\\frac{L^{ei+ee}\\ell ^{ee}}{\\ell ^{ee} - L^{ei+ee}f^{ei + ee}}\\right) .$ At the level of 4 moments, we find $S_\\kappa /S_\\sigma = 0.9318$ and with the Spitzer values for $f^{ei+ee}$ , $L^{ei}$ , and $L^{ei + ee}$ , we obtain $S_\\kappa /S_\\sigma = 0.9502$ .",
"Note that both of these results, for $\\ell ^{ei+ee}$ or $S_\\kappa /S_\\sigma $ , were obtained in a parallel approach to that taken in the main text within the QLB framework, and are completely general in the non-degenerate limit."
]
] | 1612.05574 |
[
[
"Fast Matrix Multiplication and Symbolic Computation"
],
[
"Abstract The complexity of matrix multiplication (hereafter MM) has been intensively studied since 1969, when Strassen surprisingly decreased the exponent 3 in the cubic cost of the straightforward classical MM to log 2 (7) $\\approx$ 2.8074.",
"Applications to some fundamental problems of Linear Algebra and Computer Science have been immediately recognized, but the researchers in Computer Algebra keep discovering more and more applications even today, with no sign of slowdown.",
"We survey the unfinished history of decreasing the exponent towards its information lower bound 2, recall some important techniques discovered in this process and linked to other fields of computing, reveal sample surprising applications to fast computation of the inner products of two vectors and summation of integers, and discuss the curse of recursion, which separates the progress in fast MM into its most acclaimed and purely theoretical part and into valuable acceleration of MM of feasible sizes.",
"Then, in the second part of our paper, we cover fast MM in realistic symbolic computations and discuss applications and implementation of fast exact matrix multiplication.",
"We first review how most of exact linear algebra can be reduced to matrix multiplication over small finite fields.",
"Then we highlight the differences in the design of approximate and exact implementations of fast MM, taking into account nowadays processor and memory hierarchies.",
"In the concluding section we comment on current perspectives of the study of fast MM."
],
[
"Our subjects",
"Matrix multiplication (hereafter we keep using the acronym MM) is fundamentally important for symbolic and numerical computations in linear algebra and for the theory of computing.",
"Efficient performance of MM depends on various factors, particularly on vectorization, data locality, and arithmetic cost (cf.",
"[71]).",
"In the first part of the paper (Sections –) we review the work on the decrease of the arithmetic cost, including purely theoretical study of MM of immense sizes (so far this part of the study has been most acclaimed and most generously supported!",
"), but we focus on feasible MM.",
"In our longest Section we discuss realistic acceleration of symbolic MM, taking into account nowadays processor and memory hierarchies.",
"In our concluding Section we comment on current perspectives of the study of fast MM."
],
[
"History of fast MM and its impacts",
"The cubic arithmetic time $2n^3-n^2$ of the straightforward algorithm for $MM(n)$ , that is, for $n\\times n$ MM, was commonly believed to be optimal until 1969, when Strassen's algorithm of [142] performed $MM(n)$ in $O(n^{\\omega })$ time for $\\omega =\\log _2(7)\\approx 2.8074$ .",
"This implied the exponent $\\log _2(7)$ also for numerous venerated computational problems in Computer Science, Linear Algebra, and Computer Algebra such as Boolean MM, parsing context-free grammars, computing paths and distances in graphs, the solution of a nonsingular linear system of equations, computation of the inverse and the determinant of a matrix, and its various factorizations (see more in Section ).",
"The worldwide interest to MM has immediately exploded,For the scientific world the news came as a miracle from the blue.",
"Most of the readers were particularly impressed by the power of the divide and conquer method (not novel in 1969) rather than by Strassen's ingenious algorithm for $2\\times 2$ MM, and many scientists, although not experts like Strassen, ignored or overlooked a minor but meaningful earlier acceleration of the straightforward MM that saved about 50% of its scalar multiplications (see Example REF in Section ).",
"and it was widely expected that new efficient algorithms would soon perform MM and solve the related computational problems in nearly quadratic time.",
"Even the exponent 2.8074, however, defied the attacks of literally all experts around the globe for almost a decade, until 1978, when the algorithm of [113] broke Strassen's record, improving the algorithm of [142] already at the level of feasible MM.",
"The mainstream research responded to that breakthrough by directing all effort to the decrease of the exponent of MM of unrestricted sizes and very soon succeeded in dramatic acceleration of infeasible MM of astronomical sizes.",
"New surprising resources have been found, sophisticated techniques have been developed, and by 1987 the exponent of infeasible MM was decreased below 2.38 (see [44]), although as of December 2016 it still has not been decreased below 2.37, that is, in the last 3 decades the progress was nominal (see [97] for the current record exponent).",
"Moreover, the study of infeasible MM has never made impact on practice of MM or any other realistic computations (cf.",
"[4], [5], [21], and our concluding section).",
"For $n$ restricted to be “moderate\", say, less than 1,000,000, the current record is 2.7734, achieved with the algorithm of [118] and unbeaten since 1982.",
"All algorithms supporting smaller exponents suffer from the curse of recursion (cf.",
"[123]): they beat the classical straightforward MM algorithm only after performing a large and typically immense number of recursive steps, with the input size growing exponentially in the number of such steps: the straightforward algorithm supersedes them until the input size by far exceeds realistic level, typically by many orders of magnitude."
],
[
"Focus of our presentation",
"In the context of this development, we refocus our presentation compared to the decades-old survey [119].",
"In Section we still pay tribute to the lasting interest to the exponent of infeasible MM, but we do not cover various amazing sophisticated techniques proposed exclusively for the acceleration of MM of immense sizes, which dominated the review of [119].",
"Instead we cover in some detail the techniques that are efficient already for MM of moderate sizes and have impacts on realistic computations beyond MM.",
"We feel that reduction of various computational problems to MM is interesting on its own right and because of potential benefits of wider application of fast or even straightforward algorithms for feasible MM.",
"Lately the study of these links was particularly intensive in the field of symbolic computation (see, e.g., Proceedings of ISSAC 2015 and ISSAC 2016).",
"We recall that historically no adequate comprehensive review of the MM subject has appeared for decades, not to the benefit of the field.",
"As we already explained, after the breakthrough of 1978, public interest to fast feasible MM was diverted by worldwide excitement about the exponent of infeasible MM, but also by some other factors.",
"In particular the advanced techniques of of [113] were much harder for non-experts to grasp than the catchy divide and conquer method, and public attention to the fast algorithm of [113] for feasible MM was also hurt by the folk “theorem\" about its alleged numerical instability.",
"This “theorem\" has somehow spread fast throughout the communities of numerical and symbolic linear algebra, before the classical paper [16] of 1980 proved that the “theorem\" was false.More precisely fast MM algorithms are slightly less stable numerically than the straightforward MM, but this instability is mild and rather little affects actual implementations of MM (see more details in [16], [73], [10], [50], and [8]).",
"The results of [16] became widely known only when the well-recognized article [50] extended them to all recursive bilinear algorithms for MM, but even in 2010 the Introduction of the important innovative paper [22] still referred to this “theorem\"and in 2016, the paper [80] still talks about “numerical stability issues with many levels of recursions”The paper [80] is not really about fast MM since it unrolls only one or two levels of recursion of Strassen's algorithm and then recomputes several times the submatrix additions to avoid using temporary buffers..",
"Moreover the paper [50] and its successor [10] attack [113] as well as all the work on fast feasible MM from another side.",
"Trying to simplify the discussion or perhaps to divert public attention from the advanced work on fast feasible MM to the domain of their own study and progress, the authors of these papers call “Strassen-like\" all known fast algorithms for MM.",
"This “innovation\" was based on ignorance: “Strassen-like\" algorithms, as [50] and [10] formally define them, have been long and well known under the much more informative name of noncommutative bilinear algorithms (see, e.g., [24]).The book [121] and MM survey articles [119] and [120] pay high respect to Strassen's fundamental contributions to fast MM (and similarly did Volker Strassen to the contribution of [111] to algebraic computations in sections “Pan's method\" of [143] and [145]), but we feel that calling all the advanced work on fast feasible MM Strassen-like, is not more fair or informative than, say, labeling Democritus-like the Faraday's constant, Mendeleev's periodic table, and the Heisenberg's principle of uncertainty.",
"Our personal communication in 2015 with the authors of [50] seems to help: the label “Strassen-like\" is not used, e.g., in [8], but unfortunately their original widely publicized contempt to the advanced results on fast feasible MM (including those completely distinct from Strassen's old algorithm of 1969 and in various important respects superseding it) has been deeply implanted into scientific community.",
"In the first part of our paper (Sections –) we review the previous study of fast MM with the focus on fast feasible MM and its impacts and applications to realistic computations beyond MM, and for example we included our novel extension of an old MM technique to the computation of the inner product of 2 vectors and the summation of integers (see our Examples REF and REF ).",
"In the second part of the paper (Section ) we discuss in some detail symbolic implementation of fast MM directed to minimizing communication cost and improving parallel implementation.",
"The basic algorithms for that study are mostly decades-old, and complementing them with some more recent advanced algorithms for feasible MM is one of the most natural directions to further progress in the field."
],
[
"Organization of our paper",
"We organize our paper as follows.",
"In Sections and we recall the 2 first accelerations of MM, in 1969 and 1978, respectively, and comment on their impacts beyond MM.",
"In Sections , , and we cover the fundamental classes of bilinear, trilinear and the so called APA algorithms, respectively, and discuss the associated fundamental techniques of the algorithm design, their impact on the acceleration of MM and their links to other areas of computing and algebra.",
"In Section we extend the APA technique to computing the inner product of 2 vectors and summation.",
"In Section we summarize the history of fast MM after 1978.",
"In Section we further comment on applications of fast MM and the impact of its study to other areas of computing and Mathematics.",
"In Section we discuss applications and implementations of exact fast MM.",
"In Section we comment on numerical implementation of fast MM and perspectives of its study.",
"In addition to the acronym $``MM\"$ for “matrix multiplication\", hereafter we write $``MI\"$ for “nonsingular matrix inversion\", $MM(m,n,p)$ for $m\\times n$ by $n\\times p$ MM, $MM(n)$ for $M(n,n,n)$ , and $MI(n)$ for $n\\times n$ $MI$ .",
"$W=(w_{i,j})_{i,j=1}^{m,n}$ denotes an $m\\times n$ matrix with the entries $w_{i,j}$ , for $i=1,\\dots , m$ and $j=1,\\dots , n$ ."
],
[
"1969: from the Exponent 3 to 2.8074 by means of $2{\\times }2$ -based recursive processes",
"We begin with recalling 3 old accelerations of the straightforward MM algorithm.",
"Example 2.1 From faster inner product to faster MM.",
"[See [152] and notice technical similarity to the algorithms for polynomial evaluation with preprocessing in [111] and [92].]",
"Observe that ${\\bf u}^T{\\bf v}=\\sum _{i=1}^{n/2}(u_{2i-1}+v_{2i})(v_{2i-1}+u_{2i})-\\sum _{i=1}^{n/2}u_{2i-1}u_{2i}-\\sum _{i=1}^{n/2}v_{2i-1}v_{2i},$ for any even $n$ .",
"Apply this identity to all $n^2$ inner products defining $n\\times n$ MM and compute it by using $0.5 n^3 +n^2$ scalar multiplications and $1.5n^3+2n^2-2n$ additions and subtractions.",
"Example 2.2 Winograd's $2\\times 2$ MM (cf.",
"[68], [24], [1], or [52]).",
"Compute the product $X=UV$ of a pair of $2\\times 2$ matrices, $U=\\begin{pmatrix}u_{11} & u_{12} \\\\u_{21} & u_{22}\\end{pmatrix},~V=\\begin{pmatrix}v_{11} & v_{12} \\\\v_{21} & v_{22}\\end{pmatrix},~X=UV=\\begin{pmatrix}x_{11} & x_{12} \\\\x_{21} & x_{22}\\end{pmatrix}$ by using the following expressions, $s_1=u_{21}+u_{22}$ , $s_2=s_1-u_{11}$ , $s_3=u_{11}-u_{21}$ , $s_4=u_{12}-s_2$ , $s_5=v_{12}-v_{11}$ , $s_6=v_{22}-s_5$ , $s_7=v_{22}-v_{12}$ , $s_8=s_6-v_{21}$ , $p_1=s_2s_6,~p_2=u_{11}v_{11},~p_3=u_{12}v_{21}$ , $p_4=s_3s_7,~p_5=s_1s_5,~p_6=s_4v_{22}$ , $p_7=u_{22}s_8$ , $t_1=p_1+p_2,~t_{2}=t_1+p_4$ , $t_{3}=t_1+p_5$ , $x_{11}=p_2+p_3,~x_{12}=t_3+p_6,~x_{21}=t_2-p_7,~x_{22}=t_2+p_5$ .",
"This algorithm performs $2\\times 2$ MM by using 7 scalar multiplications and 15 scalar additions and subtractions instead of 8 and 4, respectively, that is, 22 versus 12.",
"Fix, however, a sufficiently large integer $h$ , then recursively $h-1$ times substitute $2\\times 2$ matrices for $u_{ij}$ , $v_{jk}$ , and $c_{ik}$ , for all subscripts $i,j,k\\in \\lbrace 1,2\\rbrace $ , and reuse the above algorithm for every $2\\times 2$ MM.",
"This computation only involves $7^h$ scalar multiplications and $7^{h-1}+15(4^h)$ additions and subtractions, which is readily extended to performing $n\\times n$ MM for any $n$ by using $c~ n^{\\log _2(7)}$ scalar arithmetic operations overall where $\\log _2(7)\\approx 2.8074$ and careful refinement and analysis yield $c<3.92$ [68].",
"Example 2.3 Strassen's $2\\times 2$ MM [142].",
"$p_1=(u_{11}+u_{22})(v_{11}+v_{22}),~p_2=(u_{21}+u_{22})v_{11},~p_3=u_{11}(v_{12}-v_{22})$ , $p_4=(u_{21}-u_{11})(v_{11}+v_{12}),~p_5=(u_{11}+u_{12})v_{22},~p_6=u_{22}(v_{21}-v_{11})$ , $p_7=(u_{12}-u_{22})(v_{21}+v_{22})$ , $x_{11}=p_1+p_6+p_7-p_5,~x_{12}=p_3+p_5,~x_{21}=p_2+p_6,~x_{22}=p_1+p_3+p_4-p_2$ .",
"This algorithm performs $2\\times 2$ MM by using 7 scalar multiplications and 18 scalar additions and subtractions, which is a little inferior to Example REF , but also initiates a similar recursive process that performs $n\\times n$ MM for any $n$ by using $c^{\\prime }\\cdot n^{\\log _2(7)}$ scalar arithmetic operations for $c^{\\prime }<4.54$ .",
"Being a little slower, this algorithm is used much less than Winograd's, but is much more celebrated because it appeared earlier.",
"In MM literature Winograd's algorithm is rarely called Winograd's, but usually “Strassen–Winograd's algorithm\" or “Winograd's variant of Strassen's algorithm\", and sometimes less competently Strassen's, Strassen-based, or Strassen-like (cf.",
"[50], [10]).",
"We can see nothing in Example REF borrowed from Example REF , but the cited impact of [50] seems to be so deeply rooted in the Community of Computer Algebra, that even the authors of the advanced works [22] and [36] did not dare to call Example REF “Winograd's algorithm\", although their own innovative algorithms extended precisely this example and not Example REF .",
"The first line of Table REF displays the estimated arithmetic cost of the recursive bilinear algorithms for $MM(n)$ , $n=2^k$ , that begin with $2\\times 2$ MM by Strassen (of Example REF ), Winograd (of Example REF ), and Cenk and Hasan (of [36]).",
"The second line shows the decreased cost bounds where recursive process begins with $k\\times k$ MM performed with straightforward algorithm, and then one of the 3 above algorithms is applied recursively.",
"The improvement from [36] is technically interesting, but its arithmetic cost bounds are still significantly inferior to those of the algorithms of [118] as well as ones of [94], whose implementation in [88] is numerically stable and is highly efficient in using memory space.",
"Table: Arithmetic Complexity of Some 2×22\\times 2-based Recursive Bilinear Algorithms."
],
[
"The class of bilinear algorithms",
"The algorithms of Examples REF and REF belong to the important class of noncommutative bilinear algorithms, to which we refer just as bilinear.",
"Such an algorithm for $MM(m,n,p)$ first computes some linear forms $l_q(U)$ and $l^{\\prime }_q(V)$ in the entries of the input matrices $U=(u_{ij})_{i,j=1}^{m,n}$ and $V=(v_{jk})_{j,k=1}^{n,p}$ and then the entries $x_{ik}=\\sum _j u_{ij} v_{jk}$ of the product $X=UV$ as the $mp$ bilinear forms, $l_q(U)=\\sum _{i,j=1}^{m,n}\\alpha _{ij}^{(q)}u_{ij},~l^{\\prime }_q(V)=\\sum _{j,k=1}^{n,p}\\beta _{jk}^{(q)}v_{jk},~q=1,\\dots ,r,$ $x_{ik}=\\sum _{q=1}^r\\gamma _{ik}^{(q)}l_q(U)l^{\\prime }_q(V),~~i=1,\\dots ,m;~k=1,\\dots ,p.$ Here $r$ is said to be the rank of the algorithm, $u_{ij}$ , $v_{jk}$ , and $x_{ik}$ are variables or block matrices, and $\\alpha _{ij}^{(q)}$ , $\\beta _{jk}^{(q)}$ , and $\\gamma _{ik}^{(q)}$ are constant coefficients for all $i,j,k$ , and $q$ .",
"They define coefficient tensors $(\\alpha _{ij}^{(q)})_{i,j,q}$ , $(\\beta _{jk}^{(q)})_{j,k,q}$ , and $(\\gamma _{ik}^{(q)})_{i,k,q}$ , which can be also viewed as coefficient matrices $A=(\\alpha _{ij}^{(q)})_{(i,j),q},~B=(\\beta _{jk}^{(q)})_{(j,k),q},~{\\rm and}~C=(\\gamma _{ik}^{(q)})_{(i,k),q}.$"
],
[
"The ranks of bilinear algorithms and the $MM$ exponents",
"Suppose $m=n=p$ , assume that $u_{ij}$ , $v_{jk}$ , and $x_{ik}$ are block matrices, and then again recursively apply the same bilinear algorithm of rank $r$ to block matrices.",
"The resulting bilinear algorithms have ranks $r^{h}=c^{\\prime }n^{h\\omega }$ for $MM(n^h)$ , $h=2,3,\\dots $ , $\\omega =\\omega _{n,r}=\\log _{n}(r)$ and a constant $c^{\\prime }$ is independent of $n$ and $h$ .",
"One can readily extend these observations to the following result.",
"Theorem 3.1 Given a bilinear algorithm of rank $r$ for $MM(n)$ for a pair of positive integers $n$ and $r$ , one can perform $MM(K)$ by using $cK^{\\omega }$ arithmetic operations for any $K$ , $\\omega =\\omega _{n,r}=\\log _{n}(r)$ , and a constant $c$ independent of $K$ .",
"Now we define (i) the exponent of MM(n), $\\omega _n=\\min _r\\log _{n}(r)$ where an integer $n>1$ is fixed and the minimum is over the ranks $r$ of all bilinear algorithms for $MM(n)$ and (ii) the exponent of MM, $\\omega =\\min _n\\omega _n$ where the minimum is over all integers $n>1$ .Here we consider MM over the fields of real and complex numbers.",
"The exponent $\\omega $ (although not the overhead constant) stays invariant over the fields having the same characteristic [133].",
"Some of our results can be used in important applications of MM over semi-rings, but generally in that case distinct techniques are used (cf.",
"[3], [54], [156], [96]).",
"(iii) The latter concept is mathematically attractive, but the highly rewarded work for its upper estimates has diverted public attention from feasible to infeasible MM.",
"To straighten the unbalanced study of this subject, one should instead estimate the exponents of feasible MM, $\\omega _{<N}=\\min _{n<N}\\omega _n,$ for realistic upper bounds $N$ on the dimension of MM inputs."
],
[
"Bilinear versus quadratic\nalgorithms for bilinear problems",
"The straightforward algorithm for $MM(m,n,p)$ is bilinear of rank $mnp$ .",
"The bilinear algorithms of Examples REF and REF for $MM(2)$ have rank 7, which turned out to be optimal (see [76], [77], [31]).",
"[112] as well as [122] and [47]) provide an explicit expression for all bilinear algorithms of rank 7 for $MM(2)$ , including the algorithms of Examples REF and REF as special cases.",
"Among them 15 scalar additions and subtractions of Example REF are optimal [127], [33].",
"One can define bilinear algorithms for any bilinear problem, that is, for the computation of any set of bilinear forms, e.g., the product of two complex numbers $(u_1+{\\bf i} u_2)(v_1+{\\bf i} v_2)=(u_1v_1-u_2v_2)+{\\bf i}(u_1v_2+u_2v_1)$ , ${\\bf i}=\\sqrt{-1}$ .",
"The straightforward bilinear algorithm has rank 4, and here is a rank-3 bilinear algorithm, $l_1l_1^{\\prime }=u_1v_1$ , $l_2l_2^{\\prime }=u_2v_2$ , $l_3l^{\\prime }_3=(u_1+u_2)(v_1+v_2)$ , $u_1v_1-u_2v_2=l_1l_1^{\\prime }-l_2l^{\\prime }_2$ , $u_1v_2+u_2v_1=l_3l_3^{\\prime }-l_1l_1^{\\prime }-l_2l_2^{\\prime }$ .",
"See [153], [65], [66], [112], [29], [78], [144], [128], [30], [31], on the early study of bilinear algorithms and see a concise exposition in [24].",
"The book [155] covers various efficient bilinear algorithms for multiplication of pairs of integers and polynomials (the latter operation is also called the convolution of the coefficient vectors), with further applications to the design of FIR-filters.",
"The minimal rank of all bilinear algorithms for a fixed bilinear problem such as $MM(m,n,p)$ is called the rank of the problem.",
"It can be bounded in terms of the minimal arithmetic cost of the solution of the problem and vice versa [112], [24].",
"The algorithm of Example REF of rank $r=r(n)=0.5n^3 +n^2$ for $MM(n)$ , however, is not (noncommutative) bilinear; such algorithms are called quadratic or commutative bilinear.",
"(See [154] on some lower bounds on the rank of such algorithms for MM.)",
"We cannot extend to them recursive processes that bound the MM exponents by $\\log _n(r)$ ,Hereafter we refer to decreasing upper bounds on the exponent of MM as decreasing the exponent, for short.",
"because MM is not commutative, e.g., the equation $u_{2i-1}u_{2i}=u_{2i}u_{2i-1}$ is invalid for matrices $u_{2i-1}$ and $u_{2i}$ .",
"The algorithm, however, was of great importance in the history of fast MM: it was the first acceleration of the straightforward MM (it saves about 50% multiplications), but most important, it motivated the effort for the design of bilinear (rather than quadratic) algorithms of rank less than $n^3$ for $MM(n)$ .",
"It “remained\" to devise such an algorithm at least for $n=2$ , and Strassen received ample recognition for his brilliant design that accomplished exactly this."
],
[
"1978: from 2.81 to 2.78 by means of trilinear aggregation",
"In 1969 the exponents below Strassen's 2.8074 became the target of literally all the leading researchers in the field, worldwide, but remained a dream for almost a decade.",
"This dream would have come true based on bilinear algorithms of rank 6 for $MM(2)$ or rank 21 for $MM(3)$ , but it was proved that the rank of $MM(2)$ exceeds 6, and it is still unknown whether $MM(3)>22$ .",
"We refer the reader to the paper [100] for the current record lower bounds on the rank of $MM(n)$ for all $n$ , to the papers [76], [77], [112], and [29], [31], [32], [20], [19], [129], [137], and [95] for some earlier work in this direction, and to the papers [53] and [136] for various lower and upper bounds on the arithmetic complexity and the ranks of rectangular MM of smaller sizes.",
"The exponent was decreased only in 1978, after almost a decade of stalemate, when the paper [113] presented a bilinear algorithm of rank 143,640 for $MM(70)$ .",
"This breakthrough implied the exponent $\\omega =\\log _{70} (143,640)<2.7962$ for $MM(n)$ , $MI(n)$ , Boolean $MM(n)$ , and a variety of other well-known computational problems.",
"The algorithm of [113] has extended an algorithm of the paper [112] of 1972, published in RussianUntil 1976 the second author lived in the Soviet Union.",
"From 1964 to 1976 he has been working in Economics in order to make his living and has written the papers [111] and [112] in his spare time.",
"and translated into English only in 2014 in [122].",
"The progress was due to the novel combination of two techniques: trilinear interpretation of bilinear algorithms and the aggregation method.",
"By following [113] we call this combination trilinear aggregation.",
"By refining this combination of 2 techniques the paper [118] accelerated MM of moderate sizes and yielded the exponent 2.7734.",
"As we already mentioned, various algorithms combining trilinear aggregation with other advanced techniques decreased the exponent below this level (and in 1986 even below 2.38), but only when they were applied to MM of immense sizes because of the curse of recursion.",
"The technique of trilinear aggregation has been recognized for its impact on the decreases of the MM exponent, but the paper [112] was also a historical landmark in the study of multilinear and tensor decompositions.",
"Such decompositions introduced by Hitchcock in 1927 received little attention except for a minor response in 1963–70 with half of a dozen papers in the psychometrics literature.",
"The paper [112] of 1972 provided the earliest known application of nontrivial multilinear and tensor decompositions to fundamental matrix computations, now a popular flourishing area in linear and multilinear algebra with a wide range of important applications to modern computing (see [148], [93], [108], [71], and the bibliography therein).",
"Nevertheless the paper [112] has rarely been cited at all and has never been cited in the papers on multilinear and tensor decompositions."
],
[
"Trilinear Decompositions and Duality",
"Next we define trilinear representation of MM, first proposed and used in the paper [112].",
"We are also going to link it to some important computations beyond MM.",
"Let $U=(u_{ij})_{i,j}$ and $V=(v_{jk})_{j,k}$ be a pair of $m\\times n$ and $n\\times p$ matrices, respectively, and let the equations $\\sum _j u_{ij} v_{jk}=\\sum _{s=1}^rw_{ik}^{(s)}l_s(U)l^{\\prime }_s(B)$ for all $i,j$ represent a bilinear algorithm of rank $r$ for the matrix product $X=UV$ .",
"Define a trilinear decomposition of rank $r$ for $trace(UVW)=\\sum _{i,j,k} u_{i,j} v_{jk}w_{ki}$ by multiplying these equations by variables $d_{ki}$ and summing the products in $i$ and $k$ .",
"Here $W=(w_{ki})_{k,i}^{n,m}$ is an auxiliary $n\\times m$ matrix and trace$(M)$ denotes the trace of a matrix $M$ .",
"(Equivalently we can decompose the tensor of the trilinear form $trace(UVW)$ into the sum of $r$ tensors of rank 1.)",
"Example 5.1 A trilinear decomposition of rank 7 for $MM(2)$ .",
"$~~~~~~~~~ \\sum _{i,j,h=1}^2u_{ij}v_{jh}w_{hi}=\\sum _{s=1}^7l_sl_s^{\\prime }l_s^{\\prime \\prime }$ , $l_1l_1^{\\prime }l_1^{\\prime \\prime }=(u_{11}+u_{22})(v_{11}+v_{22})(w_{11}+w_{22})$ , $l_2l_2^{\\prime }l_2^{\\prime \\prime }=(u_{21}+u_{22})v_{11}(w_{21}-w_{22})$ , $l_3l_3^{\\prime }l_3^{\\prime \\prime }=u_{11}(v_{12}-v_{22})(w_{12}+w_{22})$ , $l_4l_4^{\\prime }l_4^{\\prime \\prime }=(u_{21}-u_{11})(v_{11}+v_{12})w_{22}$ , $l_5l_5^{\\prime }l_5^{\\prime \\prime }=(u_{11}+u_{12})v_{22}(w_{12}-w_{11})$ , $l_6l_6^{\\prime }l_6^{\\prime \\prime }=u_{22}(v_{21}-v_{11})(w_{11}+w_{21})$ , $l_7l_7^{\\prime }l_7^{\\prime \\prime }=(u_{12}-u_{22})(v_{21}+v_{22})w_{11}$ .",
"Conversely, we can come back to the original bilinear algorithm for the matrix product $X=UV$ if we interpret both sides of any decomposition of the trilinear form $trace(UVW)$ as linear forms in the variables $w_{ih}$ and equate the coefficients of these variables on both sides of the decomposition.",
"More generally, [112] states the equivalence of a bilinear algorithm of rank $r$ for $MM(m,n,p)$ to a trilinear decomposition of rank $r$ for the associated trilinear form and its tensor.",
"Instead of equating the variables $w_{ij}$ on both sides of the trilinear decomposition, we can equate the coefficients of all variables $u_{ij}$ or all variables $v_{jh}$ and then arrive at 2 other dual bilinear algorithms of the same rank for the problems $M(n,p,m)$ and $M(p,m,n)$ .",
"By interchanging the subscripts of the variables, we arrive at the dual bilinear algorithms of the same rank for the problems $MM(m,p,n)$ , $MM(n,m,p)$ , and $MM(p,n,m)$ as well (cf.",
"[112], [29], [78], [128]).",
"The latter extension from triples to 6-tuples is pertinent to MM, because it uses the double subscripts for the variables, but the triples of bilinear algorithms can be generated from their common trilinear representation for any bilinear computational problem, e.g., for multiplication of 2 complex numbers in the following example.",
"Example 5.2 A trilinear decomposition of rank 3 for multiplication of 2 complex numbers.",
"$u_1v_1w_1-u_2v_2w_1+u_1v_2w_2+u_2v_1w_2=u_1v_1(w_1-w_2)-u_2v_2(w_1+w_2)+(u_1+u_2)(v_1+v_2)w_2.$ For a sample application of the duality technique, one can readily deduce the following result (part 1 of [112]).",
"Theorem 5.1 Given a bilinear or trilinear algorithm of rank $r$ for $MM(m,n,p)$ and any 4-tuple of integers $r$ , $m$ , $n$ , and $p$ such that $mnp>1$ , one can perform $MM(K)$ by using $cK^{\\omega }$ arithmetic operations for any $K$ , $\\omega =\\omega _{m,n,p,r}=3\\log _{mkn}(r)$ , and a constant $c$ independent of $K$ .",
"For further applications of the duality technique, see efficient bilinear algorithms for FIR-filters and multiplication of complex numbers and polynomials in [155]."
],
[
"Trilinear Aggregation",
"Aggregation technique is well-known in business, economics, computer science, telecommunication, natural sciences, medicine, and statistics.",
"The idea is to mass together or cluster independent but similar units into much fewer aggregates.",
"Their study is simpler, but its results are supposed to characterize all these units either directly or by means of special disaggregation techniques.",
"Such aggregation/disaggregation processes proposed in [101] became a basis for creating the field of Algebraic Multigrid, now quite popular.",
"Aggregation/disaggregation techniques are behind the acceleration of MM in Example REF , which was preceded by similar application of this technique to polynomial evaluation with preprocessing of coefficients [111], [92].",
"In that example one first rewrite $u_{2i}v_{2i}=v_{2i}u_{2i}$ by using commutativity of multiplication (which does not hold if $u_{2i}$ and $v_{2i}$ are matrices), then aggregates the terms $u_{2i-1}v_{2i-1}$ and $v_{2i}u_{2i}$ into the single term $(u_{2i-1}+v_{2i})(v_{2i-1}+u_{2i})$ , thus saving 50% of scalar multiplications, that is, $0.5 n^3$ for $n\\times n$ MM.",
"For disaggregation one subtracts the correction terms $u_{2i-1}u_{2i}$ and $v_{2i}v_{2i-1}$ .",
"$n\\times n$ MM involves $0.5n^2$ pairs of such products, and so correction involves just $n^2$ scalar multiplications overall, which is a small sacrifice compared to saving $0.5n^3$ scalar multiplications.",
"The papers [112] and [113] strengthen aggregation based on restating MM as the problem of the decomposition of the trilinear form $trace(UVW)$ , so that some additional links among the subscripts of the input and output entries enable stronger aggregation and faster MM.",
"Various implementations of this technique appeared in [113], [114], [115], and a number of subsequent papers.",
"For demonstration we apply it to Disjoint MM where we compute two independent matrix products $AB$ and $UV$ by decomposing the trilinear form $trace(XYZ+UVW)=\\sum _{i,j,k=1}^{m,n,p}(x_{ij}y_{jk}z_{ki}+u_{jk}v_{ki}w_{ij})$ into the sum of rank-1 tensors.",
"Let $T=\\sum _{i,j,k=1}^{m,n,p}(x_{ij}+u_{jk})(y_{jk}+v_{ki})(z_{ki}+w_{ij})$ denote a trilinear aggregate made up of the 2 monomials $x_{ij}y_{jk}z_{ki}$ and $u_{jk}v_{ki}w_{ij}$ and let $T_1=\\sum _{i,j=1}^{m,n}x_{ij}s_{ij}w_{ij}$ , $T_2=\\sum _{j,k=1}^{n,p}u_{jk}y_{jk}r_{jk}$ , and $T_3=\\sum _{k,i=1}^{p,m}q_{ik}v_{ki}z_{ki}$ denote 3 groups of correction terms, where $q_{ik}=\\sum _{j=1}^{k}(u_{ij}+u_{jk})$ , $s_{ij}=\\sum _{k=1}^{n}(y_{jk}+v_{ki})$ , and $r_{jk}=\\sum _{i=1}^{m}(z_{ki}+w_{ij})$ .",
"Then the equation $trace(XYZ+UVW)=T-T_1-T_2-T_3$ defines a trilinear decomposition of rank $mnp+mn+np+pm$ (rather than the straightforward $2mnp$ ).",
"Table REF displays this aggregation/disaggregation technique.",
"Table: Aggregation/disaggregation of a pair of terms.The product of the 3 sums of pairs on input entries in each of the 3 columns of the table is an aggregate.",
"The 2 products of triples of entries of each of the 2 rows are the output terms $x_{ij}y_{jk}z_{ki}$ and $u_{jk}v_{ki}w_{ij}$ .",
"The cross-products of other triples of the table define 6 correction terms.",
"Their sum over all $n^3$ triples of indices $i,j$ and $k$ has rank $2(mn+np+pm)$ .",
"By subtracting this sum from the sum of all $mnp$ aggregates, we decompose $2mnp$ terms of $trace(XYZ+UVW)$ into the sum of $mnp+2(mn+np+pm)$ terms.",
"For $m=n=p=34$ this implies a decomposition of rank $n^3+6n^2$ for a pair of disjoint $MM(n)$ .",
"Demonstration of the power of trilinear aggregation can be made most transparent for Disjoint MM, whose natural link to trilinear aggregation has been shown in [119], [120], [121], and [94].",
"Such constructions for Disjoint MM, however, can frequently be extended to $MM(n)$ .",
"In particular, by playing with odd and even subscripts of the matrix entries, the paper [112] obtained a trilinear decomposition of rank $0.5 n^3+3n^2$ for $MM(n)$ and any even $n$ by means of extending the above decomposition of $trace(XYZ+UVW)$ .",
"This implied the $MM$ exponent $\\log _n (0.5n^3+3n^2)$ , which is less than 2.85 for $n=34$ .",
"The paper [113] defined a trilinear decomposition and bilinear algorithms of rank $(n^3-4n)/3+6n^2$ for $MM(n)$ , $n=2s$ , and any positive integer $s$ .",
"Substitute $n=70$ and obtain the MM exponent 2.7962.",
"Then again it is convenient to demonstrate this design for Disjoint MM associated with a decomposition of the trilinear form $trace(XYZ+UVW+ABC)$ .",
"The basic step is the aggregation/disaggregation defined by Table REF .",
"Table: Aggregation/disaggregation of a triple of terms.Sum the $mkn$ aggregates $(x_{ij}+u_{jk}+a_{ki})(y_{jk}+v_{ki}+b_{ij})(z_{ki}+w_{ij}+c_{jk})$ , subtract order of $n^2$ correction terms, and obtain a decomposition of rank $n^3+O(n^2)$ for $trace(XYZ+UVW+ABC)$ , versus the straightforward $3n^3$ .",
"The trace represents 3 disjoint problems of $MM(n)$ , that is, the computation of the 3 independent matrix products $XY$ , $UV$ , and $AB$ of size $n\\times n$ (and can be readily extended to 3 MM products of sizes $m\\times n$ by $n\\times p$ , $n\\times p$ by $p\\times m$ , and $p\\times m$ by $m\\times n$ ), and we obtain a bilinear algorithm of rank $n^3+O(n^2)$ for this bilinear task.",
"With a little more work one obtains a similar trilinear decomposition of rank $(n^3-4n)/3+6n^2$ for $MM(n)$ , $n=2s$ , and any positive integer $s$ (see [113]).",
"For $n=70$ we arrive at an upper bound 2.7962 on the MM exponent.",
"By refining this construction the algorithm of [118] decreased the upper bound below 2.7734."
],
[
"APA Algorithms and Bini's Theorem",
"The technique of Any Precision Approximation (hereafter we use the acronym APA) was another basic ingredient of the algorithms supporting the decrease of the exponent of MM.",
"The paper [15] achieved the first APA acceleration of MM, by yielding the exponent 2.7799.",
"According to [130], this came from computer search for partial $MM(2)$ where the goal was the computation of only 3 entries of $2 \\times 2$ matrix product.",
"Next we demonstrate the combination of APA and trilinear aggregation, which is more transparent and immediately produces the exponent 2.66.",
"Consider the following table.",
"Table: APA aggregation/disaggregation of a pair of terms.It defines the aggregate $(x_{ij}+\\lambda u_{jk})(y_{jk}+\\lambda v_{ki})(\\lambda ^2 z_{ki}+w_{ij})$ and 3 correction terms, similarly to Table REF , but with a decisive difference – the term $\\lambda ^3 (x_{ij}+u_{jk})v_{ki}z_{ki}$ has a smaller order of magnitude as $\\lambda \\rightarrow 0$ .",
"Therefore we arrive at trilinear decomposition $trace(XYZ+UVW)=T-T_1-T_2+O(\\lambda )$ where $T=\\lambda ^{-1} \\sum _{i,j,k=1}^{m,k,n}(x_{ij}+\\lambda u_{jk})(y_{jk}+\\lambda v_{ki})(\\lambda ^2 z_{ki}+w_{ij}),~T_1=\\sum _{i,j=1}^{m,k}x_{ij}s_{ij}w_{ij},~T_2=\\sum _{j,k=1}^{k,n}u_{jk}y_{jk}r_{jk},$ $s_{ij}=\\sum _{k=1}^{n}(y_{jk}+\\lambda v_{ki})$ , and $r_{jk}=\\sum _{i=1}^{m}(\\lambda ^2 z_{ki}+w_{ij})$ .",
"Drop the terms of order $\\lambda $ , and obtain a decomposition for Disjoint $MM(m,n,p)$ having border rank $mnp+mn+np$ .",
"For $m=p=7$ , $n=1$ this implies an APA exponent of MM $\\omega =3\\log _{49}31.5<2.66$ .",
"(Here we use Schönhage's result of [133] that deduces the MM exponent from Disjoint MM.)",
"The above APA decomposition using $mnp+mn+np$ terms is numerically unstable.",
"Indeed we would corrupt the output if we drop the summands $\\lambda u_{jk}$ and $\\lambda v_{ki}$ in the sums $x_{ij}+\\lambda u_{jk}$ and $y_{jk}+\\lambda v_{ki}$ in the aggregate $(x_{ij}+\\lambda u_{jk})(y_{jk}+\\lambda v_{ki})(\\lambda ^2 z_{ki}+w_{ij})$ , but keeping these summands doubles the precision required for the representation of these sums.",
"Similarly all other known bilinear APA algorithms are prone to numerical stability problems if their rank exceeds their border rank.",
"In [14], however, Bini proved that, for the purpose of decreasing the exponent of MM, this deficiency is immaterial if we allow unrestricted recursive processes, that is, if we ignore the curse of recursion.",
"Namely he proved that Theorem REF holds even if border rank replaces rank in its statement.",
"Bini proved this result for MM, but Schönhage in [133] extended it to Disjoint MM.",
"Both proofs yield acceleration of straightforward MM only where its input size becomes huge because of the curse of recursion.",
"Theorem 7.1 We can perform $MM(K)$ by using $\\bar{c}K^{\\omega }$ arithmetic operations for any $K$ , where $\\bar{c}$ is a constant independent of $K$ and $\\omega =\\omega _{m,n,p,r}=3\\log _{mkn}(r)$ , provided that we are given a bilinear or trilinear APA algorithm having a border rank $r$ for $MM(m,n,p)$ and a 4-tuple of integers $r$ , $m$ , $n$ , and $p$ such that $mnp>1$ .",
"First observe that by means of interpolation we can extend any APA algorithm of border rank $r$ using a polynomial $q(\\lambda )$ in $\\lambda $ , of a degree $d$ to a $\\lambda $ -free bilinear algorithm for $MM(m,n,p)$ of rank $(2d+1)r$ .",
"Now apply such an APA algorithm recursively.",
"Notice that every recursive step squares the original problem size $mnp$ but only doubles the degree of $\\lambda $ .",
"After $h$ steps, we arrive at an APA algorithm of degree $2^hd$ for the MM problem of size $(mnp)^{2^h}$ , and then (cf.",
"Theorem REF ) the interpolation factor $2(2^hd+1)$ only implies an increase of the MM exponent $\\omega $ by a tiny factor, which converges to 1 as the number of recursive steps grows to the infinity."
],
[
"Inner Product Computation and\nSummation by Means of APA Techniques\n",
"Next we cover an application of APA techniques beyond MM.",
"Recall the APA algorithm of Section , let the entries $x_{ij}$ , $y_{jk}$ , $u_{jk}$ , and $v_{ki}$ be integers in the range $[0,2^d)$ , and choose $\\lambda =2^d$ .",
"Notice that the product $(x_{ij}+\\lambda y_{jk})(u_{jk}+\\lambda v_{ki})$ fits the length $L$ of the computer word if $L\\ge 4d$ .",
"Moreover if the ratio $L/d$ is large enough, we can perform the APA computations of Section within the precision $L$ .",
"[121] exploits such observations further and devise efficient algorithms for multiplication of vectors and matrices filled with bounded integers.",
"Next we recall that technique and in Example REF show its surprising extension.",
"Suppose that the coefficient vector of a polynomial $v(\\lambda )=\\sum _{i=0}^{n-1} v_i\\lambda ^{i}$ is filled with integers from the semi-open segment $[0,2^d)$ of the real axis for a positive integer $d$ .",
"Represent this vector by the $2^d$ -ary integer $v(2^d)=\\sum _{i=0}^{n-1} v_i2^{di}$ .",
"Generally the interpolation to a polynomial of degree $n-1$ requires its evaluation at $n$ knots, but in the above special case we only need the evaluation at the single knot $2^d$ .",
"Now suppose that all coefficients $v_i$ are integers from the semi-open segment $[q,r)$ for any pair of integers $q$ and $r$ , $q<r$ .",
"Then we can apply the above recipe to compute the shifted vector ${\\bf u}=(u_i)_{i=0}^{n-1}=(v_i-q)_{i=0}^{n-1}$ , having all its components in the semi-open segment $[0,s)$ for $s=r-q$ .",
"We can finally recover the vector ${\\bf v}$ from ${\\bf u}$ .",
"By following [116] and [17], we call this technique binary segmentation.",
"Its history can be traced back to [67], and one can even view it as an application of the Kronecker map, although having specific computational flavor.",
"Next we follow [121] to compute the inner product of two integer vectors, then extend the algorithm to summation, and finally list various other applications of binary segmentation.",
"Example 8.1 (The inner product of two integer vectors, cf.",
"[121].)",
"Assume two nonnegative integers $g$ and $h$ and two vectors ${\\bf u}=(u_i)_{i=0}^{n-1}$ and ${\\bf v}=(v_i)_{i=0}^{n-1}$ with nonnegative integer coordinates in two semi-open segments, namely, $[0,2^g)$ for the coordinates of ${\\bf u}$ and $[0,2^h)$ for the coordinates of ${\\bf v}$ .",
"The straightforward algorithm for the inner product ${\\bf u}^T{\\bf v}=\\sum _{i=0}^{n-1} u_iv_i$ first computes the $n$ products $u_iv_i$ for $i=0,1,\\dots ,n-1$ and then sums them.",
"This involves $n$ multiplications and $n-1$ additions.",
"Instead, however, we can just multiply a pair of bounded nonnegative integers, apply binary segmentation to the product, and output the desired inner product.",
"Namely, introduce the two polynomials $u(x)=\\sum _{i=0}^{n-1} u_ix^i$ and $v(x)=\\sum _{i=0}^{n-1} v_ix^{n-1-i}$ .",
"Their product is the polynomial $q(x)=u(x)v(x)=\\sum _{i=0}^{2n-2} q_ix^i$ with integer coefficients in the segment $[0,2^k)$ for $k=g+h+\\lceil \\log _2 n\\rceil $ .",
"The coefficient $q_{n-1}=\\sum _{i=0}^{n-1} u_iv_i$ is precisely the inner product ${\\bf u}^T{\\bf v}$ .",
"Represent the polynomials $u(x)$ and $v(x)$ by their integer values $u(2^k)$ and $v(2^k)$ at the point $2^k$ .",
"Clearly, they lie in the semi-open segments $r_u=[0,2^{nk+g})$ and $r_v=[0,2^{nk+h})$ , respectively.",
"Now compute the integer $q(2^k)=u(2^k)v(2^k)$ , lying in the segment $[0,2^{2nk+g+h})$ , and recover the coefficient $q_{n-1}={\\bf u}^T{\\bf v}$ by applying binary segmentation.",
"Remark 8.1 We only seek the coefficient $q_{n-1}$ of the median term $q_{n-1}x^{n-1}$ of the polynomial $u(x)v(x)$ .",
"This term lies in the segment $[2^{(n-1)k},2^{(n-1)k+g+h})$ , and the next challenge is to optimize its computation.",
"Is such a more narrow task substantially simpler than the multiplication of two integers lying in the segments $r_u$ and $r_v$ ?",
"Example 8.2 (Summation of bounded integers.)",
"For ${\\bf u}=(1)_{i=0}^{n-1}$ , $g=0$ , and $k=h+\\lceil \\log _2 n\\rceil $ or ${\\bf v}=(1)_{i=0}^{n-1}$ , $h=0$ , and $k=g+\\lceil \\log _2 n\\rceil $ , the algorithm of Example REF outputs the sum of $n$ integers.",
"Remark 8.2 In the same way as for polynomial interpolation in the beginning of this section, we can relax the assumption of Examples REF and REF that the input integers are nonnegative.",
"Moreover, the summation of integers can be extended to the fundamental problem of the summation of binary numbers truncated to a fixed precision.",
"In Examples REF and REF , multiplication of two long integers followed by binary segmentation replaces either $2n-1$ or $n$ arithmetic operations, respectively.",
"This increases the Boolean (bit-wise operation) cost by a factor depending on the Boolean cost of computing the product of 2 integers or, in view of Remark REF , of computing the median segment in the binary representation of the product.",
"The increase is minor if we multiply integers in nearly linear Boolean time (see the supporting algorithms for such multiplication in [135], [1], [69]), but grows if we multiply integers by applying the straightforward algorithm, which uses quadratic Boolean time.",
"Nonetheless, in both cases one could still benefit from using Example REF if the necessary bits of the output integer fit the computer word (i.e.",
"the bits of the middle coefficient are not part of the overflow of the product), as long as the representation of the vector as an integer requires no additional cost.",
"If the output integer does not fit the word length, we can apply the same algorithms to the subproblems of smaller sizes, e.g., we can apply the algorithms of Examples REF and REF to compute the inner products of some subvectors or partial sums of integers, respectively.",
"Other applications of binary segmentation include polynomial multiplication (that is, the computation of the convolution of vectors) [67], [134], some basic linear algebra computations [121], polynomial division [17], [134], computing polynomial GCD [37], and discrete Fourier transform [134].",
"Binary segmentation can be potentially efficient in computations with Boolean vectors and matrices.",
"E.g., recall that Boolean MM is reduced to MM whose input and output entries are some bounded nonnegative integers (see [1]).",
"Quantized tensor decompositions is another promising application area (cf.",
"[148], [106], [107], [91], [109], [72])."
],
[
"Summary of the Study of the MM Exponents after 1978",
"In 1979–81 and then again in 1986 the exponent of infeasible MM was significantly decreased based on combination of trilinear aggregation, Disjoint MM, and APA techniques with unrestricted use of recursion.",
"All supporting algorithms have been built on the top of the techniques of the preceding papers.",
"More and more lenient basic bilinear/trilinear decompositions of small rank were chosen for Disjoint MM of small sizes and since 1986 for small bilinear/trilinear problems similar to Disjoint MM.",
"Transition back to MM relied on nested recursion, consistently intensified; consequently acceleration of the straightforward algorithm began only with MM of astronomical sizes.",
"By 1987 the power of these techniques seems to be exhausted, and then the progress has stopped until 2010.",
"Since then it is moving from the bound 2.376 of [44] towards 2.37 with the snail's speed.",
"That direction was prompted by the cited results of the seminal papers [14] by Dario Bini and [133] by Arnold Schönhage.",
"Schönhage, however, has concluded the introduction of [133] with pointing out that all new exponents of MM were just \"of theoretical interest\" because they were valid only for the inputs \"beyond any practical size\" and that \"Pan's estimates of 1978 for moderate\" input sizes were \"still unbeaten\".",
"Actually, as we can see in Figure REF , the exponent 2.7962 of 1978 for $MM(n)$ restricted to $n\\le 1,000,000$ has been successively decreased in [114], [115], [117], and [118] (cf.",
"also [119]), although by small margins.",
"As of December 2016, the exponent 2.7734 of [118] is still record low for $MM(n)$ with $n\\le 1,000,000$ .",
"Figures REF and REF display chronological decrease of the exponents of $MM(n)$ for $n\\le 1,000,000$ and for unrestricted $n$ , respectively.",
"The supporting algorithms of Figure REF rely solely on trilinear aggregation, and the associated overhead constants are small.",
"Some of these algorithms have been refined in [119], [94] and implemented in [87] and [88].",
"All other algorithms of Figure REF (not supporting Figure REF ) employ trilinear aggregation as well (cf.",
"[44]), but also employ other techniques and suffer from the curse of recursion.",
"The figures link each exponent to its recorded publication in a journal, a conference proceedings, or as a research report.",
"As we already mentioned, the MM exponent of [113] significantly decreased in 1979–1981 and 1986.",
"It has been updated at least 4 times during the single year of 1979: reaching below the values 2.801 in February in Research Report [115]; 2.7799 (as an APA MM exponent) in [15] in June and (as an MM exponent) in [14]; 2.548 in [133], and 2.522 in [117].",
"Both of the latter two exponents appeared in the book of abstracts of the conference on the Computational Complexity in Oberwolfach, West Germany, organized by Schnorr, Schönhage and Strassen in October (cf.",
"[119] and [133]).",
"The exponent 2.496 of [43] was reported in October 1981 at the IEEE FOCS'81, and in Figure REF we place it after the exponent 2.517 of the paper [131] of 1982, which was submitted in March 1980.",
"The Research Report version of the paper [44] appeared in August of 1986, but in Figure REF we place [44] after the paper [146], published in October of 1986 in the Proceedings of the IEEE FOCS, because the paper [146] has been submitted to FOCS'86 in the Spring of 1986 and has been widely circulated afterwards.",
"One could complete the historical account of Figure REF by including the exponents 2.7804 (announced in the Fall of 1978 in [113] and superseded in February 1979 when the paper was submitted [115]) and 2.5218007, which decreased the exponent 2.5218128 of [114] and appeared at the end of the final version of [133] in 1981, that is, before the publication, but after the submission of the exponent 2.517 of [131].",
"We refer the reader to [41], [99], [42], [79], [89], [96], and the references therein for similar progress in asymptotic acceleration of rectangular MM.",
"Figure: MM(n)MM(n) exponents for n≤1,000,000n\\le 1,000,000.Figure: MM(n)MM(n) exponents for unrestricted nn."
],
[
"Applications of Fast MM, Its Links to Other Subject Areas, Impacts of Its Study,\nand Implementation (Briefly)",
"The decrease of the exponent of MM implies theoretical acceleration of the solution of a number of important problems in various areas of computations in Algebra and Computer Science, such as Boolean MM, computation of paths and distances in graphs, parsing context-free grammars, the solution of a nonsingular linear system of equations, computations of the inverse, determinant, characteristic and minimal polynomials, and various factorizations of a matrix.",
"See [142], [34], [1], [24], [18], [3], [79], [48], [98], [160], [157], [158], [159], [86], [25], [89], [6], [54], [156], [132], [96], [4], [138], [140], [103], [104], [105], [125], and the bibliography therein and notice that some new important applications have been found very recently, e.g., in 4 papers at ISSAC 2016.",
"The reduction of other computational problems to MM increases the overhead, which is already immense for the algorithms supporting the record exponent of MM.",
"Such a reduction, however, can still be valuable because it reveals some links of independent interest and because by applying fast algorithms or even the straightforward algorithm for feasible MM one can take advantage of using block matrix software and parallel acceleration.",
"Research on fast feasible MM had a variety of feasible links to other subject areas.",
"The work on bilinear and trilinear decompositions for fast MM was an important part of the study of such decompositions in the general area of Algebraic Computations and has led to new insights and techniques.",
"We have already cited historical importance of the demonstration in 1972 in [112] of the power of tensor decompositions and valuable applications of duality to the design of efficient bilinear algorithms.",
"Trilinear aggregation was also a surprising demonstration of the power of aggregation/disaggregation methods.",
"More applications of this kind can follow in the future, such as a quite unexpected application of APA techniques to the computation of inner products and summation presented in Section .",
"It may be surprising, but apart from trilinear aggregation and the APA method, the advanced and amazing techniques developed for decreasing the exponent of infeasible MM had no theoretical (as well as practical) applications to other areas of computations or algebra (and have made no impacts on actual work on MM as well).",
"Of course, such impacts on the practice of performing this fundamental operation of modern computations are the main motivation and goal of the study of fast MM, and indeed recursive bilinear algorithms based on $2\\times 2$ Strassen's and particularly Winograd's brilliant designs of Examples REF and REF , respectively, are a valuable part of modern software for MM.",
"In Section REF we commented on numerical stability issues for fast feasible MM, and we refer the reader to [12], [73], [52], [61], [62], [28], [45], [71], [13], [8], [11], and the bibliography therein for their previous and current numerical implementation.",
"In the next section we discuss symbolic application of the latter algorithm (WRB-MM) in some detail.",
"It is very encouraging to observe dramatic increase of the activity in numerical and symbolic implementation of fast MM in recent years, towards decreasing communication cost and improving parallel implementation, in good accordance with the decrease of the arithmetic cost."
],
[
"Fast methods in exact computational linear algebra",
"Next we discuss applications and implementations of exact fast matrix multiplication (MM).",
"As we mentioned in Section REF , we first review how most of the exact linear algebra can be reduced to MM over small finite fields.",
"Then we highlight the differences in the design of approximate and exact implementations of MM taking into account nowadays processor and memory hierarchies."
],
[
"Acceleration of computations via reductions to MM",
"The design of matrix multiplication routines over a word size finite field is the main building block for computations in exact dense linear algebra, represented with coefficient domains of two kinds: word-size discrete entries (mainly in finite fields of small cardinality) and variable size entries (larger finite fields, integers, or polynomials).",
"Indeed, efficient methods for the latter task usually reduce it to the former one: reductions are made by means of evaluation/interpolation (Chinese remaindering over the integers) or by means of lifting small size solutions (via Hensel-like or high-order lifting [139]), see, e.g., [85], [63], [124] for recent surveys.",
"Over word-size finite fields, efficient implementations of MM have been obtained by means of effective algorithmic reductions originated in the complexity analysis.",
"In this case reductions have been obtained directly to MM.",
"Already Strassen in 1969 extended MM to matrix inversion (we sketch this in Examples REF and REF below), then Bunch and Hopcroft [34] extended MM to invertible LUP factorization, and further extensions followed in the eighties, for instance, in [82] to the computation of the matrix rank and of Gaussian elimination.",
"Efficient algorithms whose complexity is sensitive to the rank or to the rank profile have only very recently been discovered [83], [64], [140].",
"One of the latest reductions to MM was that of the characteristic polynomial, which has extended the seminal work of [90] more than thirty years afterwards [126].",
"Example 11.1 Triangular system solving by means of reduction to matrix multiplication.",
"Denote by $X=TRSM(U,B)$ the solution to the linear matrix equation $UX=B$ with a matrix $B$ on the right-hand side and an upper triangular invertible matrix $U$ .",
"Recursively cut the matrices $U$ , $B$ and $X$ in halves as follows: $U=\\begin{bmatrix} U_1 & V\\\\ & U_2\\end{bmatrix}$ , $X=\\begin{bmatrix} X_1 \\\\ X_2\\end{bmatrix}$ , and $B= \\begin{bmatrix} B_1 \\\\ B_2\\end{bmatrix}$ ; then obtain an efficient reduction of the solution to MM by means of the following algorithm: Recursively compute $X_2=TRSM(U_2,B_2)$ ; Compute $B^{\\prime }_1=B_1-VX_2$ ; // via fast MM Recursively compute $X_1=TRSM(U_1,B^{\\prime }_1)$ .",
"The only operations performed are fast MMs.",
"Asymptotically the low complexity is preserved, and in practice this reduction can be made very efficient, even in the case of exact computations, where intermediate reductions might occur, as shown, e.g., in [62].",
"Example 11.2 Reduction of LU factorization to MM.",
"For simplicity, consider an invertible matrix $A$ having generic rank profile (that is, having all its leading principal minors also invertible).",
"Recursively cut $A$ into halves in both dimensions, that is, represent it as $2\\times 2$ block matrix, $A=\\begin{bmatrix} A_1 & A_2 \\\\ A_3 & A_4 \\end{bmatrix}$ .",
"Then an efficient triangularization $A=LU$ can be computed by means of the following algorithm: Recursively compute $L_1 U_1 = A_1$ ; // Triangularization of the upper left block Compute $G=TRSM(U_1,A_3)$ ; // $G$ is such that $GU_1=A_3$ Compute $H=TRSM(L_1,A_2)$ ; // $H$ is such that $L_1H=A_2$ Compute $Z=A_4-GH$ ; // via fast MM Recursively compute $L_2,U_2=Z$ ; Return $L=\\begin{bmatrix} L_1 &\\\\ G & L_2\\end{bmatrix}$ and $U=\\begin{bmatrix} U_1 & H\\\\ & U_2\\end{bmatrix}$ .",
"Once again, the only operations performed in this algorithm are MMs, making it efficient as long as MM is efficient.",
"This reduction to MMs would remain efficient in the extension to the more general case where rank deficiencies are allowed and pivoting is applied [64].",
"Example 11.3 Sketch of linear system solving in arbitrary precision.",
"Next we accelerate the solution of a linear system of equations with arbitrary precision by using the two previous reductions to exact MM.",
"The idea is to solve the linear system modulo a small prime $p$ , by intensively using fast exact MM, and then to reuse this factorization in Hensel-like $p$ -adic lifting producing iterative refinement of the solution modulo $p^k$ .Hensel-like $p$ -adic lifting is a major tool of computer algebra, which has striking similarity with the classical algorithm of iterative refinement in numerical linear algebra.",
"This leads us to the following algorithm: Successively compute $L_p,U_p \\equiv A \\mod {p}$ ; // Triangularization modulo a small prime $x_0 \\equiv TRSM(U_p, TRSM(L_p, b) \\mod {p}) \\mod {p}$ ; $b_1 = \\frac{b-Ax_0}{p}$ ; // this computation is over $\\mathbb {Z}$ $x_1 \\equiv TRSM(U_p, TRSM(L_p, b_1) \\mod {p}) \\mod {p}$ ; $b_2 = \\frac{b_1-Ax_1}{p}$ ; // this computation is over $\\mathbb {Z}$ $x_2 \\equiv TRSM(U_p, TRSM(L_p, b_2) \\mod {p}) \\mod {p}$ ; $\\ldots $ One can easily show that $b \\equiv A( \\sum _{i=0}^{k-1} x_i p^i ) \\mod {p}^k$ .",
"Now recall that we can recover unique rational number $r=\\frac{a}{b}$ from its $p$ -adic series representation truncated at $p^k$ as soon as $k$ is such that $p^k$ is larger than $2ab$ and the denominator $b$ is coprime to $p$ .The recovery of a rational $r$ from its $p$ -adic truncated series is called rational number reconstruction which is an accelerated Euclidean algorithm, see, e.g., [70] for more details.",
"Namely, by combining Cramer's rule and Hadamard's bound on determinants, we represent solution values $x_i$ as rational numbers with bounded denominators and then recover them from their truncated $p$ -adic series representation by applying sufficiently many steps of iterative refinement.",
"More details on actual values of $p$ and $k$ can be found in Dixon's seminal paper [51].",
"We can prove (and this is important for practical application) that the overhead of this exact iterative refinement is quite a small constant.",
"We show this in Figure REF , for which we used the LinBoxhttps://github.com/linbox-team/linbox exact linear algebra library: despite quite large coefficient growths (for instance, up to forty thousand bits for the solution to a $8000\\times {}8000$ random integer matrix with 32 bits entries), approximate and exact arbitrary precision times remain essentially proportional.",
"Figure: Comparison of approximate and arbitrary precision linear system solvingon an Intel Xeon W3530 @2.80GHz.We achieve good practical performance of computations in linear algebra by extensively applying reductions to MM, which we mostly perform exactly over small finite fields.",
"On the one hand, nowadays machine architecture, with their memory hierarchy, is also well-adapted to the highly homogeneous structure of the straightforward MM.",
"This is true for numerical routines, where the stability issues have been worked out for fast MM in [16], [73], [50], [8].",
"This is also true for exact routines where, symmetrically, the costly reductions (modular or polynomial) have to be delayed as much as possible.",
"On the other hand, this straightforward algorithm remains faster in practice only for small matrices, but the larger memory capacity and the increase in the number of computing units makes it possible to handle, on desktop computers, dense matrices of dimensions in several hundred thousands.",
"As the practical threshold between straightforward and fast algorithms is often for matrices of dimensions about 500, several fast routines can be very effectively combined, yielding the fastest implementations to date, as explained in the next section."
],
[
"Design of fast exact matrix multiplication over word-size prime\nfields",
"The design of matrix multiplication routines over a word size finite field is the main building block for computations in exact dense linear algebra.",
"In practice, a turning point has been the introduction of the following principles in [58]: finite field arithmetic is reduced to integer arithmetic with delayed or simultaneous modular reductions; integer arithmetic is performed by floating point units (in order to take advantage of SIMD instructions and of numerical routines development – BLAS); computations are structured in blocks in order to optimize the use of the memory hierarchy of current architectures; asymptotically fast algorithms are used, mostly recursive bilinear algorithm for MM based on Winograd's $2\\times 2$ MM of Example REF (see also [52], [58], [28], hereafter we denote this algorithm WRB-MM), but also Kaporin's [88] and Bini-Capovani-Lotti-Romani's [15], [26] algorithms are used.",
"The idea is to convert a finite field matrix into its integer representation, then perform the multiplication over the integers (potentially using, exactly, a floating point representation) and convert back the result into the finite field.",
"First, a floating point representation allows us to use a sustainable code that rely on more widely supported numerical linear algebra routines.",
"Further, on the one hand, machine division (for reductions) and integer units are slower than the respective floating point operations (for instance, not all integral operations have yet SIMD support [74]).",
"But, on the other hand, as the computed results are exact, one can apply the fastest implementations of the asymptotically fast algorithms not worrying about numerical stability problems (which, even though never serious for fast MM, as this has been proved in [16], [50], can require some extra work).",
"Over finite fields, with arbitrary precision or with polynomials, it is nowadays much faster to perform additions or even multiplications than modular or polynomial reductions.",
"To take the most of the machine representation, reduction is delayed or grouped using the following techniques: Definition 11.1 Delayed reduction is the action of replacing a classical dot product with reductions, $\\sum _{i=0}^n Reduce(a_i b_i)$ , by an algorithm that reduces only from time to time, for instance, when the internal representation cannot hold the growth anymore: Compute $s=\\sum _{i=0}^{k} a_i b_i$ ; $Reduce(s)$ ; For $j=1$ to $n/k-1$ compute $s = s + \\sum _{i=jk+1}^{(j+1)k} a_ib_i$ ; $Reduce(s)$ ; done; Simultaneous modular reduction is the action of performing a single reduction on several coefficients at once: in other words, the idea is to replace independent reductions (for instance $Reduce(a)$ ;$Reduce(b)$ ;$Reduce(c)$ ) by a single reduction on grouped elements (use $t=Group(a,b,c)$ ;$Reduce(t)$ ;$a,b,c=Ungroup(t)$ instead), see, e.g., [56] for more details.",
"Moreover, in practice, for exact computations, recursion is important when fast algorithms are used: while tiling allows a static placement of blocks adapted to the memory hierarchy, larger blocks allow faster acceleration but also more delays in the application of reductions (modular reduction, polynomial reduction, etc.) [60].",
"Further, the fast algorithms can be used for a few levels of recursion and then switch back to the, statically placed, straightforward algorithm on smaller matrices.",
"Eventually (nowadays, on a current personal computer, this can typically be between n=500 and n=1000), the exact algorithm sketched above is based on the numerical BLAS and then applied without any further recursion.",
"There is actually a cascade of algorithms where, at each recursive cutting, a new choice of the best suited variant is performed before the final switch [68], [27].",
"This cascade is usually non-stationary, i.e., a different scheme, bilinear or not, can be chosen at every level of recurrence.",
"For instance, a meta-algorithm starts by partitioning the matrices into four blocks followed by application of WRB-MM.",
"This meta-algorithm is called on each of the seven multiplications of the blocks; at the recursive call the meta-algorithm can decide whether to re-apply itself again to a $2\\times 2$ block matrix or to switch either to a (2,2,3) APA algorithms (which in turns will also call this meta algorithm recursively) or to the straightforward algorithm, etc.",
"This impacts the frequency at which the modular reductions can be delayed.",
"For instance, with a classical MM and elements of a prime field modulo $p>2$ represented as integers in $\\lbrace \\frac{1-p}{2}\\ldots \\frac{p-1}{2}\\rbrace $ , on a type with a mantissa of $m$ bits, the condition is that the modular reduction in a scalar product of dimension $k$ can be delayed to the end as long as $k\\left(\\frac{p-1}{2}\\right)^2<2^m$ .",
"When applying $\\ell $ recursive levels of WRB-MM algorithm, it can be showed instead that some intermediate computations could grow above this bound [62], and the latter condition becomes $9^\\ell \\lfloor \\frac{k}{2^\\ell }\\rfloor \\left(\\frac{p-1}{2}\\right)^2<2^m$ .",
"This requires to perform by a factor of about $(9/2)^\\ell $ more modular reductions.",
"Example of sequential speed obtained by the fgemm routine algorithm of the LinBox libraryWithin its FFLAS-FFpack module [62], https://github.com/linbox-team/fflas-ffpack [57] is shown in Table REF .",
"Table: Effective Gfops (2n 3 /time/10 9 2n^3/time/10^9) of matrixmultiplications: LinBox fgemm vs OpenBLAS (s|d)gemm on one core of a Xeon E5-4620 @2.20GHzThe efficiency of the fgemm routine is largely due to the efficiency of the BLAS, but as the latter are not using fast algorithms, exact computations can be faster.",
"Modulo 37 elements are stored over single precision floats, and the sgemm subroutine can be used, whereas modulo 131071 elements are stored using double precision float, and the dgemm subroutine is used.",
"Table REF first shows that the overhead of performing the modular reductions in the $\\mathcal {O}\\left(n^3\\right)$ implementations is very limited if the matrix is large enough.",
"Then, when enabling WRB-MM $\\mathcal {O}\\left(n^{2.81}\\right)$ algorithm, a speed-up of up to $40\\%$ can be attained in both single and double precision arithmetic.",
"More recently, it has been shown also how algorithm [15] by Bini et al.",
"could be efficiently put into practice [26], also offering some interesting speed-up for prime fields of size near 10 bits."
],
[
"Memory efficient schedules",
"WRB-MM algorithm requires external temporary memory allocations in order to store the intermediate linear combinations of blocks.",
"With large matrices and current architectures, this can be penalizing because the memory availability and access to it dominate the overall computational costs.",
"It is therefore crucial to reduce as much as possible the extra memory requirements of fast methods.",
"This can be done via a careful scheduling of the steps of WRB-MM algorithm: it is not mandatory to perform 8 pre-additions, 7 multiplications and 7 post-additions in that order, with temporary memory allocations for each of these 22 steps.",
"Depending on the associated dependency graph, one can choose instead to reduce the number of allocation by following this graph and overwriting already allocated memory when the associated variable is not used any more.",
"For the product of $n\\times {}n$ matrices, without accumulation, $C\\leftarrow {}A \\times {} B$ , [52] proposed a schedule requiring, apart from $C$ , two extra temporary blocks of size $\\frac{n}{2}\\times \\frac{n}{2}$ at the first recursive levels, two extra temporary blocks of size $\\frac{n}{4}\\times \\frac{n}{4}$ for each of the seven recursive calls, etc.",
"Overall, the needed extra memory is bounded by $\\frac{2}{3}n^2$ .",
"For the product with accumulation, $C\\leftarrow {} C+ A \\times {} B$ , for more than ten years the record was three temporaries with an extra memory bounded by $n^2$ [81], but this was recently improved to two temporaries in [28].",
"Notice that [28] proposed also some schemes requiring smaller extra memory (that can actually be made quite close to zero), with the same asymptotic complexity as WRB-MM algorithm, although with a larger constant overhead factor in arithmetic operations.",
"Recently M. Bodrato proposed a variant of WRB-MM algorithm, which is symmetric and more suitable to squaring matrices, but which uses similar schedules, and therefore keeps the extra requirements [22].",
"This is not the case in [80] where no extra memory is required if only one or two recursive levels of fast MM are used, but at the cost of recomputing many additions.",
"Finally, in the case of the APA algorithm [15] by Bini et al., the requirements are also of two temporaries in the product without accumulation [26].",
"From [81], new schedules have usually been discovered by hand, but with the help of a pebble game program, which discards rapidly the wrong schedules and verifies formally the correct ones."
],
[
"Tiny finite fields",
"The practical efficiency of MM depends greatly on the representation of field elements.",
"Thus we present three kinds of compact representations for the elements of a finite field with very small cardinality: bit-packing (for ${\\mathbb {F}}_2$ ), bit-slicing (say, for ${\\mathbb {F}}_3, {\\mathbb {F}}_5, {\\mathbb {F}}_7, {\\mathbb {F}}_{2^3}$ , or ${\\mathbb {F}}_{3^2}$ ), and Kronecker substitution.",
"These representations are designed to allow efficient linear algebra operations, including MM: Definition 11.2 Compact representations for small finite fields.",
"Over ${\\mathbb {F}}_2$ , the method of the four Russians [7], also called Greasing, can be used as follows [2]: A 64-bit machine word can be used in order to represent a row vector of dimension 64.",
"Multiplication of an $m\\times k$ matrix $A$ by an $k\\times n$ matrix $B$ can be done by first storing all $2^k$ $k$ -dimensional linear combinations of rows of $B$ in a table.",
"Then the $i$ -th row of the product is copied from the row of the table indexed by the $i$ -th row of $A$ .",
"By ordering indices of the table according to a binary Gray Code, each row of the table can be deduced from the previous one, using only one row addition.",
"This decreases the bit-operation count for building the table from $k2^kn$ to $2^kn$ .",
"Choosing $k=\\log _2n$ in this method implies $MM(n)=\\mathcal {O}\\left(n^3/\\log n\\right)$ over ${\\mathbb {F}}_2$ .",
"In practice, the idea is once again to use a cascading algorithm: at first some recursive steps of fast MM is performed, and then, at a size small enough, one should switch to the greasing.",
"Bit-slicing consists in representing an $n$ -dimensional vector of $k$ -bit sized coefficients by using $k$ binary vectors of dimension $n$ [23].",
"In particular, one can apply Boolean word instruction in order to perform arithmetic on 64 dimensional vectors.",
"Over ${\\mathbb {F}}_3$ , the binary representation $0 \\equiv [0,0], 1\\equiv [1,0], -1 \\equiv [1, 1]$ allows us to add and subtract two elements in 6 Boolean operations: $\\begin{array}{ll}\\text{Add}([x_0,x_1],[y_0,y_1]) : & s \\leftarrow x_0\\oplus y_1 , t \\leftarrow x_1 \\oplus y_0\\\\& \\text{Return} (s\\wedge t, (s\\oplus x_1) \\vee (t\\oplus y_1))\\\\\\text{Sub}([x_0,x_1],[y_0,y_1]) : & t\\leftarrow x_0\\oplus y_0 \\\\& \\text{Return}(t\\vee (x_1\\oplus y_1), (t\\oplus y_1)\\wedge (y_0\\oplus x_1))\\end{array}$ Over ${\\mathbb {F}}_5$ (resp.",
"${\\mathbb {F}}_7$ ), a redundant representation $x=x_0+2x_1+4x_2 \\equiv [x_0,x_1,x_2]$ allows us to add two elements by using 20 (resp.",
"17) Boolean operations, negate in 6 (resp.",
"3) Boolean operations, and double by using 5 (resp.",
"0) Boolean operations.",
"Table: Boolean operation counts for basic arithmetic by using bit-slicing Bit-packing consists in representing a vector of field elements as an integer that fits in a single machine word by using a $2^k$ -adic representation: $(x_0,\\dots ,x_{n-1})\\in {\\mathbb {F}}_q^n \\equiv X=x_0+2^kx_1+\\dots +(2^k)^{n-1}x_{n-1} \\in \\mathbb {Z}_{2^{64}}$ Elements of extension fields are viewed as polynomials and stored as the evaluation of this polynomial at the characteristic of the field.",
"The latter evaluation is called Kronecker substitution [55].",
"Once we can pack and simultaneously reduce coefficients of the finite field in a single machine word, the obtained parallelism can be used for MM.",
"Depending on the respective sizes of the matrices in the multiplication, one can pack only the left operand, only the right one, or both [56].",
"Then, over the field extensions, fast floating point operations can also be used on the Kronecker substitution of the elements.",
"All these methods improve in fact the base case of dense linear algebra, when fast methods are not competitive anymore.",
"As already mentioned, the generic cascading idea applies: perform at first some recursive steps fast, decreasing the matrix dimension, and, at a small enough size, switch to the (improved) classical triple loop method."
],
[
"Parallelization",
"Now, we focus on the design of a parallel MM routine, which computes the matrix product $AB$ based on the WRB-MM sequential algorithm.",
"In order to parallelize the computation at the coarsest grain, the best approach is to apply first a classical block algorithm, generating a prescribed number of independent tasks, and then each of them will use the sequential WRB-MM algorithm [60], [59].",
"There, the parallel algorithm is recursive and splits the largest of either the row dimension of $A$ or the column dimension of $B$ , to form two independent tasks.",
"The granularity of the split is recursive and terminates whenever the number of created tasks becomes larger than the number of computing resources (e.g., the total number of cores).",
"This maximizes the size of the blocks, and therefore the benefit of WRB-MM algorithm, while ensuring a large enough number of tasks for the computing resources.",
"Figure: Speed of exact and numerical matrix multiplication routineson a 32 cores Intel Xeon E5-4620 2.2Ghz (Sandy Bridge) with 16384KBL3 cache .Figure REF shows the computation time of various MM algorithms: the numerical dgemm implementation of Plasma-Quark, OpenBLAS and Intel-MKL as well as the implementation of pfgemm of LinBox using the OpenMP-4.0 data-flow model.",
"Contrary to MKL, OpenBLAS or Plasma-Quark, this pfgemm routine uses the above sketched splitting strategy with WRB-MM.",
"This implementation is run over the finite field $\\mathbb {Z}/131071\\mathbb {Z}$ or with real double floating point numbers.",
"We first notice that most routines perform very similarly.",
"More precisely, Intel-MKL dgemm is faster on small matrices, but the effect of WRB-MM algorithm makes pfgemm faster on larger matrices, even in the finite field where additional modular reductions occur."
],
[
"Some Research Challenges",
"The field is still in progress, and here are some current observations.",
"(i) The implementations of fast trilinear aggregation algorithms by Kaporin in [87] and [88] are relatively little applied in practice so far, although they have important advantages versus other algorithms now in use: being at least as fast and more stable numerically and allowing highly efficient parallel implementation, they require much less memory.",
"This is because the algorithms of [87] and [88] are defined by trilinear decompositions with supersparse coefficient matrices $A=(\\alpha _{ij}^{(q)})_{(i,j),q}$ , $B=(\\beta _{jk}^{(q)})_{(j,k),q}$ , and $C=(\\gamma _{ik}^{(q)})_{(i,k),q}$ of (REF ), which is a general property of competently implemented trilinear aggregation algorithms.",
"The implementations in [87] and [88] were intended to be initial rather than final steps and were supposed to leave some room for further amelioration.",
"In particular the algorithm of [118], also relying on trilinear aggregation, has similar features, but supports even a smaller exponent, and can be a basis for further progress in the implementation of fast feasible MM.",
"(ii) The more recent algorithms of [136], obtained by means of computer search, are highly promising because they beat the exponent $\\log _2(7)$ already for various small problems $MM(m,n,p)$ where $\\min \\lbrace m,n,p\\rbrace =2$ and $\\max \\lbrace m,n,p\\rbrace \\le 6$ .",
"Their coefficient matrices $A$ , $B$ , and $C$ of (REF ) are quite dense, however, and their tested performance is inferior to recursive bilinear algorithms based on Examples REF and REF .",
"This leads to the challenge of devising MM algorithms that would combine the best features of the algorithms of [87], [88], and [136].",
"(iii) Numerical instability of APA algorithms does not prevent them from being efficient in symbolic computations, but so far only the rudimentary algorithm of [15] has been implemented [26], while much simpler and much more efficient ones are ignored (cf., e.g., our algorithm in Section ).",
"Some researchers in matrix computations still view decreasing the current record MM exponent of about 2.37 towards the lower bound 2 as the major theoretical challenge.",
"For the state of affairs in this area we refer the reader to our Figure REF and the mostly negative results in [4], [5], and [21].",
"We, however, consider breaking the barrier of 2.7733 for the realistic exponent of $MM(n)$ , $n\\le 1,000,000$ , a more important challenge.",
"The exponent 2.773 stays unbeaten since 1982 (that is, longer than Coppersmith–Winograd's barrier of 1986, broken by Stothers in 2010),The title of [149] is a little deceptive.",
"and its decrease should require more powerful tools than the acceleration of infeasible MM because of the limitation on the use of recursive bilinear algorithms.",
"We hope that this important challenge will be met in reasonable time, possibly based on combination of human ingenuity and computer search,Computer search has already helped the authors of [15], [22], and [149] in devising their algorithms.",
"and then significant impact on present day computations should be expected, whereas reaching the exponent 2 for MM of infeasible sizes per se would hardly make such an impact.",
"In view of microscopic progress in the decrease of the exponent of infeasible MM, the present day research directions towards that goal seem to lead to various dead ends, and any potential progress shall most likely rely on radically new insights, ideas and techniques, such as aggregation and mapping the input and output of MM to another dimension (cf.",
"[112], [118], [119] and [94]).",
"In the area of the implementation of MM, further progress with recursive bilinear algorithms based on fast $2\\times 2$ MM is a highly important challenge, and the recent advances by Bodrato [22] and Cenk and Hasan [36] are very encouraging, but it would greatly benefit the field if the researchers will not confine themselves to the geocentric viewpoint of 1969, restricted to $2\\times 2$ -based bilinear recursion, and will also explore heliocentric point of view of XXI century, by opening themselves to the benefits of trilinear world, aggregation, APA, and possibly some other new powerful Strassen-free techniques for fast feasible MM, yet to appear."
],
[
"Acknowledgments",
"Jean-Guillaume Dumas's work is partially supported by the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541).",
"Victor Pan's work has been supported by NSF Grants CCF–1116736 and CCF–1563942 and by PSC CUNY Award 68862–00 46.",
"Furthermore he is grateful to A. Bostan, I.V.",
"Oseledets and E.E.",
"Tyrtyshnikov for their pointers to recent works on MM and the bibliography on quantized tensor decompositions, respectively, to Franklin Lee, Tayfun Pay, and Liang Zhao for their assistance with creating Figures REF and REF , to Igor Kaporin for sharing his expertise on various issues of practical MM and providing information about his papers [87] and [88], and to Ivo Hedtke for his extensive comments to the preprint [123]."
]
] | 1612.05766 |
[
[
"A Generalized Approximation Framework for Fractional Network Flow and\n Packing Problems"
],
[
"Abstract We generalize the fractional packing framework of Garg and Koenemann to the case of linear fractional packing problems over polyhedral cones.",
"More precisely, we provide approximation algorithms for problems of the form $\\max\\{c^T x : Ax \\leq b, x \\in C \\}$, where the matrix $A$ contains no negative entries and $C$ is a cone that is generated by a finite set $S$ of non-negative vectors.",
"While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles.",
"For each of these oracles, we present positive results for the approximability of the packing problem.",
"In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort.",
"In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows.",
"Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof."
],
[
"Introduction",
"In a fractional linear packing problem, one seeks to find a solution to the problem $\\max \\lbrace c^Tx : Ax \\le b, x \\ge 0 \\rbrace $ , where the matrix $A \\in \\mathbb {N}^{m \\times n}_{\\ge 0}$ contains no negative entries and, without loss of generality, the vectors $c \\in \\mathbb {N}_{> 0}^n$ and $b \\in \\mathbb {N}_{> 0}^m$ have positive entries.",
"Many problems can be formulated as packing problems, possibly the most intuitive being the fractional knapsack problem (cf.",
"[27]).",
"More than this, many network flow problems can be seen as fractional packing problems if one allows exponential sized representations.",
"For example, the traditional maximum flow problem can be seen as the problem of packing flows on $s$ -$t$ -paths without violating the capacities of the edges.",
"A large number of authors presented approximation frameworks for such fractional packing problems, including [32], [18], [19], [38], and [3] (cf.",
"[2], [17] for an overview of these results).",
"One of the most powerful frameworks among these was developed by [17], who have shown that a $(1 - \\varepsilon )$ -approximate solution to a general fractional packing problem of the above form can be computed efficiently provided we are able to “handle” the dual problem appropriately.",
"More precisely, in the dual formulation $\\min \\lbrace b^T y: A^T y \\ge c, y \\ge 0\\rbrace $ , we need to be able to determine a most violated dual constraint efficiently: For some given infeasible solution $y$ to the dual, we need to find a dual constraintWe use the notation $B_{l\\cdot }$ for a matrix $B$ to denote the $l$ -th row of the matrix.",
"$(A^T)_{l\\cdot } y \\ge c_l$ that minimizes the value $\\frac{(A^T)_{l\\cdot } y}{c_l}$ among all dual constraints with a positive right-hand side value.",
"This constraint reveals the largest degree of violation.",
"Since the result may even hold if the number of variables in the primal formulation is of exponential size, the authors were able to provide efficient FPTASs for network flow problems such as multicommodity flow problems by using (exponential-sized) path-based formulations of the corresponding problems (cf.",
"[17]).",
"[14] later showed that it suffices to determine an approximately most violated dual constraint, whose fraction $\\frac{(A^T)_{l\\cdot } y}{c_l}$ may be up to a factor $1 + \\varepsilon $ away from the largest violation.",
"In this paper, we generalize the result of [17] to the case of packing problems over polyhedral cones.",
"More precisely, we are interested in approximate solutions to problems of the form $\\max \\ & c^T x \\\\\\text{s.t.",
"}\\ & Ax \\le b, \\\\& x \\in C,$ where the matrix $A \\in \\mathbb {N}^{m \\times n}_{\\ge 0}$ contains no negative numbers and where the cone $C$ is finitely generated by a set $S$ of non-negative vectors.",
"While the vector $b \\in \\mathbb {N}^m_{> 0}$ is still assumed to contain positive entries (without loss of generality), we allow the entries of the vector $c \\in \\mathbb {N}^n$ to have arbitrary signs.",
"We will thereby combine a large set of well-known techniques such as the fractional packing framework of [17], the extension of [14], the parametric search technique of [29], geometric-mean binary search due to [21], and transformation strategies for fractional objectives as described in [28].",
"The chosen formulation is motivated by the following observation: Network flow problems often allow a characterization by some kind of flow decomposition, i.e., each feasible flow is representable by the sum of flows on much simpler structures, which we call basic components in the following.",
"Viewed from the other side, we can express each such feasible flow as a conic combination of flows on these basic components.",
"Hence, if $C$ describes the cone that is generated by flows on basic components, we can express structural properties of each flow by the containment in the cone.",
"What usually remains are packing constraints that bound the total flow on each edge, the flow that leaves some node, or the overall costs of the flow.",
"As it will be shown, most common network flow problems can be modeled in such a way.",
"One major advantage of the presented framework is that we do not assume the cone $C$ or the set $S$ that generates it to be given explicitly.",
"Instead, we only assume that we have some kind of oracle access to the cone, which allows us to derive polynomial-time algorithms for problems that require cones with an exponential number of extreme rays.",
"In addition to this benefit, our framework provides the following advantages: The framework allows to derive fast and simple combinatorial FPTASs for a large class of packing problems and network flow problems.",
"In many cases, we are even able to derive first strongly polynomial-time FPTASs.",
"As our problem is formulated as a packing problem, the addition of further budget-constraints does not influence the applicability of the procedure, whereas such constraints usually make the design of exact algorithms significantly harder.",
"The application of the framework only requires two “ingredients”, namely the existence of some kind of flow decomposition theorem and a decent amount of control over the resulting basic components.",
"As the framework is based on the approximation scheme of [17] in its core, it works without considering some kind of residual network in case of network flow problems.",
"As a consequence, the framework can be applied to problems that do not offer a concept of residual networks.",
"One example is the maximum flow problem in generalized processing networks that will be discussed later.",
"Moreover, it maintains properties of the underlying network such as cycle freeness or signs of costs.",
"In contrast to the framework of [17], our formulation allows to stick to the natural edge-based formulations of the corresponding network flow problems and does not require an explicit reformulation of the problem as a packing problem.",
"Moreover, the presented framework is the first application of the procedure of [17] that natively supports the use of arbitrary linear objective functions, which allows the application to minimum cost flow problems.",
"To the best of our knowledge, all prior applications instead transformed the objective functions into budget-constraints and searched for the optimal budget, which requires the restriction to positive costs and which results in weakly polynomial running times (cf.",
"[17], [14], [13]).",
"The paper is structured as follows: In Section , we briefly introduce the concepts that are inherent to our framework such as the procedure of [17] and Megiddo's parametric search technique [29], [30].",
"In Section , we reformulate the given problem () as a packing problem and identify a subproblem that needs to be solved in each iteration of the algorithm.",
"Moreover, we introduce three oracle types that will be investigated in the rest of the paper: a minimizing oracle returning a cost-minimal vector in the ground set, a sign oracle only returning a vector with the same sign as a cost minimal vector, and a separation oracle either returning a vector with negative costs or stating that there is no such vector.",
"Based on these considerations, we describe the general procedure in Section and show that we can approximate problem () efficiently if we are able to find a sufficiently good initial lower bound on the most violated dual constraint.",
"In Section , we provide both weakly polynomial-time and strongly polynomial-time approaches to find such a lower bound.",
"Finally, we apply our framework to a large class of network flow and packing problems in Section , including the maximum/minimum cost flow problem, generalized minimum cost flow problem, and the maximum/minimum cost flow problem in processing networks as well as budget-constrained versions of these.",
"Moreover, we apply our framework to the maximum (weighted) spanning tree packing problem and the maximum (weighted) matroid base packing problem as examples of “pure” combinatorial problems.",
"In Table REF , we give an overview of the results that will be derived in Section .",
"Table: A summary of our results.",
"Here, mm and nn denote the number of edges and nodes, respectively.",
"The variable kk denotes the number of commodities and CC denotes the largest ratio of any two weights of commodities.",
"MM is the largest integer given in the input.",
"The notation 𝒪 ˜(·)\\mathcal {\\widetilde{O}}(\\cdot ) ignores poly-logarithmic factors in mm, so 𝒪 ˜(f(n,m))=𝒪(f(n,m)·log q m)\\mathcal {\\widetilde{O}}(f(n,m)) = \\mathcal {O}(f(n,m) \\cdot \\log ^q m) for some constant qq.",
"α(m,n)\\alpha (m,n) denotes the inverse Ackermann function.",
"F(m)F(m) is the running time of a independence testing oracle for a given matroid with mm elements in its ground set.An algorithm $A$ is called a (polynomial-time) approximation algorithm with performance guarantee $\\alpha \\in [1,\\infty )$ or simply an $\\alpha $ –approximation for a maximization problem $\\Pi $ with objective function $c$ if, for each instance $I$ of $\\Pi $ with optimum solution $x^*$ , it computes a feasible solution $x$ with objective value $c(x) \\ge \\frac{1}{\\alpha } c(x^*)$ in polynomial time.",
"An algorithm $A$ that receives as input an instance $I \\in \\Pi $ and a real number $\\varepsilon \\in (0,1)$ is called a polynomial-time approximation scheme (PTAS) if, on input $(I,\\varepsilon )$ , it computes a feasible solution $x$ with objective value $c(x) \\ge (1 - \\varepsilon ) \\cdot c(x^*)$ with a running-time that is polynomial in the encoding size $|I|$ of $I$ .",
"If this running-time is additionally polynomial in $\\frac{1}{\\varepsilon }$ , the algorithm is called a fully polynomial-time approximation scheme (FPTAS)."
],
[
"Garg and Koenemann's Fractional Packing Framework",
"Consider a fractional packing problem of the form $\\max \\lbrace c^T x : Ax \\le b, x \\ge 0\\rbrace $ with non-negative entries in the matrix $A \\in \\mathbb {N}^{m \\times n}_{\\ge 0}$ .",
"For example, we can model the maximum flow problem in such a way by interpreting the variables as flows on $s$ -$t$ -paths and requiring that the sum of flows on all paths that share some specific edge $e$ is bounded by the edge's capacity.",
"Hence, the vector $b$ corresponds to the capacities of the edges and the vector $c$ equals the all-one vector.",
"Note that we need to stick to the path-based formulation of the problem since we are not allowed to introduce flow conservation constraints as they require negative coefficients.",
"The dual formulation of the general primal problem is given as $\\min \\lbrace b^T y : A^T y \\ge c, y \\ge 0\\rbrace $ .",
"In the example, the dual problem is to find small edge-lengths such that each path has length at least one.",
"Although both the primal and the dual formulation of the problem are of exponential size in general, the fractional packing framework of [17] allows us to obtain approximate solutions for the primal problem by using these formulations only implicitly, which will be shown in the following.",
"Suppose that we want to find an $(1-\\varepsilon )$ -approximate solution for the primal problem with $\\varepsilon \\in (0,1)$ and let $\\varepsilon ^{\\prime } \\frac{\\varepsilon }{2}$ .",
"The procedure described in [17] starts with the feasible primal solution $x = 0$ and the infeasible dual solution given by $y_i \\frac{\\delta }{b_i} > 0$ for each $i \\in \\lbrace 1,\\ldots ,m\\rbrace $ , where $\\delta \\frac{(1+\\varepsilon ^{\\prime })}{\\left((1+\\varepsilon ^{\\prime })m \\right)^{\\frac{1}{\\varepsilon ^{\\prime }}}}$ .",
"In each step of the algorithm, the most violated dual constraint is determined based on the current dual solution $y$ , i.e., we determine a row $j$ in the dual formulation that minimizes $\\frac{(A^T)_{j\\cdot } y}{c_j}$ among all rows with negative right-hand side value.",
"For example, although there are exponentially many constraints in the dual formulation of the maximum flow problem, we can find the most violated constraint in $\\mathcal {O}(m + n\\log n)$ time by computing a shortest $s$ -$t$ -path with edge lengths given by the dual vector $y$ .",
"We then increase $x_j$ by the (in terms of the primal problem) maximum allowed value $\\nu \\min _{i \\in \\lbrace 1,\\ldots ,m\\rbrace : A_{ij} > 0} \\frac{b_i}{A_{ij}}$ (i.e., in the example, we send $\\nu $ units of flow on the shortest path without considering flow that has been sent in previous iterations), which will most likely make the primal solution infeasible.",
"At the same time, each variable $y_i$ will be multiplied by a factor of $\\left(1 + \\varepsilon ^{\\prime } \\cdot \\frac{\\nu }{\\frac{b_i}{A_{ij}}} \\right)$ .",
"Intuitively, for the maximum flow problem, the “congested” edges will get “longer” over time and will, thus, be used less likely in future iterations, which somehow balances the flow among all paths.",
"The algorithm stops as soon as the dual solution fulfills $\\sum _{i \\in \\lbrace 1,\\ldots ,m\\rbrace } b_i \\cdot y_i \\ge 1$ .",
"As noted above, the primal solution will most likely be infeasible since, in each iteration, the primal variables are increased regardless of the previous values.",
"However, [17] show that we obtain a feasible primal solution by scaling down the solution $x$ by $\\log _{1+\\varepsilon ^{\\prime }} \\frac{1+\\varepsilon ^{\\prime }}{\\delta }$ and that this solution is within a factor $(1 - 2\\varepsilon ^{\\prime })=(1-\\varepsilon )$ of the optimal solution.",
"Moreover, they prove that the described procedure terminates within $\\frac{1}{\\varepsilon ^{\\prime }} \\cdot m \\cdot (1 + \\log _{1+\\varepsilon ^{\\prime }} m) = \\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m \\right)$ iterations.",
"We refer to [17], [14], [13] for further details on the procedure.",
"In a follow-up publication, [14] showed that it suffices to determine an approximately most violated dual constraint in each iteration of the problem: The claimed time bound and approximation guarantee continue to hold even if we choose a dual constraint $(A^T)_{j\\cdot } y \\ge c_j$ only fulfilling $\\frac{(A^T)_{j\\cdot } y}{c_j} \\le (1 + \\varepsilon ) \\cdot \\min _{l \\in \\lbrace 1,\\ldots ,n\\rbrace : c_l > 0} \\frac{(A^T)_{l\\cdot } y}{c_l}$ .",
"We will make use of this observation in Section ."
],
[
"Megiddo's Parametric Search Technique",
"In Section , we make use of Megiddo's parametric search technique (cf.",
"[29]).",
"Since we will go into the very heart of the method, we will briefly describe its idea in the following.",
"We refer the reader to [29] and [30] for further details on the method.",
"Assume that we want to solve an optimization problem $\\Pi $ for which we already know an (exact) algorithm $A$ that solves the problem, but in which some of the input values are now linear parametric values that depend linearly on some real parameter $\\lambda $ .",
"Moreover, suppose that an algorithm $C$ is known (in the following called callback) that is able to decide if some candidate value for $\\lambda $ is smaller, larger, or equal to the value $\\lambda ^*$ that leads to an optimum solution to the underlying problem $\\Pi $ .",
"The idea of the parametric search technique is to extend the input of $A$ from constants to affine functions depending on $\\lambda $ and to simulate the execution of algorithm $A$ step by step.",
"Note that each variable remains its affine structure as long as the algorithm only performs additions, subtractions, multiplications with constants, and comparisons.",
"We call such an algorithm strongly combinatorial in the following.",
"Throughout the execution, an interval $I$ is maintained that is known to contain the optimal value $\\lambda ^*$ .",
"As soon as the simulation reaches the comparisons of two linear parametric values, since both values depend linearly on $\\lambda $ , it either holds that one of the variables is always larger than or equal to the other one in $I$ (in which case the result of the comparison is independent from $\\lambda $ ) or that there is a unique intersection point $\\lambda ^{\\prime }$ .",
"For this intersection point, we evaluate the callback $C$ in order to determine if $\\lambda ^{\\prime } < \\lambda ^*$ , $\\lambda ^{\\prime } > \\lambda ^*$ , or $\\lambda ^{\\prime } = \\lambda ^*$ and, thus, resolve the comparison, update the interval $I$ , and continue the execution.",
"Hence, as soon as the simulation of $A$ finishes, we have obtained an optimum solution to $\\Pi $ .",
"The overall running-time is given by the running-time of $A$ times the running-time of $C$ and can be further improved using parallelization techniques described in [30].",
"We refer to [29], [30] for details on the parametric search technique.",
"Further applications and extensions of parametric search techniques can moreover be found in [8], [34], [35].",
"In this section, we transform the problem () to a general (possibly exponential-sized) fractional packing problem in a first step and then reduce this problem to a more simple subproblem by incorporating the fractional packing framework of [17].",
"Moreover, we introduce three types of oracles that enable us access to the cone $C$ and that will be used throughout the rest of the paper.",
"Let $S \\lbrace x^{(1)}, \\ldots , x^{(k)} \\rbrace $ denote a finite set of $k$ non-negative $n$ -dimensional vectors $x^{(l)} \\in \\mathbb {R}^n_{\\ge 0}$ with $x^{(l)} \\ne 0$ .",
"The cone that is spanned by these vectors is given by $C \\left\\lbrace x \\in \\mathbb {R}^n : x = \\sum _{l=1}^k \\alpha _l \\cdot x^{(l)} \\text{ with } \\alpha _l \\ge 0 \\text{ for all } l \\in \\lbrace 1,\\ldots ,k\\rbrace \\right\\rbrace .$ The main result of this paper is that we are able to compute $(1-\\varepsilon )$ -approximate solutions for the problem to maximize a linear function over the cone $C$ subject to a set of packing constraints under specific assumptions that will be investigated in the following.",
"As we will see in Section , this extended framework is especially useful in the case of network flow problems for which some kind of flow decomposition theorem is known.",
"Note that we do neither require the set $S$ to be of polynomial size nor assume the set $S$ or the cone $C$ to be given explicitly.",
"Instead, as it is common when dealing with implicitly given polyhedra, we only assume the cone to be well-described, which implies that it has an encoding length of at least $n+1$ (cf.",
"[20] for further details).",
"Moreover, we make decisions over $S$ and $C$ via a given oracle $\\mathcal {A}$ (to be specified later) that yields information about $S$ based on a given cost vector $d$ .",
"We assume that the running-time $T_\\mathcal {A}$ of each oracle $\\mathcal {A}$ fulfills $T_\\mathcal {A} \\in \\Omega (n)$ since it would not be able to investigate each component of $d$ or return a vector $x \\in C$ otherwise.",
"In the following, let $A \\in \\mathbb {N}_{\\ge 0}^{m \\times n}$ denote a constraint matrix with non-negative entries, $b \\in \\mathbb {N}_{> 0}^m$ a positive right-hand side vector, and $c \\in \\mathbb {Z}^n$ a cost vector with arbitrary signs.",
"Without loss of generality, we assume that at least one entry in each row and each column of $A$ is positive.",
"Moreover, we define $N$ to be the number of non-zero entries contained in the matrix $A$ .",
"As described above, the problem we want to approximate is given as follows: $\\max \\ & c^T x \\\\\\text{s.t.",
"}\\ & Ax \\le b, \\\\& x \\in C. $ Using the definition of the cone $C$ based on equation (REF ), we obtain the following equivalent formulation of the problem (): $\\max \\ & c^T \\sum _{l=1}^k \\alpha _l \\cdot x^{(l)} \\\\\\text{s.t.",
"}\\ & A \\left( \\sum _{l=1}^k \\alpha _l \\cdot x^{(l)} \\right) \\le b, \\\\& \\alpha _l \\ge 0 && \\text{ for all } l \\in \\lbrace 1, \\ldots , k \\rbrace .$ In particular, we replaced the original variables $x$ by the weight vector $\\alpha $ and, in doing so, incorporated the constraints of the cone.",
"As noted above, this formulation might be of exponential size.",
"However, in the following, we will never need to state it explicitly but will derive results based on its implicit structure.",
"By rearranging the objective function and the packing constraints, we obtain the following equivalent formulation of the problem: $\\max \\ & \\sum _{l=1}^k \\alpha _l \\cdot \\left( c^T x^{(l)} \\right) \\\\\\text{s.t.",
"}\\ & \\sum _{l=1}^k \\alpha _l \\cdot \\left( A_{i\\cdot } x^{(l)} \\right) \\le b_i && \\text{ for all } i \\in \\lbrace 1,\\ldots ,m\\rbrace , \\\\& \\alpha _l \\ge 0 && \\text{ for all } l \\in \\lbrace 1, \\ldots , k \\rbrace .$ Clearly, we can neglect vectors $x^{(l)}$ for which $c^T x^{(l)} \\le 0$ since, without loss of generality, it holds that $\\alpha _l = 0$ for each such $l$ in an optimal solution.",
"Hence, for the moment, we restrict our considerations on vectors $x^{(l)}$ with $c^T x^{(l)} > 0$ such that the primal problem () becomes in fact a fractional packing problem (again, possibly of exponential size).",
"The dual formulation of this problem is given as follows: We will see how we can “filter out” vectors $x^{(l)}$ with negative costs in the following sections.",
"$\\min \\ & \\sum _{i=1}^m y_i \\cdot b_i \\\\\\text{s.t.",
"}\\ & \\sum _{i=1}^m y_i \\cdot \\left( A_{i\\cdot } x^{(l)} \\right) \\ge c^T x^{(l)} && \\text{ for all } l \\in \\lbrace 1,\\ldots ,k\\rbrace \\text{ with } c^T x^{(l)} > 0, \\\\& y_i \\ge 0 && \\text{ for all } i \\in \\lbrace 1, \\ldots , m \\rbrace .$ As it was shown in Section REF , we can apply the fractional packing framework of [17] to the original problem () provided we are able to determine the most violated dual constraint in equation (REF ) efficiently.",
"Hence, given a dual solution $y > 0$ , we need to be able to solve the following subproblem in polynomial time: $&\\min _{\\genfrac{}{}{0.0pt}{}{l \\in \\lbrace 1,\\ldots ,k\\rbrace }{c^T x^{(l)} > 0}} \\dfrac{\\sum _{i=1}^m y_i \\cdot \\left( A_{i\\cdot } x^{(l)} \\right)}{c^T x^{(l)}} = \\min _{\\genfrac{}{}{0.0pt}{}{l \\in \\lbrace 1,\\ldots ,k\\rbrace }{c^T x^{(l)} > 0}} \\dfrac{\\sum _{i=1}^m y_i \\cdot \\sum _{j=1}^n A_{ij} \\cdot x_j^{(l)}}{c^T x^{(l)}} \\\\=&\\min _{\\genfrac{}{}{0.0pt}{}{l \\in \\lbrace 1,\\ldots ,k\\rbrace }{c^T x^{(l)} > 0}} \\dfrac{\\sum _{j=1}^n x_j^{(l)} \\cdot \\sum _{i=1}^m y_i \\cdot A_{ij}}{c^T x^{(l)}}.$ With $a_j \\sum _{i=1}^m y_i \\cdot A_{ij}$ for $j \\in \\lbrace 1,\\ldots ,n\\rbrace $ and $a = (a_1,\\ldots ,a_n)^T$ , this subproblem reduces to $\\min _{\\genfrac{}{}{0.0pt}{}{l \\in \\lbrace 1,\\ldots ,k\\rbrace }{c^T x^{(l)} > 0}} \\dfrac{a^T x^{(l)}}{c^T x^{(l)}}.",
"$ Note that the vector $a$ depends on $y$ and, thus, changes throughout the course of the procedure of [17].",
"However, it always holds that $a_j > 0$ for each $j \\in \\lbrace 1,\\ldots ,n\\rbrace $ since $y_i > 0$ for each $i \\in \\lbrace 1,\\ldots ,m\\rbrace $ throughout the procedure and since the matrix $A$ has at least one positive and no negative entry in each row as assumed above.",
"Since $x^{(l)} \\ne 0$ and $x^{(l)} \\in \\mathbb {R}^n_{\\ge 0}$ for each $l \\in \\lbrace 1,\\ldots ,k\\rbrace $ , this also yields that $a^T x^{(l)} > 0$ , so the minimum in equation (REF ) is always strictly positive.",
"Observation 1 It always holds that $a_j > 0$ for each $j \\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"Moreover, $a^T x^{(l)} > 0$ for each $x^{(l)} \\in S$ .",
"$\\lhd $ Clearly, if the vectors in $S$ are given explicitly, we immediately obtain an FPTAS for the original problem () using the arguments given in Section REF .",
"In the following, we discuss the more elaborate case that we can access the set $S$ and the cone $C$ only via given oracles.",
"Throughout this paper, we investigate three kinds of oracles with decreasing strength.",
"The most powerful oracle to be considered can be defined as follows: Definition 2 (Minimizing Oracle) For a given vector $d \\in \\mathbb {R}^n$ , a minimizing oracle for the set $S$ returns a vector $x^{(l^*)} \\in S$ that minimizes $d^T x^{(l)}$ among all vectors $x^{(l)} \\in S$ .",
"$\\lhd $ Clearly, the notion of minimizing oracles requires very powerful algorithms.",
"For example, if $S$ is the set of unit-flows on simple cycles in a given graph $G$ , a minimizing oracle would need to be able to determine a most negative simple cycle, which is $\\mathcal {NP}$ -complete in general (see Section REF ).",
"In many cases, it suffices to consider a much weaker type of oracle given as follows: Definition 3 (Sign Oracle) For a given vector $d \\in \\mathbb {R}^n$ , a sign oracle for the set $S$ returns a vector $x^{(l)} \\in S$ with $\\operatorname{sgn}d^T x^{(l)} = \\operatorname{sgn}d^T x^{(l^*)}$ , where $x^{(l^*)}$ minimizes $d^T x^{(i)}$ among all vectors in $S$ .",
"$\\lhd $ The sign function $\\operatorname{sgn}\\colon \\mathbb {R} \\mapsto \\lbrace -1,0,1\\rbrace $ returns $-1$ , 0, or 1 depending on whether the argument is negative, zero, or positive, respectively.",
"Rather than determining a vector in $S$ with minimum cost, a sign oracle only returns any vector whose cost have the same sign as a minimum-cost vector, which may be much easier to find.",
"In the example above, we can easily find a cycle with the same costs as a most negative cycle by computing a minimum mean cycle in $\\mathcal {O}(nm)$ time (cf.",
"[26] and Section REF ).",
"An even simpler kind of oracle is given as follows: Definition 4 (Separation Oracle) For a given vector $d \\in \\mathbb {R}^n$ , a separation oracle for the set $S$ either states that $d^T x^{(i)} \\ge 0$ for all vectors $x^{(i)} \\in S$ or returns a certificate $x^{(l)} \\in S$ that fulfills $d^T x^{(l)} < 0$ .",
"$\\lhd $ Clearly, the notion of separation oracles yields the least powerful yet most natural definition of an oracle.",
"The name “separation oracle” is based on the fact that such an oracle can be seen as a traditional separation oracle for the dual cone $C^* \\lbrace w \\in \\mathbb {R}^n: w^T x \\ge 0 \\text{ for all } x \\in C \\rbrace $ of the cone $C$ (cf.",
"[20]).",
"Note that each minimizing oracle also induces a sign oracle and that each sign oracle induces a separation oracle, so the considered oracles have in fact decreasing strength.",
"In particular, each approximation algorithm that is based on a sign oracle (separation oracle) is also valid for the case of a minimizing oracle (sign oracle).",
"For the special case of uniform costs $c^T x^{(l)}$ for all vectors $x^{(l)} \\in S$ , we get the first approximation result based on the procedure of [17]: Theorem 5 Suppose that $c^T x^{(l)} = \\hat{c}$ for all $x^{(l)} \\in S$ and some constant $\\hat{c} > 0$ .",
"Given a minimizing oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time, a $(1 - \\varepsilon )$ -approximate solution for the problem () can be computed in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot (N + T_\\mathcal {A}) \\right)$ time.",
"Since $c^T x^{(l)} = \\hat{c}$ for each $x^{(l)} \\in S$ , the subproblem given in equation (REF ) reduces to the problem of finding a vector $x^{(l)}$ with minimum cost $\\left(\\frac{1}{\\hat{c}} \\cdot a\\right)^T x^{(l)}$ among all vectors in $S$ .",
"Using the minimizing oracle, we can compute a minimizer for (REF ) in $\\mathcal {O}(T_\\mathcal {A})$ time based on the cost vector $d \\frac{1}{\\hat{c}} \\cdot a$ .",
"Note that this cost vector can be built in $\\mathcal {O}(N)$ time as each entry $a_j$ of $a$ is defined to be $\\sum _{i=1}^m y_i \\cdot A_{ij}$ , where each $y_i$ stems from the framework of [17].",
"Consequently, we need look at each of the $N$ entries of $A$ once in order to build $d$ .",
"Hence, we are able to determine a most violated dual constraint of () in $\\mathcal {O}(N + T_\\mathcal {A})$ time, so the claim follows immediately from the arguments outlined in Section REF .",
"Note that the vector $d$ that is constructed in the above procedure is always positive in each component according to Observation REF .",
"As a consequence, if for example we use the vector $d$ to denote the length of edges in a graph, we can use [12]'s ([12]) algorithm to compute a shortest path.",
"This will be used in Section .",
"In the following sections, we will focus on the more general cases in which the costs are no longer uniform and in which we may only have access to the cone via weaker types of oracles."
],
[
"General Algorithm",
"Throughout this section, we assume that there is a separation oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time.",
"Hence, the presented results are valid for the case of minimizing oracles and sign oracles as well.",
"In the subsequent section, we will see where the different strengths of the oracles come into play.",
"The procedure of the upcoming algorithm is based on an idea introduced by [14], which was originally developed for the maximum multicommodity flow problem: For $\\lambda ^*$ to denote the optimal value of the most violated dual constraint in equation (REF ), we let $\\underline{\\lambda }$ denote a positive lower bound for $\\lambda ^*$ .",
"We will show in Section how we can find a good initial value for this lower bound efficiently.",
"In each iteration of the procedure of [17] as described in Section REF , we need to determine an approximately most violated dual constraint corresponding to some vector $x^{(j)} \\in S$ fulfilling $\\frac{a^T x^{(j)}}{c^T x^{(j)}} \\le (1 + \\varepsilon ) \\cdot \\lambda ^* = (1 + \\varepsilon ) \\cdot \\min _{\\genfrac{}{}{0.0pt}{}{l \\in \\lbrace 1,\\ldots ,k\\rbrace }{c^T x^{(l)} > 0}} \\dfrac{a^T x^{(l)}}{c^T x^{(l)}}.$ For $\\lambda \\in \\mathbb {R}$ , let $d(\\lambda ) a - \\lambda c$ and $D(\\lambda ) \\min \\lbrace d(\\lambda )^T x^{(l)} : l \\in \\lbrace 1,\\ldots ,k\\rbrace \\text{ with } c^T x^{(l)} > 0 \\rbrace $ .",
"Similar to the minimum ratio cycle problem [28], [29], [30], we get the following characterization of the relation between the sign of $D(\\lambda )$ and the sign of $\\lambda ^* - \\lambda $ : Lemma 6 For some given value of $\\lambda \\in \\mathbb {R}$ , it holds that $\\operatorname{sgn}(D(\\lambda )) = \\operatorname{sgn}(\\lambda ^* - \\lambda )$ .",
"Let $L \\lbrace l \\in \\lbrace 1,\\ldots ,k\\rbrace : c^T x^{(l)} > 0 \\rbrace $ .",
"First, consider the case that $D(\\lambda ) > 0$ .",
"The claim follows by simple arguments: $D(\\lambda ) > 0 &\\Longleftrightarrow d(\\lambda )^T x^{(l)} > 0 &&\\text{ for all } l \\in L \\\\&\\Longleftrightarrow (a - \\lambda c)^T x^{(l)} > 0 &&\\text{ for all } l \\in L \\\\&\\Longleftrightarrow \\frac{a^T x^{(l)}}{c^T x^{(l)}} > \\lambda &&\\text{ for all } l \\in L \\\\&\\Longleftrightarrow \\lambda ^* > \\lambda .$ Conversely, if $D(\\lambda ) < 0$ , we get the following equivalences by similar arguments: $D(\\lambda ) < 0 &\\Longleftrightarrow d(\\lambda )^T x^{(l)} < 0 &&\\text{ for some } l \\in L \\\\&\\Longleftrightarrow (a - \\lambda c)^T x^{(l)} < 0 &&\\text{ for some } l \\in L \\\\&\\Longleftrightarrow \\frac{a^T x^{(l)}}{c^T x^{(l)}} < \\lambda &&\\text{ for some } l \\in L \\\\&\\Longleftrightarrow \\lambda ^* < \\lambda .$ Finally, in the remaining case $D(\\lambda ) = 0$ , it follows by continuity that $\\lambda ^* = \\lambda $ , which shows the claim.",
"Lemma REF implies that $\\lambda ^*$ is the maximum value of $\\lambda $ such that $D(\\lambda ) \\ge 0$ , i.e., such that $d(\\lambda )^T x^{(l)} \\ge 0$ for each $x^{(l)} \\in S$ with $c^T x^{(l)} > 0$ .",
"In each iteration of our general procedure, we call the given separation oracle $\\mathcal {A}$ with the vector $d((1+\\varepsilon )\\underline{\\lambda })$ .",
"We distinguish between the two possible outcomes of one such call: Case 1: The oracle returns some certificate $x^{(l)} \\in S$ with $d((1+\\varepsilon )\\underline{\\lambda })^T x^{(l)} < 0$ .",
"In this case, we get that $&& \\left( a - (1 + \\varepsilon ) \\cdot \\underline{\\lambda } \\cdot c \\right)^T x^{(l)} &< 0 \\\\\\Longleftrightarrow && a^T x^{(l)} &< (1 + \\varepsilon ) \\cdot \\underline{\\lambda } \\cdot c^T x^{(l)} \\\\\\Longrightarrow && \\frac{a^T x^{(l)}}{c^T x^{(l)}} &< (1 + \\varepsilon ) \\cdot \\underline{\\lambda } \\\\\\Longrightarrow && \\frac{a^T x^{(l)}}{c^T x^{(l)}} &< (1 + \\varepsilon ) \\cdot \\lambda ^*.$ The third inequality follows from the fact that $a^T x^{(l)} > 0$ (cf.",
"Observation REF ) and that $\\underline{\\lambda } > 0$ such that it also holds that $c^T x^{(l)} > 0$ .",
"The returned vector $x^{(l)}$ yields an approximately most violated dual constraint.",
"We use this dual constraint and continue the procedure of [17].",
"Note that, during an iteration of the procedure, it holds that $c^T x^{(j)}$ remains constant for each vector $x^{(j)} \\in S$ (since it does not depend on the dual solution $y$ ) and that $a^T x^{(l)}$ does not decrease (since both the entries in $x^{(l)}$ and the entries in $A$ are non-negative).",
"Hence, $\\underline{\\lambda }$ continues to be a lower bound for the (possibly increased) new value $\\lambda ^*$ of (REF ).",
"Case 2: The oracle states that all vectors $x^{(l)}$ fulfill $d((1+\\varepsilon )\\underline{\\lambda })^T x^{(l)} \\ge 0$ .",
"It now holds that $&& \\left( a - (1 + \\varepsilon ) \\cdot \\underline{\\lambda } \\cdot c \\right)^T x^{(l)} &\\ge 0 && \\forall x^{(l)} \\in S \\\\\\Longleftrightarrow && a^T x^{(l)} &\\ge (1 + \\varepsilon ) \\cdot \\underline{\\lambda } \\cdot c^T x^{(l)} && \\forall x^{(l)} \\in S \\\\\\Longleftrightarrow && \\frac{a^T x^{(l)}}{c^T x^{(l)}} &\\ge (1 + \\varepsilon ) \\cdot \\underline{\\lambda } && \\forall x^{(l)} \\in S \\text{ with } c^T x^{(l)} > 0 \\\\\\Longleftrightarrow && \\lambda ^* &\\ge (1 + \\varepsilon ) \\cdot \\underline{\\lambda }.$ In this case, we can update the lower bound $\\underline{\\lambda }$ to $(1 + \\varepsilon ) \\cdot \\underline{\\lambda }$ and continue.",
"Hence, in each iteration of the algorithm, we either proceed in the procedure of [17] or we increase the lower bound by a factor of $1 + \\varepsilon $ .",
"Again, we want to stress that $\\underline{\\lambda }$ continues to be a lower bound for $\\lambda ^*$ after an iteration of the above algorithm.",
"The following theorem shows that this yields an efficient approximation algorithm for the problem $(\\ref {eqn:OriginalProblem})$ provided we are given a sufficiently good initial estimate for $\\lambda ^*$ : Theorem 7 Given a separation oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time and given an initial lower bound $\\underline{\\lambda }$ for the initial value of $\\lambda ^*$ fulfilling $\\underline{\\lambda } \\le \\lambda ^* \\le m^{\\frac{1}{\\varepsilon } m} \\cdot \\underline{\\lambda }$ , a $(1 - \\varepsilon )$ -approximate solution for problem () can be determined in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m \\cdot (N + T_\\mathcal {A}) \\right)$ time.",
"The correctness of the procedure follows from the arguments outlined in Section REF , the preceding discussion, and the fact that the initial value of $\\underline{\\lambda }$ is a valid lower bound for $\\lambda ^*$ .",
"In each step of the algorithm, we evaluate the given separation oracle and – based on its result – either perform one iteration of the procedure of [17] and [14] or update the lower bound $\\underline{\\lambda }$ .",
"As noted in Section REF , the former case occurs up to $\\mathcal {O}(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m)$ times.",
"In order to determine the number of updates to $\\underline{\\lambda }$ , let $(\\lambda ^*)^{(k)}$ , $\\underline{\\lambda }^{(k)}$ , $y_i^{(k)}$ denote the values of the corresponding variables after the $k$ -th iteration of the overall algorithm and let $\\tau $ denote the number of iterations until the algorithm stops.",
"Note that the procedure stops as soon as $\\sum _{i=1}^m b_i \\cdot y^{(k)}_i \\ge 1$ .",
"Hence, after the $(\\tau -1)$ -th iteration, it holds that $y^{(\\tau -1)}_i < \\frac{1}{b_i}$ for each $i \\in \\lbrace 1,\\ldots ,m\\rbrace $ .",
"Since each variable $y^{(\\tau -1)}_i$ will be increased by a factor of at most $1 + \\varepsilon $ in the final iteration, it holds that $y^{(\\tau )}_i < (1 + \\varepsilon ) \\cdot \\frac{1}{b_i}$ .",
"Since the initial value of each variable $y_i$ was set to $y^{(0)}_i \\frac{\\delta }{b_i}$ , every dual variable increases by a factor of at most $\\frac{1 + \\varepsilon }{\\delta }$ during the execution of the algorithm, so $y^{(\\tau )}_i \\le \\frac{1 + \\varepsilon }{\\delta } \\cdot y^{(0)}$ .",
"However, this also implies that $(\\lambda ^*)^{(\\tau )} \\le \\frac{1 + \\varepsilon }{\\delta } \\cdot (\\lambda ^*)^{(0)}$ .",
"Since the lower bound $\\underline{\\lambda }$ remains to be a lower bound after every step of the algorithm as discussed above, it holds that $\\underline{\\lambda }^{(\\tau )} \\le (\\lambda ^*)^{(\\tau )} \\le \\frac{1 + \\varepsilon }{\\delta } \\cdot (\\lambda ^*)^{(0)} \\le \\frac{1 + \\varepsilon }{\\delta } \\cdot m^{\\frac{1}{\\varepsilon } m} \\cdot \\underline{\\lambda }^{(0)},$ where the third inequality follows from the requirement that $(\\lambda ^*)^{(0)} \\le m^{\\frac{1}{\\varepsilon } m} \\cdot \\underline{\\lambda }^{(0)}$ .",
"Since $\\underline{\\lambda }$ is increased by a factor of $1 + \\varepsilon $ in each update step, we get that the number of such steps is bounded by $\\log _{1 + \\varepsilon } \\dfrac{\\underline{\\lambda }^{(\\tau )}}{\\underline{\\lambda }^{(0)}}&\\le \\log _{1 + \\varepsilon } \\left( \\dfrac{1 + \\varepsilon }{\\delta } \\cdot m^{\\frac{1}{\\varepsilon } m} \\right)= \\log _{1 + \\varepsilon } \\dfrac{1 + \\varepsilon }{\\delta } + \\log _{1 + \\varepsilon } m^{\\frac{1}{\\varepsilon } m} \\\\&= \\log _{1 + \\varepsilon } \\dfrac{1 + \\varepsilon }{\\frac{1+\\varepsilon }{((1+\\varepsilon )m)^{\\frac{1}{\\varepsilon }}}} + \\frac{1}{\\varepsilon } \\cdot m \\log _{1 + \\varepsilon } m \\\\&= \\log _{1 + \\varepsilon } ((1 + \\varepsilon ) m)^\\frac{1}{\\varepsilon } + \\frac{1}{\\varepsilon } \\cdot m \\log _{1 + \\varepsilon } m \\\\&= \\frac{1}{\\varepsilon } \\cdot (1 + \\log _{1 + \\varepsilon } m) + \\frac{1}{\\varepsilon } \\cdot m \\log _{1 + \\varepsilon } m \\\\&\\in \\mathcal {O}\\left( \\frac{1}{\\varepsilon } \\cdot m \\log _{1+\\varepsilon } m \\right)= \\mathcal {O}\\left( \\frac{1}{\\varepsilon } \\cdot m \\cdot \\frac{\\log m}{\\log (1+\\varepsilon )} \\right)= \\mathcal {O}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m \\log m \\right)$ and, thus, matches the number of iterations of the procedure of [17].",
"The claim then follows by the fact that, in each step of the algorithm, we need $\\mathcal {O}(N)$ time to compute the entries of the vector $d((1+\\varepsilon )\\underline{\\lambda })$ and $T_\\mathcal {A}$ time to evaluate the oracle.",
"Note that the allowed deviation of the initial lower bound $\\underline{\\lambda }$ to $\\lambda ^*$ in Theorem REF is chosen in a way such that the number of update steps to $\\underline{\\lambda }$ does not dominate the $\\mathcal {O}(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m)$ steps of the overall procedure."
],
[
"Determining a lower bound",
"The proof of Theorem REF shows that the strongly polynomial number of oracle calls depends on the assumption that the initial value for the lower bound $\\underline{\\lambda }$ is not “too far away” from the real value $\\lambda ^*$ of the most violated dual constraint.",
"In this section, we present a weakly polynomial-time and a strongly polynomial-time approach to find such a sufficiently good initial value."
],
[
"Weakly Polynomial-Time Approach for Separation Oracles",
"We start by providing a general approach that is valid for all three types of oracles.",
"The running time will depend (logarithmically) on the largest number given in the input, denoted by $M \\max \\lbrace (\\max _i b_i), (\\max _j c_j), (\\max _{i,j} A_{i,j}), n, m\\rbrace \\in \\mathbb {N}$ .",
"Lemma 8 Suppose that we are given a separation oracle $\\mathcal {A}$ running in $T_\\mathcal {A}$ time.",
"An initial lower bound $\\underline{\\lambda }$ for $\\lambda ^*$ fulfilling $\\underline{\\lambda } \\le \\lambda ^* \\le m^{\\frac{1}{\\varepsilon } m} \\cdot \\underline{\\lambda }$ can be determined in weakly polynomial time $\\mathcal {O}((T_\\mathcal {A} + N) \\cdot (\\log \\log M - (\\log \\frac{1}{\\varepsilon } + \\log m + \\log \\log m)))$ .",
"Let $x^{(l)} \\in S$ denote a vector with $\\frac{a^T x^{(l)}}{c^T x^{(l)}} = \\lambda ^*$ that determines the minimum in equation (REF ).",
"Using that $y_i \\frac{b_i}{\\delta }$ for each $i \\in \\lbrace 1,\\ldots ,m\\rbrace $ at the beginning of the procedure, we get that $\\frac{a^T x^{(l)}}{c^T x^{(l)}}&= \\frac{\\sum _{j=1}^n a_j \\cdot x^{(l)}_j}{\\sum _{j=1}^n c_j \\cdot x_j^{(l)}}= \\frac{\\sum _{j=1}^n \\left( \\sum _{i=1}^m y_i \\cdot A_{ij} \\right) \\cdot x^{(l)}_j}{\\sum _{j=1}^n c_j \\cdot x_j^{(l)}}= \\frac{\\sum _{j=1}^n \\sum _{i=1}^m \\frac{\\delta }{b_i} \\cdot A_{ij} \\cdot x^{(l)}_j}{\\sum _{j=1}^n c_j \\cdot x_j^{(l)}}.",
"$ Without loss of generality, we can assume the separation oracle $\\mathcal {A}$ to always return a vector $x^{(l)} \\in S$ with $\\max _{j \\in \\lbrace 1,\\ldots ,n\\rbrace } x_j = 1$ (whenever it returns a vector at all): For each $x \\in C$ , it also holds that $\\beta \\cdot x \\in C$ for some positive constant $\\beta $ .",
"Hence, if the oracle does not fulfill the required property, we can wrap it into a new oracle $\\mathcal {A^{\\prime }}$ which divides the vector returned by $\\mathcal {A}$ by $\\max _{j \\in \\lbrace 1,\\ldots ,n\\rbrace } x_i > 0$ .",
"Using this fact in equation (REF ), we get the following lower and upper bound on $\\lambda ^*$ : $\\lambda ^* &\\ge \\frac{\\delta }{M} \\cdot \\frac{\\sum _{j=1}^n \\left(\\sum _{i=1}^m A_{ij} \\right) \\cdot x^{(l)}_j}{\\sum _{j=1}^n M \\cdot x_j^{(l)}}\\ge \\frac{\\delta }{M} \\cdot \\frac{\\sum _{j=1}^n 1 \\cdot x^{(l)}_j}{n \\cdot M}\\ge \\frac{\\delta }{n \\cdot M^2} \\ge \\frac{\\delta }{M^3} \\lambda _1\\multicolumn{2}{l}{\\text{and}}\\\\\\lambda ^* &\\le \\frac{\\delta }{1} \\cdot \\frac{\\sum _{j=1}^n \\left(\\sum _{i=1}^m A_{ij} \\right) \\cdot x^{(l)}_j}{\\sum _{j=1}^n 1 \\cdot x_j^{(l)}}\\le \\delta \\cdot \\frac{\\sum _{j=1}^n m \\cdot M \\cdot x^{(l)}_j}{1}\\le \\delta \\cdot nm \\cdot M \\le \\delta \\cdot M^3 \\lambda _2.$ According to Lemma REF , each feasible lower bound $\\underline{\\lambda }$ for $\\lambda ^*$ is characterized by the fact that $D(\\underline{\\lambda }) \\ge 0$ , so an oracle call with the vector $d(\\underline{\\lambda })$ results in the answer that there are no vectors in $S$ with negative costs.",
"Since $\\underline{\\lambda }$ is required to fulfill $\\underline{\\lambda } \\le \\lambda ^* \\le m^{\\frac{1}{\\varepsilon } m} \\cdot \\underline{\\lambda }$ , we only need to consider values for $\\underline{\\lambda }$ of the form $\\lambda _1 \\cdot (m^{\\frac{1}{\\varepsilon } m})^k$ in $[\\lambda _1,\\lambda _2]$ for integral values of $k$ .",
"Moreover, since the oracle returns a vector if and only if $\\underline{\\lambda } > \\lambda ^*$ , we can perform a binary search on these values in order to find the best possible lower bound for $\\lambda ^*$ .",
"In total, we get the following number of iterations: $\\mathcal {O}\\left(\\log \\log _{m^{\\frac{1}{\\varepsilon } m}} \\frac{\\lambda _2}{\\lambda _1} \\right)&= \\mathcal {O}\\left(\\log \\log _{m^{\\frac{1}{\\varepsilon } m}} \\dfrac{\\delta \\cdot M^3}{\\frac{\\delta }{M^3}} \\right)= \\mathcal {O}\\left(\\log \\log _{m^{\\frac{1}{\\varepsilon } m}} M \\right) \\\\&= \\mathcal {O}\\left(\\log \\frac{\\log M}{\\log m^{\\frac{1}{\\varepsilon } m}} \\right)= \\mathcal {O}\\left(\\log \\frac{\\log M}{\\frac{1}{\\varepsilon } \\cdot m \\log m} \\right) \\\\&= \\mathcal {O}\\left(\\log \\log M - \\left(\\log \\frac{1}{\\varepsilon } + \\log m + \\log \\log m \\right) \\right).$ In combination with the overhead of $N + T_\\mathcal {A}$ to call the oracle (as in the proof of Theorem REF ), we get the claimed time bound.",
"Note that the time bound given in Lemma REF is in fact only weakly polynomial for very large values of $M$ : The binary search only has an effect on the overall running time if the encoding length $\\log M$ of $M$ fulfills $\\log M \\in \\omega (\\frac{1}{\\varepsilon } \\cdot m \\log m)$ , i.e., if $M$ is exponential in $\\frac{1}{\\varepsilon } \\cdot m \\log m$ .",
"Theorem REF in combination with Lemma REF yields the following theorem: Theorem 9 Given a separation oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time, a $(1-\\varepsilon )$ -approximate solution for problem () can be determined in weakly polynomial time $\\mathcal {O}((T_\\mathcal {A} + N) \\cdot (\\frac{1}{\\varepsilon ^2} \\cdot m \\log m + \\log \\log M - (\\log \\frac{1}{\\varepsilon } + \\log m + \\log \\log m)))$ .$\\Box $ In particular, if the oracle $\\mathcal {A}$ runs in polynomial time, we immediately obtain an FPTAS for problem () according to Theorem REF ."
],
[
"Strongly Polynomial-Time Approach for Sign Oracles",
"In the previous subsection, we introduced a method to determine an initial lower bound for $\\lambda ^*$ that is valid for each of the investigated types of oracles.",
"However, although the general procedure that was described in Section performs a strongly polynomial number of steps, the overall procedure would not yield a strongly polynomial FPTAS, in general, even if the oracle runs in strongly polynomial time due to the weakly polynomial overhead of the binary search.",
"In this section, we present an alternative method for minimizing and sign oracles running in strongly polynomial time.",
"In the subsequent subsection, we generalize the result to separation oracles.",
"According to Lemma REF , we can decide about the direction of the deviation between some candidate value $\\lambda $ and the desired value $\\lambda ^*$ , if we are able to determine the sign of $D(\\lambda )$ .",
"Clearly, this task is strongly related to the definition a sign oracle for $S$ .",
"However, the value $D(\\lambda )$ is defined to be the minimum of $d(\\lambda )^T x^{(l)}$ among all vectors $x^{(l)}$ that additionally fulfill $c^T x^{(l)} > 0$ whereas the sign oracle is not required to consider only such vectors according to Definition REF .",
"Nevertheless, as it will be shown in the following lemma, we can neglect this additional restriction when evaluating the sign oracle: Lemma 10 For any positive value of $\\lambda $ , it holds that $\\operatorname{sgn}(D(\\lambda )) = \\operatorname{sgn}(d(\\lambda )^T x^{(l)})$ where $x^{(l)}$ is a vector returned by a sign oracle for $S$ .",
"First consider the case that $\\operatorname{sgn}(d(\\lambda )^T x^{(l)}) = -1$ , i.e., that $d(\\lambda )^T x^{(l)} < 0$ .",
"Using the definition of $d(\\lambda )$ , we get that $(a - \\lambda c)^T x^{(l)} = a^T x^{(l)} - \\lambda \\cdot c^T x^{(l)} < 0$ .",
"Since both $a^T x^{(l)} > 0$ according to Observation REF and $\\lambda > 0$ , it must hold that $c^T x^{(l)} > 0$ as well.",
"Thus, we can conclude that $D(\\lambda ) \\le d(\\lambda )^T x^{(l)} < 0$ .",
"Now consider the case that $\\operatorname{sgn}(d(\\lambda )^T x^{(l)}) = 0$ .",
"According to Definition REF , it holds that there are no vectors $x^{(j)} \\in S$ with $d(\\lambda )^T x^{(j)} < 0$ , so $D(\\lambda ) \\ge 0$ .",
"As in the previous case, we get that $(a - \\lambda c)^T x^{(l)} = a^T x^{(l)} - \\lambda \\cdot c^T x^{(l)} = 0$ if and only if $c^T x^{(l)} > 0$ since both $a^T x^{(l)} > 0$ and $\\lambda > 0$ .",
"Hence, we also get that $D(\\lambda ) \\le d(\\lambda )^T x^{(l)} = 0$ , so $D(\\lambda ) = 0$ .",
"Finally, if $\\operatorname{sgn}(d(\\lambda )^T x^{(l)}) = 1$ , there are no vectors $x^{(i)} \\in S$ with $d(\\lambda )^T x^{(i)} \\le 0$ .",
"Thus, it also holds that $D(\\lambda ) > 0$ , which shows the claim.",
"Lemma REF now allows us to determine a sufficiently good initial lower bound $\\underline{\\lambda }$ .",
"In fact, as it will be shown in the following lemma, we are even able to determine an exact most violated dual constraint in each iteration of the procedure: Lemma 11 Given a strongly combinatorial and strongly polynomial-time sign oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time, a most violated dual constraint can be determined in $\\mathcal {O}\\left(N + T^2_\\mathcal {A} \\right)$ time.",
"Lemma REF and Lemma REF imply that $\\lambda ^*$ is the unique value for $\\lambda $ for which the sign oracle returns a vector $x^{(l)} \\in S$ with $d(\\lambda )^T x^{(l)} = 0$ .",
"In particular, the returned vector $x^{(l)}$ is a minimizer for (REF ).",
"Hence, since the values $a_i$ can be computed in $\\mathcal {O}(N)$ time, we are done if we are able to determine such a vector $x^{(l)}$ in $\\mathcal {O}(T^2_\\mathcal {A})$ time.",
"Let $d(\\lambda )$ be defined as above, where $\\lambda $ is now treated as a symbolic value that is known to be contained in an interval $I$ .",
"Initially, we set $I$ to $(0,+\\infty )$ since the optimal value $\\lambda ^*$ is known to be strictly positive (cf.",
"equation (REF )).",
"Note that the costs $(d(\\lambda ))_i = a_i - \\lambda \\cdot c_i$ fulfill the linear parametric value property.",
"We simulate the execution of the sign oracle $\\mathcal {A}$ at input $d(\\lambda )$ using [29]'s ([29]) parametric search technique as described in Section REF .",
"The underlying idea is to “direct” the control flow during the execution of $\\mathcal {A}$ in a way such that it eventually returns the desired vector minimizing (REF ).",
"Whenever we need to resolve a comparison between two linear parametric values that intersect at some point $\\lambda ^{\\prime } \\in I$ , we call the sign oracle itself with the cost vector $d d(\\lambda ^{\\prime })$ in order to determine the sign of $D(\\lambda ^{\\prime })$ .",
"If $D(\\lambda ^{\\prime }) = 0$ , we found a minimizer for equation (REF ) and are done.",
"If $D(\\lambda ^{\\prime }) < 0$ ($D(\\lambda ^{\\prime }) > 0$ ), the candidate value $\\lambda ^{\\prime }$ for $\\lambda ^*$ was too large (too small) according to Lemma REF and Lemma REF such that we can update the interval $I$ to $I \\cap (-\\infty , \\lambda ^{\\prime })$ ($I \\cap (\\lambda ^{\\prime }, +\\infty )$ ), resolve the comparison, and continue the simulation of the oracle algorithm.",
"After $\\mathcal {O}(T_\\mathcal {A})$ steps, the simulation terminates and returns a vector $x^{(l)} \\in S$ that fulfills $d(\\lambda ^*)^T x^{(l)} = 0$ for the desired value $\\lambda ^* \\in I$ .",
"Hence, this vector yields a most violated constraint in (REF ).",
"Since the described simulation runs in $\\mathcal {O}(T_\\mathcal {A}^2)$ time, the claim follows.",
"Note that we actually still obtain a polynomial running-time of the above algorithm even if we do not assume the sign oracle to run in strongly polynomial time but only to run in weakly polynomial time.",
"However, the running-time of the resulting algorithm might exceed the stated time bound since the candidate values $\\lambda ^{\\prime }$ that determine the input to the callback oracle are rational numbers whose representation might involve exponential-size numbers of the form $H^{T_\\mathcal {A}}$ for some $H$ with polynomial encoding length.",
"Although the running-time of a weakly polynomial-time oracle algorithm depends only logarithmically on the size of these numbers, the running-time might still increase by a large (polynomial) factor.",
"Lemma REF can be incorporated into the procedure of [17] to obtain an FPTAS for problem () running in $\\mathcal {O}(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m \\cdot (N + T^2_\\mathcal {A}))$ time.",
"However, it can also be used to find an initial lower bound $\\underline{\\lambda }$ for $\\lambda ^*$ (which, in fact, equals $\\lambda ^*$ ), which yields the following theorem in combination with Theorem REF : Theorem 12 Given a strongly combinatorial and strongly polynomial-time sign oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time, there is a strongly polynomial FPTAS for the problem () running in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot \\left(N + T_\\mathcal {A} \\right) + T^2_\\mathcal {A} \\right)$ time.",
"$\\Box $"
],
[
"Strongly polynomial-time approach for Separation Oracles",
"Although separation oracles are probably the most natural kind of oracle, they are also the weakest of the considered oracle types.",
"The proof of Lemma REF relies on the fact that we are able to decide if some candidate value $\\lambda $ is too small, too large, or equal to the desired value.",
"In the case of a separation oracle, however, the case that $d^T x^{(i)} \\ge 0$ for all vectors $x^{(i)} \\in S$ does no longer include the information whether there is a vector $x^{(l)} \\in S$ with $d^T x^{(l)} = 0$ (in which case we have found the desired vector in the parametric search as described above) or if $d^T x^{(i)} > 0$ for all $x^{(i)} \\in S$ .",
"For example, if we come across a comparison of the form $a_0 + \\lambda \\cdot a_1 \\le b_0 + \\lambda \\cdot b_1$ during the simulation where $a_1 > b_1$ , we are actually interested in the information whether or not the optimal value $\\lambda ^*$ fulfills $\\lambda ^* \\le \\lambda ^{\\prime } \\frac{b_0 - a_0}{a_1 - b_1}$ .",
"However, if we use the separation oracle with the cost vector $d(\\lambda ^{\\prime })$ , we only obtain the information whether $\\lambda ^* < \\lambda ^{\\prime }$ (in case that the oracle returns a certificate) or if $\\lambda ^* \\ge \\lambda ^{\\prime }$ .",
"Hence, in the latter case, the outcome of the comparison is not yet resolved since we need the additional information whether or not $\\lambda ^* = \\lambda ^{\\prime }$ , so we cannot continue the simulation without any further ado.",
"Nevertheless, as it will be shown in the following lemma, we can gather this additional information by a more sophisticated approach: Lemma 13 Given a strongly combinatorial and strongly polynomial-time separation oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time, a most violated dual constraint can still be determined in $\\mathcal {O}\\left(N + T^2_\\mathcal {A} \\right)$ time.",
"The claim directly follows from Lemma REF if we can show that we can extend the given separation oracle into a sign oracle for $S$ .",
"As in the proof of Lemma REF , we simulate the execution of the separation oracle using the parametric cost vector $d(\\lambda ) a - \\lambda c$ .",
"Assume that the execution halts at a comparison that needs to be resolved, resulting in a candidate value $\\lambda ^{\\prime }$ for the desired value $\\lambda ^*$ .",
"We invoke the separation oracle with the cost vector $d d(\\lambda ^{\\prime })$ .",
"Clearly, if the oracle returns a certificate $x^{(l)}$ with $d^T x^{(l)} < 0$ , we can conclude that $D(\\lambda ^{\\prime }) < 0$ such that the value $\\lambda ^{\\prime }$ was too large according to Lemma REF and the result of the comparison is determined.",
"Conversely, if the oracle states that $d^T x^{(i)} \\ge 0$ for all $x^{(i)} \\in S$ , we can conclude that $D(\\lambda ^{\\prime }) \\ge 0$ .",
"However, we may not yet be able to resolve the comparison since its result may rely on the additional information whether $D(\\lambda ^{\\prime }) = 0$ or $D(\\lambda ^{\\prime }) > 0$ as shown above.",
"Nevertheless, we can extract this information by one additional call to the oracle as it will be shown in the following.",
"First suppose that $\\lambda ^{\\prime } = \\lambda ^*$ .",
"In this situation, it holds that $d(\\lambda ^{\\prime })^T x^{(i)} \\ge 0$ for all $x^{(i)} \\in S$ and there is at least one vector $x^{(l)} \\in S$ that fulfills $d(\\lambda ^{\\prime })^T x^{(l)} = 0$ .",
"Since all the functions $f^{(i)}(\\lambda ) d(\\lambda )^T x^{(i)} = a^T x^{(i)} - \\lambda \\cdot c^T x^{(i)}$ are linear functions of $\\lambda $ with negative slope (in case that $c^T x^{(i)} > 0$ ; otherwise, the function has no positive root at all), it holds that several functions $f^{(l)}$ evaluate to zero at $\\lambda ^{\\prime }$ while every other function attains its root at a larger value for $\\lambda $ (cf.",
"Figure REF ).",
"Hence, for every larger value of $\\lambda $ , the separation oracle changes its outcome and returns a certificate.",
"In particular, for a sufficiently small but positive value of $\\delta $ , the separation oracle called with the cost vector $d(\\lambda ^{\\prime } + \\delta )$ returns a vector $x^{(l)} \\in S$ with $d(\\lambda ^{\\prime } + \\delta ) x^{(l)} < 0$ that additionally fulfills $d(\\lambda ^{\\prime })^T x^{(l)} = 0$ (so $x^{(l)}$ yields a most violated constraint in the overall procedure).",
"Clearly, the value of $\\delta $ must be small enough to guarantee that we do not reach the root of another function $f^{(i)}$ (i.e., smaller than the distance between the dashed and the dotted line in Figure REF ).",
"Figure: Illustration of the two cases that may occur during the simulation of the separation oracle in case that the separation oracle did not return a certificate when evaluated for a candidate value λ ' \\lambda ^{\\prime }.Now suppose that $\\lambda ^{\\prime } < \\lambda ^*$ (cf.",
"Figure REF ).",
"In this case, for a sufficiently small but positive value of $\\delta $ , the separation oracle returns the same answer when called with the cost vector $d(\\lambda ^{\\prime } + \\delta )$ as long as $\\lambda ^{\\prime } + \\delta \\le \\lambda ^*$ (i.e., as long as $\\delta $ is smaller than the distance between the dotted and the dashed line in Figure REF ).",
"Consequently, in order to separate this case from the former case, it suffices to specify a value for $\\delta $ that is smaller than the distance between any two roots of the functions that occur both in the instance and during the simulation of $\\mathcal {A}$ .",
"We can then use a second call to the decision oracle in order to decide whether a candidate value $\\lambda ^{\\prime }$ is smaller than or equal to the optimal value $\\lambda ^*$ .",
"First note that the root of each function $f^{(i)}$ is given by the rational number $\\frac{a^T x^{(i)}}{c^T x^{(i)}}$ .",
"Since the coefficients $c_j$ are part of the instance $I$ of the problem () and since the values $a_j = \\sum _{i=1}^m y_i \\cdot A_{ij}$ are generated within the framework of [17], the encoding length of both values is polynomial in the problem size.",
"Similarly, as noted in Section , we can assume that the encoding lengths of all vectors $x^{(i)}$ returned by the oracle are in $\\Omega (n)$ and, since the oracle runs in polynomial time, polynomially bounded in the instance size.",
"Consequently, there is some bound $M_f$ with polynomial encoding length such that the root of each function $f^{(i)}$ can be represented by a fraction $\\frac{p_i}{q_i}$ with $p_i,q_i \\in \\mathbb {N}$ and $q_i \\le M_f$ .",
"Now consider the root $-\\frac{a_0 - b_0}{a_1 - b_1}$ of some function $g$ of the form $g(\\lambda ) (a_0 - b_0) + \\lambda \\cdot (a_1 - b_1)$ that stems from a comparison of two linear parametric values of the forms $a_0 + \\lambda \\cdot a_1$ and $b_0 + \\lambda \\cdot b_1$ .",
"Assume that we are in the $k$ -th step of the simulation.",
"Since the oracle algorithm is strongly combinatorial, the values $a_0 + \\lambda \\cdot a_1$ and $b_0 + \\lambda \\cdot b_1$ result from one or more of the input values $d_j a_j - \\lambda \\cdot c_j$ (which are the only linear parametric values at the beginning of the simulation) as well as a sequence of up to $k$ additions or subtractions with other linear parametric values and multiplications with constants.",
"Hence, since $k \\in \\mathcal {O}(T_\\mathcal {A})$ and $\\mathcal {A}$ runs in (strongly) polynomial time, there is a bound $M_g$ with polynomial encoding length such that the root $-\\frac{a_0 - b_0}{a_1 - b_1}$ of each such function $g$ considered up to the $k$ -th step of the simulation can be represented by a fraction of the form $\\frac{p}{q}$ with $p,q \\in \\mathbb {N}$ and $q \\le M_g$ .",
"Now let $\\mu _1 = \\frac{p_1}{q_1}$ and $\\mu _2 = \\frac{p_2}{q_2}$ with $\\mu _1 \\ne \\mu _2$ denote the roots of two of the above functions of the form $f^{(i)}$ or $g$ .",
"Since $q_1,q_2 \\le M_f \\cdot M_g$ , we get that $\\left| \\mu _1 - \\mu _2 \\right| = \\left| \\frac{p_1}{q_1} - \\frac{p_2}{q_2} \\right| = \\left| \\frac{p_1 \\cdot q_2 - p_2 \\cdot q_1}{q_1 \\cdot q_2} \\right| \\ge \\frac{1}{M_f^2 \\cdot M_g^2} \\mu $ Hence, choosing $\\delta \\frac{\\mu }{2}$ , we are able to differentiate between the three cases $D(\\lambda ) < 0$ , $D(\\lambda ) = 0$ , and $D(\\lambda ) > 0$ .",
"Moreover, by returning any vector in $S$ in the case of $D(\\lambda ) > 0$ and returning the certificate in every other case, the separation oracle is extended into a sign oracle and the correctness follows from the proof of Lemma REF .",
"Note that the running time remains to be $\\mathcal {O}\\left(N + T^2_\\mathcal {A} \\right)$ (as in the case of a sign oracle in Lemma REF ) since the encoding length of the number $\\delta $ is polynomially bounded and since the oracle algorithm is assumed to run in strongly polynomial time.",
"Actually, since we do not have direct access to the set $S$ , we need to obtain such a vector via an oracle access.",
"However, by calling the oracle once more with a very large value for $\\lambda $ or by returning some vector found before, we obtain a certificate in $S$ , which we can return.",
"Lemma REF now yields one of the main results of this paper: Theorem 14 Given a strongly combinatorial and strongly polynomial-time sign oracle $\\mathcal {A}$ for $S$ running in $T_\\mathcal {A}$ time, there is a strongly polynomial FPTAS for the problem () running in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot \\left(N + T_\\mathcal {A} \\right) + T^2_\\mathcal {A} \\right)$ time.",
"$\\Box $"
],
[
"Applications",
"In this section, we present several applications of the introduced framework.",
"We will be able to derive new strongly polynomial-time FPTASs for several well-known network flow and packing problems and complement or even improve upon well-known results.",
"All graphs considered in this section are assumed to be connected, such that the number of nodes $n$ fulfills $n \\in \\mathcal {O}(m)$ ."
],
[
"Budget-Constrained Maximum Flows",
"In the budget-constrained maximum flow problem, the aim is to determine a flow with maximum value in an $s$ -$t$ -network that is additionally restricted by a budget-constraint of the form $\\sum _{e \\in E} b_e \\cdot x_e \\le B$ for non-negative integers $b_e \\in \\mathbb {N}$ for each $e \\in E$ a budget $B \\in \\mathbb {N}$ .",
"The problem is known to be efficiently solvable by combinatorial algorithms, both in weakly polynomial-time [1], [4], [5], [6] and in strongly polynomial-time [23].",
"In the following, we present a strongly polynomial-time FPTAS for the problem, which is both much more simple and efficient than the exact strongly polynomial-time algorithm.",
"In order to apply our framework, we need to show that each feasible solution is decomposable in some kind of basic component and that we are able to handle these components appropriately.",
"Without loss of generality, since each budget-constrained maximum flow $x$ is also a traditional $s$ -$t$ -flow and since flows on cycles do not contribute to the flow value, it holds that $x$ can be decomposed into $m^{\\prime } \\le m$ flows $\\overline{x}^{(j)}$ on $s$ -$t$ -paths $P_j$ such that $x = \\sum _{j=1}^{m^{\\prime }} \\overline{x}^{(j)}$ .",
"Hence, if $x^{(l)}$ denotes the flow with unit flow value on some path $P_l$ in the set of $s$ -$t$ -paths $\\lbrace P_1,\\ldots ,P_k\\rbrace $ , it holds that each (budget-constrained) maximum flow $x$ is contained in the cone $C$ that is generated by the vectors in the set $S \\lbrace x^{(l)} : l \\in \\lbrace 1,\\ldots ,k\\rbrace \\rbrace $ .",
"Consequently, we can formulate the budget-constrained maximum flow problem as follows: $\\max \\ & \\sum _{e \\in E} c_e \\cdot x_e \\\\\\text{s.t.",
"}\\ & \\sum _{e \\in E} b_e \\cdot x_e \\le B, \\\\& x_e \\le u_e && \\text{for each } e \\in E, \\\\& x \\in C,$ where $c_e = 1$ if $e \\in \\delta ^-(t)$ , and $c_e = 0$ , else.",
"Note that the flow conservation constraints are now modeled by the containment in the cone $C$ , such that a packing problem over a polyhedral cone remains, i.e., a problem of the form ().",
"In the above formulation, it holds that $c^T x^{(l)} = \\hat{c} 1$ for each $x^{(l)} \\in S$ since each $s$ -$t$ -path contributes equally to the value of the flow.",
"Hence, we can apply Theorem REF if we can show that there is a minimizing oracle for $S$ , i.e., that we can determine a vector $x^{(l)}$ minimizing $d^T x^{(l)}$ for a given cost vector $d$ .",
"This simply reduces to the determination of a shortest $s$ -$t$ -path with respect to the edge lengths $d$ .",
"Note that, since the vector $a$ is always positive in each component according to Observation REF and since $\\hat{c} = 1$ , we need to search for a shortest path with non-negative edge lengths in $\\operatorname{SP}(m,n) \\in \\mathcal {O}(m + n\\log n)$ time according to the proof of Theorem REF .",
"Thus, we get that there is an FPTAS for the budget-constrained maximum flow problem running in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot \\operatorname{SP}(m,n) \\right)$ time since the number $N$ of non-zero entries in the constraint matrix in (REF ) is bounded by $2m \\in \\mathcal {O}(SP(m,n))$ .",
"Note that this running time is still obtained even if we add (a constant number of) different budget-constraints.",
"We want to stress that our framework allows to stick to the commonly used edge-based formulation of the problem, in which there is a linear number of variables defining the flow on single edges.",
"In contrast, one is required to use the path-based formulation of the problem when using the original framework of [17]: The flow conservation constraints, which define the “shape” of a feasible flow, cannot be directly used in a formulation as a packing problem.",
"These constraints, however, are now modeled by the containment in the cone $C$ .",
"Moreover, note that the only ingredients that we used are that (1) each flow decomposes into flows on some type of basic components ($s$ -$t$ -paths) and (2) that we are able to handle these basic components efficiently, which allowed us to apply the framework."
],
[
"Budget-Constrained Minimum Cost Flows",
"In the budget-constrained minimum cost flow problem, the aim is to determine a minimum cost flow subject to a budget constraint of the form $\\sum _{e \\in E} b_e \\cdot x_e \\le B$ , similarly to the budget-constrained maximum flow problem that was studied above.",
"The problem is known to be efficiently solvable in weakly and strongly polynomial-time [22], [23].",
"In [22], a strongly polynomial-time FPTAS was presented for the budget-constrained minimum cost flow problem, which runs in $ \\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m \\cdot (nm \\log m \\log \\log m + n^3 \\log n + nm \\log ^2 n \\log \\log n) \\right) $ time and which uses similar ideas as the ones presented above.",
"In the following, we improve upon this result.",
"When considering the (equivalent) circulation based version of the problem in which there are no demands and flow conservation holds at each node, it is easy to see that each optimal flow can be decomposed into flows on simple cycles.",
"Hence, we can restrict our considerations to flows that are contained in the cone $C$ that is spanned by flows on simple cycles with unit flow value.",
"The result of Theorem REF cannot be applied to this problem for two reasons: On the one hand, since we are dealing with arbitrary costs, it clearly does no longer hold that $c^T x^{(l)}$ is constant among all flows on cycles with unit flow value.",
"On the other hand, any minimizing oracle would be required to return a vector $x^{(l)}$ that minimizes $d^T x^{(l)}$ for a given cost vector $d$ , so it would be necessary to find a most negative cycle $C^*$ in the underlying graph.",
"However, this problem is known to be $\\mathcal {NP}$ -complete in general since finding a most negative simple cycle in a graph with edge costs $d_e = -1$ for each $e \\in E$ is equivalent to deciding if the graph contains a Hamiltonian cycle (cf.",
"[16]).",
"Nevertheless, we are able to determine a cycle $C$ with the same sign as the most negative cycle $C^*$ efficiently by computing a minimum mean cycle in $\\mathcal {O}(nm)$ time (cf.",
"[26]).",
"Hence, we can apply both Theorem REF and Theorem REF to the budget-constrained minimum cost flow problem in order to obtain a weakly polynomial-time FPTAS running in $\\mathcal {O}\\left(nm \\cdot \\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m + \\log \\log M - \\log \\frac{1}{\\varepsilon } - \\log m - \\log \\log m \\right) \\right)$ time and, since the minimum mean cycle algorithm of [26] is both strongly polynomial and strongly combinatorial, a strongly polynomial-time FPTAS with a time bound of $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot nm + (nm)^2 \\right) = \\mathcal {O}\\left(nm \\cdot \\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m + nm \\right) \\right).$ The latter running time can be improved by making use of the following observation: As it was shown in Lemma REF , the sign oracle is incorporated into Megiddo's parametric search in order to determine a minimizer of $\\min _{\\genfrac{}{}{0.0pt}{}{l \\in \\lbrace 1,\\ldots ,k\\rbrace }{c^T x^{(l)} > 0}} \\dfrac{a^T x^{(l)}}{c^T x^{(l)}} $ for a positive cost vector $a$ and a vector $c$ .",
"In the case of the budget-constrained minimum cost flow problem, this reduces to the determination of a minimum ratio cycle $C$ .",
"[30] derived an algorithm that determines a minimum ratio cycle in a simple graph in $\\mathcal {O}(n^3 \\log n + n m \\log ^2n \\log \\log n)$ time by making use of a parallel algorithm for the all-pairs shortest path problem in combination with [26]'s minimum mean cycle algorithm [26] as a negative cycle detector in his parametric search.",
"This running time was later improved by [10] to $\\mathcal {O}(n^3 \\log n + n m \\log ^2n)$ .",
"Hence, the strongly polynomial FPTAS can be improved to run in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot nm + n^3 \\log n + n m \\log ^2n \\right) = \\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot n m^2 \\log n \\right)$ time on simple graphs.",
"In the case of multigraphs, one can use a technique introduced in [22] in order to transform the graph into an equivalent simple graph in $\\mathcal {O}(nm \\log m \\log \\log m)$ time before applying Cole's minimum ratio cycle algorithm, yielding an FPTAS running in $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot nm + nm \\log m \\log \\log m + n^3 \\log n + n m \\log ^2n \\right) = \\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot n m^2 \\log m \\right)$ time.",
"Hence, in both cases, the strongly polynomial-time FPTAS dominates the FPTASs introduced above.",
"The claimed running time holds even if we add up to $\\mathcal {O}(n)$ different budget constraints to the problem."
],
[
"Budget-Constrained Minimum Cost Generalized Flows",
"The generalized minimum cost flow problem is an extension of the minimum cost flow problem, in which each edge $e \\in E$ is denoted with an additional gain factor $\\gamma _e$ .",
"The flow that enters some edge $e$ is multiplied by $\\gamma _e$ as soon as it leaves the edge (cf.",
"[36]).",
"In the budget-constrained minimum cost generalized flow problem, the flow is additionally restricted by a budget-constraint of the form $\\sum _{e \\in E} b_e \\cdot x_e \\le B$ as above.",
"The traditional minimum cost generalized flow problem (without an additional budget constraint) is known to be solvable by combinatorial algorithms in weakly polynomial-time [37].",
"Moreover, there is a strongly polynomial-time FPTAS running in $\\mathcal {\\widetilde{O}}\\left( \\log \\frac{1}{\\varepsilon } \\cdot n^2 m^2 \\right)$ time presented by [37].",
"However, this algorithm makes use of the inherent structure of traditional generalized flows and cannot be extended to the budget-constrained case without further ado.",
"Earlier, [31] presented an FPTAS for the related generalized minimum cost maximum flow problem with non-negative edge costs with a weakly polynomial running time of $\\mathcal {\\widetilde{O}}\\left(\\frac{1}{\\varepsilon ^2} \\cdot \\log \\frac{1}{\\varepsilon } \\cdot n^2 m^2 \\cdot \\log mCU \\right)$ , which, as well, cannot be easily extended to the budget-constrained case.",
"Another weakly polynomial-time FPTAS for this problem running$M$ denotes the largest absolute value of each number given in the problem instance, assuming gain factor are given as ratios of integers.",
"in $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot n m^2 \\cdot (\\log \\frac{1}{\\varepsilon } + \\log \\log M) \\right)$ time was presented by [13], which is also based on the procedure of [17] and which can be extended to the budget-constrained version of the problem.",
"Using our framework, we present two much simpler FPTASs that work for the generalized minimum cost flow with arbitrary edge costs and that complement the above ones by achieving better time complexities in specific cases.",
"Again, we consider the circulation based version of the problem in which the excess is zero at each node $v \\in V$ .",
"As it was shown in [37], every such generalized circulation $x$ can be decomposed into at most $m$ flows on unit-gain cycles and bicycles, i.e., flows on cycles $C$ with $\\prod _{e \\in C} \\gamma _e = 1$ and flows on pairs of cycles $(C_1,C_2)$ with $\\prod _{e \\in C_1} \\gamma _e < 1$ and $\\prod _{e \\in C_2} \\gamma _e > 1$ that are connected by a path, respectively.",
"Hence, every generalized circulation lies in the cone $C$ that is generated by flows on such unit-gain cycles and bicycles: $\\max \\ & \\sum _{e \\in E} c_e \\cdot x_e \\\\\\text{s.t.",
"}\\ & \\sum _{e \\in E} b_e \\cdot x_e \\le B, \\\\& x_e \\le u_e && \\text{for each } e \\in E, \\\\& x \\in C.$ Note that this formulation does not differ from the models in the previous applications.",
"Instead, the “structural complexity” of the problem that comes with the introduction of gain factors is modeled by the containment in the cone $C$ .",
"We are done if we are able to find a separation oracle for the set that generates this cone.",
"[37] shows that there is a unit-gain cycle or bicycle with negative costs in a given network if and only if a specialized system with two variables per inequality (2VPI) is infeasible.",
"Among others, [9] present a procedure that checks the feasibility of such a system and, in case that it is infeasible, provides a “certificate of infeasibility”, which corresponds to a negative cost unit-gain cycle/bicycle [37].",
"This procedure runs in $\\mathcal {\\widetilde{O}}(n)$ time on $\\mathcal {O}(nm)$ processors.",
"Hence, when used as a separation oracle, we are able to apply Theorem REF .",
"This yields an FPTAS running in $\\mathcal {\\widetilde{O}}\\left(n^2 m \\cdot \\left(\\frac{1}{\\varepsilon ^2} \\cdot m + \\log \\log M^{\\prime } - \\log \\frac{1}{\\varepsilon } \\right) \\right)$ time, where $M^{\\prime }$ is an upper bound on the absolute costs $c_e$ , fees $b_e$ , and capacities $u_e$ of the edges $e \\in E$ – independent of the numbers involved to represent the gain factors.",
"Moreover, since the separation oracle is both strongly polynomial and strongly combinatorial [37], we can apply Theorem REF in order to obtain a strongly polynomial-time FPTAS.",
"Using parallelization techniques that are common when using Megiddo's parametric search [30], the time that is necessary to find an initial most violated dual constraint using Lemma REF can be improved from $\\mathcal {\\widetilde{O}}((nm)^2)$ to $\\mathcal {\\widetilde{O}}(n \\cdot (nm + nm \\log (nm) + \\log (nm) \\cdot (n^2m))) = \\mathcal {\\widetilde{O}}(n^3 m)$ .",
"This yields an FPTAS with a strongly polynomial running time in $\\mathcal {\\widetilde{O}}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m \\cdot n^2 m + n^3 m \\right) = \\mathcal {\\widetilde{O}}\\left(\\frac{1}{\\varepsilon ^2} \\cdot n^2 m^2 \\right).$ This algorithm embodies the first strongly polynomial-time FPTAS for the budget-constrained generalized minimum cost flow problem and improves upon the running time of the weakly polynomial-time FPTAS.",
"Moreover, this FPTAS outperforms both the algorithm of [31] and, for large values of $M$ or small values of $\\varepsilon $ , the algorithm of [13]."
],
[
"Maximum Flows in Generalized Processing Networks",
"Generalized processing networks extend traditional networks by a second kind of capacities, so called dynamic capacities, that depend on the flow itself.",
"More precisely, the flow on each edge $e=(v,w) \\in E$ is additionally constrained to be at most $\\alpha _e \\cdot \\sum _{e^{\\prime } \\in \\delta ^+(v)} x_{e^{\\prime }}$ for some edge-dependent constant $\\alpha _e \\in (0,1]$ , i.e., the flow on $e$ may only make up a specific fraction $\\alpha _e$ of the total flow leaving the starting node $v$ of $e$ .",
"This extension allows to model manufacturing and distillation processes, in particular (cf.",
"[24]).",
"Similar to $s$ -$t$ -paths, the “basic component” in the field of generalized processing networks is the notion of so-called basic flow distribution schemes.",
"For each node $v \\in V$ with $\\delta ^+(v) \\ne \\emptyset $ , such a basic flow distribution scheme $\\beta $ is a function that assigns a value in $[0,\\alpha _e]$ to each edge $e \\in \\delta ^+(v)$ such that $\\sum _{e \\in \\delta ^+(v)} \\beta _e = 1$ and at most one edge $e \\in \\delta ^+(v)$ fulfills $\\beta _e \\in (0,\\alpha _e)$ .",
"Intuitively, a basic flow distribution scheme describes how flow can be distributed to the outgoing edges at each node without violating any dynamic capacity constraint.",
"In [24], the authors show that each flow in a generalized processing network can be decomposed into at most $2m$ flows on basic flow distribution schemes.",
"Hence, we can conclude that each maximum flow in a generalized processing network is contained in the cone $C$ that is generated by unit-flows on basic flow distribution schemes.",
"Moreover, for the problem on acyclic graphs and for a given cost vector $d$ , we can determine a basic flow distribution scheme $\\beta $ that allows a unit-flow $x$ with minimum costs $d(x) \\sum _{e \\in E} d_e \\cdot x_e$ in linear time $\\mathcal {O}(m)$ (cf.",
"[24]).",
"By using Theorem REF , we get an FPTAS for the maximum flow problem in generalized processing networks with a strongly polynomial running-time of $\\mathcal {O}(\\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m)$ .",
"This result is in particular interesting since it is unknown whether an exact solution can be determined in strongly polynomial time since the problem is at least as hard to solve as any linear fractional packing problem (cf.",
"[24] for further details)."
],
[
"Minimum Cost Flows in Generalized Processing Networks",
"Similar to the previous problem, each minimum cost flow in a generalized processing network is contained in the cone that is generated by flows with unit flow value on basic flow distribution schemes.",
"On acyclic graphs, we have the same minimizing oracle as described above.",
"Since the costs $c_e$ are now arbitrary, we can no longer apply Theorem REF .",
"Nevertheless, since each minimizing oracle induces a sign oracle, we are able to apply Theorem REF , which yields an FPTAS for the problem running in strongly polynomial-time $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m \\cdot m + m^2 \\right) = \\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m \\right).$ This matches the time complexity of the maximum flow variant of the problem described in Section REF ."
],
[
"Maximum Concurrent Flow Problem",
"The maximum concurrent flow problem is a variant of the maximum multicommodity flow problem, in which a demand $d_j$ is given for each commodity $j$ with source-sink-pair $(s_j,t_j) \\in V \\times V$ .",
"The task is to determine the maximum value of $\\lambda $ such that a fraction $\\lambda $ of all demands is satisfied without violating any edge capacity.",
"While several FPTASs emerged for this problem, the best time bound at present is given by $\\mathcal {\\widetilde{O}}\\left(\\frac{1}{\\varepsilon ^2} \\cdot (m^2 + kn) \\right)$ due to [25], where $k \\in \\mathcal {O}(n^2)$ denotes the number of commodities.",
"The problem can be approximated efficiently with our framework by using the following novel approach: In order to improve the objective function value by one unit, we need to send $d_j$ units of each commodity.",
"Hence, each concurrent flow decomposes into basic components of the following type: A set of flows on $k$ paths, containing a flow with value $d_j$ on an $(s_j,t_j)$ path for each commodity $j$ .",
"For a given (positive) cost vector, a basic component with minimum costs can be found by determining a shortest path between each commodity.",
"Since [12]'s ([12]) algorithm computes the shortest paths from one node to every other node, we only need to apply it $\\min \\lbrace k,n\\rbrace $ times (once for each of the distinct sources of all commodities), which yields a minimizing oracle running in $\\mathcal {O}(\\min \\lbrace k,n\\rbrace \\cdot (m + n\\log n))$ time and an FPTAS running in $\\mathcal {\\widetilde{O}}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m^2 \\cdot \\min \\lbrace k,n\\rbrace \\right)$ time according to Theorem REF .",
"This algorithm has a worse time complexity than the one of [25].",
"Nevertheless, the application of the presented framework is much simpler than the algorithm given in [25] (and even matches its time complexity in the case of sparse graphs with a large number of commodities) and inherently allows the incorporation of additional budget-constraints."
],
[
"Maximum Weighted Multicommodity flow Problem",
"The maximum weighted multicommodity flow problem is a generalization of the maximum multicommodity flow problem, in which a positive weight $c_j$ is denoted with each commodity and the aim is to maximize the weighted flow value.",
"The problem is known to be solvable in $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\min \\lbrace \\log C, k\\rbrace \\right)$ time as shown by [14], where $C$ denotes the largest ratio of any two weights of commodities.",
"Similar to the multicommodity flow problem, each feasible flow decomposes into flows with unit flow value on single $(s_j,t_j)$ paths.",
"Moreover, the determination of such a path with minimal costs reduces to $\\min \\lbrace k,n\\rbrace $ shortest path computations with possibly negative costs, similar to the maximum concurrent flow problem considered above.",
"Using similar ideas as in the case of the budget-constrained minimum cost flow problem (Section REF ), this would yield an FPTAS with a running time in $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot \\min \\lbrace n,k\\rbrace \\cdot nm + \\min \\lbrace n,k\\rbrace \\cdot n^3 \\right)$ .",
"This running time can be improved as follows: As above, we can consider the cone $C$ to be spanned by flows on $(s_j,t_j)$ paths for each commodity, but where each flow between any $(s_j,t_j)$ -pair now has flow value $\\frac{1}{c_j}$ .",
"Each vector in the ground set $S$ then has uniform costs.",
"In order to apply Theorem REF , we need to be able to determine a cost-minimal vector with respect to a given positive cost vector $d$ .",
"One straight-forward way to obtain such a minimizing oracle is to compute a shortest path for each commodity $j$ using the edge lengths $\\frac{d_e}{c_j}$ for each $e \\in E$ and to choose a shortest path among all commodities.",
"This would result in a $\\mathcal {\\widetilde{O}}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m^2 \\cdot k \\right)$ time FPTAS, similar to the previous application.",
"However, it suffices to compute only $\\min \\lbrace n,k\\rbrace $ shortest paths per iteration, which can be seen as follows: For each node $s$ out of the set of the $\\min \\lbrace k,n\\rbrace $ distinct source nodes, we perform two steps: We first compute the shortest path distance to every other node using [12]'s ([12]) algorithm.",
"Afterwards, for each node that corresponds to the sink $t_j$ of a commodity $j$ with source $s_j=s$ , we multiply the distance from $s_j$ to $t_j$ by $\\frac{1}{c_j}$ .",
"By repeating this procedure for each source and keeping track of the overall shortest path, we obtain a minimizing oracle.",
"This yields an FPTAS running in $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\cdot \\min \\lbrace n,k\\rbrace \\right)$ time, which complements the result of [14].",
"This example shows that more sophisticated definitions of the ground set $S$ and the cone $C$ may improve the running time of the procedure.",
"Finally, using this approach, we can even further improve the algorithm to obtain a time bound of $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\right)$ using an idea that was applied by [14] to the (unweighted) multicommodity flow problem: For an initially tight lower bound $\\underline{L}$ on the length of a shortest path for any commodity (which can be computed in $\\mathcal {\\widetilde{O}}(\\min \\lbrace n,k\\rbrace \\cdot m)$ time as above at the beginning), we can stick to one commodity $j$ in each iteration of the overall procedure and compute a single shortest path from the source $s_j$ to the sink $t_j$ .",
"Once the length of this shortest path multiplied by $\\frac{1}{c_j}$ becomes as large as $(1 + \\varepsilon ) \\cdot \\underline{L}$ , we go on to the next commodity and continue the procedure.",
"After each commodity was considered, we update $\\underline{L}$ to $(1 + \\varepsilon )\\cdot \\underline{L}$ and continue with the first commodity.",
"Following the lines of [14], this yields an FPTAS running in $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot (m^2 + km) \\right)$ time as there are $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot k \\right)$ shortest path computations that lead to a change of the commodity.",
"However, since [12]'s ([12]) algorithm computes the distance to every other node, we only need to consider $\\min \\lbrace k,n\\rbrace $ different nodes by grouping commodities with the same source as above, which reduces the running time to $\\mathcal {\\widetilde{O}}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\right)$ (see [14] for details on the algorithm).",
"Although [14] both considered this technique and introduced the maximum weighted multicommodity flow problem, she refrained from applying this procedure to the problem."
],
[
"Maximum Spanning Tree Packing Problem",
"In the maximum spanning tree packing problem, one is given an undirected graph $G=(V,E)$ with positive edge capacities $u_e$ .",
"Let $\\mathcal {T}$ denote the set of all spanning trees in $G$ .",
"The aim is to find a solution to the problem $\\max \\ & \\sum _{T \\in \\mathcal {T}} x_T \\\\\\text{s.t.",
"}\\ & \\sum _{T \\in \\mathcal {T}: e \\in T} x_T \\le u_e && \\forall \\ e \\in E, \\\\& x_T \\ge 0 && \\forall T \\in \\mathcal {T},$ i.e., one seeks to pack as many spanning trees as possible (in the fractional sense) without violating any edge capacity.",
"While the problem was investigated in a large number of publications, the fastest (exact) algorithm for the problem is due to [15] and runs in $\\mathcal {O}\\left( n^3 m \\log \\frac{n^2}{m} \\right)$ time.",
"Let $S$ denote the set of incidence vectors $\\chi _T$ of spanning trees $T \\in \\mathcal {T}$ , where $(\\chi _T)_e = 1$ if $e \\in T$ and $(\\chi _T)_e = 0$ else.",
"Since each spanning tree contains exactly $n-1$ edges, the problem can be stated in an equivalent edge-based fashion as follows: $\\max \\ & \\frac{1}{n-1} \\cdot \\sum _{e \\in E} x_e \\\\\\text{s.t.",
"}\\ & x_e \\le u_e && \\forall \\ e \\in E, \\\\& x \\in C.$ In order to apply Theorem REF (which is eligible since each spanning tree contributes equally to the objective function value), we need a minimizing oracle for the set $S$ .",
"However, this simply reduces to the determination of a minimum spanning tree, which can be done in $\\mathcal {O}(m \\cdot \\alpha (m,n))$ time, where $\\alpha (m,n)$ denotes the inverse Ackermann function (cf.",
"[7]).",
"This yields a strongly polynomial-time FPTAS for the problem running in $\\mathcal {O}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m \\cdot \\alpha (m,n)\\right)$ time.",
"Our framework also applies to a weighted version of the problem: Assume that each edge is labeled with an additional cost $c_e$ (with arbitrary sign) and assume that the weight $c(T)$ of each spanning tree $T \\in \\mathcal {T}$ is defined to be the sum of the weights of its edges, i.e., $c(T) \\sum _{e \\in E} c_e$ .",
"The aim is then to maximize the objective function $\\sum _{T \\in \\mathcal {T}} c(T) \\cdot x_T$ .",
"As above, we can stick to an equivalent edge-based formulation using the objective function $\\frac{1}{n-1} \\cdot \\sum _{e \\in E} c_e \\cdot x_e$ .",
"The minimum spanning tree algorithm can then be used as a sign oracle, which allows us to apply Theorem REF to the problem.",
"This yields an FPTAS for the maximum weighted spanning tree packing problem running in strongly polynomial time $ \\mathcal {O}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m \\cdot \\alpha (m,n) + m^2 \\cdot \\alpha ^2(m,n) \\right) = \\mathcal {O}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m \\cdot \\alpha (m,n) \\right).",
"$ To the best of our knowledge, this is the first combinatorial approximation algorithm for this problem."
],
[
"Maximum Matroid Base Packing Problem",
"Having a closer look at the results of Section REF , one might expect that they can be generalized to matroids: As spanning trees form the bases of graphic matroids, the presented ideas suggest that the framework can also be applied to packing problems over general matroids.",
"In the maximum matroid base packing problem, a matroid $M(S,\\mathcal {I})$ with ground set $S \\lbrace 1,\\ldots ,m\\rbrace $ and independent sets in $\\mathcal {I}$ is given as well as a positive capacity $u_i \\in \\mathbb {N}_{> 0}$ for each $i \\in S$ .",
"For $r$ to be the rank function of $M$ , let $\\mathcal {B} \\subset \\mathcal {I}$ denote the set of bases such that $I \\in \\mathcal {B}$ if and only if $r(I) = r(S)$ .",
"The aim of the problem is to pack as many bases of $M$ as possible (in the fractional sense) without violating any capacity constraints: $\\max \\ & \\sum _{I \\in \\mathcal {B}} x_I \\\\\\text{s.t.",
"}\\ & \\sum _{I \\in \\mathcal {B}: i \\in I} x_I \\le u_i && \\forall \\ i \\in S, \\\\& x_I \\ge 0 && \\forall I \\in \\mathcal {B}.$ As it is common when dealing with matroids, we assume that the matroid is described by an independence testing oracle, which checks if some set $S^{\\prime } \\subseteq S$ is independent in $M$ (cf.",
"[33]).",
"Let $F(m)$ denote the running time of this oracle.",
"As it is shown in [33], the problem can be solved in $\\mathcal {O}(m^7 \\cdot F(m))$ time using a result derived by [11].",
"As it was the case in the maximum spanning tree packing problem in Section REF , the problem can be formulated in an equivalent element-based fashion as follows: $\\max \\ & \\frac{1}{r(S)} \\cdot \\sum _{i \\in S} x_i \\\\\\text{s.t.",
"}\\ & x_i \\le u_i && \\forall \\ i \\in S, \\\\& x \\in C,$ where the cone $C$ is spanned by the incidence vectors of bases in $\\mathcal {B}$ .",
"In order to apply our framework, we need to be able to handle these bases efficiently.",
"However, as we are dealing with matroids, we can find a cost-minimal basis $I \\in \\mathcal {B}$ of $M$ with respect to a given cost vector $d$ just by applying the Greedy algorithm (cf.",
"[33]): Starting with $I = \\emptyset $ , we sort the elements by their costs and iteratively add each element in the sorted sequence unless the independence test fails.",
"This yields a minimizing oracle for $\\mathcal {B}$ running in $\\mathcal {O}(m \\cdot F(m) + m\\log m)$ time.",
"Hence, we immediately get an FPTAS for the maximum matroid base packing problem running in strongly polynomial time $\\mathcal {O}\\left(\\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m \\cdot (F(m) + \\log m) \\right)$ according to Theorem REF .",
"As it was the case in Section REF , we can also extend our results to a weighted version of the problem: Assume we are additionally given costs $c_i \\in \\mathbb {Z}$ and want to maximize $\\sum _{I \\in \\mathcal {B}} c(I) \\cdot x_I$ , where $c(I) \\sum _{i \\in I} c_i$ .",
"Equivalently, we can also maximize $\\frac{1}{r(S)} \\cdot \\sum _{i \\in S} c_i \\cdot x_i$ in the element-based formulation of the problem.",
"Using the above minimizing oracle as a sign oracle, we can apply both Theorem REF and, in case that the independence testing oracle is strongly polynomial and strongly combinatorial, Theorem REF to the problem.",
"This yields two FPTASs for the problem running in $ \\mathcal {O}\\left( (m \\cdot F(m) + m \\log m) \\cdot \\left(\\frac{1}{\\varepsilon ^2} \\cdot m \\log m + \\log \\log M - \\log \\frac{1}{\\varepsilon } - \\log m - \\log \\log m \\right) \\right) $ and $ \\mathcal {O}\\left( \\frac{1}{\\varepsilon ^2} \\cdot m^2 \\log m \\cdot (F(m) + \\log m) + (m \\cdot F(m) + m \\log m)^2 \\right) $ time, respectively.",
"To the best of our knowledge, no other polynomial-time algorithm is known for this problem."
],
[
"Conclusion",
"We investigated an extension of the fractional packing framework by [17] that generalizes their results to fractional packing problems over polyhedral cones.",
"By combining a large diversity of known techniques, we derived a framework that can be easily adopted to a large class of network flow and packing problems.",
"This framework may in particular be applicable even if the cone has an exponential-sized representation as it only relies on a strongly polynomial number of oracle calls in order to gather information about the cone.",
"In many cases, its application allows the derivation of approximation algorithms that are either the first ones with a strongly polynomial running time or the first combinatorial ones at all.",
"For a large variety of applications, we were even able to complement or improve existing results.",
"The presented paper raises several questions for future research.",
"On the one hand, we believe that our results can be applied to a much larger set of problems and can be used to obtain combinatorial FPTASs for complex problems without much effort.",
"It may also be possible that the results continue to hold for even weaker kinds of oracles.",
"On the other hand, as our framework is based on the one of [17] in its core, all of the derived approximation algorithms have a running time in $\\Omega (\\frac{1}{\\varepsilon ^2} \\cdot m\\log m \\cdot n)$ and, in particular, have a quadratic dependency on $\\frac{1}{\\varepsilon }$ .",
"It may be possible to achieve a subquadratic dependency on $\\frac{1}{\\varepsilon }$ by relying on other approaches such as the one of [3].",
"Nevertheless, it seems that this trade comes with a worse dependence on other parameters, a worse practical performance, or a worse generality of the presented results."
]
] | 1612.05474 |
[
[
"Magnetar central engine and possible gravitational wave emission of\n nearby short GRB 160821B"
],
[
"Abstract GRB 160821B is a short gamma-ray burst (GRB) at redshift $z=0.16$, with a duration less than 1 second and without detection of any \"extended emission\" up to more than 100 seconds in both {\\em Swift}/BAT and {\\em Fermi}/GBM bands.",
"An X-ray plateau with a sharp drop 180 seconds after the BAT trigger was observed with {\\em Swift}/XRT.",
"No supernova or kilo-nova signature was detected.",
"Assuming the central engine of this SGRB is a recently born supra-massive magnetar, we can explain the SGRB as jet radiation and its X-ray plateau as the internal energy dissipation of the pulsar wind as it spins down.",
"We constrain its surface magnetic field as $B_{\\rm p}<3.12\\times 10^{16}$ G and initial spin period as $P_0< 8.5\\times 10^{-3}$ seconds.",
"Its equation of state is consistent with the GM1 model with $M_{\\rm TOV} \\sim 2.37 M_\\odot$ and ellipticity $\\epsilon<0.07$.",
"Its gravitational wave (GW) radiation may be detectable with the future Einstein Telescope, but is much weaker than the current detectability limit of advanced-LIGO.",
"The GW radiation of such an event would be detectable by advanced-LIGO if it occurred at a distance of 100 Mpc ($z=0.023$)."
],
[
"Introduction",
"The progenitors of short gamma-ray bursts (SGRBs), which have a hard spectrum and short duration (Kouveliotou et al.",
"1993), remain elusive (Zhang 2011).",
"Several lines of observational evidence, e.g.",
"low level of star formation (Barthelmy et al.",
"2005; Berger et al.",
"2005; Gehrels et al.",
"2005), a large offset from the center of the host galaxy (e.g.",
"Fox et al.",
"2005; Fong et al.",
"2010), as well as the non-association of bright supernovae (SNe) with short GRBs (Berger 2014 and references therein), suggest that SGRBs may form in compact star mergers, such as neutron star$-$ neutron star mergers (NS$-$ NS, Paczýnski 1986; Eichler et al.",
"1989), neutron star$-$ black hole mergers (NS$-$ BH, Paczýnski 1991), or black hole$-$ black hole mergers (BH$-$ BH; Zhang 2016).",
"The coalescence of two compact stars is also expected to be a strong source of gravitational wave (GW) radiation, and such systems are the main targets of the advanced Laser Interferometer Gravitational-wave Observatory (LIGO)/Virgo detectors.",
"Two GW events (GW 150914 and GW 151226) and one GW candidate (LVT 151012) were detected with LIGO and are proposed black hole binary mergers (Abbott et al.",
"2016a,b).",
"Electromagnetic (EM) transients associated with gravitational wave bursts (GWBs) have not been confidently detected, although associations of weak EM counterparts with these GW events were claimed (Connaughton et al.",
"2016).",
"Since the two GW events are believed to be from BH-BH systems, it is still highly debated whether or not the merger of a BH$-$ BH system can be accompanied by an EM counterpart (Zhang 2016; Zhang et al 2016; Connaughton et al.",
"2016; Xiong 2016).",
"Further observations are required to confirm the existence of BH-BH EM counterparts.",
"NS-NS mergers as the progenitors of SGRBs have been extensively studied.",
"Depending on the nascent NS mass ($M_{\\rm p}$ ), two possible outcomes of the merger are expected.",
"One possibility is a black hole, which forms when $M_{\\rm p}$ is much greater than the maximum non-rotating mass ($M_{\\rm TOV}$ , Rosswog et al.",
"2003; Rezzolla et al.",
"2011; Ravi & Lasky 2014).",
"Another possibility is a rapidly spinning, strongly magnetized neutron star (“millisecond magnetar”), in the case where $M_{\\rm p}$ is less than $M_{\\rm TOV}$ but greater than $M_{\\rm max}$ (the maximum gravitational mass) (Usov 1992; Thompson 1994; Dai & Lu 1998a,b; Zhang & Mészáros 2001; Metzger et al.",
"2008, 2011; Bucciantini et al.",
"2012).",
"The post-merger evolution of magnetars also depends on the mass lying between $M_{\\rm p}$ and $M_{\\rm TOV}$ .",
"One possible channel is a magnetar in an equilibrium state which injects energy from the magnetar wind via loss of rotation energy for $M_{\\rm p}\\le M_{\\rm TOV}$ (Giacomazzo & Perna 2013).",
"This well explains the long lasting energy injection phase observed with the Swift X-Ray Telescope (XRT; Burrows et al.",
"2004).",
"Another evolving channel is that of a supra-massive NS, which may survive if $M_{\\rm TOV}<M_{\\rm p}< M_{\\rm max}$ , when magnetic braking and viscosity compel the star into uniform rotation.",
"As the period of the magnetar decreases via rotational energy loss, the maximum gravitational mass decreases.",
"The magnetar collapses into a black hole when its centrifugal force cannot balance the gravitational force (Duez et al.",
"2006; Ravi & Lasky 2014).",
"Theoretically, it is expected that a Poynting-flux dominated outflow is driven by the injected wind as the magnetar spins down (e.g.",
"Dai & Lu 1998a; Zhang & Mészáros 2001).",
"The observed X-ray “internal plateau” (the rapid flux drop off at the end of the plateau emission with a decay slope $\\alpha >3$ )Throughout the paper we adopt the convention $F_\\nu \\propto t^{-\\alpha } \\nu ^{-\\beta }$ .",
"with XRT in a few long and short GRBs (Troja et al.",
"2007; Lyons et al.",
"2010; Rowlinson et al.",
"2010, 2013; Lü et al.",
"2015; Du et al.",
"2016) may be evidence for this evolution channel.",
"The rapid decay following the plateau cannot be accommodated in any external shock model and can be attributed to internal dissipation of a central engine wind, which is likely a signature of the collapse of a supra-massive magnetar central engine into a black hole (Troja et al.",
"2007; Liang et al.",
"2007; Lyons et al.",
"2010; Lü & Zhang 2014; Lü et al.",
"2015).",
"It has also been proposed that this phenomenon may be accompanied by a fast radio transient, i.e., fast radio burst (FRB, Lorimer et al.",
"2007; Zhang 2014).",
"In NS-NS merger models, it is predicted that EM signals can not be avoided after the merger due to the high magnetic field strength at the NS surface (Metzger & Berger 2012).",
"In addition, a NS-NS merger would also lose energy via gravitational wave quadrupole emission (Fan et al.",
"2013; Lasky et al.",
"2014; Lasky & Glampedakis 2016).",
"Therefore, hunting for possible associations of SGRBs with GW events is interesting.",
"The LIGO team has searched for such associations for many years, but no events have been reported.",
"Comparing the BH-BH mergers with the associated GWBs detected with the advanced-LIGO, the energy lost via GWB in these systems would be much larger than that expected in NS-NS merger systems (Corsi & Mészáros 2009; Hild et al.",
"2011; Fan et al.",
"2013).",
"Therefore, the advanced-LIGO detection rate for NS-NS mergers should be much lower than that of BH-BH merger systems.",
"GRB 160821B is a nearby bright SGRB with a redshift of $z=0.16$ .",
"This paper dedicates analysis of its multi-wavelength data and constrains the properties of its central engine as well as its possible GW radiation.",
"We present our data reduction from Swift and Fermi observations in §2.",
"In §3, we compare the properties of GRB 160821B with other SGRBs.",
"The derived parameters for a magnetar central engine and the equation of state of newly-born NSs are presented in §4.",
"In §5, we present a constraint on the ellipticity of the NS-NS system and the probability of detectable gravitational wave radiation.",
"Conclusions are drawn in §6 with some additional discussion.",
"Throughout the paper, a concordance cosmology with parameters $H_0=71~\\rm km~s^{-1}~Mpc^{-1}$ , $\\Omega _M=0.30$ , and $\\Omega _{\\Lambda }=0.70$ is adopted.",
"GRB 160821B triggered the Burst Alert Telescope (BAT) at 22:29:13 UT on 2016 August 21 (Siegel et al.",
"2016).",
"We developed an IDL script to automatically download the Swift BAT data.",
"We use the standard HEASOFT tools (version 6.12) to process the data.",
"We run bateconvert from the HEASOFT software release to obtain the energy scale for the BAT events.",
"The light curves and spectra are extracted by running batbinevt (Sakamoto et al.",
"2007).",
"Then, we calculate the cumulative distribution of the source counts using the arrival time of a fraction between 5 and 95 per cent of the total counts to define $T_{90}$ .",
"The time bin size is fixed to 64 ms in this case due to the short duration.",
"The background is extracted using two time intervals, one before and one after the burst.",
"We model the background as Poisson noise, which is the standard background model for prompt emission.",
"We invoked Xspec to fit the spectra.",
"For technical details please refer to Sakamoto et al.",
"(2007).",
"XRT began observing the field 57 seconds after the BAT trigger (Siegel et al.",
"2016).",
"We made use of the public data from the Swift archive$\\rm http://www.swift.ac.uk/xrt_curves/00709357$.",
"The Ultra-Violet Optical Telescope (UVOT; Roming et al.",
"2005) observed the field at $T_0+76$ s, but no optical afterglow was consistent with the XRT position (Evans et al.",
"2016).",
"There was also no detection in the initial UVOT exposures (Xu et al.",
"2016).",
"Preliminary, 3$\\sigma $ upper limits data are obtained by using the UVOT photometric system for the first finding chart (FC) exposure (Breeveld et al.",
"2016).",
"On the other hand, $r$ - and $z$ -band afterglow images were obtained by using William Herschel Telescope on La Palma (Levan et al.",
"2016).",
"In the spectrum of the candidate host galaxy several prominent emission lines were found (H$\\beta $ , [O$III$ ] and H$\\alpha $ ), at a redshift of $z=0.16$ .",
"The physical offset of the afterglow from the candidate host galaxy is approximately 15 kpc (Levan et al.",
"2016)."
],
[
"Fermi Gamma-ray Burst Monitor (GBM) triggered and located GRB 160821B at 22:29:13.33 UT on 21 August 2016 (Stanbro et al.",
"2016).",
"GBM has 12 sodium iodide (NaI) and two bismuth germanate (BGO) scintillation detectors, covering the energy range 8 keV to 40 MeV (Meegan et al.",
"2009).",
"We downloaded GBM data of this GRB from the public science support center at the official Fermi web sitehttp://fermi.gsfc.nasa.gov/ssc/data/.",
"Each of the GBM detectors collected the data with three different types: CTIME, CSPEC, and TTE.",
"Then, we extracted the light curves and performed spectral analysis based on the package gtBurst.",
"By invoking the heasoft command fselect and the ScienceTools command gtbin, we extracted light curves with a time-bin of 64 ms in a user-specified energy band from the GBM.",
"We clicked “Tasks $\\rightarrow $ Make spectra for XSPEC” in $gtBurst$ to extract the source spectrum of the GBM data.",
"The background spectra are extracted from the time intervals before and after the prompt emission phase, modeled with a polynomial function, and the source spectrum is extracted by applying the background model to the prompt emission phase.",
"This GRB occurred right at the edge of the Large Area Telescope (LAT) field-of-view (Atwood et al.",
"2009), about 61 degrees from boresight.",
"So we do not expect to detect the high-energy signal if it exists."
],
[
"As shown in Fig.1, the BAT light curve shows a single short peak with duration $T_{90}=0.48\\pm 0.07$ seconds, and there is no evidence of extended emission detected in the BAT energy range up to 100 s after the BAT trigger ($T_0$ ).",
"The time-integrated BAT spectrum can be fit by a single power law with photon index $\\Gamma _{\\gamma }=1.98\\pm 0.11$ .",
"The BAT band (15$-$ 150 keV) peak flux is $(1.7\\pm 0.2)~\\rm photons~cm^{-2}~s^{-1}$ , and the total fluence is $(1.1\\pm 0.1)\\times 10^{-7} ~\\rm erg ~cm^{-2}$ (Palmer et al.",
"2016).",
"The GBM light curve of GRB 160821B is also shown in Figure 1 with a 64 ms time-bin.",
"The profile of light curve is similar with BAT data, has a bright peak with duration $T_{90}\\sim 1.2$ s in 8$-$ 1000 keV range.",
"The GBM spectra can be fit by a power law function due to lack high energy photonsStanbro et al.",
"(2016) find that the spectrum can be fit by a power law function with an exponential high-energy cutoff.",
"The power law index is $\\Gamma _{\\rm GBM}=-1.31\\pm 0.6$ , and the cutoff energy $E_{\\rm p}=84\\pm 19~\\rm keV$ ..",
"Here, we joint fit the time-averaged spectra of Fermi/GBM+Swift/BAT with a power-law model, and found $\\Gamma _{\\rm BAT+GBM}=1.88\\pm 0.12$ with $\\chi ^{2}=0.84$ (Fig.",
"2).",
"The total fluence in 8-10000 keV is $(2.52\\pm 0.19)\\times 10^{-6} ~\\rm erg ~cm^{-2}$ .",
"The isotropic energy $E_{\\rm \\gamma , iso}=(2.1\\pm 0.2)\\times 10^{50}~\\rm erg$ .",
"Norris et al (2000) discovered an anti-correlation between GRB peak luminosity and the delay time ($t_{\\rm lag}$ ) in different energy bands, meaning softer photons usually arrive later than hard photons.",
"This spectral lag is always significant in long-duration GRBs (Norris et al.",
"2000; Gehrels et al.",
"2006; Liang et al.",
"2006), but is typically negligible in short-duration GRBs (Norris & Bonnell 2006; Zhang et al.",
"2009).",
"We extracted 4 ms binned light curves in the following three BAT energy bands: 15-25 keV, 25-50 keV, 50-100 keVThe signal in 100-150 keV is too weak to be extracted, so we do not consider the emission in this energy band.. Then, we used the cross-correlation function method (CCF; Norris et al.",
"2000; Ukwatta et al.",
"2010) to calculate the lags between 25-50 keV and 50-100 keV light curves.",
"In order to address the questions of whether the short GRB 160821B is consistent with typical Type I and other short-hard GRBs with in the spectral lag distribution.",
"Figure 4 shown the peak luminosity as function of spectral lag for typical Type II, Type I, other short-hard, and GRB 160821B.",
"Here, Type II GRBs are corresponding to confirmed supernova (SN) association, or have a high specific star formation rate (SSFR) and do not have a large offset from the galaxy center.",
"On the contrary, Type I GRBs are occured in elliptical or early type host galaxy without SN signature, or has a relatively low local SSFR and large offset from the host galaxy center.",
"For the other short-hard GRBs, do not satisfy neither of the two criteria of the Type I sample, and do not have their host galaxy identified, but with a short duration, hard spectral (Zhang et al.",
"2009).",
"Type II GRBs sample are from Norris & Bonnell (2006), and give the best power-law model fitting with $2\\sigma $ region of the fitting.",
"Type I and other short-hard GRBs are collected from Zhang et al (2009).",
"For GRB 160821B, we found $t_{\\rm lag}=(10\\pm 6)$ ms, which is consistent with result of Palmer et al (2016).",
"It is deviated from $2\\sigma $ region of the Type II GRBs fitting, but both peak luminosity and lag value of this case is comparable with Type I GRBs.",
"The initial X-ray light curve is best fit by a broken power law, which reads $F=F_0 \\left[\\left(\\frac{t}{t_{\\rm b}}\\right)^{\\omega {\\alpha _1}}+\\left(\\frac{t}{t_{\\rm b}}\\right)^{\\omega {\\alpha _2}}\\right]^{-1/\\omega },$ where $\\omega $ describes the sharpness of the break and is taken to be 3 in this analysis (Liang et al.",
"2007).",
"There is an initial decay slope $\\alpha _1=0.21\\pm 0.14$ , followed by a steeper decay of $\\alpha _2=4.52\\pm 0.45$ with a break time $t_{b}=180\\pm 46$ seconds after the BAT trigger.",
"No significant X-ray flare was detected during the observational time.",
"The X-ray spectrum in the 0.3$-$ 10 keV energy band is best fit by an absorbed power law with $\\Gamma _X=1.95^{+0.21}_{-0.08}$ and column density $N_{\\rm H} = (7.5\\pm 2.1)\\times 10^{20} ~\\rm cm^{-2}$ .",
"The X-ray light curve along with the $\\Gamma _{\\rm X}$ evolution is shown in Fig.3.",
"About 1000 seconds after the BAT trigger, another component emerged which is likely a normal decay and post-jet break.",
"We used a broken power law to fit this component and found $\\alpha _3 \\sim 0.45$ , $\\alpha _4 \\sim 3.5$ with break time around 35000 seconds.",
"We follow the method discussed in Zhang et al.",
"(2007) to calculate $E_{\\rm K,iso}$ , which is almost constant during the normal decay phase in X-ray afterglow.",
"We assume that it is in the $\\nu > \\rm max(\\nu _{\\rm m}, \\nu _{\\rm c})$ region, where the afterglow flux expression does not depend on the medium density.",
"In our calculations, the microphysics parameters of the shock are assigned standard values derived from observations, i.e.",
"$\\varepsilon _e=0.01$ and $\\varepsilon _B=0.001$ , and thus $E_{\\rm K, iso}\\sim 8\\times 10^{52} ~\\rm erg$ .",
"If the later break is assumed to be a post-jet breakThe error bar of the last X-ray data point is large and thus it is difficult to identify where the jet break occurs.",
"However, it is possible to provide a lower limit to the jet opening angle if we assume that it is a jet break., one can estimate the jet opening angle $\\theta _j\\sim 0.063~\\rm rad\\sim 3.6$ degrees with medium density $n=0.1~\\rm cm^{-3}$ .",
"The XRT light curve of short GRB 160821B that has a short plateau emission following an abrupt decay is unusual, but not odd.",
"This temporal behavior is similar to short GRB 090515 (Rowlinson et al.",
"2010).",
"In Figure 5(a), we collect all of the short GRB light curves without extended emission to compare with the X-ray emission of GRB 160821B.",
"We found that most short GRBs appeared to have a steeper decay around several hundred seconds.",
"Particularly, the X-ray emission behavior of GRB 160821B is similar to GRB 090515, which is the first short GRB claimed to have a magnetar central engine origin (Rowlinson et al.",
"2010).",
"Also, the plateau flux of GRB 160821B is the highest compared to other short GRBs.",
"In Figure 5(b), we show the fluence in the BAT band (15$-$ 150 keV) and flux in the XRT band (0.3$-$ 10 keV) at $T_0+100$ s for all the short GRBs in the Swift sample.",
"GRB 160821B is shown with a filled circle.",
"As expected, the higher fluence GRBs tend to have higher flux X-ray afterglows.",
"The abrupt decay following the bright X-ray plateau observed in GRB 160821B is difficult to explain by invoking the external shock model of the black hole central engine.",
"It must invoke the contributions from internal dissipation of a central engine.",
"In this section, we propose the use of the millisecond magnetar central engine model to explain the abrupt decay behavior in the X-ray afterglow emission, and constrain the parameters of the magnetar."
],
[
"Magnetar central engine",
"According to Zhang & Mészáros (2001), the characteristic spin down luminosity $L_0$ and time scale $\\tau $ are written: $L_0 = 1.0 \\times 10^{49}~{\\rm erg~s^{-1}} (B_{\\rm p,15}^2 P_{0,-3}^{-4} R_6^6),$ $\\tau = 2.05 \\times 10^3~{\\rm s}~ (I_{45} B_{\\rm p,15}^{-2} P_{0,-3}^2 R_6^{-6}),$ where $I$ is the moment of inertia of a typical NS with mass $M_{\\rm NS}=1.4 M_{\\odot }$ , $P_0$ is the initial spin period, $B_{\\rm p}$ is the magnetic field strength, $R$ is the radius of the NS, and the convention $Q=10^x Q_x$ is adopted in cgs units for all other parameters throughout the paper.",
"The spin-down time scale can be generally identified as the lower limit of the observed break time, i.e.",
"$\\tau > t_{b}/(1+z),$ where $t_{\\rm b}$ is the break time after the internal plateau found using a broken power-law function fitting.",
"A redshift $z=0.16$ is adopted.",
"The bolometric luminosity at the break time $t_{\\rm b}$ is: $L_b = 4\\pi D_L^2 F_b \\cdot k,$ where $F_b=(1.6\\pm 0.82)\\times 10^{-9}~\\rm erg~cm^{-2} s^{-1}$ is the X-ray flux at $t_{b}$ , $D_L^2$ is luminosity distance, and $k$ is $k$ -correction factor.",
"The characteristic spin-down luminosity is essentially the plateau luminosity, which may be estimated as $L_0 \\simeq L_{\\rm b}=(1.8\\pm 0.6)\\times 10^{48}~\\rm erg~s^{-1}.$ Based on Equation (REF ) and Equation (REF ), one can derive the magnetar parameters $B_{\\rm p}$ and $P_0$ : $B_{\\rm p,15} = 2.05(I_{45} R_6^{-3} L_{0,49}^{-1/2} \\tau _{3}^{-1})~\\rm G,$ $P_{0,-3} = 1.42(I_{45}^{1/2} L_{0,49}^{-1/2} \\tau _{3}^{-1/2})~\\rm s.$ Using the lower limit of $\\tau $ we derive upper limits for $P_0$ and $B_{\\rm p}$ , which are respectively $P_0 < 8.5 \\times 10^{-3}$ s and $B_{\\rm p}< 3.12 \\times 10^{16}$ G. Figure 6a shows the $B_{\\rm p}-P_0$ diagram for GRB 160821B, and compares other short GRBs."
],
[
"Equation of state of NS",
"Another relevant timescale is the collapse time of a supra-massive magnetar, $t_{\\rm col}$ .",
"The post-internal plateau decay slope $\\alpha _2$ is steeper than 2, which is the standard spin down luminosity evolution with time (Zhang & Mészáros, 2001).",
"The break time is therefore defined by the collapse time $t_{\\rm col}$ , and one can write $\\tau \\simeq t_{\\rm b}.$ The maximum gravitational mass ($M_{\\rm max}$ ) depends on spin period which increases with time.",
"Using the same method description in Lasky et al.",
"(2014) and Lü et al.",
"(2015), we can write down $t_{\\rm col}$ as a function of $M_{\\rm p}$ $t_{\\rm col} &=& \\frac{3c^{3}I}{4\\pi ^{2}B_{\\rm p}^{2}R^{6}}[(\\frac{M_{\\rm p}-M_{\\rm TOV}}{\\hat{\\alpha } M_{\\rm TOV}})^{2/\\hat{\\beta }}-P_{0}^{2}]\\nonumber \\\\&=&\\frac{\\tau }{P_{\\rm 0}^{2}}[(\\frac{M_{\\rm p}-M_{\\rm TOV}}{\\hat{\\alpha } M_{\\rm TOV}})^{2/\\hat{\\beta }}-P_{0}^{2}].$ where $\\hat{\\alpha }$ , $\\hat{\\beta }$ , and $M_{\\rm TOV}$ are dependent on the equation of state.",
"Therefore we can use $t_{\\rm col}$ to constrain the NS equation of state (EOS).",
"Here, we only consider five EOS (SLy, APR, GM1, AB-N, and AB-L) for the given proto-magnetar mass distribution derived from the total mass distribution of Galactic NS$-$ NS binary systems (Fig.6b).",
"(1) SLy: is effective nuclear interaction by neutron rich matter with $M_{\\rm TOV}=2.05 M_{\\odot }$ and $R=9.97 \\rm km$ .",
"(2) APR: assume that the inner material is included both dense nucleon admixture of quark materr, with $M_{\\rm TOV}=2.20M_{\\odot }$ and $R=10.00 \\rm km$ .",
"(3) GM1: by relating scalar and vector couplings of the hyperons for saturated nuclear matter with $M_{\\rm TOV}=2.37 M_{\\odot }$ and $R=12.05 \\rm km$ .",
"(4) AB-N: neutrons nuclear attraction due to pion exchange tensor with $M_{\\rm TOV}=2.67 M_{\\odot }$ and $R=12.90 \\rm km$ .",
"(5) AB-L: neutrons nuclear attraction due to scalar exchange with $M_{\\rm TOV}=2.71 M_{\\odot }$ and $R=13.70 \\rm km$ (Lasky et al.",
"2014).",
"Our results show that the GM1 model gives an $M_{\\rm p}$ band falling within the $2\\sigma $ region of the protomagnetar mass distribution, such that the GM1 EOS is the best candidate for a non-rotating NS with maximum mass $M_{\\rm TOV}=2.37 M_{\\odot }$ ."
],
[
"The energy budget of magnetar",
"One of the most important necessary conditions of a magnetar central engine candidate for GRBs is that the sum of the prompt emission energy ($E_{\\rm \\gamma , iso}$ ), internal plateau energy ($E_{\\rm X, iso}$ ), and kinetic energy ($E_{\\rm K,iso}$ ) after jet correction should be less than the total rotation energy (energy budget of magnetar) if we assume the magnetar wind is isotropic.",
"The total rotation energy of the millisecond magnetar is $E_{\\rm rot} = \\frac{1}{2} I \\Omega _{0}^{2}\\simeq 2 \\times 10^{52}~{\\rm erg}~M_{1.4} R_6^2 P_{0,-3}^{-2},$ where $\\Omega _0 = 2\\pi /P_0$ is the initial angular frequency of the neutron star.",
"The total energy of the prompt emission is $E_{\\rm \\gamma , iso}=(2.1\\pm 0.2)\\times 10^{50}~\\rm erg$ within the energy range 8-1000 keV.",
"The X-ray internal plateau energy can be roughly estimated using the break time and break luminosity (Lü & Zhang 2014), i.e.",
"$E_{\\rm X, iso} &\\simeq & L_b \\cdot \\frac{t_b}{1+z} \\nonumber \\\\&\\simeq & (2.79\\pm 0.9)\\times 10^{49}~\\rm erg.$ To estimate the kinetic energy $E_{\\rm K, iso}$ , which is used in the standard forward afterglow model, one has $E_{\\rm K, iso}\\sim 8\\times 10^{52} ~\\rm erg$ (see section 2.3).",
"Therefore $E_{\\rm rot}\\gg \\frac{1}{2}\\theta ^{2}_{\\rm j}(E_{\\rm \\gamma , iso}+E_{\\rm X, iso}+E_{\\rm K, iso})$ , which satisfies the magnetar central engine energy budget requirement."
],
[
"Ellipticity constraints of newly-born NS",
"The coalescence of double neutron stars is believed to be one of the most likely sources for powering gravitational wave radiation with associated EM signals.",
"These events have promising detectability prospects with current and future gravitational wave detectors like advanced LIGO/Virgo (Zhang 2013; Gao et al.",
"2013; Yu et al.",
"2013; Fan et al.",
"2013).",
"If indeed a magnetar drove GRB 160821B, why was the total rotation energy of the magnetar much larger than the sum of the prompt emission energy, internal energy and kinetic energy?",
"Several possible reasons may be used in interpreting the gap in the energy compared to the energy budget.",
"One is that the efficiency is as low as $\\sim 0.01$ , such low efficiency may disfavor the magnetic energy dissipation process (Fan et al.",
"2013).",
"Another possibility is the missing energy must have been carried away by non-electromagnetic gravitational wave radiation (Fan et al.",
"2013; Lasky & Glampedakis 2016; Ho 2016), or carried to the black hole before spin down.",
"Following Fan et al.",
"(2013) and Lasky & Glampedakis (2016), a magnetar loses rotational energy through two channels: magnetic dipole torques ($L_{\\rm m}$ ) gravitational wave radiation ($L_{\\rm w}$ ) $-dE_{\\rm rot}/dt &=& L_{\\rm m} + L_{\\rm w} \\nonumber \\\\&=& \\frac{B^2_{\\rm p}R^{6}\\Omega ^{4}}{6c^{3}}+\\frac{32GI^{2}\\epsilon ^{2}\\Omega ^{6}}{5c^{5}},$ where $\\epsilon =2(I_{\\rm xx}-I_{\\rm yy})/(I_{\\rm xx}+I_{\\rm yy})$ is the ellipticity in terms of the principal moments of inertia, assuming the magnetar has a triaxial shape.",
"Following the method of Lasky & Glampedakis (2016), GW radiation can be more efficient than magnetic dipole radiation because of its stronger dependence on the neutron star spin rate $\\Omega $ , i.e.",
"$\\Omega ^{6}$ and $\\Omega ^{4}$ respectively.",
"The upper limit on the ellipticity ($\\epsilon $ ) can be expressed simply with a dependence on observed plateau luminosity and break time (Lasky & Glampedakis 2016), $&\\epsilon _{\\rm obs}& \\le \\biggl (\\frac{15c^{5}\\eta I}{512G L^{2}_{0} t^{3}_{\\rm b}}\\biggr )^{1/2}\\nonumber \\\\&=& \\!0.33\\eta \\biggl (\\frac{I}{10^{45}~\\rm g\\,cm^{2}}\\biggr )^{\\!1/2}\\!\\biggl (\\frac{L_{0}}{10^{49}~\\rm erg\\,s^{-1}}\\biggr )^{\\!-1}\\!\\biggl (\\frac{t_{\\rm b}}{100~\\rm s}\\biggr )^{\\!-3/2}.\\nonumber \\\\$ Using the typical NS mass and radius, $\\eta =0.1$ , $L_{0}\\sim 5.38\\times 10^{47}~\\rm erg~s^{-1}$ , and $t_{\\rm b}\\sim 180$ s, one has $\\epsilon _{\\rm obs}< 0.07$ ."
],
[
"Detection probability of gravitational wave",
"If most of the rotation energy is released via gravitational wave radiation with a frequency $f$ , the gravitational wave strain for a rotating neutron star at distance $D_{\\rm L}$ can be expressed as $h(t)=\\frac{4G I \\epsilon }{D_{\\rm L}c^{4}} \\Omega (t)^{2}.$ The noise power spectral density of the detector, $S_{h}(f)$ , and the stationary phase approximation implies $\\tilde{h}(f)^{2}=h(t)^{2}|dt/df|$ , where $\\tilde{h}(f)$ is the Fourier transform of $h(t)$ .",
"Following method of Lasky & Glampedakis (2016), $\\tilde{h}(f)$ can be expressed as $\\tilde{h}(f) &=& \\frac{1}{D_{\\rm L}}\\sqrt{\\frac{5GI}{2c^{3}f}} \\nonumber \\\\&\\approx & 2.6\\times 10^{-25} \\biggl (\\frac{I}{10^{45}~\\rm g\\,cm^{2}}\\frac{1~\\rm kHz}{f}\\biggr )^{\\!1/2}\\biggl (\\frac{D_{\\rm L}}{100~\\rm Mpc}\\biggr )^{\\!-1}.",
"\\nonumber \\\\$ So $\\tilde{h}(f)$ is independent of the neutron star ellipticity, but depends on the angular frequency evolution with time.",
"The characteristic amplitude $h_{\\rm c}=f h(t)\\sqrt{dt/df}=f\\tilde{h}(f)$ (Corsi & Mészáros 2009; Hild et al.",
"2011) is $h_{\\rm c} &=& \\frac{f}{D_{\\rm L}}\\sqrt{\\frac{5GI}{2c^{3}f}} \\nonumber \\\\&\\approx & 8.22\\times 10^{-24} \\biggl (\\frac{I}{10^{45}~\\rm g\\,cm^{2}}\\frac{f}{1~\\rm kHz}\\biggr )^{\\!1/2}\\biggl (\\frac{D_{\\rm L}}{100~\\rm Mpc}\\biggr )^{\\!-1}~\\rm .\\nonumber \\\\$ For GRB 160821B, its redshift $z=0.16$ corresponds to $D_{\\rm L}=765~\\rm Mpc$ .",
"Using this and $f=1000~\\rm Hz$ , one can estimate the maximum value of the strain $h_{\\rm c}$ , which is less than $1.1\\times 10^{-24}$ .",
"In Fig.7, we plot the gravitational wave strain sensitivity for advanced-LIGO and Einstein Telescope (ET), from Figure 3 of Lasky & Glampedakis (2016).",
"It is clear that the strain of GRB 160821B is below the initial LIGO or advanced-LIGO noise curve.",
"However, it is comparable to the proposed detectability limit of ET and such a signal may be detected by ET in the future.",
"On the other hand, keeping the total energy constant and moving the event to a lower redshift allows one to estimate the minimum detectability distance of such an event.",
"The gravitational wave strain amplitude will be stronger if the event occurs at a lower redshift.",
"One can estimate the cosmological distances the gravitational wave signal can be detected by current advanced-LIGO.",
"We simulate this source at different distances and calculate the GW strain amplitude.",
"Then we compare that value with the current sensitivity of advanced-LIGO.",
"We find that this GW signal could be detected if shifted to about $100~\\rm Mpc$ , which corresponds to redshift $z\\sim 0.023$ (Fig.",
"7)."
],
[
"Conclusions and Discussion",
"GRB 160821B is a short gamma-ray burst (GRB) of duration less than 1 second, at redshift $z=0.16$ , observed by Swift and Fermi.",
"We presented a broadband analysis of its prompt and afterglow emission and found that there is no evidence to for any “extended emission” up to more than 100 seconds in Swift/BAT and Fermi/GBMPresence or absence of extended emission of short GRB may be related to different physics process.. More interestingly, the X-ray plateau was followed by an extremely steep decay as observed by Swift/XRT but which is not unique in the Swift era, i.e.",
"it is similar to GRB 090515 (Rowlinson et al.",
"2010), which was the first short GRB with such behavior.",
"This behavior is very difficult to explain with the standard external shock model of black hole central engine, but could be consistent with the prediction of a magnetar central engine.",
"It is likely that it formed into supra-massive NS initially and collapsed into black hole after several hundred seconds.",
"This event is thus one important probe for studying the physical properties of the central engine and progenitor of GRBs.",
"Our analysis shows the initial short plateau emission in its X-ray lightcurve, which is consistent with energy injection from the magnetar wind of a supra-massive magnetar losing rotation energy, and followed by a steeper decay due to the magnetar collapsing to a black hole.",
"The derived magnetar surface magnetic field $B_{\\rm p}$ and the initial spin period $P_0$ fall into a reasonable range, i.e.",
"$B_{\\rm p}<3.12\\times 10^{16}$ and G$P_0 < 8.5\\times 10^{-3}$ s. Using the collapse time to constrain the equation of state of the neutron star shows consistency with the GM1 model with $M_{\\rm TOV} \\sim 2.37 M_\\odot $ .",
"The total isotropic-equivalent electromagnetic energy ($\\gamma $ -ray energy, internal plateau energy, and kinetic energy) is much less than the energy budget of the magnetar (a few $\\times 10^{52}~\\rm erg$ ), suggesting that the missing energy of the supra-massive magnetar may be radiated via gravitational waves, or carried into the black hole before spin down.",
"If it is indeed that the energy dissipated via gravitational waves, one can constrain the ellipticity of the NS to $\\epsilon <0.07$ .",
"Also, the upper limit of the gravitational wave strain can be estimated as $h_{\\rm c}\\approx 1.1\\times 10^{-24}$ at $f=1000~\\rm Hz$ , which is below the advanced-LIGO noise curve, but may be detectable by Einstein Telescope in the future.",
"If we shift this source to $\\sim 100~\\rm Mpc$ cosmological distance ($z\\sim 0.023$ ), then the gravitational wave signal could be detected by the current advanced-LIGO.",
"The event rate density of SGRBs depends on the minimum luminosity threshold.",
"Given the detectability horizon of advanced-LIGO, i.e.",
"a distance of 100 $\\rm Mpc$ , all the observed SGRBs are above the BAT sensitivity.",
"Following the method of Sun et al.",
"(2015), if we consider the minimum isotropic luminosity of the observed SGRBs, which gives an event rate of $4.2^{+1.3}_{-1.0}~\\rm Gpc^{-3}~yr^{-1}$ above $7\\times 10^{49}~\\rm erg~s^{-1}$ and varies by a factor less than two for different delay timescale models, we estimate there are 2 SGRBs every one hundred years within 100 $\\rm Mpc$ .",
"This is quite small and consistent with the non-detection of any SGRBs accompanying detected GW events.",
"It is also possible that there may be low-luminosity SGRBs extending to a luminosity of $10^{47}~\\rm erg~s^{-1}$ , which is the detection limit for Swift/BAT for SGRBs at 100 $\\rm Mpc$ .",
"The estimated event rate density above this luminosity threshold is much larger than that of the $7\\times 10^{49}~\\rm erg~s^{-1}$ luminosity threshold.",
"In this case, one may expect one low luminosity SGRB every two years.",
"However, this is quite speculative as we have not seen any low luminosity SGRBs yet.",
"We have already had a few such cases for LGRBs.",
"Both of the two cases can be tested with future detections.",
"For current circumstances, the first scenario is preferred.",
"On the other hand for NS-NS mergers a more isotropic, sub-relativistic outflow could be ejected during a neutron-rich merger which can synthesize heavier radioactive elements via r-process.",
"A thermal UV-optical transient may be powered by radioactive decay except the short GRB and its X-ray afterglow (Li & Paczynski 1998; Rezzolla et al.",
"2011; Yu et al.",
"2013).",
"However, if the post-merger product is a supra-massive NS supported by rigid rotation, e.g.",
"GRB 160821B, the spin-down magnetic dipole radiation of the NS remnant provides an additional energy source to the ejecta.",
"This optical transient (Li-Paczynski-nova, macro-nova, kilo-nova, merger-nova, r-process) emission component would be significantly enhanced since it is heated by the magnetar wind and could easily exceed the r-process power (Li & Paczynski 1998; Tanvir et al.",
"2013; Berger et al.",
"2013; Yu et al.",
"2013; Yang et al.",
"2015; Gao, Zhang & Lü 2016; Gao et al.",
"2016).",
"From the theoretical point of view, it is expected that the optical or near-infrared bump is detected at late time or an excess of flux would be visible in the spectral energy distribution.",
"One can use the properties of observed merger-nova to constrain the parameters of the central engine.",
"However, due to lack of optical observations, catching the possible merger-nova is expected by following up with an optical instrument in the future, i.e.",
"Hubble Space Telescope (HST).",
"We acknowledge the use of the public data from the Swift, Fermi data archive, and the UK Swift Science Data Center.",
"We also thank the anonymous referees for helpful comments.",
"This work is supported by the National Basic Research Program (973 Programme) of China 2014CB845800, the National Natural Science Foundation of China (Grant No.11603006, 11533003, 11503011), the One-Hundred-Talents Program of Guangxi colleges, Guangxi Science Foundation (grant No.",
"2016GXNSFCB380005, 2013GXNSFFA019001), Scientific Research Foundation of GuangXi University (Grant No XGZ150299).",
"Figure: The Swift/BAT and Fermi/GBM lightcurve of GRB 160821Bin different energy bands with 64 ms time bin.Figure: Joint fit of the time-averaged spectra of BAT (green points)and GBM (red and black points) data with a power-law model.Figure: The Swift/XRT light curve of GRB 160821B (black points).",
"Thelower plot shows the photon index evolution.",
"The red solid line and blue dashed line show thebroken power-law fit to the light curve.Figure: Luminosity-spectral lag diagram.",
"The red star indicates GRB 160821B.Grey dots, green squares, and blue diamonds are indicated Type II, Type I, and other-short hard GRBs, respectively.The solid black line and two dash lines are represented the best linear fit to type II GRBs and 2σ2\\sigma region.Figure: (a): Comparing the X-ray light curve (0.3--10 keV) of GRB 160821Bwith other short GRBs without extended emission.",
"(b): The fluence in BAT energy band (15--150keV) versus the flux in XRT band (0.3--10 keV) for all Swift SGRBs which wereobserved 100s after the triggertime.",
"The red star marks the location of GRB 160821B.Figure: (a): Inferred magnetar parameters, initial spin period P 0 P_0 vs.surface polar cap magnetic field strength B p B_p derived for GRB 160821B (red star),compared with other short GRBs (grey triangle) and GRB 090515 (blue point).",
"Thevertical solid line is the break-up spin period limit for a neutron star (Lattimer &Prakash 2004).",
"(b): Collapse time as a function of the protomagnetar mass of GRB 160821B fordifferent EOS: SLy (black), APR (red), GM1 (green), AB-N (blue), and AB-L (cyan).The horizontal dotted line is the observed collapse time.Figure: Gravitational wave strain evolution with frequency for GRB 160821B,at distance D L =765 Mpc D_{\\rm L}=765~\\rm Mpc (black solid line), 200 Mpc 200~\\rm Mpc (pink dot line),100 Mpc ( bluedash - dotline )100~\\rm Mpc (blue dash-dot line).",
"The grey region is the strain of GW 150914 between35 Hz 35~\\rm Hz and 250 Hz 250~\\rm Hz.The black dotted line and red dashed line are the sensitivity limits for aLIGO and the EinsteinTelescope, respectively."
]
] | 1612.05691 |
[
[
"Parameterization of Coarse-grained Molecular Interactions through\n Potential of Mean Force Calculations and Cluster Expansions Techniques"
],
[
"Abstract We present a systematic coarse-graining (CG) strategy for many particle molecular systems based on cluster expansion techniques.",
"We construct a hierarchy of coarse-grained Hamiltonians with interaction potentials consisting of two, three and higher body interactions.",
"The accuracy of the derived cluster expansion based on interatomic potentials is examined over a range of various temperatures and densities and compared to direct computation of pair potential of mean force.",
"The comparison of the coarse-grained simulations is done on the basis of the structural properties, against the detailed all-atom data.",
"We give specific examples for methane and ethane molecules in which the coarse-grained variable is the center of mass of the molecule.",
"We investigate different temperature and density regimes, and we examine differences between the methane and ethane systems.",
"Results show that the cluster expansion formalism can be used in order to provide accurate effective pair and three-body CG potentials at high $T$ and low $\\rho$ regimes.",
"In the liquid regime the three-body effective CG potentials give a small improvement, over the typical pair CG ones; however in order to get significantly better results one needs to consider even higher order terms."
],
[
"Introduction",
"The theoretical study of complex molecular systems is a very intense research area due to both basic scientific questions and technological applications.",
"[1] A main challenge in this field is to provide a direct quantitative link between chemical structure at the molecular level and measurable macroscopic quantities over a broad range of length and time scales.",
"Such knowledge would be especially important for the tailored design of materials with the desired properties, over an enormous range of possible applications in nano-, bio-technology, food science, drug industry, cosmetics etc.",
"A common characteristic of all complex fluids is that they exhibit multiple length and time scales.",
"Therefore, simulation methods across scales are required in order to study such systems.",
"On the all-atom level description, classical atomistic models have successfully been used in order to quantitatively predict the properties of molecular systems over a considerable range of length and time scales.",
"[2], [1], [3], [4] However, due to the broad spectrum of characteristic lengths and times involved in complex molecular systems it is desirable to reduce the required computational cost by describing the system through a small number of degrees of freedom.",
"Thus, coarse-grained (CG) models have been used in order to increase the length and time scales accessible by simulations.",
"[1], [5], [6], [7], [8], [9], [3], [10], [11], [12], [13], [5], [14], [15], [16], [17], [18], [19], [20], [21], [22] From a mathematical point of view, coarse-graining is a sub-field of dimensionality reduction;[4] there are several statistical methods for the reduction of the degrees of freedom under consideration in a deterministic or stochastic model, such as principal component analysis, polynomial chaos and diffusion maps.",
"[20] Here we focus our discussion on CG methods based on a combination of recent computational methods and old theoretical tools from statistical mechanics.",
"Such CG models, which are developed by lumping groups of atoms into CG particles and deriving the effective CG interaction potentials directly from more detailed (microscopic) simulations, are capable of predicting quantitatively the properties of specific molecular systems (see for example refs.",
"[5], [6], [7], [8], [9], [11], [12], [13], [23], [15], [17], [18], [19], [24], [25] and references therein).",
"The most important part in all systematic CG models, based on detailed atomistic data, is to develop rigorous all-atom to CG methodologies that allow, as accurate as possible, estimation of the CG effective interaction.",
"With such approaches the combination of atomistic and hierarchical CG models could allow the study of a very broad range of length and time scales of specific molecular systems without adjustable parameters, and by that become truly predictive.",
"[14], [11], [15] There exists a variety of methods that construct a reduced CG model that approximates the properties of molecular systems based on statistical mechanics.",
"For example: (a) In structural, or correlation-based, methods the main goal is to find effective CG potentials that reproduce the pair radial distribution function $g(r)$ , and the distribution functions of bonded degrees of freedom (e.g.",
"bonds, angles, dihedrals) for CG systems with intramolecular interaction potential.",
"[21], [22], [6], [7], [10], [9] The CG effective interactions in such methods are obtained using the direct Boltzmann inversion, or reversible work, method [26], [10], [27], [28] or iterative techniques, such as the iterative Boltzmann inversion, IBI [29], [7], and the inverse Monte Carlo, IMC, (or inverse Newton) [22], [30] approach.",
"(b) Force matching (FM) or multi-scale CG (MSCG) methods [31], [5], [14], [32], [16], [33] is a mean least squares problem that considers as observable function the total force acting on a coarse bead.",
"(c) The relative entropy (RE) [8], [18], [34] method employs the minimization of the relative entropy, or Kullback-Leibler divergence, between the microscopic Gibbs measure $\\mu $ and $\\mu ^{\\theta }$ , representing approximations to the exact coarse space Gibbs measure.",
"In this case, the microscopic probability distribution can be thought as the observable.",
"The minimization of the relative entropy is performed through Newton-Raphson approaches and/or stochastic optimization techniques.",
"[35], [19] In practice, all above numerical methods are employed to approximate a many body potential of mean force (PMF), $U_{\\text{PMF}}$ , describing the equilibrium distribution of CG particles observed in simulations of atomically detailed models.",
"Besides the above numerical parametrization schemes, more analytical approaches have also been developed for the approximation of the CG effective interaction, based on traditional liquid state theory and on pair correlation functions.",
"[36], [37], [38], [39], [40], [41], [42] Here we discuss an approach for estimating $U_{\\text{PMF}}$ , and the corresponding effective CG non-bonded potential, based on cluster expansion methods.",
"Such methods originate from the works of Mayer and collaborators [43] in the 40's.",
"In the 60's numerous approximate expansions have been further developed [44], [45] for the study of the liquid state.",
"Later, with the advancement of powerful computational machines, the main focus has been directed on improving the computational methods such as Monte Carlo and molecular dynamics.",
"However, the latter are mostly bulk calculations and they get quite slow for large systems.",
"Reducing the degrees of freedom by coarse-graining has been a key strategy to construct more efficient methods, but with many open questions with respect to error estimation, transferability and adaptivity of the suggested methods.",
"Based on recent developments of the mathematical theory of expansion methods in the canonical ensemble [46], our purpose is to combine the two approaches and obtain powerful computational methods, whose error compared to the target atomistic calculations can be quantified via rigorous estimates.",
"In principle, the validity of these methods is limited to the gas regime.",
"Here we examine the accuracy of these methods in different state points.",
"This attempt consists of the following: a priori error estimation of the approximate schemes depending on the different regimes, a posteriori error validation of the method from the coarse-grained data and design of related adaptive methods.",
"In previous years, we have carried out this program for the case of lattice systems, obtaining higher order schemes and a posteriori error estimates [47], for both short and long range interactions [48] and designing adaptive methods [49] and investigating possible strategies for reconstruction of the atomistic information.",
"[50] This is very much in the spirit of the polymer science literature [51], [10], [11] and in this paper we get closer by considering off-lattice models.",
"The proposed approach is based on typical schemes that are based on isolated molecules.",
"[27], [52], [26] Here we extend such approaches using cluster expansion tools for deriving CG effective potentials.",
"We start from typical 2-body (pair) effective interaction, but some results can be extended to many-body interactions as well.",
"We also present a detailed theoretical investigation about the effect of higher order terms in obtaining CG effective interaction potentials for realistic molecular systems.",
"Then, we show some first results from the implementation of three-body terms on the effective CG potential; a more detailed work on the higher order terms will be given in a forthcoming work.",
"[53] The structure of the paper is as follows: In Section , we introduce the atomistic molecular system and its coarse-graining via the definition of the CG map, the n-body distribution function and the corresponding n-body potential of mean force.",
"The cluster expansion based formulation of the CG effective interaction is presented in Section .",
"Details about the model systems (methane and ethane) and the simulation considered here are discussed in Section .",
"Results are presented in Section .",
"Finally, we close with Section summarizing the results of this work.",
"Here we give a short description of the molecular model in the microscopic (all-atom) and mesoscopic (coarse-grained) scale.",
"Assume a system of N (classical) atoms (or molecules) in a box $\\Lambda (\\ell ):=(-\\frac{\\ell }{2},\\frac{\\ell }{2}]^d\\subset \\mathbb {R}^d$ (for some $\\ell >0$ ), at temperature $T$ .",
"We will also denote the box by $\\Lambda $ when we do not need to explicit the dependence on $\\ell $ .",
"We consider a configuration $ \\mathbf {q}\\equiv \\lbrace q_1,\\ldots ,q_N\\rbrace $ of $N$ atoms, where $q_i$ is the position of the $i^{th}$ atom.",
"The particles interact via a pair potential $V:\\mathbb {R}^d\\rightarrow \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ , which is stable and tempered.",
"Stability means that there exists a constant $B\\ge 0$ such that: $\\sum _{1\\le i<j \\le N} V(q_i-q_j) \\ge -BN,$ for all $N$ and all $q_1,...,q_N$ .",
"Moreover, temperedness requires that $C(\\beta ):= \\int _{\\mathbb {R}^d} |e^{-\\beta V(r)}-1| dr <\\infty .$ where $\\beta =\\frac{1}{k_{B}T}$ and $k_{B}$ is Boltzmann's constant.",
"The canonical partition function of the system is given by $Z_{\\beta ,\\Lambda ,N}:=\\frac{1}{N!",
"}\\int _{\\Lambda ^N} dq_1\\,\\ldots dq_N \\,e^{-\\beta H_{\\Lambda }(\\mathbf {q})},$ where $H_{\\Lambda }$ is the Hamiltonian (total energy) of the system confined in a domain $\\Lambda $ : $H_{\\Lambda }(\\mathbf {p}, {\\mathbf {q}}):=\\sum _{i=0}^{N}\\frac{p^2_i}{2m}+U({\\bf q}).$ By $U(\\mathbf {q})$ we denote the total potential energy of the system: $U( {\\mathbf {q}}) :=\\sum _{1\\le i<j \\le N} V(q_i-q_j), $ where for simplicity we assume periodic boundary conditions on $\\Lambda $ .",
"Integrating over the momenta in (REF ), we get: $Z_{\\beta ,\\Lambda ,N}=\\frac{\\lambda ^N}{N!",
"}\\int _{\\Lambda ^N} dq_1\\,\\ldots dq_N \\,e^{-\\beta U(\\mathbf {q})}=:\\lambda ^N Z_{\\beta ,\\Lambda ,N}^U,$ where $\\lambda :=(\\frac{2m\\pi }{\\beta })^{d/2}$ .",
"In the sequel, for simplicity we will consider $\\lambda =1$ and identify $Z_{\\beta ,\\Lambda ,N}\\equiv Z_{\\beta ,\\Lambda ,N}^U$ .",
"Fixing the positions $q_1$ and $q_2$ of two particles, we define the two-point correlation function : $\\rho _{N, \\Lambda }^{(2), at}(q_1,q_2):=\\frac{1}{(N-2)!",
"}\\int dq_3 \\ldots dq_N\\frac{1}{Z_{\\beta ,\\Lambda ,N}}e^{-\\beta U(\\mathbf {q})}.$ It is easy to see that in the thermodynamic limit the leading order is $\\rho ^2$ , where $\\rho =\\frac{N}{|\\Lambda |}$ and $|\\Lambda |$ is the volume of the box $\\Lambda $ .",
"Thus, it is common to define the following order one quantity $g(r):=\\frac{1}{\\rho ^{2}}\\rho _{N, \\Lambda }^{(2), at}(q_1,q_2)$ , for $r=|q_1-q_2|$ .",
"More generally, for $n\\le N$ , we define the $n$ -body version $g^{(n)}(q_1,\\dots ,q_n) = \\frac{1}{(N-n)!",
"\\rho ^n}\\int _{\\Lambda ^{N-n}} dq_{n+1}\\dots dq_N\\!\\frac{1}{Z_{\\beta ,\\Lambda ,N}}e^{-\\beta U(\\mathbf {q})},$ and from that the order $n$ potential of mean force (PMF), $U_{\\text{PMF}}(q_1,\\dots ,q_n)\\!$ , [54], [55] given by $U_{\\text{PMF}}(q_1,\\dots ,q_n) := -\\frac{1}{\\beta } \\log g^{(n)}(q_1,\\dots ,q_n) .$ We define the coarse-graining map $T:(\\mathbb {R}^d)^N\\rightarrow (\\mathbb {R}^d)^M$ on the microscopic state space, given by $T: \\mathbf {q}\\mapsto T(\\mathbf {q})\\equiv (T_1(\\mathbf {q}),\\ldots , T_M(\\mathbf {q}))\\in \\mathbb {R}^M$ , which determines the $M$ $(M<N)$ CG degrees of freedom as a function of the atomic configuration $\\bf q$ .",
"We call “CG particles” the elements of the coarse space with positions $\\mathbf {r}\\equiv \\lbrace r_1,\\ldots ,r_M\\rbrace $ .",
"The effective CG potential energy is defined by $U_{\\text{eff}}(r_1,\\ldots ,r_M):=-\\frac{1}{\\beta }\\log \\int _{\\lbrace T\\mathbf {q}=\\mathbf {r}\\rbrace }dq_1\\,\\ldots dq_N \\,e^{-\\beta U(\\mathbf {q})},$ where the integral is over all atomistic configurations that correspond to a specific CG one using the coarse-graining map.",
"In the example we will deal with later, the configuration $\\mathbf {r}$ will represent the centers of mass of groups of atomistic particles.",
"This coarse graining gives rise to a series of multi-body effective potentials of one, two, up to $M$ -body interactions, which are unknown functions of the CG configuration.",
"Note also that by the construction of the CG potential in (REF ) the partition function is the same: $Z_{\\beta ,\\Lambda ,N}=\\int dr_1\\ldots dr_M\\int _{\\lbrace T\\bf q=\\bf r\\rbrace }d\\mathbf {q}e^{-\\beta U(\\mathbf {q})}=\\int dr_1\\ldots dr_Me^{-\\beta U_{\\text{eff}}(r_1,\\ldots ,r_M)}=:Z^{cg}_{\\beta ,\\Lambda ,M}$ The main purpose of this article is to give a systematic way (via the cluster expansion method) of constructing controlled approximations of $U_{\\text{eff}}$ that can be efficiently computed and at the same time we have a quantification of the corresponding error for both “structural\" and “thermodynamic\" quantities.",
"By structural we refer to $g(r)$ , while by thermodynamic to the pressure and the free energy.",
"Note that both depend on the partition function, but they can also be related [55] to each other as follows: $\\beta p=\\rho -\\frac{\\beta }{6}\\rho ^2 \\int _0^{\\infty }r u^{\\prime }(r)g(r)4\\pi r^2 dr,$ at least for the case of pair-interaction potentials."
],
[
"Coarse-grained approximations",
"As mentioned above there are several methods in the literature that give approximations to the effective (CG) interaction potential $U_{\\text{eff}}$ as defined in (REF ).",
"Below we list some of them without claim of being exhaustive: (a) The `correlation-based (eg.",
"DBI, IBI and IMC) methods that use the pair radial distribution function $g(r)$ , related to the two-body potential of mean force for the intermolecular interaction potential, as well as distribution functions of bonded degrees of freedom (e.g.",
"bonds, angles, dihedrals) for CG systems with intramolecular interaction potential.",
"[21], [22], [6], [7], [10], [9] These methods will be further discussed below.",
"(b) Force matching (FM) methods [31], [5], [16] in which the observable function is the average force acting on a CG particle.",
"The CG potential is then determined from atomistic force information through a least-square minimization principle, to variationally project the force corresponding to the potential of mean force onto a force that is defined by the form of the approximate potential.",
"(c) Relative entropy (RE)[8], [18], [19] type methods that produce optimal CG potential parameters by minimizing the relative entropy, Kullback-Leibler divergence between the atomistic and the CG Gibbs measures sampled by the atomistic model.",
"In addition to the above numerical methods, analytical works for the estimation of the effective CG interaction, based on integral equation theory, have also been developed [40].",
"A brief review and categorization of parametrization methods at equilibrium is given in references [17], [56].",
"The correlation-based iterative (e.g.",
"IBI and IMC) methods use the fact that for a pair interaction $u(r)$ , by plugging the virial expansion of $p$ in powers of $\\rho $ into (REF ) and comparing the orders of $\\rho $ , one obtains that $g(r)=e^{-\\beta u(r)}\\gamma (r),\\quad \\gamma (r)=1+c_1(r) \\rho + c_2(r) \\rho ^2+\\ldots $ Given the atomistic “target\" $g(r)$ from a free (i.e., without constraints) atomistic run, by inverting (REF ) and neglecting the higher order terms of $\\gamma (r)$ one can obtain a first candidate for a pair coarse-grained potential $u(r)$ .",
"Then, one calculates the $g(r)$ that corresponds to the first candidate and by iterating this procedure eventually obtains the desired two-body coarse-grained potential.",
"This iteration should in principle converge since there exists a pair interaction that can be reconstructed from a given correlation function .",
"However, this is only an approximation (accounting for the neglected terms of order $\\rho $ and higher in the expansion of $\\gamma (r)$ ) since we know that the “true\" CG interaction potential should be multi-body, as a result of integrating atomistic degrees of freedom.",
"Hence, having agreement on $g(r)$ does not secure proper thermodynamic behaviour and several methods have been employed towards this direction, see for example refs [7], [57], [40] and the references within.",
"In order to maintain the correct thermodynamic properties, our approach in this paper is based on cluster expanding (REF ) with respect to some small but finite parameter $\\epsilon $ depending on the regime we are interested in.",
"For technical reasons we will focus on low density - high temperature regime.",
"As it will be explained in detail in the next section, the resulting cluster expansion provides us with a hierarchy of terms: $U_{\\text{eff}}=U^{(2)}+U^{(3)}+O(\\epsilon ^3),\\qquad U^{(2)}(r_1,\\ldots , r_M):=\\sum _{i,j}W^{(2)}(r_i,r_j),\\quad U^{(3)}(r_1,\\ldots , r_M):=\\sum _{i,j,k}W^{(3)}(r_i,r_j,r_k),\\quad \\text{etc},$ together with the corresponding error estimates.",
"The above terms can in principle be calculated independently via fast atomistic simulations of 2, 3, etc.",
"molecules, in the spirit of the conditional reversible work CRW method.",
"In more detail, the effective non-bonded (two-body) CG potential can be computed as follows: (a) One method is by fixing the distance $r_{1,2}:=r_1-r_2$ between two molecules and perform molecular dynamics with such forces that maintain the fixed distance $r_{1,2}$ .",
"In this way we sample over the constrained phase space and obtain the conditioned partition function as in (REF ).",
"Then, by integration of the constrained force the two-body effective potential can be obtained.",
"(b) Alternatively, by inverting $g(r)$ in (REF ) for two isolated molecules, the two-body effective potential can be directly obtained, since for such a system $\\gamma (r)=1$ .",
"Here we examine both methods, see Figure REF .",
"Note also that the validity of cluster expansion provides rigorous expansions for $g(r)$ , the pressure and the other relevant quantities.",
"Hence, with this approach we can have a priori estimation of the errors made in (REF ).",
"Another benefit of the cluster expansion is that the error terms can be written in terms of the coarse-grained quantities allowing for a posteriori error estimates and the design of adaptive methods [49]; see also discussion in Section ."
],
[
"Cluster expansion",
"The cluster expansion method originates from the work of Mayer and collaborators, see ref.",
"[43] for an early review, and consists of expanding the logarithm of the partition function in an absolutely convergent series of an appropriately chosen small but finite parameter.",
"Here we will adapt this method to obtain an expansion of the conditioned partition function (REF ).",
"For the purpose of this article we assume that the CG map $T$ is a product $T=\\otimes _{i=1}^M T^i$ creating $M$ groups of $l_1, \\ldots , l_M$ particles each.",
"We index the particles in the $i^{th}$ group of the coarse-grained variable $r_i$ by $k^i_1, \\ldots , k^i_{l_i}$ .",
"We also denote them by $\\mathbf {q}^i:=(q_{k^i_1}, \\ldots , q_{k^i_{l_i}})$ , for $i=1,\\ldots ,M$ .",
"Then (REF ) can be written as: $U_{\\text{eff}}(r_1,\\ldots ,r_M):=-\\frac{1}{\\beta }\\log \\prod _{i=1}^M \\lambda ^i(\\lbrace T^i\\mathbf {q}^i=r_i\\rbrace )-\\frac{1}{\\beta }\\log \\int \\prod _{i=1}^M\\mu (d\\mathbf {q}^i;r_i)e^{-\\beta U(\\mathbf {q})},$ where, for simplicity, we have introduced the normalized conditional measure: $\\mu (d\\mathbf {q}^i;r_i):=\\frac{1}{l_i!",
"}dq_{k^i_1}\\ldots dq_{k^i_{l_i}}\\frac{\\mathbf {1}_{\\lbrace T^i\\mathbf {q}^i=r_i\\rbrace }}{\\lambda ^i(\\lbrace T^i\\mathbf {q}^i=r_i\\rbrace )},$ and by $\\lambda ^i$ we denote the measure $\\frac{1}{l_i!",
"}dq_{k^i_1}\\ldots dq_{k^i_{l_i}}$ .",
"To perform a cluster expansion in the second term of (REF ) we rewrite the interaction potential as follows: $U(\\mathbf {q})=\\sum _{i<j} \\bar{V}(\\mathbf {q}^i,\\mathbf {q}^j),\\quad \\text{where}\\quad \\bar{V}(\\mathbf {q}^i,\\mathbf {q}^j):=\\sum _{m=1}^{l_i}\\sum _{m^{\\prime }=1}^{l_{j}} V(|q_{k^i_m}-q_{k^j_{m^{\\prime }}}|).$ Then, we have $e^{-\\beta U(\\mathbf {q})} & = &\\prod _{i<j} \\left(1+ e^{-\\beta \\bar{V}(\\mathbf {q}^i,\\mathbf {q}^j)}-1\\right) \\nonumber \\\\&=&\\sum _{\\begin{array}{c}V_1,\\ldots ,V_m \\\\ |V_i|\\ge 2, V_i\\subset \\lbrace 1,\\ldots ,N\\rbrace \\end{array}}\\prod _{l=1}^m\\sum _{g\\in \\mathcal {C}_{V_l}}\\prod _{\\lbrace i,j\\rbrace \\in E(g)} f_{i,j}(\\mathbf {q}^i,\\mathbf {q}^j),\\quad \\text{where}\\quad f_{i,j}(\\mathbf {q}^i,\\mathbf {q}^j):=e^{-\\beta \\bar{V}(\\mathbf {q}^i,\\mathbf {q}^j)}-1,$ where for $V\\subset \\lbrace 1,\\ldots ,N\\rbrace $ , we denote by $\\mathcal {C}_{V}$ the set of connected graphs on the set of vertices with labels in $V$ .",
"Furthermore, for $g\\in \\mathcal {C}_{V}$ , we denote by $E(g)$ the set of its edges.",
"Figure: Visualization of the partition in () for non-intersecting sets V 1 ={2,3,4,5}V_1=\\lbrace 2,3,4,5\\rbrace , V 2 ={6,7}V_2=\\lbrace 6,7\\rbrace , V 3 ={9,10,11}V_3=\\lbrace 9,10,11\\rbrace in each of which we display by solid lines the connected graphs g i ∈𝒞 V i g_i\\in \\mathcal {C}_{V_i}, i=1,2,3i=1,2,3.Since $\\mu $ in (REF ) is a normalized measure, from (REF ) we obtain: $U_{\\text{eff}}(r_1,\\ldots ,r_M) & = &-\\frac{1}{\\beta }\\log \\prod _{i=1}^M \\lambda ^i(\\lbrace T^i\\mathbf {q}^i=r_i\\rbrace )-\\frac{1}{\\beta }\\log \\sum _{\\begin{array}{c}V_1,\\ldots ,V_m \\\\ |V_i|\\ge 2, V_i\\subset \\lbrace 1,\\ldots ,N\\rbrace \\end{array}}\\prod _{l=1}^m\\zeta (V_i)\\nonumber \\\\&=&-\\frac{1}{\\beta }\\log \\prod _{i=1}^M \\lambda ^i(\\lbrace T^i\\mathbf {q}^i=r_i\\rbrace )-\\frac{1}{\\beta }\\sum _{V\\subset \\lbrace 1,\\ldots ,N\\rbrace }\\zeta (V)+\\frac{1}{\\beta }\\sum _{\\begin{array}{c}V,V^{\\prime }:\\\\ V\\cap V^{\\prime }=\\varnothing \\end{array}}\\zeta (V)\\zeta (V^{\\prime })+\\ldots ,$ where $\\zeta (V):=\\int \\sum _{g\\in \\mathcal {C}_{V}}\\prod _{\\lbrace i,j\\rbrace \\in E(g)} f_{i,j}(\\mathbf {q}^i,\\mathbf {q}^j) d\\mathbf {q}_V$ with $\\mathbf {q}_V:=\\lbrace \\mathbf {q}^i\\rbrace _{i\\in V}$ , is a function over the atomistic details of the system.",
"Note that the above expression involves a sum over all possible pairs, triplets etc.",
"which is a convergent series for values of the density $\\rho =\\frac{N}{|\\Lambda |}$ and of the inverse temperature $\\beta $ such that $\\rho C(\\beta )<c_0$ , where $C(\\beta )$ is defined in (REF ) and $c_0$ is a known small positive constant.",
"[46] If we simplify the sum in (REF ) one can obtain [46] expansion (REF ) where $W^{(2)}(r_1,r_2):=-\\frac{1}{\\beta }\\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\,f_{1,2}(\\mathbf {q}^1,\\mathbf {q}^2)$ and $W^{(3)}(r_1,r_2,r_3):=-\\frac{1}{\\beta }\\int \\mu (d{\\bf q}^1;r_1)\\,\\mu (d{\\bf q}^2;r_2)\\,\\mu (d{\\bf q}^3;r_3)\\,f_{1,2}(\\mathbf {q}^1,\\mathbf {q}^2)\\,f_{2,3}(\\mathbf {q}^2,\\mathbf {q}^3)\\,f_{3,1}(\\mathbf {q}^3,\\mathbf {q}^1).$ Recall also the definition of $f_{i,j}$ in (REF )."
],
[
"Full calculation of the PMF",
"Notice that the potentials $W^{(2)}$ and $W^{(3)}$ in (REF ) and (REF ), respectively, have been expressed via the Mayer functions $f_{i,j}$ .",
"However, the full effective interaction potential between two CG particles can be directly defined as the (conditional) two-body PMF given by $W^{(2), \\text{full}}(r_1,r_2):=-\\frac{1}{\\beta }\\log \\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\,e^{-\\beta \\bar{V}({\\bf q}^1,{\\bf q}^2)}.$ By adding and subtracting 1, we can relate it to (REF ): $-\\beta W^{(2), \\text{full}}(r_1,r_2) & = &\\log \\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\,e^{-\\beta \\bar{V}({\\bf q}^1,{\\bf q}^2)} =\\log (1+\\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\,f_{1,2}(\\mathbf {q}^1,\\mathbf {q}^2))\\nonumber \\\\& = &\\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\,f_{1,2}(\\mathbf {q}^1,\\mathbf {q}^2)-\\frac{1}{2}\\left(\\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\,f_{1,2}(\\mathbf {q}^1,\\mathbf {q}^2)\\right)^{2}+\\ldots $ Higher order terms in the above equation are expected to be less/more important in high/low temperature.",
"Similarly, for three CG degrees of freedom $r_1, r_2, r_3$ , the full PMF is given by $W^{(3), \\text{full}}(r_1,r_2,r_3):=-\\frac{1}{\\beta }\\log \\int \\mu (d{\\bf q}^1;r_1)\\, \\mu (d{\\bf q}^2;r_2) \\mu (d{\\bf q}^3;r_3)\\,e^{-\\beta \\sum _{1\\le i < j \\le 3}\\bar{V}({\\bf q}^i,{\\bf q}^j)}.$ By adding and subtracting 1 we can relate it to (REF ) and (REF ) (in the following we simplify notation by not explicitly showing the dependence on the atomistic configuration and neglecting the normalized conditional measure): $e^{-\\beta W^{(3),\\text{full}}} & = & \\int e^{-(V_{12}+V_{13}+V_{23})}\\nonumber \\\\& = & 1+ \\int f_{12}+\\int f_{13}+\\int f_{23}+\\int f_{12}f_{13}+\\int f_{13}f_{23}+\\int f_{12}f_{23}+\\int f_{12}f_{23} f_{13},$ which implies that $W^{(3),\\text{full}} & = & -\\frac{1}{\\beta }\\left[\\int f_{12}+\\int f_{13}+\\int f_{23}+\\int f_{12}f_{23} f_{13}+\\right.\\nonumber \\\\&& \\left.\\int f_{12}f_{13}+\\int f_{13}f_{23}+\\int f_{12}f_{23}-(\\int f_{12} \\int f_{13}+\\int f_{13}\\int f_{23}+\\int f_{12}\\int f_{23})\\right]+\\ldots $ In principle, we can rewrite (REF ) with respect to $W^{(2),\\text{full}}$ and $W^{(3),\\text{full}}$ .",
"Note however, that both of these terms contain the coarse-grained two-body interactions, hence in order to avoid double-counting, when we use both, we have to appropriately subtract the two-body contributions.",
"For some related results, see also the discussion about Figure REF ."
],
[
"Thermodynamic consistency",
"As already mentioned, several coarse-graining strategies lack of thermodynamic consistency, see also the discussion by Louis [58] and Guenza [40].",
"On the other hand, by construction, the cluster expansion approach gives quantified approximations to the correct thermodynamic behaviour.",
"Hence, from (REF ), by considering only the two-body contribution, for the finite volume free energy we have that $-\\frac{1}{\\beta |\\Lambda |}\\log Z_{\\beta ,\\Lambda ,N} =-\\frac{1}{\\beta |\\Lambda |}\\log \\int dr_1\\ldots dr_Me^{-\\beta U^{(2)}}+\\frac{1}{\\beta |\\Lambda |} O(\\epsilon ^3)),$ where the error is uniform in $N$ and $|\\Lambda |$ and negligible in the limit.",
"Thus, the approximation $U^{(2)}$ of the CG Hamiltonian implies a good approximation of the free energy.",
"Similarly, for the pressure as a function of the activity $z$ , we have: $\\frac{1}{\\beta |\\Lambda |}\\log \\sum _{N\\ge 0}z^N Z_{\\beta ,\\Lambda ,N}=\\frac{1}{\\beta |\\Lambda |}\\log \\sum _{N\\ge 0}z^N\\int dr_1\\ldots dr_Me^{-\\beta U^{(2)}}+\\frac{1}{\\beta |\\Lambda |}O(\\epsilon ^3).$ Both quantities have limits given by absolutely convergent series with respect to $\\rho =N/|\\Lambda |$ for the first and $z$ or $\\rho $ for the second.",
"As a side remark, let us mention that in order to compute them we have two options: the first is to use (REF ) and calculate the integral $\\int dr_1\\ldots dr_M e^{-\\beta U^{(2)}}$ using molecular dynamics.",
"Alternatively, we can use the corresponding expansions - e.g.",
"for the free energy we would obtain [59] $-\\frac{1}{\\beta |\\Lambda |}\\log Z_{\\beta ,\\Lambda ,N}=\\rho (\\log \\rho -1)+\\sum _{n\\ge 1} \\beta _{\\Lambda }\\rho ^n+\\text{finite volume errors}$ - and compute the coefficients $\\beta _{\\Lambda }$ .",
"The latter are not bulk computations as they involve 2, 3, etc particles so they are rather efficient, at least up to some order."
],
[
"Pair correlation function",
"Recalling the coarse-grained map $T$ from the previous section, we fix two centers of mass $r_1$ and $r_2$ and integrate over all atomistic configurations so that the first two groups $\\mathbf {q}^1$ and $\\mathbf {q}^2$ of atomistic configurations have the above fixed centers of mass.",
"Partitioning the $N$ particles into $M$ groups of $l_1,\\ldots ,l_M$ particles and choosing two of them (indexed by 1 and 2) to be the fixed ones, we define the “projected” correlation function at the coarse-grained scale as follows: $\\rho _{N,\\Lambda }^{(2), proj}(r_1, r_2) & := &\\int _{\\lbrace T_1({\\mathbf {q}}^1)=r_1, \\, T_2({\\bf q}^2)=r_2\\rbrace }\\prod _{i=1}^M \\lambda ^i(d\\mathbf {q}^i) \\,\\frac{1}{Z_{\\beta ,\\Lambda ,N}}e^{-\\beta U(\\mathbf {q})}\\nonumber \\\\&=& \\int dr_3 \\ldots dr_M\\int \\prod _{i=1}^M\\mu (d\\mathbf {q}^i;r_i)\\frac{1}{Z_{\\beta ,\\Lambda ,N}}e^{-\\beta U(\\mathbf {q})}\\nonumber \\\\& = &\\int dr_3 \\ldots dr_M\\frac{1}{Z^{cg}_{\\beta ,\\Lambda ,M}}e^{-\\beta U_{\\text{eff}}(r_1,\\ldots ,r_M)}.$ Hence, using (REF ) we can construct coarse-grained approximations for the correlation functions as well.",
"Alternatively, as a corollary of the cluster expansion, we can write (REF ) as a convergent power series with respect to the density.",
"These are old results [45] for which the convergence has also been proved recently in the context of the canonical ensemble.",
"[60] In the limit $N\\rightarrow \\infty $ , $\\Lambda \\rightarrow \\mathbb {R}^d$ such that $\\frac{N}{|\\Lambda |}=\\rho $ , we obtain: $g(r)=e^{-\\beta \\bar{V}({\\bf q}^1- {\\bf q}^2)}\\left[1+\\rho C_3({\\bf q}^1,{\\bf q}^2)+\\rho ^2 C_4({\\bf q}^1,{\\bf q}^2)+\\ldots \\right],\\qquad r:=T({\\bf q}^1)-T({\\bf q}^2),$ where $C_3({\\bf q}^1,{\\bf q}^2):= \\int _{\\Lambda }d{\\bf q}_3 \\, f_{1,3} f_{3,2},\\qquad f_{i,j}:=e^{-\\beta \\bar{V}({\\bf q}_i-{\\bf q}_j)}-1$ and $C_4({\\bf q}^1,{\\bf q}^2): = \\int dq_3\\, dq_4 \\, f_{1,3} f_{3,4} f_{4,2}+ 4 \\int dq_3\\, dq_4 \\, f_{1,3} f_{3,4} f_{1,4} f_{4,2}+ \\int dq_3 \\, dq_4 f_{1,3} f_{3,2} f_{1,4} f_{4,2}+\\int dq_3 \\, dq_4 \\, f_{1,3} f_{1,4} f_{2,3} f_{2,4} f_{3,4}$ Note that this formula could also be used at the coarse-grained level with the pair coarse-grained potential $W^{(2)}$ , giving an alternative way to compute it."
],
[
"The model",
"A main goal of this work, as mentioned before, is to examine the parameterization of a coarse-grained model using the cluster expansion formalism described above for simple realistic molecular systems; in this work we study liquid methane and ethane.",
"In more detail, we consider $N$ molecules of $CH_4$ and we denote as $\\bar{\\bf q}\\equiv \\lbrace \\bar{q}_1,\\ldots ,\\bar{q}_N\\rbrace $ to be the positions of the $N$ many carbons and ${\\bf q}_i\\equiv \\lbrace q_{i,1},\\ldots , q_{i,4}\\rbrace $ be the positions of the 4 hydrogens that correspond to the $i^{th}$ carbon.",
"We have two types of interactions, namely the bonded with (many body) interaction potential $V_b$ and the non-bonded with pair interaction potential $V_{nb}$ .",
"The latter are of Lennard-Jones type between all possibilities: $C-C$ , $C-H$ and $H-H$ (with different coefficients), i.e., $V_{nb}=V_{CC}+V_{CH}+V_{HH}$ .",
"In the model used here the non-bonded interactions within the same $CH_4$ molecule are excluded.",
"The microscopic canonical partition function is given by $Z_{CH_4}=\\frac{1}{N!",
"}\\int _{\\Lambda ^N} d\\bar{\\bf q}\\,(\\frac{1}{4!",
"})^N\\int _{\\Lambda ^{4N}}\\prod _{i=1}^N d{\\bf q}_ie^{-\\beta \\left(\\sum _{i=1}^N V_b(\\bar{q}_i,{\\bf q}_i)+U_{nb}(\\bar{\\bf q},{\\bf q}_1,\\ldots ,{\\bf q}_N)\\right)},$ where $U_{nb}$ is a pair potential of all possible pairs among $\\bar{\\bf q},{\\bf q}_1,\\ldots ,{\\bf q}_N$ , all of L-J type (eventually with different parameters).",
"Note also that since only the 4 particles of $H$ are indistinguishable, we have introduced the factor $1/4!$ for each molecule.",
"We are interested in computing the effective Hamiltonian when only the centers of mass of the $N$ many molecules are prescribed.",
"Hence, let us introduce a map $T:\\Lambda ^5\\rightarrow \\Lambda $ which gives the center of mass of a molecule consisting of an atom of $C$ together with the prescribed 4 atoms of $H$ which are linked to $C$ by the bonded interactions, i.e., by denoting ${\\bf \\bar{q}}_i\\equiv (\\bar{q}_i,{\\bf q}_i)$ we have: $T({\\bf \\bar{q}}_i):=\\frac{1}{m_C+4 m_H}(m_C \\bar{q}_i+ m_H\\sum _{j=1}^4 q_{i,j}).$ We introduce the variables $r_1,\\ldots , r_N$ for the centers of mass.",
"Our goal is to find the effective potential $U_{\\text{eff}}(r_1,\\ldots ,r_N)$ .",
"We define the “bonded\" (normalized) prior measure by $d\\hat{\\mu }_b({\\bf \\bar{q}}_i;r_i):=\\frac{1}{Z_b(r_i)}d{\\bf \\bar{q}}_i\\mathbf {1}_{T({\\bf \\bar{q}}_i)=r_i}e^{-\\beta V_b({\\bf \\bar{q}}_i)},\\quad Z_b(r_i):=\\frac{1}{4!",
"}\\int _{\\Lambda ^5}d{\\bf \\bar{q}}_i\\mathbf {1}_{T({\\bf \\bar{q}}_i)=r_i}e^{-\\beta V_b({\\bf \\bar{q}}_i)}.$ Note that here we could have also included possible non-bonded interactions between atoms of the same molecule.",
"This would be important for the case of coarse-graining a molecule with intra-molecular non-bonded interactions; for the methane molecule studied here such interactions do not exist.",
"Then, from (REF ) we obtain: $Z_{CH_4} = \\frac{1}{N!",
"}\\int _{\\Lambda ^N}dr_1\\ldots dr_N\\,\\prod _{i=1}^N Z_b(r_i)\\int \\prod _{i=1}^N d\\hat{\\mu }_b({\\bf \\bar{q}}_i;r_i)e^{-\\beta U_{nb}({\\bf \\bar{q}}_1,\\ldots ,{\\bf \\bar{q}}_N)}.$ The effective free energy is defined by: $e^{-\\beta U_{\\text{eff}}(r_1,\\ldots ,r_M)}:=\\prod _{i=1}^N Z_b(r_i)\\int \\prod _{i=1}^N d\\hat{\\mu }_b({\\bf \\bar{q}}_i;r_i)e^{-\\beta U_{nb}({\\bf \\bar{q}}_1,\\ldots ,{\\bf \\bar{q}}_N)},$ for which we can construct approximations following formula (REF ).",
"A similar analysis holds for ethane as well.",
"The total (atomistic) potential energy $V({q})$ , for both methane and ethane, is defined by $V({q}) = V_{bond}({q}) + V_{angle}({q}) + V_{LJ}({q}) \\ .$ where $V_{bond}({q}), V_{angle}({q})$ are quadratic intramolecular potential functions of the bonds and angles respectively.",
"$V_{LJ}({q})$ is the non-bonded potential as defined in the previous subsection.",
"The parameters values of $CH_4$ are summarized in Table REF .",
"Table: Non-bonded LJLJ coefficients as well as bond and angle coefficients for methane.",
"The more simple, non-spherically symmetric ethane molecule consists of one rigid bond connecting two united atom $CH_{3}$ beads.",
"Table REF summarizes this model.",
"Table: Non-bonded LJLJ coefficients for ethane."
],
[
"Simulations",
"The simplest system to simulate is the one with only two interacting methane, or ethane, molecules in vacuum.",
"This is a reference system for which the many-body PMF is equal to the two-body one.",
"In addition we have also simulated the corresponding liquid systems.",
"The atomistic and CG model methane systems were studied through molecular dynamics and Langevin dynamics (LD) simulations.",
"All simulations were conducted in the NVT ensemble.",
"For the MD simulations the Nose-Hoover thermostat was used.",
"Langevin dynamics models a Hamiltonian system which is coupled with a thermostat.",
"[63] The thermostat serves as a reservoir of energy.",
"The densities of both liquid methane and ethane systems were chosen as the average values of NPT runs at atmospheric pressure.",
"NVT equilibration and production runs of few $ns$ followed and the size of the systems were 512 $CH_{4}$ and 500 $CH_3-CH_3$ molecules.",
"We note here that the BBK integrator used for Langevin dynamics exhibits pressure fluctuations of the order of $\\pm 40$ atm in the liquid phase, whereas temperature fluctuations have small variance and the system is driven to the target temperature a lot faster than with conventional MD.",
"In order to compute the effective non-bonded coarse-grained potential, different simulation runs have been used which are discussed below."
],
[
"Constrained runs",
"The first method which we use in order to estimate the effective CG potential is by constraining the intermolecular distance between two molecules, $r=r_{1,2}$ , in order to compute the constrained partition function (REF ).",
"We call it “constrained run” of two methane, or ethane, molecules and special care had to be taken in order to avoid long sampling of the low probability short distances.",
"This method is very similar to the typical conditional reversible work methods in which CG degrees of freedom are constrained at a fixed values for deriving CG potentials, as well as in free energy calculations.",
"Technically, we pin the centres of mass (COM) of each CG particle in space and, on every step throughout the Langevin dynamics trajectory, we subtract the total force acting on each COM.",
"Hence, we allow the atoms to move, resulting in rotations but not translations of the CG degrees of freedom ($\\textrm {CH}_4$ , COM).",
"During these runs the constraint forces are recorded.",
"The mean value $\\langle f \\rangle _{r_{12}=r}$ is calculated in the same manner and we get $W^{(2), \\text{full, f}}(r)$ , from $f =-\\nabla W$ .",
"Both $W^{(2), \\text{full, f}}(r)$ and $W^{(2), \\text{full, u}}(r)$ are based on the same trajectory.",
"Then, the effective potential is calculated by numerical integration of the constraint force $\\langle f \\rangle _{r_{12}=r}$ from $r_{min}$ up to $r_{max}$ .",
"The constrained run technique described above, accelerates the sampling for short distances but there is a caveat; the ensemble average at very short distances (left part of the potential well) is strongly affected by the non-bonded forces on specific atoms between the two molecules.",
"For example, the two $CH_{4}$ molecules are oriented according to the highly repulsive forces and rotate around the axis connecting the two COM's.",
"Due to this specific reason, we utilized stochastic (Langevin) dynamics in order to better explore the subspace of the phase space, as a random kick breaks this alignment.",
"We determine the minimum amount of steps needed for the ensemble average to converge, in a semi-empirical manner upon inspection of the error-bars."
],
[
"Geometric direct computation of PMF",
"In order to further accelerate the sampling and alleviate the noise problems at high energy regions, that might become catastrophic in the case of the non-symmetric $CH_{3}-CH_{3}$ model, we have also calculated the two-body PMF (constraint partition function) directly, through “full sampling\" of all possible configurations using a geometrical method proper for rigid bodies.",
"In more detail, the geometric averaged constrained two-body effective potential ${W}^{(2), geom}(r)$ , is obtained by rotating the two $CH_4$ molecules around their COM's, through their Eulerian angles and taking account of all the possible (up to a degree of angle discretization) orientations.",
"The main idea is to cover every possible (discretized) orientation and associate it with a corresponding weight.",
"The Euler angles proved to be the easiest way to implement this; each possible orientation is calculated via a rotation matrix using three (Euler) angles in spherical coordinates.",
"The above way of sampling is more accurate (less noisy) than constrained MD and considerably faster.",
"In addition, the nature of the computations allows massive parallelization of the procedure.",
"We used a ZYZ rotation with $d\\phi = d\\psi = d\\theta = \\pi /20$ for $CH_{4}$ and simple spherical coordinate sampling with $d\\phi = \\pi /20, d\\theta = \\pi /45$ for $CH_{3}-CH_{3}$ (as it is diagonally symmetric in the united atom description).",
"Note however, that in this case the molecules are treated as rigid bodies; i.e., bond lengths and bond angles are kept fixed, essentially it is assumed that intra-molecular degrees of freedom do not affect the intermolecular (non-bonded potential) ones.",
"The advantage of this method is that we avoid long (and more expensive) molecular simulations of the canonical ensemble, which might also get trapped in local minima and inadequately sample the phase space.",
"We should also state that this method is very similar to the one used by McCoy and Curro in order to develop a $CH_4$ united-atom model from all-atom configurations.",
"[52] All atomistic and coarse-grained simulations have been performed using a home-made simulation package, whereas all analysis has been executed through home-made codes in Matlab and Python.",
"Figure: Snapshot of model systems in atomistic and coarse-grained description.",
"(a-b) Two and three methanes used for the estimation of the CG effective potential from isolated molecules.",
"(c) Bulk methane liquid."
],
[
"Calculation of the effective two-body CG potential",
"First, we present data related to the calculation of the two-body potential of mean force for the ideal system of two (isolated) molecules.",
"For such a system the conditional M-body CG PMF is a 2-body one, i.e., the pair approximation in the effective CG interaction is exact.",
"In more detail, in Figures REF a and REF b we provide data for the CG effective interaction between two methane and ethane molecules, through the following methods: (a) A direct calculation of the PMF, $W^{(2),geom}$ , using a geometrical approach as described in Section REF that involves the direct calculation of the constraint partition function, treating the two molecules as rigid bodies.",
"Note that in this case in the all-atom description bond lengths and bond angles are kept fixed.",
"(b) A calculation of the PMF using the constraint force approach, $W^{(2), \\text{full, f}}$ , as described in section REF .",
"In this case the constraint force required to keep two methane molecules fixed at a specific distance is computed.",
"Then through a numerical integration the effective potential between the two molecules (CG particles), $ U^{\\text{PMF}}_{\\text{CF}}$ , is computed.",
"This is a method that has been extensively used in the literature to estimate effective pair CG interaction between two molecules, as well as differences in the free energy between two states.",
"Alternatively, through the same set of atomistic configurations the two-body PMF, $W^{(2), \\text{full, u}}$ , can be directly calculated through Eq (REF ).",
"(c) DBI method: The CG effective potential, $W^{(2), g(r)}$ , is obtained by inverting the pair (radial) correlation function, $g(r)$ , computed through a stochastic LD run with only two methane (or ethane) molecules in the simulation box.",
"The pair correlation function, $g(r)$ , of the two methane molecules is also shown in Figure REF a.",
"The first two of the above methods refer to the direct calculation of the constrained partition function (REF ) with constrained forces and canonical sampling, while the third uses the “Direct Boltzmann Inversion\" approach.",
"All above data correspond to temperatures in which both methane and ethane are liquid at atmospheric pressure (values of $-k_BT$ are also shown in Figure REF ).",
"First, for the case of the two methane molecules (Figure REF a) we see very good agreement between the different methods.",
"As expected, slightly more noisy is the $W^{(2), \\text{full, u}}(r_{12})$ curve as fluctuations in the $\\langle e^{-\\beta u} \\rangle $ term for a given $r_{12}$ distance in equation (REF ), are difficult to cancel out.",
"The small probability configurations in high potential energy $u$ regimes having a large impact in the average containing the exponent, hence the corresponding plot is not as smooth as the others are.",
"In addition, as previously mentioned, $W^{(2), \\text{full, f}}$ comes from the same trajectory (run) but the integration of the $\\langle f \\rangle _{r_{12}}$ from $r_{\\text{cutoff}}$ up to $r_{12}$ washes out any non-smoothness.",
"Note, that for the same system recently CG effective potentials based on IBI, force matching and relative entropy methods have been derived and compared against each other.",
"[56] Second, for the case of the two ethane molecules (Figure REF b) we see a good, but not perfect, agreement between the different sets of data, especially in the regions of high potential energy (short distances).",
"This is not surprising if we consider that high energy data from any simulation technique that samples the canonical ensemble, exhibit large error bars, due to difficulties in sampling.",
"The latter is more important for ethane compared to methane case due to its molecular structure; indeed the atomistic structure of methane approximates much better the spherical structure of CG particles than ethane.",
"The only method that provides a “full\", within the numerical discretization, sampling at any distance is the geometric one; however as discussed before (see Section ) such a method neglects the bond lengths and bond angle fluctuations.",
"Figure: Relation of the PMF through cluster expansions and energy averaging at high temperatures, i.e., W (2) (r 1 ,r 2 )W^{(2)}(r_{1}, r_{2}) and W (2),full (r 1 ,r 2 )W^{(2),\\text{full}}(r_{1}, r_{2}) through expansion over β\\beta for CH 4 CH_{4} at T=300KT=300K (left panel) and CH 3 -CH 3 CH_{3}-CH_{3} at T=650KT=650K (right panel).",
"As expected from the analytic form and the relation between the two formulas, W (2) W^{(2)} and W (2),full W^{(2),\\text{full}} tend to converge to the same effective potential.Next, we also examine an alternative method for the computation of the effective CG potential, by calculating the approximate terms from the cluster expansion approach.",
"For the latter we use the data from the constraint runs of two methane molecules integrated over all atomistic degrees of freedom, as given in formula (REF ).",
"In Figures REF a and b we demonstrate the Potential of Mean Force through cluster expansions and the effect of higher order terms as shown in equation (REF ), of the two isolated molecules, for $CH_{4}$ and $CH_{3}-CH_{3}$ respectively.",
"As discussed in the Section cluster expansion is expected to be more accurate at high temperatures and/or lower densities.",
"For this, we examine both systems at higher temperatures, than of the data shown in Figure REF ; Values of $-k_BT$ are shown with full lines.",
"Both systems show the same behavior.",
"First, it is clear that the agreement between $W^{(2)}$ and the (more accurate) $W^{(2),\\text{full}}$ is very good only to long distances, whereas there are strong discrepancies in the regions where the potential is minimum as well as in the high energy regions (short distances).",
"Second, it is evident that adding terms up to the second order with respect to $\\beta $ , we obtain a better approximation of $W^{(2),\\text{full}}$ .",
"Next, we further examine the dependence of the PMF, for the two isolated methanes, on the temperature, by studying the molecules at $T=80K$ , $120K$ , $300K$ and $900K$ .",
"In more detail, in Figures REF a and b we compare the difference between $W^{(2)}$ and $W^{(2),\\text{full}}$ at different temperatures.",
"As discussed in Section , the cluster expansion method is valid only in the high temperature regime.",
"This is directly observed in Figure REF a; at high temperatures, $W^{(2)}$ is very close to $W^{(2),\\text{full}}$ , which is exact for the system consisting of two molecules.",
"Note the small differences at short distances, which, as also discussed in the previous subsection, are even smaller if higher order terms are included in the calculation of $W^{(2)}$ ; see also Figure REF .",
"On the contrary, at low temperatures there is a strong discrepancy around the potential well as shown in Figure REF b.",
"In fact, for values of $r$ close to the potential well and for rather high values of $\\beta $ the contribution to the integral (REF ) is large and the latter can exceed one, rendering the expansion in (REF ) not valid.",
"In Figure REF b we see that the term (REF ) is not small so the expansion (REF ) is not valid.",
"The case for ethane is qualitatively similar.",
"For completeness, we also plot the potential of mean force at different temperatures for the system of two $CH_{4}$ molecules, see Figure REF .",
"In principle, equation (REF ) is a calculation of free energy, hence it incorporates the temperature of the system and thus both approximations to the exact two-body PMF, $W^{(2)}$ and $W^{(2),\\text{full}}$ , are not transferable.",
"Indeed, we observe slight differences in the CG effective interactions (free energies) for the various temperatures, which become larger for the highest temperature.",
"Figure: Potential of mean force at different temperatures (geometric averaging).",
"Two CH4 molecules at T=80K, 100K, 120K, 300K"
],
[
"Bulk CG CH4 runs using a pair potential",
"In the next stage, we examine quantitatively the accuracy of the effective CG interaction potential (approximation of the two-body PMF), in the liquid state based on structural properties like $g(r)$ .",
"Here we use the different CG models (approximated pair CG interaction potentials) derived above, to predict the properties of the bulk CG methane and ethane liquids.",
"In all cases we compare with structural data obtained from the reference all-atom bulk system, projected on the CG description.",
"In Figures REF a and b we assess the discrepancy between the CG (projected) pair distribution function, $g(r)$ , taken from an atomistic run, and the one obtained from the corresponding CG run based on $W^{(2),\\text{full}}$ as already seen in Figure REF of methane and ethane respectively.",
"Note that $g(r)$ is directly related to the effective CG potentials ($N=2$ in Eq (REF )).",
"It is clear that for methane (Figure REF a) the CG model with the $W^{(2),\\text{full}}$ potential gives a g(r) very close to the one derived from the analysis of the all-atom data.",
"This is not surprising if we consider that for most molecular systems small differences in the interaction potential lead to even smaller differences in the obtained pair correlation function.",
"Interestingly the CG model with the $W^{(2)}$ is also in good agreement to the reference one, despite the small differences in the CG interaction potential discussed above (see Figures REF and REF ).",
"As expected, the difference comes from the missing higher order terms of eq (REF ).",
"The fact that the CG effective potential, which is derived from two isolated methane molecules, give a very good agreement for the methane structure in the liquid state is not surprising if we consider the geometrical structure of methane, which is rather close to the spherical one, and the typical van der Waals type of interactions between methane molecules.",
"On the contrary, for the case for ethane (Figure REF b) predictions of $g(r)$ using pair CG potential are much different compared to the atomistic one, especially for the short distances.",
"Even larger differences would be expected for more complex systems with long-range interactions, such as water.",
"[56] Similar is the case also for the other temperatures (T = 80K) studied here (data not shown here).",
"Figure: RDF from atomistic and CG using pair potential, W (2) W^{(2)}, for CH 4 CH_{4} system at T=80KT=80K (left panel) and CH 3 -CH 3 CH_{3}-CH_{3} at T=150T=150(right panel).",
"Spherical CG approximation to the non-symmetric ethane molecule induces discrepancy implies there is more room for improvement."
],
[
"Effect of temperature-density",
"We further study the structural behavior of the CG systems at different state points; i.e., temperature/density conditions, compared to the atomistic ones.",
"First, we examine the temperature effect by simulating the systems discussed above (see Figure REF ) at higher temperatures; however keeping the same density.",
"In Figures REF a,b we present the RDF of methane from atomistic and CG runs using pair potential at $T=300K$ , and $T=900K$ respectively.",
"It is clear that the analysis of the CG runs using the $W^{(2),\\text{full}}$ potential gives a pair distribution function $g(r)$ close to the atomistic one for both (high) temperatures, similar to the case of the $T=100K$ shown above.",
"In addition, the CG model with the $W^{(2)}$ potential is in very good agreement with the atomistic data at high temperature (Figure REF b), whereas there are small discrepancies at lower temperatures (Figure REF a), in particular at the maximum of $g(r)$ .",
"This is shown in the inset of Figures REF a,b.",
"Note also that in this high temperature the incorporation of the higher order terms in $W^{(2)}$ leads to very similar potential as the $W^{(2),\\text{full}}$ (see also Fig.",
"REF ), and consequently to very accurate structural $g(r)$ data as well.",
"Figure: RDF from atomistic data, and CG models using pair potential at different temperatures: (a) T=300K, (b) T=900K.",
"In both cases the density is 0.3799 gr cm 3 \\frac{gr}{cm^3}.Figure: RDF from atomistic and CG using pair potential at different densities ρ 1 <ρ 2 \\rho _1<\\rho _2, (a) T=300K, (b) T=900K.Next, we examine the structural behavior of the CG systems at different densities.",
"In Figure REF a we present the $g(r)$ from atomistic and CG runs using pair potential at different densities ($\\rho _1= 0.3799 \\frac{gr}{cm^3}$ and $\\rho _2= 0.0395 \\frac{gr}{cm^3}$ and $T=300K$ , and $T=900K$ ).",
"There is apparent discrepancy from the reference (atomistic) system in both densities in agreement to the data discussed above in Figure REF a.",
"For the case of higher temperature data ($T=900K$ ) and the same densities, as shown in Figure REF b, the pair distribution function, $g(r)$ , obtained from the CG model with the $W^{(2)}$ effective interaction is very close to the data derived from the $W^{(2),\\text{full}}$ one, and in very good agreement to the reference, all-atom, data.",
"This is not surprising since, as discussed before, at high temperatures the cluster expansion is expected to be more accurate, since cluster expansions hold for high $T$ and low $\\rho $ .",
"From Figure REF a we deduce that despite the different potentials $W^{(2)}, W^{(2),\\text{full}}$ (Figure REF ), we obtain the same $g(r)$ for the liquid case, as a result of the close packing and frequent collisions.",
"Overall, the higher the temperature the better the agreement in the $g(r)$ derived from the CG models using any of $W^{(2)}$ and $W^{(2),\\text{full}}$ .",
"These data are in better agreement with the atomistic data as well."
],
[
"Effective three-body potential",
"In the last part of this work we briefly discuss the direct computation of the three-body effective CG potential and its implementation in a (stochastic) dynamic simulation.",
"More results about the three-body terms will be presented in a future work.",
"[53]"
],
[
"Calculation of the effective three-body potential",
"In the following we present data for the 3-body potential of mean force estimated from simulation runs and geometric computations involving three isolated molecules.",
"We have two suggestions for the 3-body PMF: (a) Formula (REF ) derived from cluster expansion formalism, which is valid for rather high temperatures and (b) another one based on the McCoy-Curro scheme given in formula (REF ).",
"Similarly to the two-body potential, the corresponding calculations can be performed by running constrained molecular dynamics (or any other method that performs canonical sampling).",
"For this one needs to calculate the derivative of the three-body potential with respect to some distance.",
"However, as previously stated, deterministic MD simulations of a constrained system might easily get trapped in local energy minima, so we utilized stochastic dynamics for the three-body case.",
"In addition, rare events (high energy, low probability configurations) induce noise to the data, despite long equilibration (burn-in) periods or stronger heat-bath coupling in the simulations.",
"Although smoothing could in principle have been applied, it would wash-out important information needed upon derivation with respect to positions ($f = - \\nabla _{\\bf {q}} W^{(3)} $ ).",
"Therefore, we choose here to present results from the “direct\" geometric averaging approach.",
"The total calculations are one order of magnitude more than the two-body ones (all possible orientations of the two molecules for one of the third one), so special care was given to spatial symmetries.",
"The new effective three-body potential, $W^{(3),\\text{full}}(r_{12}, r_{13}, r_{23})$ , incorporates three intermolecular distances: $r_{12}, r_{13}, r_{23}$ .",
"The discretization of the COM's in space is on top of the angular discretization mentioned in Section REF and relates to the above three distances.",
"In more detail, in Figures REF a-d we present simulations based on the effective three body potential $W^{(3), \\text{full}}$ and the sum $\\sum W^{(2),\\text{full}}$ (geometric averaging) for $CH_{4}$ at $T=80K$ for different COM distances [$Å$ ]: (a) $r_{12}=3.9$ , $r_{13}=3.9$ , (b)$r_{12}=4.0$ , $r_{13}=4.0$ , (c) $r_{12}=4.3$ , $r_{13}=4.0$ , (d) $r_{12}=3.8$ , $r_{13}=5.64$ .",
"In each case the sum of the corresponding two-body terms is also shown.",
"At smaller distances, the potential of the triplet deviates from the sum of the three pairwise potentials and this is where improvement in accuracy can be obtained.",
"As shown in Figure REF improvement is needed for close distances around the (3 dimensional) well.",
"We used a 3-dimensional cubic polynomial to fit the potential data (conjugate gradient method) which means that 20 constants should be determined.",
"A lower order polynomial cannot capture the curvature of the forces upon differentiation.",
"The benefit of this fitting methodology (over partial derivatives for instance) is the analytical solution of the forces with respect to any of $r_{12}, r_{13}, r_{23}$ in contrast to tabulated data that induce some small error.",
"Figure: Effective potential comparison between the W (3),full W^{(3),\\text{full}} 3-body and ∑W (2),full \\sum W^{(2),\\text{full}} simulations (geometric averaging) for CH 4 CH_{4} at T=80KT=80K for different COM distances [ÅÅ] .",
"(a) r 12 =3.9,r 13 =3.9r_{12}=3.9, r_{13}=3.9 (b)r 12 =4.0,r 13 =4.0r_{12}=4.0, r_{13}=4.0 (c) r 12 =4.3,r 13 =4.0r_{12}=4.3, r_{13}=4.0, (d)r 12 =3.8,r 13 =5.64r_{12}=3.8, r_{13}=5.64 .Overall, there are clear differences between the 3-body PMF, $W^{(3),\\text{full}}$ , and the sum of three two-body interactions, $\\sum W^{(2),\\text{full}}$ , at short $r_{12}$ , $r_{13}$ and $r_{23}$ distances.",
"On the contrary, for larger distances the sum of two-body interactions seems to represent the full three-body PMF very accurately.",
"This is a clear indication of the rather short range of the three-body terms.",
"Based on the above data, the range of the 3-body terms for this system (methane at $T=80K$ ) is: $r_{12}\\in [3.8:4.1]Å, r_{13}\\in [3.8:4.1]Å$ and $ r_{23}\\in [3.8:5] Å$ ; hence, the maximum distance for which three-body terms were considered, is $r_{\\text{cut-off},3}$ =5$Å$ .",
"In practice we need to identify all possible triplets within $r_{\\text{cut-off},3}$ .",
"Naturally, by including higher-order terms the computational cost has increased as well.",
"More information about the numerical implementation of the three-body CG effective potential and its computational efficiency will be given elsewhere.",
"[53] We should state here that in order to keep constant the temperature (in the BBK algorithm) due to the extra three-body terms in the CG force field a larger coupling constant value for the heat bath was required."
],
[
"CG Runs with the effective three-body potential",
"Next we examine the effect of the 3-body term on the CG model by performing bulk CG stochastic dynamics simulations using the new CG model with the 3-body terms described above.",
"Results about the pair distribution function, $g(r)$ , for bulk (liquid) methane at $T=80K$ are shown in in Figure REF .",
"In this graph data from the atomistic MD runs (projected in the CG description), the CG model involving only pair CG potentials, and the new CG model that also involves 3-body terms are shown.",
"First, it is clear that $g(r)$ data derived from the CG model that involves only pair CG potentials show clear deviations, compared to the reference all-atom data.",
"Note, that these differences are slightly larger than the ones discussed before (see Figure REF ), in which data at a higher temperature are presented.",
"Second, the incorporation of the three-body terms in the effective CG potential slightly improves the prediction of the $g(r)$ , mainly in the first maximum regime.",
"Figure: RDF from atomistic and CG using pair, W (2),full W^{(2),\\text{full}}, and three-body, W (3),full W^{(3),\\text{full}}, potential for CH 4 CH_{4} (T=80KT=80K).",
"Three dimensional cubic polynomial was used for the fitting.",
"\"Clustered\" means that one particular CG atom belongs to one triplet in comparison to all possible triplets."
],
[
"Discussion and conclusions",
"In recent years we have experienced an enormous increase of computational power due to both hardware improvements and clever CPU-architecture.",
"However, atomistic simulations of large complex molecular systems are still out of reach in particular when long computational times are desirable.",
"A generic strategy in order to improve efficiency of the computational methods is to reduce the dimensionality (degrees of freedom) by considering systematic coarse-grained models.",
"There have been many suggestions on how to compute the relevant CG effective interactions in such models; a main issue here is that even if in the microscopic (atomistic) level there are only pair interactions, after coarse-graining a multi-body effective potential (many-body PMF) is derived, which for realistic molecular complex systems cannot be calculated.",
"Therefore, a common trend has been to approximate them by an “effective” pair potential by comparing the pair correlation function $g(r)$ .",
"This seems reasonable since given the correlation function one can solve the “inverse problem” [64] and find an interaction to which it corresponds.",
"But, this is an uncontrolled approximation without thermodynamic consistency.",
"Instead, here we suggest to explicitly compute the constrained configuration integral over all atomistic configurations that correspond to a given coarse-grained state and from that suggest approximations with a quantifiable error.",
"This is similar to the virial expansion where one needs to integrate over all positions of particles that correspond to a fixed density and it is based on the recent development of establishing the cluster expansion in the canonical ensemble.",
"[46]; see also Ref.",
"[60], [55] for the corresponding (in the canonical ensemble) expansions for the correlation functions and the Ornstein-Zernike equation.",
"The main drawback that limits the applicability of these expansions is that they are rigorously valid only in the gas phase.",
"To extend them to the liquid state is an outstanding problem and even several successful closures like the Percus-Yevick are not rigorously justified.",
"Therefore, there is need of further developing these methods and relate them to computational strategies.",
"In this paper we extend the above methods by presenting an approach based on cluster expansion techniques and numerical computations of isolated molecules.",
"As a first test we presented a detailed investigation of the proposed methodology to derive CG potentials for methane and ethane molecular systems.",
"Each CG variable corresponds to the center-of-mass for each molecule.",
"Below, we summarize our main findings: (a) The hierarchy of the cluster expansion formalism allowed us to systematically define the CG effective interaction as a sum of pair, triplets, etc.",
"interactions.",
"Then, CG effective potentials can be computed as they arise from the cluster expansion.",
"(b) The two-body coarse-grained potentials can be efficiently computed via the cluster expansion giving comparable results with the existing methods, such as the conditional reversible work.",
"In addition we present a more efficient direct geometric computation of the constrained partition function.",
"(c) The obtained pair CG potentials were used to model the corresponding liquid systems and the derived $g(r)$ data were compared against the all-atom ones.",
"Clear differences between methane and ethane systems were observed; For the (almost spherical) methane, pair CG potentials seems to be a very good approximation, whereas much larger differences between CG and atomistic distribution functions were observed for ethane.",
"(d) We further investigated different temperature and density regimes, and in particular cases where the two-body approximations are not good enough compared to the atomistic simulations.",
"In the latter case, we considered the next term in the cluster expansion, namely the three-body effective potentials and we found that they give a small improvement over the pair ones.",
"Overall, we conjecture that the cluster expansion formalism can be used in order to provide accurate effective pair and three-body CG potentials at high $T$ and low $\\rho $ regimes.",
"In order to get significantly better results in the liquid regime one needs to consider even higher order terms, which are in general more expensive to be computed and more difficult to be treated.",
"A more detailed analysis of the higher-order terms will be a part of a future work.",
"[53] Finally, another future goal is to extend this investigation in larger molecules (e.g.",
"polymeric chains) that involve intra-molecular CG effective interactions as well, and to systems with long range (e.g.",
"Coulombic) interactions.",
"We acknowledge support by the European Union (ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the NSRF-Research Funding Programs: THALES and ARISTEIA II."
]
] | 1612.05429 |
[
[
"Improper Signaling in Two-Path Relay Channels"
],
[
"Abstract Inter-relay interference (IRI) challenges the operation of two-path relaying systems.",
"Furthermore, the unavailability of the channel state information (CSI) at the source and the limited detection capabilities at the relays prevent neither eliminating the interference nor adopting joint detection at the relays nodes.",
"Improper signaling is a powerful signaling scheme that has the capability to reduce the interference impact at the receiver side and improves the achievable rate performance.",
"Therefore, improper signaling is adopted at both relays, which have access to the global CSI.",
"Then, improper signal characteristics are designed to maximize the total end-to-end achievable rate at the relays.",
"To this end, both the power and the circularity coefficient, a measure of the impropriety degree of the signal, are optimized at the relays.",
"Although the optimization problem is not convex, optimal power allocation for both relays for a fixed circularity coefficient is obtained.",
"Moreover, the circularity coefficient is tuned to maximize the rate for a given power allocation.",
"Finally, a joint solution of the optimization problem is proposed using a coordinate descent method based on alternate optimization.",
"The simulation results show that employing improper signaling improves the achievable rate at medium and high IRI."
],
[
"Introduction",
"Next generation wireless communication adopts technologies that extends the network coverage and improve the data rate.",
"One of the candidate technologies is full-duplex relaying that targets to double the spectral efficiency.",
"On the other hand, cooperative communication is an interesting technology to improve the data rate and extend the communication range.",
"Full-duplex relaying is employed to extend the network coverage while improving the link quality.",
"Despite of the promising performance that full-duplex can achieve, replacing all half-duplex nodes by full-duplex ones is not possible to be done immediately.",
"During the roll-out phase, half-duplex nodes are used to support full-duplex services.",
"Two path relaying, which is also known as, alternate relaying, is a distributed realization of full-duplex relaying.",
"Full-duplex relaying suffers from self-interference, whereas the two-path relaying suffers from inter-relay interference (IRI).",
"Therefore, different interference mitigation techniques need to be adopted to relief the effect of the interference [1].",
"Improper signaling is used to mitigate the interference impact on communication systems.",
"It is an asymmetric Gaussian signaling scheme that assumes unequal power of the real and imaginary components and/or dependent real and imaginary components.",
"It is used in underlay cognitive radio [2], [3], [4], [5], [6], overlay cognitive radio [7], full-duplex relaying [8], Z-interference channel [9], [10] and asymmetric hardware distortions [11].",
"Recently, we considered the two-path relaying network and showed that improper signaling can be advantageous over proper signaling to mitigate the IRI [12], [13].",
"Specifically, in [12], improper signaling is adopted in two-path relaying system, where only the same circularity coefficient, a measure of the degree of impropriety of the signal, for both relays is optimized to mitigate the interference while the relays use their maximum power.",
"On the other hand, in [13], we considered the same problem but with different circularity coefficients at the relays.",
"Moreover, we considered asymmetric time allocation for the two transmission phases while the relays use their maximum power.",
"In this paper, we take the problem in [12] further and optimize both the relay power and circularity coefficient, which measures the degree of impropriety of the transmit signal, to maximize the end-to-end achievable rate of the two-path relaying system.",
"First, we consider proper signaling and introduce optimal relays power allocation for the system.",
"In the case of using improper signaling, we allocate the relays power with a fixed circularity coefficient.",
"Moreover, we tune the circularity coefficient while fixing the transmit power.",
"Then, we jointly optimize the relays power and circularity coefficient via a coordinate descend based method by iterating between the optimal solutions of the individual problems till a convergence obtained.",
"Finally, we investigate through numerical results the merits that can be reaped if the relays use improper signals using different strategies."
],
[
"System Model",
"We consider here an alternate two-path relaying network consisting of one source node, $S$ , two half-duplex relay nodes, $R_1$ and $R_2$ , and one destination node, $D$ , as shown in Fig.",
"REF .",
"We adopt decode-and-forward protocol at both relays.",
"Moreover, the relays transmit and receive in turn, i.e., in one time slot one relay receives and the other relay transmits, and in the next time slot the other way around.",
"Let $h_i$ and $g_i$ , $i \\in \\lbrace 1, 2\\rbrace $ , denote the channel between $S$ and $R_i$ and the channel between $R_i$ and $D$ , respectively.",
"We assume channel reciprocity for the inter-relay channel which is denoted by $f$ .",
"Moreover, let us assume that the source transmit power is $p_{\\rm {s}}$ , the relay transmit power is $p_{\\rm {r}}$ , and the noise variance at each receiving node is $\\sigma _{\\rm {n}}^2$ .",
"The transmit powers are limited to a power budget of $p_{\\rm {max}}$ .",
"First, we give the following definitions of improper random variables (RV).",
"Definition 1 [14] The complementary (pseudo-) variance of a zero mean complex random variable $x$ is defined as $\\tilde{\\sigma }_x^2=E\\lbrace x^2\\rbrace $ , where $\\mathbb {E}\\lbrace .\\rbrace $ denotes the expectation operator.",
"If $\\tilde{\\sigma }_x^2=0$ , then $x$ is called proper signal, otherwise it is called improper.",
"Definition 2 [2] The circularity coefficient of the signal $x$ is a measure of its impropriety degree and is defined as $\\mathcal {C}_x ={|\\tilde{\\sigma }_x^2|}/{\\sigma _x^2}$ , where ${\\sigma }_x^2=E\\lbrace |x|^2\\rbrace $ is the conventional variance and $|.|$ is the absolute value operation.",
"The circularity coefficient satisfies $0\\le \\mathcal {C}_x \\le 1$ .",
"In particular, $\\mathcal {C}_x = 0$ and $\\mathcal {C}_x = 1$ correspond to proper and maximally improper signals, respectively .",
"We assume that no channel state information is available at $S$ which necessitates the use of proper signaling at $S$ and also makes dirty paper coding of no benefit to fully cancel the IRI.",
"Also, we assume that no direct link is available between $S$ and $D$ .",
"For simplicity and tractability, we consider a yet illustrative scenario by assuming equal power and same circularity coefficient for the relays which may not be optimal.",
"However, as it will be shown in the simulation results, though these sub-optimal assumptions, improper signals show a significantly better performance than proper signaling.",
"Furthermore, we expect even better performance if we increase the degrees of freedom by letting different power and circularity coefficient at the relays.",
"Also, we assume the receivers use the simple practical decoding techniques by treating the interference as a Gaussian noise.",
"Figure: Two-Path Relay Channel.",
"The blue solid lines represent the signal links and the red dashed lines represent the IRI.During time slot $k$ , the signal received at $R_i$ with $i=2-\\mod {(}k,2)$ is given byFor the rest of the paper, we let $i,j \\in \\lbrace 1,2\\rbrace $ , $i \\ne j$ $y_i[k] = \\sqrt{p_s} h_i[k] s[k]+\\sqrt{p_r}f[k] x_{j}[k]+n_i[k],$ where $s[k]$ is the transmit proper signal by $S$ in time slot $k$ and $n_i[k]$ is the additive noise at $R_i$ with variance $\\sigma _{\\rm {n}}^2$ .",
"$x_j[k]$ is the improper signal, with circularity coefficient $\\mathcal {C}_x$ , transmitted by $R_j$ with $j=1+\\mod {(}k,2)$ .",
"The received signal at $D$ from $R_i$ in time slot $k+1$ is given by $y_D[k+1] = \\sqrt{p_r}g_i[k+1] x_{i}[k+1]+n[k+1],$ where $n[k+1]$ is the additive noise at $D$ .",
"In the following, we assume the channels to be quasi-static block flat fading channels and therefore we drop the time index $k$ for notational convenience.",
"The additive noise at the receivers is modeled as a white, zero-mean, circularly symmetric, complex Gaussian with variance $\\sigma _{{n}}^2$ .",
"The alternating two-path relaying system mimics a full-duplex system by transferring the data through two Z-interference channels, where two transmitters ($S$ and $R_i$ ) are sending messages each intended for one of the two receivers ($R_j$ and $D$ ) as shown in Fig.",
"REF .",
"Hence, as a result of using improper signals at $R_j$ and proper signals at $S$ while treating the interference as a Gaussian noise, the achievable rate of the first hop of the $i$ th path ($S-R_i$ ) can be expressed after some simplification steps as [15] ${\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) = {\\log _2}\\left( {1 + \\frac{{{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2}}}{{{p_{\\rm {r}}}{{\\left| f \\right|}^2} + \\sigma _n^2}}} \\right)\\hspace{-3.0pt} + \\hspace{-3.0pt} \\frac{1}{2}{\\log _2}\\left( {\\frac{{1 - {\\cal C}_{{{{y}}_i}}^2}}{{1 - {\\cal C}_{{\\cal {I}}_i}^2}}} \\right),$ where ${\\cal C}_{{{{y}}_i}}$ and ${\\cal C}_{{\\cal I}_i}$ are the circularity coefficients of the received and interference-plus-noise signals at $R_i$ , respectively, which can be calculated as ${{\\cal C}_{{{{y}}_i}}} = \\frac{{{p_{\\rm {r}}}{{\\left| f \\right|}^2}{{\\cal C}_x}}}{{{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2} + {p_{\\rm {r}}}{{\\left| f \\right|}^2} + \\sigma _n^2}},\\;\\;{{\\cal C}_{{\\cal I}_i}} = \\frac{{{p_{\\rm {r}}}{{\\left| f \\right|}^2}{{\\cal C}_x}}}{{{p_{\\rm {r}}}{{\\left| f \\right|}^2} + \\sigma _n^2}}.$ Hence, (REF ) can be simplified to ${\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) &= \\frac{1}{2} \\times \\nonumber \\\\& {\\log _2}\\left( 1 + \\frac{{2{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2}\\left( {{p_{\\rm {r}}}{{\\left| f \\right|}^2} + \\sigma _n^2} \\right) + p_{\\rm {s}}^2{{\\left| {{h_i}} \\right|}^4}}}{{\\left( {1 - {\\cal C}_x^2} \\right)p_{\\rm {r}}^2{{\\left| f \\right|}^4} + 2{p_{\\rm {r}}}{{\\left| f \\right|}^2}\\sigma _n^2 + \\sigma _n^4}} \\right).$ Similarly, the achievable rate of the second hop of the $i$ th path can be obtained from (REF ) as ${\\mathcal {R}_{i,2}}\\left( {{{p_{\\rm {r}},\\cal C}_x}} \\right) = {\\log _2}\\left( {1 + \\frac{{{p_{\\rm {r}}}{{\\left| {{g_i}} \\right|}^2}}}{{\\sigma _n^2}}} \\right) + \\frac{1}{2}{\\log _2}\\left( {\\frac{{1 - {\\cal C}_{{y_{{D}}}}^2}}{{1 - {\\cal C}_{{{\\cal I}_{{D}}}}^2}}} \\right),$ where ${\\cal C}_{{{{y}}_D}}$ and ${\\cal C}_{{\\cal I}_D}$ are the circularity coefficients of the received and interference-plus-noise signals at $D$ , respectively, which can be computed as ${{\\cal C}_{{y_{{D}}}}} = \\frac{{{p_{\\rm {r}}}{{\\left| {{g_i}} \\right|}^2}{{\\cal C}_x}}}{{{p_{\\rm {r}}}{{\\left| {{g_i}} \\right|}^2} + \\sigma _n^2}},\\quad {{\\cal C}_{{{\\cal I}_{{D}}}}} = 0.$ Then, (REF ) reduces to ${\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) = \\frac{1}{2}{\\log _2}\\left( {1 + \\frac{{2{p_{\\rm {r}}}{{\\left| {{g_i}} \\right|}^2}}}{{\\sigma _n^2}} + \\frac{{p_{\\rm {r}}^2{{\\left| {{g_i}} \\right|}^4}\\left( {1 - {\\cal C}_x^2} \\right)}}{{\\sigma _n^4}}} \\right).$ Hence, the end-to-end achievable rate of the $i$ th path can be calculated from ${\\mathcal {R}_{{_i}}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) = \\min \\Big \\lbrace {{\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right),{\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)} \\Big \\rbrace .$ Accordingly, the overall end-to-end achievable rate of the two-path relaying system, for sufficiently large number of time slotsOne slot is missed at the start of the transmission without delivering information from $S$ to $D$ ., is expressed as the arithmetic mean of ${{\\mathcal {R}_{{_i}}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)}$ ${\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) = \\frac{1}{2}\\sum \\limits _{i = 1}^2 {{\\mathcal {R}_{{_i}}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)}.$ Remark 1 One can notice that if $\\mathcal {C}_x=0$ in (REF ), we obtain the conventional expression for the total achievable rate of the two-path relaying system under the use of proper signals as $&{\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}},0} \\right) = \\frac{1}{2}\\sum \\limits _{i = 1}^2 {{\\mathcal {R}_{{_i}}}\\left( {{p_{\\rm {r}}},0} \\right)}= \\frac{1}{2} \\times \\nonumber \\\\&\\sum \\limits _{i = 1}^2 \\min \\Bigg \\lbrace {{\\log }_2}\\left( {1 + \\frac{{{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2}}}{{\\sigma _n^2 + {p_{\\rm {r}}}{{\\left| f \\right|}^2}}}} \\right), {{\\log }_2}\\left( {1 + \\frac{{{p_{\\rm {r}}}{{\\left| {{g_i}} \\right|}^2}}}{{\\sigma _n^2}}} \\right) \\Bigg \\rbrace .$"
],
[
"Improper Gaussian Signaling Design for Two-Path Relaying Systems",
"In this section, we aim at optimizing the relays signal parameters represented in the relay's transmit power $p_{\\rm {r}}$ and the circularity coefficient $\\mathcal {C}_x$ in order to maximize the instantaneous end-to-end achievable rate of the system.",
"First, the intuition behind the benefit of using improper signals at the relays is that it provides an additional degree of freedom that can be optimized in order to alleviate the effect of the IRI on the relays or, in the worst case, kept at the same performance as proper signaling, i.e., $\\mathcal {C}_x=0$ .",
"Moreover, improper signaling has the ability to control the interference signal dimension, and it is one form of interference alignment [10], [16].",
"Furthermore, when using proper signals, $R_i$ can improve the rate of the second hop of the $i$ th by boosting its transmit power.",
"However, this will deteriorate the rate of the first hop of the $j$ th path and here improper signaling attains its benefit.",
"By increasing the asymmetry of the relay's transmit signal, by boosting the circularity coefficient, the relay can increase its power and has a less adverse effect on the other one.",
"Now, in order to reap the benefits of improper signaling, we design the power and circularity coefficient of the relays.",
"For this purpose, we formulate the following optimization problem ${\\bf {P1}}:&\\mathop {\\max }\\limits _{{p_{\\rm {r}}},\\mathcal {C}_x}\\qquad {\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x} \\right) \\nonumber \\\\&\\;{\\rm {s}}{\\rm {.t}}{\\rm {.",
"}}\\quad \\quad \\;\\;0 < {p_{\\rm {r}}} \\le {p_{\\max }},\\nonumber \\\\& \\qquad \\qquad \\; 0\\le \\mathcal {C}_x \\le 1.$ Solving ${\\bf {P1}}$ optimally is difficult as it is a non-convex optimization problem.",
"Here, we propose a coordinate-descent (CD) based method in which we consider two problems, optimizing the relays transmit power for a fixed circularity coefficient and optimizing the circularity coefficient for a fixed transmit power.",
"Finally, we perform alternate optimization of the optimal solutions of the two problems till we get convergence.",
"Remark 2 [17] The CD method is popular for its efficiency, simplicity and scalability.",
"Moreover, it is guaranteed to converge to a local solution if the global optimal solution is attained for each of the sub-problems.",
"However, it does not necessarily converge to the global optimal solution as the objective function is non-convex.",
"Following Remark REF , we will show the optimal solutions of the two sub-problems.",
"First, for notational convenience, we give the following definitions.",
"Definition 3 Let $\\pi $ denote the permutation of $\\lbrace 1,2\\rbrace $ that sets the points ${z_i} \\in \\mathbb {R}_{++}$ in an increasing order such that $z_{\\pi _1} \\le z_{\\pi _2}$ .",
"Also, let $\\mathcal {F}_{i,j}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) ={\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right) +{\\mathcal {R}_{j,2}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)$ and $k_i\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)=\\mathop {\\arg \\min }\\limits _{a \\in \\lbrace 1,2\\rbrace } {\\mathcal {R}_{i,a}}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)$ .",
"Sub-problem 1) Relays Transmit Power Optimization Problem In this part, we optimize the relays transmit power for a fixed circularity coefficient $\\mathcal {C}_x^o$ .",
"The corresponding optimization problem is given by ${\\bf {P2}}\\left(\\mathcal {C}_x^o\\right):&\\mathop {\\max }\\limits _{{p_{\\rm {r}}}}\\qquad {\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o} \\right)\\nonumber \\\\&\\;{\\rm {s}}{\\rm {.t}}{\\rm {.",
"}}\\quad \\quad \\;\\; 0 < {p_{\\rm {r}}} \\le {p_{\\max }}.$ It can be verified that ${\\bf {P2}}$ is a non-convex optimization problem which makes it hard, in general, to find its optimal solution.",
"Also, due to the coupling between the achievable rates of the two paths in terms of $p_{\\rm {r}}$ , maximizing the rates of each individual path with respect to $p_{\\rm {r}}$ and taking the arithmetic mean is not optimal.",
"However, thanks to some special monotonicity properties of the objective function, we show that the optimal solution of ${\\bf {P2}}$ lies either at the intersection between ${\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)$ and ${\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)$ , if exists or one of the stationary points of the $\\mathcal {F}_{i,j}\\left( {{p_{\\rm {r}}},{{\\cal C}_x^o}} \\right)$ with respect to $p_{\\rm {r}}$ , if exists or the power budget $p_{\\rm {max}}$ .",
"Next, we will compute the intersection and stationary points.",
"Proposition 1 There exists at most one intersection point, ${p_i}$ , between ${\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)$ and ${\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)$ over the feasible interval $0 < p_{\\rm {r}} \\le p_{\\rm {max}}$ .",
"Moreover, this intersection point can be obtained by solving the quartic equationThe quartic equation can be solved by Ferrari's method [18].",
"However, since the roots derived from this quartic equation are extremely complex and lengthy, we omit them due to the space limitations.in (REF ).",
"Figure: NO_CAPTIONBy equating ${\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)$ and ${\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)$ , we obtain (REF ).",
"Then, by arranging the coefficients of the quartic equation in a descending order, the signs of theses coefficients, according to the sign of the linear term is either $\\lbrace +,+,+,+,-\\rbrace $ or $\\lbrace +,+,+,-,-\\rbrace $ .",
"In both cases, there is only one change of signs.",
"For our real quartic polynomial, this determines the number of positive roots to be exactly one root over $\\mathbb {R}_{++}$ by using Descartes rule of signs [19].",
"Hence, there exists at most one intersection point over the feasible interval.",
"Remark 3 For the case of using proper signals at the relays i.e., $\\mathcal {C}_x^o=0$ , the quartic equation reduces to the following quadratic equation $p_{_i}^2 + \\frac{{\\sigma _n^2}}{{{{\\left| f \\right|}^2}}}{p_i} - \\frac{{{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2}\\sigma _n^2}}{{{{\\left| {{g_i}} \\right|}^2}{{\\left| f \\right|}^2}}} = 0,$ which can be solved to obtain the intersection point as ${p_i} = \\frac{1}{{2\\left| f \\right|}}\\left( {\\sqrt{\\frac{{\\sigma _n^4}}{{{{\\left| f \\right|}^2}}} + \\frac{{4{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2\\sigma _n^2}}}{{{{\\left| {{g_i}} \\right|}^2}}}} - \\frac{{\\sigma _n^2}}{{\\left| f \\right|}}} \\right).$ Now, We can divide $\\mathbb {R}_{++}$ into three intervals where $\\lbrace 0, p_{\\pi _1}, p_{\\pi _2}, \\infty \\rbrace $ are the boundaries for these intervals.",
"From (REF ), ${\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o} \\right)$ can be reformulated in each interval as $&{\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o} \\right) = \\frac{1}{2} \\times \\nonumber \\\\& \\left\\lbrace {\\begin{array}{*{20}{c}}{\\sum \\limits _{i = 1}^2 {\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right), }&{\\rm {if}}&{0 < {p_{\\rm {r}}} \\le p_{\\pi _1}}\\\\{\\mathcal {R}_{\\pi _1,1}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right)+{\\mathcal {R}_{\\pi _2,2}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right),&{\\rm {if}}&{p_{\\pi _1} < {p_{\\rm {r}}} \\le p_{\\pi _2} }\\\\{\\sum \\limits _{i = 1}^2 {\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}},\\mathcal {C}_x^o}\\right), }&{\\rm {if}}&{p_{\\pi _2} < {p_{\\rm {r}}} < \\infty }\\end{array}} \\right..$ There are at maximum five stationary points $p^{(n)}_{{\\rm {st}}_i} \\in \\mathbb {C}$ , $n \\in \\lbrace 1,2,3,4,5\\rbrace $ of $\\mathcal {F}_{i,j}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)$ which can be calculated by finding the roots of the derivative of $\\mathcal {F}_{i,j}\\left( {{p_{\\rm {r}}},{{\\cal C}_x}} \\right)$ with respect to $p_{\\rm {r}}$ over the interval $0 < p_{\\rm {r}} \\le p_{\\rm {max}}$ .",
"The resulting equation is a quintic equationThe feasible roots of the quintic equation can be obtained numerically., which is very lengthy and we omit it due to space limitation.",
"Remark 4 When using proper signals at the relays, the quintic equation reduces to the quadratic equation $\\frac{{{{\\left| {{g_j}} \\right|}^2}}}{{\\sigma _n^2}}{\\left( {\\sigma _n^2 + {p_{{{\\mathrm {st}}}_i}}{{\\left| f \\right|}^2}} \\right)^2} = {p_{\\rm {s}}}{\\left| {{h_i}} \\right|^2}\\left( {{{\\left| f \\right|}^2} - {{\\left| {{g_j}} \\right|}^2}} \\right),$ which can be solved to get only one possible stationary point as ${p_{{{\\mathrm {st}}}_i}} = \\sqrt{\\frac{{\\sigma _n^2{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2}}}{{{{\\left| {{g_j}} \\right|}^2}{{\\left| f \\right|}^4}}}\\left( {{{\\left| f \\right|}^2} - {{\\left| {{g_j}} \\right|}^2}} \\right)} - \\frac{{\\sigma _n^2}}{{{{\\left| f \\right|}^2}}},$ in which it can be easily shown that ${p_{{{\\mathrm {st}}}_i}} \\in \\mathbb {R}_{++}$ if and only if ${\\left| f \\right|^2} - {\\left| {{g_j}} \\right|^2} > \\frac{{\\sigma _n^2{{\\left| {{g_j}} \\right|}^2}}}{{{p_{\\rm {s}}}{{\\left| {{h_i}} \\right|}^2}}}.$ Before introducing the optimal solution of ${\\bf {P2}}$ , let us give the following definition Definition 4 Let the set of feasible transmit powers $\\mathcal {P}_{\\rm {int}}=\\left\\lbrace p_i \\mid {0 < p_i\\le p_{\\rm {max}} } \\right\\rbrace $ .",
"Also, the set of feasible stationary points $\\mathcal {P}_{\\rm {st}}=\\left\\lbrace p^{(n)}_{{\\rm {st}}_i} ,\\mid {p_{\\pi _1} < p^{(n)}_{{\\rm {st}}_i}\\le p_{\\pi _2} }\\; \\&\\; p^{(n)}_{{\\rm {st}}_i} \\le p_{\\rm {max}}\\right\\rbrace $ .",
"From Definition REF , $\\mathcal {P}_{\\rm {int}}$ and $\\mathcal {P}_{\\rm {st}}$ can be empty sets.",
"Based on the aforementioned analysis, the optimal solution of ${\\bf {P2}}$ can be found from the following theorem.",
"Theorem 1 In a two-path relaying system, where the two relays transmit improper signals and by treating interference as a Gaussian noise, the optimal power allocation, at a fixed circularity coefficient, that maximizes the total achievable rate constrained by a power budget $p_{\\rm {max}}$ can be obtained as $\\begin{array}{l}p_{\\rm {r}}^* = \\mathop {\\arg \\max }\\limits _{p \\in \\mathcal {P}_{{\\rm {T}}}} {\\mathcal {R}_{{\\rm {T}}}}\\left( {p,\\mathcal {C}_x^o} \\right),\\end{array}$ where $\\mathcal {P}_{{\\rm {T}}}=\\mathcal {P}_{\\rm {int}} \\cup \\mathcal {P}_{\\rm {st}} \\cup p_{\\rm {max}}$ .",
"From the definition of the total rate function in (REF ), it can be readily verified that the function in the first interval, i.e., ${0 < {p_{\\rm {r}}} \\le p_{\\pi _1} }$ , is monotonically increasing in $p_{\\rm {r}}$ , thus the optimal solution of $\\bf {P2}$ in this interval is $p_{\\pi _1}$ .",
"Moreover, the function in (REF ) in the last interval, i.e., ${p_{\\pi _2} < {p_{\\rm {r}}} < \\infty }$ , is monotonically decreasing in $p_{\\rm {r}}$ and hence the optimal solution in this interval is $p_{\\pi _2}$ .",
"If the maximum of ${R_{{\\rm {T}}}}$ is in the middle interval, it must occur at a stationary point.",
"Finally, we limit these points by the power budget and this concludes the proof.",
"Sub-problem 2) Circularity Coefficient Optimization Problem Now, we optimize the impropriety of the relays transmit signal, measured by the circularity coefficient, assuming a fixed transmit power ${p_{\\rm {r}}^o}$ .",
"To this end, we formulate the following optimization problem.",
"${\\bf {P3}}\\left(p_{\\rm {r}}^o\\right):&\\mathop {\\max }\\limits _{\\mathcal {C}_x}\\qquad {\\mathcal {R}_{{\\rm {T}}}}\\left( {{p_{\\rm {r}}^o},\\mathcal {C}_x} \\right)\\nonumber \\\\&\\; {\\rm {s}}{\\rm {.t}}{\\rm {.}}",
"\\quad \\quad \\;\\; 0\\le \\mathcal {C}_x \\le 1.$ This problem has been addressed in our work [12] and the optimal solution is given in the following theorem.",
"Theorem 2 [12] In a two-path relaying system, where the two relays transmit improper signals and by treating interference as a Gaussian noise, the optimal circularity coefficient, at a fixed relay transmit power, that maximizes the total achievable rate can be obtained as Case 1: no intersection points $\\mathcal {C}_x^*={\\left\\lbrace \\begin{array}{ll}{0},&{\\rm {if}}\\quad k_1\\left( {\\mathcal {C}}\\right)=k_2\\left( {\\mathcal {C}}\\right)=2,\\; 0 \\le \\mathcal {C} \\le 1\\\\{1},&{\\rm {if}}\\quad k_1\\left( {\\mathcal {C}}\\right)=k_2\\left( {\\mathcal {C}}\\right)=1,\\; 0 \\le \\mathcal {C} \\le 1\\\\{\\mathop {\\arg \\max }\\limits _{\\mathcal {C}_x \\in {\\lbrace {0,\\mathcal {C}}_{{\\rm {st}}_i},1\\rbrace }} \\mathcal {F}_{i,j}\\left({p_{\\rm {r}}^o} ,{\\mathcal {C}_x}\\right)},&{\\rm {if}} \\quad k_1\\left( {\\mathcal {C}}\\right)=i, k_2\\left( {\\mathcal {C}}\\right)=j,0 \\le \\mathcal {C} \\le 1\\end{array}\\right.",
"}.$ Case 2: one intersection point, ${{\\cal C}_{{i}}}$ $\\mathcal {C}_x^*={\\left\\lbrace \\begin{array}{ll}{\\mathop {\\arg \\max }\\limits _{\\mathcal {C}_x \\in {\\lbrace {{\\cal C}_{{i}}},{\\mathcal {C}}_{{\\rm {st}}_j},1\\rbrace }} \\mathcal {F}_{j,i}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right)},&{\\rm {if}}\\quad {k_j\\left( {\\mathcal {C}}\\right)=1,\\;0 \\le \\mathcal {C} \\le 1}\\\\{\\mathop {\\arg \\max }\\limits _{\\mathcal {C}_x \\in {\\lbrace 0,{\\mathcal {C}}_{{\\rm {st}}_i},{{\\cal C}_{{i}}}\\rbrace }} \\mathcal {F}_{i,j}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right)},&{\\rm {if}}\\quad {k_j\\left( {\\mathcal {C}}\\right)=2,\\;0 \\le \\mathcal {C} \\le 1}\\end{array}\\right.",
"}.$ Case 3: two intersection points, ${\\left(\\mathcal {C}_{\\pi _1},\\mathcal {C}_{\\pi _2} \\right)}$ $\\mathcal {C}_x^*=\\mathop {\\arg \\max }\\limits _{\\mathcal {C}_x \\in {\\lbrace \\mathcal {C}_{\\pi _1},{\\mathcal {C}}_{\\rm {st}_{\\pi _2}},\\mathcal {C}_{\\pi _2} \\rbrace }} \\mathcal {F}_{\\pi _2,\\pi _1}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right).$ where $\\mathcal {C}_i$ and ${\\mathcal {C}}_{{\\rm {st}}_i}$ are the intersection between ${\\mathcal {R}_{i,1}}\\left( {{p_{\\rm {r}}^o},\\mathcal {C}_x}\\right)$ and ${\\mathcal {R}_{i,2}}\\left( {{p_{\\rm {r}}^o},\\mathcal {C}_x}\\right)$ and the stationary point for $\\mathcal {F}_{i,j}\\left({p_{\\rm {r}}^o},\\mathcal {C}_x\\right)$ with respect to $\\mathcal {C}_x$ , over the feasible interval $0 < \\mathcal {C}_x \\le 1$ , respectivelyFor more about the existance and uniqueness of $\\mathcal {C}_i$ and ${\\mathcal {C}}_{{\\rm {st}}_i}$ , please refer to [12].. An extended version of the proof in [12] is provided in the appendix.",
"Coordinate Descent: Joint Optimization Problem Here, we aim at optimizing jointly the relays power and circularity coefficient in order to maximize the total rate of the two-path relaying system via CD, in which we implement alternate optimization of $p_{\\rm {r}}$ and $\\mathcal {C}_x$ .",
"In this method, we optimize the transmit power for a fixed circularity coefficient.",
"Then, we use the optimal power in the previous step to optimize for the circularity coefficient and iterate between the optimal solutions till a stopping criterion is satisfied.",
"For this purpose, we develop Algorithm I to obtain the optimization parameters of $\\bf {P1}$ .",
"algorithm Algorithm I: Joint Alternate Optimization of the power and circularity coefficient based on the CD method.",
"[1] Input $h_i$ , $g_i$ , $f$ , ${\\sigma _n^2}$ , $p_{\\rm {max}}$ , $\\epsilon _{\\rm {max}}$ , $0 < {p_{\\rm {r}}^o} \\le p_{\\rm {max}} $ , $0 \\le \\mathcal {C}_x^o \\le 1$ .",
"Initialize ${p_{\\rm {r}}}\\leftarrow {p_{\\rm {r}}^o}$ , $\\mathcal {C}_x\\leftarrow \\mathcal {C}_x^o$ and $\\epsilon \\leftarrow \\infty $ $\\epsilon > \\epsilon _{\\rm {max}}$ Compute $\\hat{p}_{\\textrm {{r}}} $ from $\\textbf {P2} \\left( \\mathcal {C}_x\\right)$ using Theorem 1 Compute $ \\hat{ \\mathcal {C}}_x $ from $\\textbf {P3} \\left(\\hat{p}_{\\textrm {{r}}} \\right)$ using Theorem 2 Set $\\epsilon =\\max \\left\\lbrace \\mid \\hat{\\mathcal {C}}_x-\\mathcal {C}_x\\mid ,\\mid \\hat{p}_{\\textrm {r}}-p_{\\textrm {r}}\\mid \\right\\rbrace $ $\\%max\\; error$ Update ${p_{\\rm {r}}}\\leftarrow {\\hat{p}_{\\rm {r}}}$ Update $\\mathcal {C}_x\\leftarrow \\hat{\\mathcal {C}}_x$"
],
[
"Numerical Results",
"In this section, we numerically evaluate the average end-to-end rate of the proposed two-path relaying system using improper signaling.",
"Throughout the following simulation scenarios, we compare between proper and improper signaling.",
"For proper based system system, we include two scenarios: maximum power allocation (MPA) and optimal power allocation (OPA).",
"On the other hand for improper based system, we include three scenarios: MPA for maximally improper relay signal, i.e., $\\mathcal {C}_x=1$ , optimized CD based method using an initial point for the power as $p^o_{\\textrm {r}}=p_{\\textrm {max}}$ and two different initial starting points for the circularity coefficient; $\\mathcal {C}_x^{0} = 0$ and $\\mathcal {C}_x^{0} = 1$ and the joint optimal allocation of $p_{\\mathrm {r}}$ and $\\mathcal {C}_x$ using a fine exhaustive grid search (GS) as a benchmark for the alternate optimization.",
"The average channel signal-to-noise ratios (SNRs) are defined as $\\gamma _{h_i} = {\\sigma ^2_{h_i}}/{\\sigma ^2_n}$ , $\\gamma _{g_i} = {\\sigma ^2_{g_i}}/{\\sigma ^2_n}$ and $\\gamma _{f} = {\\sigma ^2_{f}}/{\\sigma ^2_n}$ .",
"The results are averaged over 10000 channel realizations and $\\epsilon _{\\rm {max}}=0.0001$ .",
"As for the simulation setup, we assume symmetric relays links with zero-mean complex Gaussian distribution and $\\gamma _{h_i} = 10 \\; \\mathrm {dB}$ , $\\gamma _{g_i} = 15 \\; \\mathrm {dB}$ , $\\gamma _{f}=20 \\; \\mathrm {dB}$ , unless otherwise specified.",
"Firstly, to explore the impact of improper signaling on two-path relaying systems, we study the average rate performance versus $\\gamma _{f}$ as can be seen in Fig REF .",
"It is clear that, proper and improper based systems suffer from a rate degradation as the interference link increases which worsen the performance of $\\mathrm {S}-\\mathrm {R}$ links and thus limits the end-to-end rate.",
"For the proper based system, we observe that optimizing the relay power reduces the IRI impact on the relays and improve the rate.",
"As for improper signaling, optimizing $\\mathcal {C}_x$ with maximum power can significantly boost the rate at mid and high interference levels.",
"At low interference levels, improper-MPA achieves better performance than proper-MPA, however it can not compete with proper-OPA as the interference is not dominant in such situation and thus proper signaling becomes preferable.",
"The same observation is observed for other improper based systems when compared with proper-MPA.",
"As for CD joint optimization solution, the proper choice of initial points in CD plays an important role in the overall performance compared with the GS solution as can be observed in Fig.",
"REF .",
"As a result, staring the CD with $\\mathcal {C}_x^{0} =1$ can converge to the GS solution while $\\mathcal {C}_x^0 =0$ improves the rate performance but it does not converge to the optimal performance.",
"This observation can be justified as the solution at high interference levels reduces to maximally improper, i.e., $\\mathcal {C}_x^*=1$ as can be seen from the improper-MPA system.",
"Figure: The average achievable end-to-end rate for proper and improper signaling with different methods versus γ f \\gamma _{f}.Secondly, we study the average end-to-end rate performance of the aforementioned system versus $\\gamma _{h_i}$ as can be shown in Fig.",
"REF .",
"At very low $\\gamma _{h_i}$ values, the first hops become a bottleneck and degrade the end-to-end average rate for both proper and improper based systems.",
"As $\\gamma _{h_i}$ increases, improper systems use more transmit powers and alleviate the IRI through the increase of the signal impropriety by boosting the circularity coefficient while proper-MPA systems use relatively less power.",
"This improvement gap remains until the value of $\\gamma _{h_i}$ becomes relatively large with respect to $\\gamma _{f}$ , and hence the proper based system starts to enhance its performance by increasing its transmit power.",
"At high $\\gamma _{h_i}$ , both systems tend to utilize the power budget and the improper solution reduces to proper.",
"From this investigation, we can state that improper signaling is preferred when the first hops become a bottleneck.",
"As expected from the previous simulation scenario at $\\gamma _{f} = 20 \\; \\mathrm {dB}$ , improper-MPA achieves a close performance to the improper-GS.",
"Figure: The average achievable end-to-end rate for proper and improper signaling with different techniques versus γ h i \\gamma _{h_i}."
],
[
"Conclusion",
"In this paper, we propose to use improper signaling in order to mitigate the inter-relay interference (IRI) in two-path relaying systems.",
"First, we formulate an optimization problem to tune the relays transmit power and the circularity coefficient, a measure of the degree of asymmetry of the signal, to maximize the total end-to-end achievable rate of the two-path relaying system considering a power budget.",
"We first introduce the optimal allocation of the relays power at a fixed circularity coefficient to maximize the achievable rate, then we optimize the circularity coefficient at a fixed relays power.",
"After that we numerically optimize the relays power and circularity coefficient jointly through a coordinate descent based method.",
"The numerical results show a significant improvement of the total rate when the relays transmit improper signals, specifically, at mid and high IRI values.",
"More generally, the merits of using improper signaling become significant when the first hop is the bottleneck of the system due to either week gains or the excess of IRI."
],
[
"Proof of Theorem 2",
"In fact, this theorem has been proved in [12], however, here we give additionally graphs of the possible configurations of the rate functions ${\\mathcal {R}_{i,j}}\\left( {{p_{\\rm {r}}^o},{{\\cal C}_x}} \\right)$ in (REF ) and (REF ).",
"These graphs makes the optimization problem more visually clear for the convenience of the reader.",
"For the first case in Fig.",
"REF , we have four different orientations for the minimum pair of rate functions for the two paths.",
"The minimum pair is the two decreasing functions ${\\mathcal {R}_{i,2}}\\left( {p_{\\rm {r}}^o},{\\mathcal {C}_x}\\right),\\forall i$ and hence, their sum will also be decreasing and the optimal solution is $\\mathcal {C}_x^*=0$ .",
"Similar argument applies if the minimum pair is the two increasing functions yielding $\\mathcal {C}_x^*=1$ .",
"If the minimum pair is of opposite monotonicity, we need to compute the stationary point of their sum because if there is a maximum on $0 < \\mathcal {C}_x<1$ , it must occur at the stationary point calculated from [12].",
"In the second case in Fig.",
"REF , the intersection point, ${{\\cal C}_{{i}}}$ , of the two hops rates of the $i$ th path, divides the $\\mathcal {C}_x$ range into two intervals.",
"In the first interval $0 < \\mathcal {C}_x\\le {{\\cal C}_{{i}}}$ , the minimum rate of the $i$ th path is ${\\mathcal {R}_{i,1}}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right)$ , and in the second interval ${{\\cal C}_{{i}}} < \\mathcal {C}_x\\le 1$ , the minimum rate of the $i$ th path is ${\\mathcal {R}_{i,2}}\\left( {p_{\\rm {r}}^o},{\\mathcal {C}_x}\\right)$ .",
"For the $j$ th path, we have two different orientations on $0 < \\mathcal {C}_x<1$ , either the minimum is the first or the second hop.",
"Hence, by a similar argument as in Case 1, the result follows directly.",
"Finally, in the third case in Fig.",
"REF , we can write the total achievable rate as $&{\\mathcal {R}_{{\\rm {T}}}}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x} \\right) = \\frac{1}{2} \\times \\\\ \\nonumber & \\left\\lbrace {\\begin{array}{*{20}{c}}{\\sum \\limits _{i = 1}^2 {\\mathcal {R}_{i,1}}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right), }&{\\rm {if}}&{0 < \\mathcal {C}_x \\le \\mathcal {C}_{\\pi _1}}\\\\{R_{\\pi _2,1}}\\left( {p_{\\rm {r}}^o},{\\mathcal {C}_x}\\right)+{R_{\\pi _1,2}}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right),&{\\rm {if}}&{\\mathcal {C}_{\\pi _1} < \\mathcal {C}_x \\le \\mathcal {C}_{\\pi _2} }\\\\{\\sum \\limits _{i = 1}^2 {\\mathcal {R}_{i,2}}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x}\\right), }&{\\rm {if}}&{\\mathcal {C}_{\\pi _2} < \\mathcal {C}_x < 1 }\\end{array}} \\right.\\hspace{-5.0pt}.$ From the definition of the total rate function in (REF ), it can be readily verified that the function in the first interval, i.e., ${0 < \\mathcal {C}_x \\le \\mathcal {C}_{\\pi _1}}$ , is monotonically increasing in $\\mathcal {C}_x$ , thus the optimal solution of in this interval is $\\mathcal {C}_{\\pi _1}$ .",
"Moreover, the function in (REF ) in the last interval, i.e., ${\\mathcal {C}_{\\pi _2} < \\mathcal {C}_x < 1 }$ , is monotonically decreasing in $\\mathcal {C}_x$ and hence the optimal solution in this interval is $\\mathcal {C}_{\\pi _2}$ .",
"If the maximum of ${\\mathcal {R}_{{\\rm {T}}}}\\left({p_{\\rm {r}}^o}, {\\mathcal {C}_x} \\right) $ , with respect to $\\mathcal {C}_x$ , is in the middle interval, it must occur at a stationary point and this concludes the proof.",
"Figure: Possibilities for the rate functions configurations in case of no intersections between the 1st and 2nd hops of both paths (solid lines for the minumum rate function).Figure: Possibilities for the rate functions configurations in case of existence of intersection between the 1st and 2nd hops of only one of the paths (solid lines for the minumum rate function).Figure: Possibilities for the rate functions configurations in case of existence of intersection between the 1st and 2nd hops of both paths (solid lines for the minumum rate function)."
]
] | 1612.05743 |
[
[
"Autonomous Localization and Mapping Using a Single Mobile Device"
],
[
"Abstract This paper considers the problem of simultaneous 2-D room shape reconstruction and self-localization without the requirement of any pre-established infrastructure.",
"A mobile device equipped with co-located microphone and loudspeaker as well as internal motion sensors is used to emit acoustic pulses and collect echoes reflected by the walls.",
"Using only first order echoes, room shape recovery and self-localization is feasible when auxiliary information is obtained using motion sensors.",
"In particular, it is established that using echoes collected at three measurement locations and the two distances between consecutive measurement points, unique localization and mapping can be achieved provided that the three measurement points are not collinear.",
"Practical algorithms for room shape reconstruction and self-localization in the presence of noise and higher order echoes are proposed along with experimental results to demonstrate the effectiveness of the proposed approach."
],
[
"introduction",
"Indoor localization has become more important in recent years as numerous applications, e.g., public safety or location based services, rely on accurate indoor localization [1].",
"As GPS signals are severely attenuated in typical indoor environment, a number of alternative technologies have been proposed for indoor localization, e.g.",
"those using WiFi [2], [3], [4], UWB signal [5], [6], [7], LED light [8], [9] , or some combination of the above.",
"These technologies inevitably require indoor geometry information.",
"There are applications where the indoor room geometry may need to be acquired concurrently with localization.",
"This is generally referred to as simultaneous localization and mapping (SLAM).",
"We comment that the so-called WiFi-SLAM still requires indoor mapping information; SLAM refers to the training process that associates mapping information with the WiFi signature [10].",
"There are also applications where mapping itself is the ultimate goal instead of self-localization [11], [12].",
"For many applications where room shape reconstruction is required, acoustic based approach is arguably more suitable as rooms are often defined by dominant sound reflectors (walls).",
"The distance measurements as measured through acoustic echoes contain rich information about the location of the measurement points as well as the room geometry.",
"A key advantage of the acoustic based approach is that no pre-established infrastructure is needed; this is in sharp contrast with other approaches which inevitably require either deployment of anchor nodes [13], [14] or the availability of ambient WiFi signals as well as preliminary maps [10].",
"This unique advantage has the potential to broaden the applications of indoor mapping and localization to systems where current technologies are either unsuitable or too expensive to implement.",
"The most prevalent acoustic based approach is to employ a single fixed loudspeaker and a microphone array, or equivalently, a fixed loudspeaker and a mobile microphone [15], [16], [17], [18], [19], [20].",
"It was shown that both the room shape and the geometry of the microphone array (or the trajectory of the mobile microphone) can be estimated by first order echoes [21].",
"Furthermore, bearing only SLAM can be achieved using a mobile microphone array [22].",
"The fact that a microphone array needs to be deployed leaves much to be desired: fully autonomous SLAM should require minimum deployment effort.",
"Ideally, a single mobile device that moves around would autonomously reconstruct the room shape while tracking its own movement within the recovered room geometry.",
"Indeed, room shape recovery using a single acoustic device has been addressed in the literature.",
"It was established that any convex polygon can be reconstructed by the entire set of both first and second order echoes collected using a fixed device with a collocated microphone and loudspeaker [18].",
"However, experimental results, including that of our own, demonstrated that higher order acoustic echoes are often difficult to recover, thus the requirement of having the entire set of second order echoes makes such an approach impractical.",
"On the other hand, given only grouped first order echoes, SLAM can be achieved for a large class of convex polygon other than parallelograms [23].",
"This result was strengthened in [24] where it was established that parallelograms are the only convex polygons that are not recoverable via grouped first order echoes.",
"Here “-grouped” means correct labeling, i.e., the correspondence between collected echoes and walls is known.",
"This paper makes further progress in overcoming the shortcomings of the approaches in [23], [24].",
"The reconstruction will again be based on first order echoes only but without the knowledge of echo labeling.",
"To overcome the ambiguity associated with parallelograms, our approach leverages the ever expanding capability of various motion sensors embedded in latest smart phones, including accelerometer, magnetometer, and gyroscope.",
"Those sensors are capable of measuring distance and direction information of a moving device [25], [26], [27].",
"However, existing results indicate that while distance measures have reasonable accuracy, direction measurement is often subject to large measurement error [28].",
"Thus our current approach only exploits the distance measurements and the key question to be addressed is how much additional information will be needed for acoustic SLAM to be able to recover all convex polygons.",
"The major contribution of the paper is to establish that with three non-collinear measurement points, SLAM can be achieved for all convex polygons using ungrouped first order echoes provided that the distances between consecutive measurement points are known.",
"Note that this additional information is much weaker than the knowledge of the complete geometry of the measurement - this is tantamount to knowing only two sides of a triangle which is inadequate to construct the triangle.",
"An added advantage of this additional distance information is that it removes the need for grouped echoes, making the scheme much more widely applicable as it can accommodate a great deal of freedom in the movement of the device.",
"Preliminary results have been reported in [29].",
"The present work, in addition to expanding on technical details, contains several new results including a more detailed analysis on exactly what is the minimum amount of distance information that is needed for SLAM.",
"Specifically, it is further established that with ungrouped echoes, a single distance measure does not suffice for parallelograms.",
"Note the subtle but important difference with that of [23], [24] in which grouped instead of ungrouped echoes are assumed.",
"The rest of the paper is organized as follows.",
"Section II introduces the indoor propagation model of acoustic signals, image source model and existing results on 2-D with a single device.",
"Theoretical guarantee of successful SLAM given distances between consecutive measurement points is provided in Section III along with a practical algorithm that handles the presence of measurement noise and higher order/spurious peaks.",
"Experiment results are provided in Section IV followed by conclusion in Section V. Acoustic signal propagation from a loudspeaker to a microphone in a room can be described by the room impulse response (RIR), which includes both line-of-sight (LOS) and reflected components.",
"If the microphone and loudspeaker are much closer to each other compared to the distance between the device and the walls, we say it is a co-located device.",
"For a co-located device at the $j$ th measurement point denoted by $O_{j}$ , the RIR is, ignoring dispersion, $h^{(j)}(t)=\\sum _{i}\\alpha _{i}^{(j)}\\delta (t-\\tau _{i}^{(j)}),$ where $\\alpha _{i}^{(j)}$ 's and $\\tau _{i}^{(j)}$ 's are path gains and delays from the transmitter to the receiver, respectively.",
"Since higher order reflective paths typically have much weaker power, $h^{(j)}(t)$ can be approximated by the first $N_{j}+1$ components including LOS and $N_{j}$ reflective paths: $h^{(j)}(t)\\approx \\sum _{i=0}^{N_{j}}\\alpha _{i}^{(j)}\\delta (t-\\tau _{i}^{(j)}),$ where we assume that the $N_{j}$ reflective paths contain all first order reflections and higher order ones that are detectable.",
"Notice that for an arbitrary convex polygon, not every measurement point has first order echoes to all the walls.",
"We refer to those measurement points can receive all first order echoes as feasible measurement points.",
"Denote by $s(t)$ the emitted signal at the speaker.",
"Then the received signal at the microphone for the $j$ th measurement point is $r^{(j)}(t) = s(t) * h^{(j)}(t)+\\omega (t),$ where $*$ denotes linear convolution and $\\omega (t)$ is the additive noise.",
"Ideally, the delays can be recovered from the received signal $r^{(j)}(t)$ if $s(t)$ behaves like a Dirac delta function [17].",
"However, this requires a wideband acoustic signal along with a wideband acoustic channel, including that of the microphone receiver.",
"A more practical alternative is to emit $s(t)$ with a desired auto-correlation function that is peaky and then implement a correlator at the microphone: $m^{(j)}(t) = r^{(j)}(t) * s(t).$ Thus, the first and dominant peak of $m^{(j)}(t)$ corresponds to the LOS components, while the remaining peaks correspond to reflective components.",
"The time difference of arrival (TDOA) in reference to the LOS component can be used for estimating the delays of different reflective paths.",
"A simple peak-detection method will be introduced in Section V.A, where the chirp signal is used for $s(t)$ because of its nice auto-correlation property.",
"Define a column vector $\\tilde{\\mathbf {r}}_{j}=\\bigg \\lbrace \\frac{(\\tau _{i}^{(j)}-\\tau _{0}^{(j)})c}{2}\\bigg \\rbrace _{i=1}^{N_{j}},$ where $c$ is the speed of sound and $\\tau _{i}^{(j)}$ is the arrival time of the $i$ th path with $\\tau _{0}^{(j)}$ corresponding to the LOS component.",
"Then $\\tilde{\\mathbf {r}}_{j}$ contains all the distances between the device and the walls, along with some higher order terms."
],
[
"Image Source Model",
"With the image source model [15], reflections within a constrained space can be viewed as free space LOS propagations from virtual sources to the receiver.",
"Let the coordinate of $O_{j}$ be denoted by $\\mathbf {o}_{j}$ .",
"As show in Fig.",
"1, the first order image source of $O_{j}$ with respect to the $i$ th wall is Figure: The image source model: 𝐨 ˜ j,i \\tilde{\\mathbf {o}}_{j,i} and 𝐨 ˜ j,k \\tilde{\\mathbf {o}}_{j,k} are first-order image sources with respect to the iith and kkth wall and 𝐨 ˜ j,ik \\tilde{\\mathbf {o}}_{j,ik} is the second-order image source with respect to the iith and kkth wall in the stated order.$\\tilde{\\mathbf {o}}_{j,i} = 2\\langle \\mathbf {p}_{i}-\\mathbf {o}_{j},\\mathbf {n}_{i}\\rangle \\mathbf {n}_{i} + \\mathbf {o}_{j},$ where $\\mathbf {p}_{i}$ is any point on the $i$ th wall, $\\mathbf {n}_{i}$ is the outward norm vector of the $i$ th wall and $\\langle \\mathbf {x},\\mathbf {y}\\rangle $ denotes the inner product between $\\mathbf {x}$ and $\\mathbf {y}$ .",
"Let $r_{j,i}$ be the distance between $O_{j}$ and the $i$ th wall, then $r_{j,i}=\\frac{1}{2}||\\tilde{\\mathbf {o}}_{j,i} - \\mathbf {o}_{j}||_{2}.$ Moreover, the second order image source of $O_{j}$ with respect to the $i$ th and the $k$ th wall is $\\tilde{\\mathbf {o}}_{j,ik}=2\\langle \\mathbf {p}_{k}-\\tilde{\\mathbf {o}}_{j,i},\\mathbf {n}_{k}\\rangle \\mathbf {n}_{k}+\\tilde{\\mathbf {o}}_{j,i}.$ Similarly, we denote by $r_{j,ik}$ the half distance between $\\mathbf {o}_{j}$ and $\\tilde{\\mathbf {o}}_{j,ik}$ .",
"Following similar steps, higher order image sources can be represented by lower order image sources.",
"Then all the elements of $\\tilde{\\mathbf {r}}_{j}$ can be represented by the real source and image sources.",
"For the rest of the paper, the term echo is used to refer to either the delay $\\tau _{i}^{(j)}$ or the corresponding elements of $\\tilde{\\mathbf {r}}_{j}$ if no ambiguity occurs."
],
[
"Two Extreme Cases",
"The most benign case is when the location of the measurement points are known, or equivalently, the distance between pairwise measurement points are given [30].",
"In this case, only room shape reconstruction is of interest and the problem becomes trivial, at least in the noiseless case.",
"It amounts to finding common tangent lines of circles centered at three non-collinear measurement points.",
"The other extreme is when the reconstruction is free of any geometry information of the measurement points.",
"In this case, both room shape and self-localization are of interest.",
"This was first investigated in [23] where it was established that a large class of convex polygons can be reconstructed by grouped first order echoes and, subsequently, the coordinates of measurement points can be also estimated.",
"An important exception is parallelograms and it was shown in [23] that unique reconstruction of parallelograms is impossible using first-order echoes alone.",
"The result was later strengthened in [24] where it was proved that all convex polygons except parallelogram can be reconstructed subject to the usual rotation and reflection ambiguities."
],
[
"SLAM with Two Path Lengths",
"Consider a convex planar $K$ -polygon.",
"As shown in Fig.2, a mobile device with co-located microphone and loudspeaker emits pulses and receives echoes at $\\lbrace O_{j}\\rbrace _{j=1}^{3}$ .",
"Without loss of generality, we assume that $O_{1}$ is the origin, $O_{2}$ lies on the $x$ -axis, and $O_{3}$ lies above the $x$ -axis.",
"Let $\\varphi =(\\pi -\\angle O_{1}O_{2}O_{3})\\in (0,\\pi )$ and the lengths of $O_{1}O_{2}$ and $O_{2}O_{3}$ be denoted by $d_{12}$ and $d_{23}$ , respectively.If $\\varphi \\in (0,2\\pi )$ , i.e.",
"we do not have control of where to place $O_{3}$ , then the reconstruction is subject to reflection ambiguity (c.f.",
"Theorem III.3).",
"Suppose the mobile device is capable of measuring its path length when moving from one place to another, i.e.",
"$d_{12}$ and $d_{23}$ are known.",
"Our goal is to simultaneously determine the room shape and the coordinate of $O_{3}$ using first-order echoes.",
"From Fig.",
"2, it is straightforward to show that $(r_{2,i}-r_{1,i})+d_{12}\\cos \\theta _{i}=0,$ $d_{23}\\cos (\\theta _{i}-\\varphi )+(r_{3,i}-r_{2,i})=0.$"
],
[
"Ideal Case",
"Let $\\mathbf {r}_{j}=\\lbrace r_{j,i}\\rbrace _{i=1}^{K}$ be a column vector with its entries defined in (REF ).",
"We assume for now that, the one-to-one mapping $f_{j}:\\tilde{\\mathbf {r}}_{j}\\mapsto \\mathbf {r}_{j}$ is known for all $j$ 's.",
"In other words, $r_{j,i}$ 's have been correctly chosen from $\\tilde{\\mathbf {r}}_{j}$ for $j=1,2,3$ and $i=1,\\ldots ,K$ .",
"For the rest of the paper, we say that the received echoes are grouped if echoes are correctly labeled.",
"The remaining problem is to determine the uniqueness of $\\theta _{i}$ 's and $\\varphi $ given (REF ) and (REF ).",
"Define $\\alpha _{ii^{\\prime }}=-\\frac{r_{2,i}-r_{1,i^{\\prime }}}{d_{12}}$ and $\\beta _{ii^{\\prime }}=-\\frac{r_{3,i^\\prime {}}-r_{2,i}}{d_{23}}$ .",
"For simplicity we denote $\\alpha _{ii}$ and $\\beta _{ii}$ by $\\alpha _{i}$ and $\\beta _{i}$ , respectively.",
"Given grouped echoes and Eqs.",
"(REF ) and (REF ), we have $\\theta _{i}=\\pm \\arccos \\alpha _{i} \\quad \\text{and}\\quad \\theta _{i}-\\varphi =\\pm \\arccos \\beta _{i},$ There are four possible sign combinations for a given $i$ , $\\theta _{i}=\\arccos \\alpha _{i} \\quad \\text{and}\\quad \\theta _{i}-\\varphi =\\arccos \\beta _{i}$ $\\theta _{i}=\\arccos \\alpha _{i} \\quad \\text{and}\\quad \\theta _{i}-\\varphi =-\\arccos \\beta _{i}$ $\\theta _{i}=-\\arccos \\alpha _{i} \\quad \\text{and}\\quad \\theta _{i}-\\varphi =\\arccos \\beta _{i}$ $\\theta _{i}=-\\arccos \\alpha _{i} \\quad \\text{and}\\quad \\theta _{i}-\\varphi =-\\arccos \\beta _{i}.$ Lemma 3.1 Suppose $O_{j}\\:(j=1,2,3)$ are feasible and not collinear.",
"Given grouped first order echoes, with probability 1, there exist exactly two sign combinations such that (REF ) and (REF ) hold simultaneously for all $i$ if $\\varphi $ and the direction of both $\\overrightarrow{O_{1}O_{2}}$ and $\\overrightarrow{O_{2}O_{3}}$ are randomly chosen.",
"The two possible sign combinations have opposite signs for $\\varphi $ and all $\\theta _{i}$ 's and correspond to reflection of each other in terms of recovered room shapes.",
"Assume without loss of generality that the ground truth of the polygon is (REF ) for all $i\\in \\lbrace 1,\\ldots ,K\\rbrace $ .",
"Note that (REF ) implies that (REF ) holds for $\\theta _{i}^{\\prime }=-\\theta _{i}$ and $\\varphi ^{\\prime }=-\\varphi <0$ for all $i$ , i.e., they correspond to reflections of each other.",
"Suppose multiple sign combinations hold for a wall.",
"Without loss of generality, let $i=1$ .",
"From (REF ) we have $\\varphi =\\arccos \\alpha _{1}-\\arccos \\beta _{1}.$ Assume that one of the following equations also holds, $\\varphi =-\\arccos \\alpha _{1}-\\arccos \\beta _{1},$ $\\varphi =\\arccos \\alpha _{1}+\\arccos \\beta _{1},$ $\\varphi =-\\arccos \\alpha _{1}+\\arccos \\beta _{1}.$ Then we have the following three cases If (REF ) and (REF ) hold, we must have $\\theta _{1}=0$ which implies that $O_{1}O_{2}$ is perpendicular to the first wall, and $\\varphi =-\\arccos \\beta _{1}$ .",
"If (REF ) and (REF ) hold, we must have $\\arccos \\beta _{1}=0$ , which implies that $O_{2}O_{3}$ is perpendicular to the first wall.",
"If (REF ) and (REF ) hold, we must have $\\varphi =0$ , which contradicts the assumption that $O_{1}$ , $O_{2}$ and $O_{3}$ are not collinear.",
"Given that the three measurement points are randomly chosen, and, subsequently, $\\varphi $ , $\\overrightarrow{O_{1}O_{2}}$ and $\\overrightarrow{O_{2}O_{3}}$ are random, the first two cases do not occur with probability one.",
"If a subset of (REF )-(REF ) holds for $i$ and $i^{\\prime }$ simultaneously, then we must have $(\\theta _{i},\\theta _{i^{\\prime }})\\in \\lbrace \\theta _{i}=0,\\theta _{i}=\\varphi ,\\varphi =0\\rbrace \\times \\lbrace \\theta _{i^{\\prime }}=0,\\theta _{i^{\\prime }}=\\varphi ,\\varphi =0\\rbrace $ , which again, do not occur due to randomly chosen measurement points.",
"Similarly, it can be shown that for more than 2 walls, (REF ) would imply none of (REF )-(REF ) holds for all walls."
],
[
"Echo Labeling",
"Since echoes may arrive in different orders at different $O_{j}$ 's and $\\tilde{\\mathbf {r}}_{j}$ contains higher order echoes if $N_{j}>K$ , $f_{j}$ is usually unknown.",
"We say the received echoes are ungrouped if $f_{j}$ is unknown for some $j$ .",
"Thus given $\\tilde{\\mathbf {r}}_j$ , our task is to first determine the mapping $f_{j}$ , i.e., label the echoes, followed by estimation of $\\theta _{i}$ 's and $\\varphi $ .",
"Lemma 3.2 With ungrouped echoes, any mapping $f_{j}^{\\prime }$ that differs from the correct mapping $f_{j}$ will result, with probability 1, the following two possible cases there exists no solution to (REF ) and (REF ) given no parallel edges, or the reconstructed room shape has larger dimension with respect to parallel edges.",
"We illustrate the proof by considering the case $K=4$ .",
"The result can be easily extended to $K=3$ and $K>4$ .",
"Suppose again that the ground truth is (REF ) for all $i$ .",
"We first consider parallelograms and exclude odd higher order echoes resulting from a pair of parallel walls.",
"The distances between $O_{j}\\:(j=1,2,3)$ and the four walls satisfy $r_{1,1}+r_{1,2}=r_{2,1}+r_{2,2}=r_{3,1}+r_{3,2}=a,$ $r_{1,3}+r_{1,4}=r_{2,3}+r_{2,4}=r_{3,3}+r_{3,4}=b.$ One can see that for some $f_{j}^{\\prime }$ 's, pairs of $\\lbrace \\alpha _{ii^{\\prime }},\\beta _{ii^{\\prime }}\\rbrace \\: (i,i^{\\prime }\\in \\lbrace 1,2,3,4\\rbrace )$ are related to each other.",
"Consider for example the $f_{j}^{\\prime }$ 's resulting in $\\lbrace \\alpha _{12},\\alpha _{21}, \\alpha _{34}, \\alpha _{43}\\rbrace $ and $\\lbrace \\beta _{12},\\beta _{21}, \\beta _{34}, \\beta _{43}\\rbrace $ .",
"Since $\\alpha _{12}+\\alpha _{21}=0$ , $\\alpha _{34}+\\alpha _{43}=0$ , $\\beta _{12}+\\beta _{21}=0$ and $\\beta _{34}+\\beta _{43}=0$ , we have $\\arccos (\\alpha _{21})=\\pi \\pm \\arccos (\\alpha _{12}),$ $\\arccos (\\alpha _{43})=\\pi \\pm \\arccos (\\alpha _{34}),$ $\\arccos (\\beta _{21})=\\pi \\pm \\arccos (\\beta _{12}),$ $\\arccos (\\beta _{43})=\\pi \\pm \\arccos (\\beta _{34}).$ Thus (REF ) reduces to two equations $\\varphi =\\pm \\arccos (\\alpha _{12})\\pm \\arccos (\\beta _{12}),$ $\\varphi =\\pm \\arccos (\\alpha _{34})\\pm \\arccos (\\beta _{34}).$ With probability 1, these two equations do not hold simultaneously as $\\alpha _{12}$ , $\\beta _{12}$ are independent of $\\alpha _{34}$ , $\\beta _{34}$ due to randomly chosen measurement points.",
"Other $f_{j}^{\\prime }(\\ne f_{j})$ 's always have at least two equations with independent choice of $\\alpha $ and $\\beta $ .",
"Hence no solution can be found for those instances.",
"Suppose $f_{j}^{\\prime }$ 's are chosen such that we have $\\alpha _{ii^{\\prime }}$ and $\\beta _{ii^{\\prime \\prime }}$ ($i\\ne i^{\\prime }, \\: i\\ne i^{\\prime \\prime }$ ).",
"For rooms with no more than one pair of parallel walls, only echoes chosen according to $f_{j}$ 's satisfy (REF ) for all $i$ .",
"This is because for those rooms, at least one of (REF ) and (REF ) does not hold.",
"Thus some $\\alpha _{ii^{\\prime }}$ 's and $\\beta _{ii^{\\prime \\prime }}$ 's are not related since $r_{1i^{\\prime }}$ , $r_{2i}$ and $r_{3i^{\\prime \\prime }}$ are randomly chosen from $\\tilde{\\mathbf {r}}_{1}$ , $\\tilde{\\mathbf {r}}_{2}$ and $\\tilde{\\mathbf {r}}_{3}$ , respectively.",
"Given parallel edges, however, higher order echoes may also satisfy (REF ) and (REF ).",
"For instance, as shown in Fig.",
"3, suppose that walls 1 and 3 are parallel.",
"Then it is easy to verify that $r_{j,131} - r_{j^{\\prime },131} = r_{j,1} - r_{j^{\\prime },1},$ $r_{j,313} - r_{j^{\\prime },313} = r_{j,3} - r_{j^{\\prime },3},$ where $j\\ne j^{\\prime }$ .",
"Hence, (REF ) and (REF ) provide the same $\\cos \\theta _{1}$ , $\\cos \\theta _{3}$ , $\\cos (\\theta _{1}-\\varphi )$ and $\\cos (\\theta _{3}-\\varphi )$ if $r_{j,1}$ and $r_{j,3}$ are replaced by $r_{j,131}$ and $r_{j,313}$ , respectively.",
"By Lemma III.1, the third order echoes resulting from a pair of parallel edges lead to a larger room with the same norm vectors.",
"Exactly the same argument applies to odd higher order echoes from a pair of parallel edges.",
"Therefore, Lemma III.2 is proved.",
"Remark 1: The ambiguities resulting from parallel edges can be easily eliminated if we always choose SLAM result with the smallest room size.",
"Given Lemma III.1 and Lemma III.2, we have the following result on the identifiability of any convex polygonal room by using only first order echoes.",
"Theorem 3.3 With probability 1, SLAM can be achieved subject to reflection ambiguity given any convex planar $K$ -polygon, by using the first order echoes received at three random points in the feasible region, with known $d_{12}$ and $d_{23}$ and unknown $\\varphi \\in (0,2\\pi )$ .",
"Figure: A room with a pair of parallel edges.",
"Here wall 1 and 3 are parallel.Remark 2: Both the room shape and the coordinate of $O_{3}$ are subject to reflection ambiguity for $\\varphi \\in (0,2\\pi )$ .",
"If, however, we can limit $\\varphi \\in (0,\\pi )$ , SLAM will be free of such ambiguity.",
"Remark 3: In reality, it is inevitable to collect reflections from the ceiling and the floor.",
"However, by Theorem III.3, if distances corresponding to these echoes are included, no polygon can be recovered provided that the trajectory of the device lies in a plane that is perpendicular to the walls."
],
[
"A Practical Algorithm",
"In a real acoustic system, $m^{(j)}(t)$ 's in (REF ) are inevitably corrupted by measurement noise leading to corrupted measurement of $\\tilde{\\mathbf {r}}_{j}$ .",
"Let the corrupted version of $\\tilde{\\mathbf {r}}_{j}$ be denoted by $\\hat{\\mathbf {r}}_{j}$ .",
"Two issues arise.",
"First, given $f_{j}$ 's, $\\varphi $ obtained by (REF ) for different $i$ 's are not necessarily identical.",
"The second issue is the possibility that the computed cosine values in (REF ) may have absolute value exceeding 1.",
"For the former, we propose a heuristic scheme of choosing the echo and sign combination that yield the smallest variance of the estimated $\\varphi $ 's across different $i$ 's.",
"Notice that in the noiseless case with perfect echo measurements, the variance of the estimated $\\varphi $ 's across different $i$ 's is 0 if the correct echo and sign combination is selected while all others will have non-zero (potentially large variance).",
"For the latter, define a feasible $\\cos \\theta _{i}$ as $\\cos \\theta _{i} = {\\left\\lbrace \\begin{array}{ll}1, & \\text{if}\\ 1\\le -\\frac{\\hat{r}_{2,i}-\\hat{r}_{1,i}}{d_{12}} < 1+\\epsilon \\\\-\\frac{\\hat{r}_{2,i}-\\hat{r}_{1,i}}{d_{12}}, & \\text{if}\\ -1<-\\frac{\\hat{r}_{2,i}-\\hat{r}_{1,i}}{d_{12}}<1 \\\\-1, & \\text{if}\\ -1-\\epsilon < -\\frac{\\hat{r}_{2,i}-\\hat{r}_{1,i}}{d_{12}} \\le -1\\end{array}\\right.",
"},$ where $\\epsilon > 0$ is a tuning parameter determined by the noise level.",
"Feasible $\\cos (\\theta _{i}-\\varphi )$ can be similarly defined.",
"The echo combination is said to be infeasible if either $|\\frac{\\hat{r}_{2,i}-\\hat{r}_{1,i}}{d_{12}}|>1+\\epsilon $ or $|\\frac{\\hat{r}_{3,i}-\\hat{r}_{2,i}}{d_{23}}|>1+\\epsilon $ .",
"Only those feasible $\\theta _{i}$ 's and $\\varphi $ will be used in computing the variance of the estimated $\\varphi $ .",
"As the number of walls for the room is not known in prior, the proposed algorithm needs to first reconstruct some room shapes with $K=3,\\ldots ,N$ walls.",
"Then the desired room shape is the feasible one with the largest number of walls.",
"In order to reconstruct a room shape with $K$ walls, the number of echo combinations that need to be exhausted is $\\binom{N_{1}}{K}\\binom{N_{2}}{K}\\binom{N_{3}}{K}(K!",
")^2.$ For simplicity assume that $N=N_{1}=N_{2}=N_{3}$ .",
"Let $V_{th}$ be the threshold of the variance.",
"The corresponding algorithm is summarized as Algorithm 1.",
"Reconstruct convex polygon given distances between consecutive measurement points [1] Set $K=3$ and $V_{th}$ .",
"$K\\le N$ Set $V_{K}=\\inf $ and the stored polygon with $K$ walls be empty.",
"$n=1:\\big (\\binom{N}{K}\\big )^{3}(K!",
")^2$ Based on the $n$ th echo combination, choose $K$ elements from $\\hat{\\mathbf {r}}_{1}$ , $\\hat{\\mathbf {r}}_{2}$ , $\\hat{\\mathbf {r}}_{3}$ , respectively.",
"Compute $\\cos \\theta _{i}$ 's and $\\cos (\\theta _{i}-\\varphi )$ for $i=1,\\ldots ,K$ .",
"$\\cos \\theta _{i}$ 's and $\\cos (\\theta _{i}-\\varphi )$ are feasible Compute $\\text{Var}[\\varphi ]$ for different sign combinations and keep the one with the smallest $\\text{Var}[\\varphi ]$ .",
"$\\text{Var}[\\varphi ]< V_{K}$ and the room shape does not fully cover the stored one with $K$ walls Keep the echo and sign combination and set $V_{K} = \\text{Var}[\\varphi ]$ for $K$ .",
"$K=K+1$ .",
"Keep the SLAM results the largest $K$ such that $V_{K} < V_{th}$ ."
],
[
"SLAM with One Path Length",
"Now that we have established that two distances between three consecutive measurement points are sufficient to overcome the drawback of using first order echoes alone, a natural question is what would be the least amount of information that is required to achieve SLAM for any convex polygons.",
"Specifically we examine the case where only one distance between a pair of measurement points is known.",
"We show that for a parallelogram, there exist multiple rooms satisfying (REF ) and (REF ) in this case, thus the answer is negative, i.e.",
"a single distance measurement is insufficient for SLAM with ungrouped first order echoes.",
"Without loss of generality, assume $d_{12}$ is known but $d_{23}$ is not.",
"As shown in Fig.",
"4, let $O_{1}$ be the origin, $O_{2}$ be on the x-axis and $O_{3}(x_{3},y_{3})\\:(y_{3}\\ne 0)$ is unknown.",
"We also assume that the direction of $\\overrightarrow{O_{1}O_{2}}$ with respect to the desired room is unknown.",
"By geometry, we have (REF ) and $(r_{3,i}-r_{1,i})+x_{3}\\cos \\theta _{i}+y_{3}\\sin \\theta _{i}=0.$ Eq.",
"(REF ) can also be rewritten in a matrix form $\\mathbf {A}[x_{3},y_{3}]^{T}=\\mathbf {b},$ where $\\mathbf {A} =\\begin{bmatrix}\\cos \\theta _{1} & \\sin \\theta _{1} \\\\\\vdots & \\vdots \\\\\\cos \\theta _{K} & \\sin \\theta _{K}\\end{bmatrix},$ and $\\mathbf {b}=[-(r_{3,1}-r_{1,1}),\\ldots ,-(r_{3,K}-r_{1,K})]^{T}.$ Figure: A mobile device is employed to measure the geometry of a room.",
"The mobile device collects echoes at O 1 O_{1}, O 2 O_{2} and O 3 O_{3} successively.",
"Only the distances between O 1 O_{1} and O 2 O_{2} (d 12 d_{12}) is known.The ground truth of a parallelogram is assumed to be $\\mathbf {A}=\\begin{bmatrix}\\cos \\theta _{1} & \\sin \\theta _{1} \\\\\\cos \\theta _{2} & \\sin \\theta _{2} \\\\\\cos \\theta _{3} & \\sin \\theta _{3} \\\\\\cos \\theta _{4} & \\sin \\theta _{4}\\end{bmatrix}\\:\\text{and}\\:\\;\\mathbf {b}=\\begin{bmatrix}-(r_{3,1}-r_{1,1}) \\\\-(r_{3,2}-r_{1,2}) \\\\-(r_{3,3}-r_{1,3}) \\\\-(r_{3,4}-r_{1,4})\\end{bmatrix},$ where $r_{1,1}+r_{1,3}=r_{2,1}+r_{2,3}=r_{3,1}+r_{3,3},$ $r_{1,2}+r_{1,4}=r_{2,2}+r_{2,4}=r_{3,2}+r_{3,4}.$ Let $\\mathbf {A}^{\\prime }=\\begin{bmatrix}\\cos \\theta _{13} & \\sin \\theta _{13} \\\\\\cos \\theta _{24} & \\sin \\theta _{24} \\\\\\cos \\theta _{31} & \\sin \\theta _{31} \\\\\\cos \\theta _{42} & \\sin \\theta _{42}\\end{bmatrix}\\mathbf {b}^{\\prime }=\\begin{bmatrix}-(r_{3,1}-r_{1,3}) \\\\-(r_{3,2}-r_{1,4}) \\\\-(r_{3,3}-r_{1,1}) \\\\-(r_{3,4}-r_{1,2})\\end{bmatrix}.$ Then $\\cos \\theta _{13}+\\cos \\theta _{31}=0\\quad \\text{and}\\quad \\cos \\theta _{24}+\\cos \\theta _{42}=0.$ Moreover, since $\\sin \\theta =\\pm \\sqrt{1-\\cos ^{2}\\theta }$ , $\\sin \\theta _{13}+\\sin \\theta _{31}=0\\quad \\text{and}\\quad \\sin \\theta _{24}+\\sin \\theta _{42}=0$ can hold if we manipulate the sign of square root properly.",
"Then $\\text{rank}(\\mathbf {A}^{\\prime })=\\text{rank}([\\mathbf {A}^{\\prime },\\mathbf {b}^{\\prime }])=2$ .",
"Thus a room shape and the coordinate of $O_{3}$ different from the ground truth and its reflection also satisfy both (REF ) and (REF )."
],
[
"Experiment Setup",
"We describe in the following some preliminary experimental results.",
"Enormous challenges exist to conduct a truly autonomous SLAM.",
"Chief among them are: the search space (number of combinations) is extremely large - using for example, some modest numbers, e.g.",
"$K=4$ and $N_{1}=N_{2}=N_{3}=8$ , the number of echo combinations exceeds $10^{7}$ , combining with the sign combinations the search space is in the billions; the measurement of motion sensors is still subject to large errors and some robustification of the reconstruction algorithm will need to be investigated if the true motion sensor measurements are used.",
"The purpose of the experimental design is thus to demonstrate the feasibility of the proposed scheme in an idealized situation with a certain degree of human intervention to alleviate the above challenges.",
"We use a laptop as a microphone and a HTC M8 phone as our loudspeaker.",
"As the loudspeaker of the cell phone is not omnidirectional and is power limited, we place the speaker of the cell phone towards each wall to ensure the corresponding first order echo is strong enough.",
"Note that the microphone will record both first order echoes and some higher order ones.",
"A chirp signal linearly sweeping from 30Hz to 8kHz is emitted by the cell phone.",
"The sample rate at the receiver is $f_{s}=96$ kHz.",
"It has been shown in [31], [32] that if the input chirp signal is correlated with its windowed version, the output may resemble a delta function, which is desirable for better delay resolution.",
"Figure: Comparison of convolution result.",
"The maximum values of the two convolution result are set to be identical.Our simulation indicates that correlating the received signals with its triangularly windowed version outperforms the correlator using the original one.",
"The comparison is shown in Fig. 5.",
"Figure: Sample of correlator output: Peaks with solid red ellipses correspond to walls while peaks with dash green ellipses correspond to either noise or higher order echoesFig.",
"6 is a sample path of the correlator output collected in the room where this experiment is conducted.",
"In Fig.",
"6, peaks marked with red ellipse are desired while those with green ellipse correspond to noise, the ceiling, the floor, higher order echoes or other spurious sources.",
"In our experiment, we use $|m^{(j)}(t)|$ rather than $m^{(j)}(t)$ since the true peaks may be either positive or negative.",
"Local maxima of $|m^{(j)}(t)|$ corresponding to Fig.",
"6 are shown in Fig. 7.",
"A heuristic way to detect peaks, summarized in Algorithm 3, is to check the slope of each local maxima.",
"Three requirements are needed for the proposed algorithm: 1) the minimum distance between the device and the walls is no less than $d_{min}$ , 2) the minimum TDOA of two detected consecutive echoes is no less than $\\Delta t$ , 3) the maximum candidate distance corresponding to detected peaks is no more than $d_{max}$ .",
"The reason for the requirements is as follows: 1) since the correlation property of the chirp signal is not ideal and the power of the LOS component is much larger than that of reflective components, the distance between the device and the walls should be large enough such that the peaks corresponding to reflective components are not overshadowed by the LOS component, 2) as the power of reflective paths decays rapidly, it is reasonable to restrict the detectable echoes within certain distances which depends on the power of loudspeaker.",
"Given $d_{min} = 0.6$ m, $d_{max}=6.5$ m and $\\Delta t = \\frac{0.5\\text{m}}{c}$ , where $c=346\\text{m/s}$ , the detection results are marked by arrows in Fig. 7.",
"We can see that the desired peaks are always detected.",
"In order to detect as less false peaks as possible, one possible modification is to apply a tapering threshold which decreases as $t$ increases.",
"Peak detection algorithm [1] find LOS peak $(t_{0}^{(j)},m_{0}^{(j)})$ .",
"find local maxima of $|m^{(j)}(t)|$ appearing from $t_{0}^{(j)}+t_{min}$ to $t_{0}^{(j)}+t_{max}$ .",
"find all peaks that are peaky and store them in $M$ set $P = {Ø}$ $|P| < |M|$ there exist peaks in $M$ whose locations are \"close\" to any peak in $P$ remove those peaks from $M$ .",
"add the peak with the largest magnitude of $M$ to $P$ .",
"Figure: Illustration of the performance of the proposed peak detection algorithm."
],
[
"SLAM Results",
"Echoes are collected at $O_{j}\\:(j=1,\\ldots ,4)$ and $d_{j,j+1}\\:(j=1,2,3)$ are measured by tape measure.",
"The proposed peak detection algorithm is used to estimate the candidate distances from received signals.",
"Note that the number of detected peaks are much larger than the number of first order echoes.",
"Heuristics are used to remove peaks (e.g.",
"those of small magnitudes) - otherwise, checking all combinations of echoes become computationally prohibitive.",
"The proposed algorithm for SLAM is verified by experiment at $O_{1}$ , $O_{2}$ , $O_{3}$ and $O_{2}$ , $O_{3}$ , $O_{4}$ .",
"Given $O_{2}$ , $O_{3}$ , $O_{4}$ , we assume that $O_{2}$ is the origin and $O_{3}$ lies on the $x$ -axis.",
"Even if some elements of $\\mathbf {r}_{j}$ have measurement errors up to 10cm, SLAM is accomplished with small error of both the room shape and the coordinates of $O_{3}$ and $O_{4}$ with only unlabeled first-order echoes.",
"In the presence of higher order echoes, the proposed algorithm may perform poorly and ambiguity may occur when the variance of $\\varphi $ is the only criterion used to determine $f_{j}$ 's.",
"With noisy measurement, it is possible that the incorrect echo combination may yield feasible $\\theta _{i}$ and $\\varphi $ with variance smaller than that of the correct echo combination.",
"Furthermore, an interesting phenomenon is that sometimes the proposed algorithm is unable to provide the correct room shape, but the estimate of $\\varphi $ is always close to the true value.",
"This means that better echo labeling approach is needed for robust SLAM.",
"As most rooms are regular, we add a heuristic constraint: all the angles of two adjacent walls are between $50^{\\circ }$ and $130^{\\circ }$ .",
"The comparison between the SLAM result and the ground truth is illustrated in Fig. 8.",
"The candidate distances are obtained by the peak detection algorithm.",
"Note that the coordinate system in Fig.",
"8(b) is a rotation of that in Fig.",
"8(a) by $135^{\\circ }$ counterclockwise.",
"The SLAM results shown in the two figures are rotational images of each other.",
"Experimental result indicate that heuristic constraints such as the above can largely eliminate incorrect combinations.",
"Figure: Comparison between the ground truth (black) and experiment result (red underlined)"
],
[
"Conclusion",
"This work makes progress in acoustic SLAM using a single mobile device with unlabeled first order echoes.",
"Theoretical guarantee of 2-D SLAM is established when two path lengths corresponding to three consecutive measurement points are available.",
"Conversely, it was also shown that with only a single distance measurement, 2-d SLAM with unlabeled first order echoes is not possible for all convex polygons.",
"The result is summarized in Table I.",
"Table: Feasibility of SLAM with unlabeled first order echoes and different geometry knowledgeWhile theoretical guarantee can be established for the noiseless case, the proposed algorithm needs to be enhanced to ensure a fully autonomous 2-D SLAM.",
"Two particular issues that need to be further addressed include the robustness with respect to measurement noise and the computational complexity when a large number of peaks are detected at each measurement location."
]
] | 1612.05793 |
[
[
"Importance sampling-based approximate optimal planning and control"
],
[
"Abstract In this paper, we propose a sampling-based planning and optimal control method of nonlinear systems under non-differentiable constraints.",
"Motivated by developing scalable planning algorithms, we consider the optimal motion plan to be a feedback controller that can be approximated by a weighted sum of given bases.",
"Given this approximate optimal control formulation, our main contribution is to introduce importance sampling, specifically, model-reference adaptive search algorithm, to iteratively compute the optimal weight parameters, i.e., the weights corresponding to the optimal policy function approximation given chosen bases.",
"The key idea is to perform the search by iteratively estimating a parametrized distribution which converges to a Dirac's Delta that infinitely peaks on the global optimal weights.",
"Then, using this direct policy search, we incorporated trajectory-based verification to ensure that, for a class of nonlinear systems, the obtained policy is not only optimal but robust to bounded disturbances.",
"The correctness and efficiency of the methods are demonstrated through numerical experiments including linear systems with a nonlinear cost function and motion planning for a Dubins car."
],
[
"lti[LTI]linear time invariant mras[MRAS]Model Reference Adaptive Search mdp[MDP]Markov decision process saop[SAOP]Sampling-based Approximate-Optimal Planning mdp[MDP]Markov decision process hjb[HJB]Hamilton-Jacobi-Bellman kl[KL]Kullback–Leibler rbf[RBF]Radial basis function mpc[MPC]Model Predictive Control asaop[ASAOP]Adaptive Search-based Approximate-Optimal Planning output-exponent-marker=$\\mathrm {e}$ compatibility=false In this paper, we propose a sampling-based planning and optimal control method of nonlinear systems under non-differentiable constraints.",
"Motivated by developing scalable planning algorithms, we consider the optimal motion plan to be a feedback controller that can be approximated by a weighted sum of given bases.",
"Given this approximate optimal control formulation, our main contribution is to introduce importance sampling, specifically, model-reference adaptive search algorithm, to iteratively compute the optimal weight parameters, i.e., the weights corresponding to the optimal policy function approximation given chosen bases.",
"The key idea is to perform the search by iteratively estimating a parametrized distribution which converges to a Dirac's Delta that infinitely peaks on the global optimal weights.",
"Then, using this direct policy search, we incorporated trajectory-based verification to ensure that, for a class of nonlinear systems, the obtained policy is not only optimal but robust to bounded disturbances.",
"The correctness and efficiency of the methods are demonstrated through numerical experiments including linear systems with a nonlinear cost function and motion planning for a Dubins car.",
"This paper presents an importance sampling based approximate optimal planning and control algorithm.",
"Optimal motion planning in deterministic and continuous systems is computationally NP-complete [1] except for linear time invariant systems.",
"For nonlinear systems, there is a vast literature on approximate solutions and algorithms.",
"In optimal planning, the common approximation scheme is discretization-based.",
"By discretizing the state and input spaces, optimal planning is performed by solving the shortest path problem in the discrete transition systems obtained from abstracting the continuous dynamics, using heuristic-based search or dynamic programming.",
"Comparing to discretization-based methods, sampling-based graph search, includes Probabilistic RoadMap (PRM) [2], RRT [3], RRT* [4], are more applicable for high-dimensional systems.",
"While RRT has no guarantee on the optimality of the path [4], RRT* compute an optimal path asymptotically provided the cost functional is Lipschitz continuous.",
"However, such Lipschitz conditions may not be satisfied for some cost functions under specific performance consideration.",
"The key idea in the proposed sampling-based planning method builds on a unification of importance sampling and approximate optimal control [5], [6].",
"In approximate optimal control, the objective is to approximate both the value function, i.e., optimal cost-to-go, and the optimal feedback policy function by weighted sums of known basis functions.",
"As a consequence, the search space is changed from infinite trajectory space or policy space to a continuous space of weight vectors, given that each weight vector corresponds to a unique feedback controller.",
"Instead of solving the approximate optimal control through training actor and critic neural networks (NNs) using trajectory data [7], [8], we propose a sampling-based method for sampling the weight vectors for a policy function approximation and searching for the optimal one.",
"This method employs mras [9], a probabilistic complete global optimization algorithm, for searching the optimal weight vector that parametrizes the approximate optimal feedback policy.",
"The fundamental idea is to treat the weight vector as a random variable over a parameterized distribution and the optimal weight vector corresponds to a Dirac's Delta function which is the target distribution.",
"The mras algorithm iteratively estimates the parameter that possesses the minimum Kullback-Leibler divergence with respect to an intermediate reference model, which assigns a higher probability mass on a set of weights of controllers with improved performance over the previous iteration.",
"At the meantime, a set of sampled weight vectors are generated using the parameterized distribution and the performance of their corresponding policies are evaluated via simulation-based policy evaluation.",
"Under mild conditions, the parameterized distribution converges, with probability one, to the target distribution that concentrates on the optimal weight vector with respect to given basis functions.",
"mras resembles another adaptive search algorithm called cross-entropy(CE) method and provides faster and stronger convergence guarantee for being less sensitive to input parameters [9], [10].",
"Previously, CE algorithm has been introduced for motion planning [11], [12] based on sampling in the trajectory space.",
"The center idea is to construct a probability distribution over the set of feasible paths and to perform the search for an optimal trajectory using CE.",
"The parameters to be estimated is either a sequence of motion primitives or a set of via-points for interpolation-based trajectory planning.",
"Differ to these methods, ours is the first to integrate importance sampling to estimate parameterization of the optimal policy function approximation for continuous nonlinear systems.",
"Since the algorithm performs direct policy search, we are able to enforce robustness and stability conditions to ensure the computed policy is both robust and approximate optimal, provided these conditions can be evaluated efficiently.",
"To conclude, the contributions of this paper are the following: First, we introduce a planning algorithm by a novel integration of model reference adaptive search and approximate optimal control.",
"Second, based on contraction theory, we introduce a modification to the planning method to directly generate stabilizing and robust feedback controllers in the presence of bounded disturbances.",
"Last but not the least, through illustrative examples, we demonstrate the effectiveness and efficiency of the proposed methods and share our view on interesting future research along this direction.",
"Notation: The inner product between two vectors $w,v\\in \\mathbf {R}^n$ is denoted $w^\\intercal v$ or $ \\langle w,v \\rangle $ .",
"Given a positive semi-definite matrix $P$ , the $P$ -norm of a vector is denoted $\\Vert x \\Vert _P = \\sqrt{x^\\intercal P x}$ .",
"We denote $\\Vert x \\Vert $ for $P$ being the identity matrix.",
"$I_{\\lbrace E\\rbrace }$ is the indicator function, i.e., $I_{E}=1$ if event $E$ holds, otherwise 0.",
"For a real $\\alpha \\in \\mathbb {R}$ , $\\lceil \\alpha \\rceil $ is the smallest integer that is greater than $\\alpha $ .",
"We consider continuous-time nonlinear systems of the form $\\begin{split}\\Sigma : \\quad & \\dot{x}(t) = f(x(t),u(t)),\\\\&x(t)\\in X, u(t)\\in U.\\end{split}$ where $x \\in X$ is the state, $u \\in U $ is the control input, $x_0\\in X$ is the initial state, and $f(x,u)$ is a vector field.",
"We assume that $X$ and $U$ are compact.",
"A feedback controller $u:X\\rightarrow U$ takes the current state and outputs a control input.",
"The objective is to find a feedback controller $u^\\ast $ that minimizes a finite-horizon cost function for a nonlinear system $\\begin{split}\\min _{u} J( x_0, u)& = \\int _{0}^{T} \\ell ( x(t), u(t))dt+ g(x(T),u(T))\\\\\\mbox{subject to: }& \\dot{x}(t) = f(x(t),u(t)),\\\\&x(t)\\in X, \\; u(t)\\in U, \\; x(0)=x_0.\\end{split}$ where $T$ is the stopping time, $\\ell : X\\times U \\rightarrow \\mathbb {R}^+ $ defines the running cost when the state trajectory traverses through $x$ and the control input $u$ is applied and $g: X\\rightarrow \\mathbb {R}^+$ defines the terminal cost.",
"As an example, a running cost function can be a quadratic cost $\\ell (x,u) = \\Vert x \\Vert _R +\\Vert u \\Vert _Q$ for some positive semi-definite matrices $Q$ and $R$ , and a terminal cost can be $g(x,u) = \\Vert x-x_f \\Vert _R$ where $x_f$ is a goal state.",
"We denote the set of feedback policies to be $\\Pi $ .",
"For infinite horizon optimal control, the optimal policy is independent of time and a feedback controller suffices to be a minimizing argument of (REF ) (see Ref.",
"[13]).",
"For finite-horizon optimal control, the optimal policy is time-dependent.",
"However, for simplicity, in this paper, we only consider time-invariant feedback policies and assume the time horizon $T$ is of sufficient length to ignore the time constraints.",
"mras algorithm, introduced in [9], aims to solve the following problem: $z^\\ast \\in \\arg \\max _{z\\in Z} H(z),\\quad z\\in \\mathbf {R}^n$ where $Z$ is the solution space and $H:\\mathbf {R}^n\\rightarrow \\mathbf {R}$ is a deterministic function that is bounded from below.",
"It is assumed that the optimization problem has a unique solution, i.e., $z^\\ast \\in Z$ and for all $z\\ne z^\\ast $ , $H(z) < H(z^\\ast )$ .",
"The following regularity conditions need to be met for the applicability of mras.",
"Assumption 1 For any given constant $\\xi < H(z^\\ast )$ , the set $\\lbrace z \\mid H(z) \\ge \\xi \\rbrace \\cap Z$ has a strictly positive Lebesgue or discrete measure.",
"This condition ensures that any neighborhood of the optimal solution $z^\\ast $ will have a positive probability to be sampled.",
"Assumption 2 For any constant $\\delta >0$ , $\\sup _{z\\in A_\\delta }H(z)< H(z^\\ast )$ , where $A_\\delta := \\lbrace z\\mid \\Vert z-z^\\ast \\Vert \\ge \\delta \\rbrace \\cap X$ , and we define the supremum over the empty set to be $-\\infty $ .",
"Selecte a sequence of reference distributions $\\lbrace g_k(\\cdot )\\rbrace $ with desired convergence properties.",
"Specifically, the sequence $\\lbrace g_k(\\cdot )\\rbrace $ will converge to a distribution that concentrates only on the optimal solution.",
"Selecte a parametrized family of distribution $f(\\cdot , \\theta )$ over $X$ with parameter $\\theta \\in \\Theta $ .",
"Optimize the parameters $\\lbrace \\theta _k\\rbrace $ iteratively by minimizing the following KL distance between $f(\\cdot , \\theta _k) $ and $g_k(\\cdot )$ .",
"$d(g_k, f(\\cdot , \\theta )):= \\int _{Z}\\ln \\frac{g_k(z)}{f(z,\\theta )}g_k(z)\\nu (dz).$ where $\\nu (\\cdot )$ is the Lebesgue measure defined over $ Z$ .",
"The sample distributions $\\lbrace f(\\cdot ,\\theta _k)\\rbrace $ can be viewed as compact approximations of the reference distributions and will converge to an approximate optimal solution as $ \\lbrace g_k(\\cdot )\\rbrace $ converges provided certain properties of $\\lbrace g_k(\\cdot )\\rbrace $ is retained in $f(\\cdot , \\theta _k)$ .",
"Selecte a sequence of reference distributions $\\lbrace g_k(\\cdot )\\rbrace $ with desired convergence properties.",
"Specifically, the sequence $\\lbrace g_k(\\cdot )\\rbrace $ will converge to a distribution that concentrates only on the optimal solution.",
"Selecte a parametrized family of distribution $f(\\cdot , \\theta )$ over $X$ with parameter $\\theta \\in \\Theta $ .",
"Optimize the parameters $\\lbrace \\theta _k\\rbrace $ iteratively by minimizing the following KL distance between $f(\\cdot , \\theta _k) $ and $g_k(\\cdot )$ .",
"$d(g_k, f(\\cdot , \\theta )):= \\int _{Z}\\ln \\frac{g_k(z)}{f(z,\\theta )}g_k(z)\\nu (dz).$ where $\\nu (\\cdot )$ is the Lebesgue measure defined over $ Z$ .",
"The sample distributions $\\lbrace f(\\cdot ,\\theta _k)\\rbrace $ can be viewed as compact approximations of the reference distributions and will converge to an approximate optimal solution as $ \\lbrace g_k(\\cdot )\\rbrace $ converges provided certain properties of $\\lbrace g_k(\\cdot )\\rbrace $ is retained in $f(\\cdot , \\theta _k)$ .",
"Note that the reference distribution $\\lbrace g_k(\\cdot )\\rbrace $ is unknown beforehand as the optimal solution is unknown.",
"Thus, the mras algorithm employs the estimation of distribution algorithms [14] to estimate a reference distribution that guides the search.",
"To make the paper self-contained, we will cover details of mras in the development of the planning algorithm.",
"In this section, we present an algorithm that uses mras in a distinguished way for approximate optimal feedback motion planning.",
"The policy function approximation $\\bar{u}: X\\rightarrow U$ is a weighted sum of basis functions, $\\bar{u}(x) = \\sum _{i=1}^N w_i \\phi _i(x)$ where $\\phi _i : X\\rightarrow \\mathbf {R}, i=1,\\ldots , N$ are basis functions, and the coefficients $w_i$ are the weight parameters, $i=1,\\ldots , N$ .",
"An example of basis function can be polynomial basis $\\phi =[1,x, x^2,x^3,\\ldots , x^N]$ for one-dimensional system.",
"A commonly used class of basis functions is rbf.",
"It can be constructed by determining a set of centers $c_i,\\ldots , c_N \\in X$ , and then constructing rbf basis functions $\\phi _i = \\exp (- \\frac{\\Vert x-c_i \\Vert ^2}{2\\sigma ^2})$ , for each center $c_i$ , where $\\sigma $ is a pre-defined parameter.",
"In vector form, a policy function approximation is represented by $\\bar{u} = \\langle w, \\phi \\rangle $ where vector $\\phi = \\left[ \\phi _1,\\ldots \\phi _N \\right]^\\intercal $ and $w = [w_1,\\ldots , w_N]^\\intercal $ .",
"We let the domain of weight vector be $W$ and denote it by $\\Pi _\\phi = \\lbrace \\langle w, \\phi \\rangle \\mid w \\in W, \\langle w, \\phi \\rangle \\in \\Pi \\rbrace $ the set of all policies that can be generated by linear combinations of pre-defined basis functions.",
"In the following context, unless specifically mentioned, the vector of basis functions is $\\phi $ .",
"Clearly, for any weight vector $w$ , $J(x_0, \\langle w, \\phi \\rangle )\\ge \\min _{u\\in \\Pi }J(x_0, u)$ .",
"Thus, we aim to solve $\\min _{w} J(x_0, \\langle w, \\phi \\rangle )$ so as to minimize the error in the optimal cost introduced by policy function approximation.",
"Definition 1 (Approximate optimal feedback policy) Given a basis vector $\\phi $ , a weight vector $w^\\ast $ with respect to $\\phi $ is optimal if and only if $\\langle w^\\ast , \\phi \\rangle \\in \\Pi _\\phi $ and for all $ w\\in W$ such that $ \\langle w, \\phi \\rangle \\in \\Pi _\\phi $ , $J(x_0, \\langle w^\\ast , \\phi \\rangle ) \\le J(x_0, \\langle w,\\phi \\rangle ).$ The approximate optimal feedback policy is $\\bar{u}^\\ast =\\langle w^\\ast , \\phi \\rangle $ .",
"By requiring $ J(x_0, \\langle w^\\ast , \\phi \\rangle ) \\le J(x_0, \\langle w,\\phi \\rangle ) $ , it can be shown that the optimal weight vector $w^\\ast $ minimizes the difference between the optimal cost achievable with policies in $\\Pi _\\phi $ and the cost under the global optimal policy.",
"For clarity in notation, we denote $J(x_0, \\langle w, \\phi \\rangle )$ by $J(x_0; w)$ as $\\phi $ is a fixed basis vector throughout the development of the proposed method.",
"Clearly, if the actual optimal policy $u^\\ast $ can be represented by a linear combination of selected basis functions, then we obtain the optimal policy by computing the optimal weight vector, i.e., $u^\\ast = \\langle w^\\ast ,\\phi \\rangle $ .",
"Here, we assume a feedback policy can be represented by $\\langle w, \\phi \\rangle $ for some weight vector $w \\in W$ .",
"In cases when the basis functions are continuous, a feedback policy must be a continuous function of the state.",
"However, this requirement is hard to satisfy for many physical systems due to, for example, input saturation.",
"In cases when a feasible controller is discontinuous, we can still use a continuous function to approximate, and then project the continuous function to the set $\\Pi $ of applicable controllers.",
"Using function approximation, we aim to solve the optimal feedback planning problem in (REF ) approximately by finding the optimal weight vector with respect to a pre-defined basis vector.",
"The main algorithm is presented next.",
"In this section, we present an adaptive search-based algorithm to compute the approximate optimal feedback policy.",
"The algorithm is “near” anytime, meaning that it returns a feasible solution after a small number of samples.",
"If more time is permitted, it will quickly converge to the globally optimal solution that corresponds to the approximate optimal feedback policy.",
"The algorithm is probabilistic complete under regularity conditions of mras.",
"We start by viewing the weight vector as a random variable $\\mathbf {w}$ governed by a multivariate Gaussian distribution with a compact support $W$ .",
"The distribution is parameterized by parameter $\\theta =(\\mu ,\\Sigma )$ , where $\\mu $ is a $N$ -dimensional mean vector and $\\Sigma $ is the $N$ by $N$ covariance matrix.",
"Recall $N$ is the number of basis functions.",
"The optimal weight vector $w^\\ast $ can be represented as a target distribution $p_\\mathsf {goal}$ as a Dirac's Delta, i.e., $p_\\mathsf {goal}( w^\\ast )=\\infty $ and $p_\\mathsf {goal}(w)=0$ for $w \\ne w^\\ast $ .",
"Dirac's Delta is a special case of multivariate Gaussian distribution with zero in the limit case of vanishing covariance.",
"Thus, it is ensured that the target distribution can be arbitrarily closely approximated by multivariate Gaussian distribution by a realization of parameter $\\theta $ .",
"Recall that the probability density of a multivariate Gaussian distribution is defined by $&p(w; \\theta ) = \\frac{1}{\\sqrt{(2\\pi )^N \\vert \\Sigma \\vert }}\\exp (-\\frac{1}{2}(x-\\mu )^\\intercal \\Sigma ^{-1}(x-\\mu )), \\\\& \\theta =(\\mu ,\\Sigma ), \\forall w\\in W,$ where $N$ is the dimension of weight vector $w\\in W$ and $\\vert \\Sigma \\vert $ is the determinant of $\\Sigma $ .",
"Now, we are ready to represent the main algorithm, called saop, which includes the following steps.",
"Initialization: The initial distribution is selected to be $p(\\cdot , \\theta _0)$ , for $\\theta _0 =( \\mu _0, \\Sigma _0)\\in \\Theta $ which can generate a set of sample to achieve a good coverage of the sample space $W$ .",
"For example, $\\mu _0 =\\mathbf { 0} \\in \\mathbf {R}^N$ and $\\Sigma _0 = \\mathbf { I}\\in \\mathbf {R}^N $ which is an identity matrix.",
"The following parameters are used in this algorithm: $\\rho \\in (0,1]$ for specifying the quantile, the improvement parameter $\\varepsilon \\in \\mathbf {R}^+$ , a sample increment percentage $\\alpha $ , an initial sample size $N_1$ , a smoothing coefficient $\\lambda \\in (0,1]$ .",
"Let $k=1$ .",
"Sampling-based policy evaluation: At each iteration $k$ , generate a set of $N_k$ samples $W_k \\subseteq W $ from the current distribution $p(\\cdot , \\theta _k)$ .",
"For each $w\\in W_k $ , using simulation we evaluate the cost $J(x_0 ; w)$ from the initial state $x_0$ and the feedback policy $u(x) =\\langle w , \\phi (x) \\rangle $ with system model in (REF ).",
"The cost $J(x_0; w)$ is determined because the system is deterministic and has a unique solution.",
"Policy improvement with Elite samples: Next, the set $\\lbrace J(x_0; w) \\mid w \\in W_k \\rbrace $ is ordered from largest (worst) to smallest (best) among given samples: $J_{k, (0)} \\ge \\ldots \\ge J_{k, (N_k)}$ We denote $\\kappa $ to be the estimated $(1-\\rho )$ -quantile of cost $J(\\cdot ; w)$ , i.e., $\\kappa = J_{k, \\lceil (1-\\rho )N_k \\rceil }$ .",
"The following cases are distinguished.",
"If $k=1$ , we introduce a threshold $ \\gamma = \\kappa $ .",
"If $k \\ne 1$ , the following cases are further distinguished: $\\kappa \\le \\gamma - \\varepsilon $ , i.e., the estimated $(1-\\rho )$ -quantile of cost has been reduced by the amount of $\\varepsilon $ from the last iteration, then let $ \\gamma =\\kappa $ .",
"Let $N_{k+1}=N_k$ and continue to step 4).",
"Otherwise $\\kappa > \\gamma - \\varepsilon $ , we find the largest $ \\rho ^{\\prime } $ , if it exists, such that the estimated $(1-\\rho ^{\\prime })$ -quantile of cost $\\kappa ^{\\prime } = J_{k, \\lceil (1-\\rho ^{\\prime } )N_k \\rceil } $ satisfies $\\kappa ^{\\prime } \\le \\gamma -\\varepsilon $ .",
"Then let $\\gamma = \\kappa ^{\\prime }$ and also let $\\rho = \\rho ^{\\prime }$ .",
"Continue to step 4).",
"However, if no such $\\rho ^{\\prime }$ exists, then there is no update in the threshold $\\gamma $ but the sample size is increased to $N_{k+1} = \\lceil (1+\\alpha ) N_k\\rceil $ .",
"Let $\\theta _{k+1} =\\theta _k$ , $k=k+1$ , and continue to step 2).",
"Parameter(Policy) update: We update parameters $\\theta _{k+1}$ for iteration $k+1$ .",
"First, we define a set $E = \\lbrace w \\mid J(x_0; w) \\le \\gamma , w \\in W_k, j=1,\\ldots , k \\rbrace $ of elite samples.",
"Note that the parameter update in $\\theta $ is to ensure a higher probability for elite samples.",
"To achieve that, for each elite sample $w\\in E $ , we associated a weight such that a higher weight is associated with a weight vector with a lower cost and a lower probability in the current distribution.",
"The next parameter $\\theta _{k+1}$ is selected to maximize the weighted sum of probabilities of elite samples.",
"To this end, we update the parameter as follows.",
"$\\theta _{k+1} ^\\ast \\\\=\\arg \\max _{\\theta \\in \\Theta }\\mathbb {E}_{\\theta _k}\\left[\\frac{S(J(x_0,w))^k}{p(w,\\theta _k)}I_{J(x_0, w)\\le \\gamma }\\ln p(w,\\theta )\\right]$ where $\\mathbb {E}_{\\theta }(\\nu )$ is the expected value of a random variable $\\nu $ given distribution $p(\\cdot , \\theta )$ , $S: \\mathbf {R}\\rightarrow \\mathbf {R}^+$ is a strictly decreasing and positive function Possible choices can be $S(x) = \\exp (-x)$ or $S(x)=\\frac{1}{x}$ if $x$ is strictly positive.. $S(J(x_0; w))^k/p(w,\\theta _k)$ is the weight for parameter $w$ .",
"Initialization: The initial distribution is selected to be $p(\\cdot , \\theta _0)$ , for $\\theta _0 =( \\mu _0, \\Sigma _0)\\in \\Theta $ which can generate a set of sample to achieve a good coverage of the sample space $W$ .",
"For example, $\\mu _0 =\\mathbf { 0} \\in \\mathbf {R}^N$ and $\\Sigma _0 = \\mathbf { I}\\in \\mathbf {R}^N $ which is an identity matrix.",
"The following parameters are used in this algorithm: $\\rho \\in (0,1]$ for specifying the quantile, the improvement parameter $\\varepsilon \\in \\mathbf {R}^+$ , a sample increment percentage $\\alpha $ , an initial sample size $N_1$ , a smoothing coefficient $\\lambda \\in (0,1]$ .",
"Let $k=1$ .",
"Sampling-based policy evaluation: At each iteration $k$ , generate a set of $N_k$ samples $W_k \\subseteq W $ from the current distribution $p(\\cdot , \\theta _k)$ .",
"For each $w\\in W_k $ , using simulation we evaluate the cost $J(x_0 ; w)$ from the initial state $x_0$ and the feedback policy $u(x) =\\langle w , \\phi (x) \\rangle $ with system model in (REF ).",
"The cost $J(x_0; w)$ is determined because the system is deterministic and has a unique solution.",
"Policy improvement with Elite samples: Next, the set $\\lbrace J(x_0; w) \\mid w \\in W_k \\rbrace $ is ordered from largest (worst) to smallest (best) among given samples: $J_{k, (0)} \\ge \\ldots \\ge J_{k, (N_k)}$ We denote $\\kappa $ to be the estimated $(1-\\rho )$ -quantile of cost $J(\\cdot ; w)$ , i.e., $\\kappa = J_{k, \\lceil (1-\\rho )N_k \\rceil }$ .",
"The following cases are distinguished.",
"If $k=1$ , we introduce a threshold $ \\gamma = \\kappa $ .",
"If $k \\ne 1$ , the following cases are further distinguished: $\\kappa \\le \\gamma - \\varepsilon $ , i.e., the estimated $(1-\\rho )$ -quantile of cost has been reduced by the amount of $\\varepsilon $ from the last iteration, then let $ \\gamma =\\kappa $ .",
"Let $N_{k+1}=N_k$ and continue to step 4).",
"Otherwise $\\kappa > \\gamma - \\varepsilon $ , we find the largest $ \\rho ^{\\prime } $ , if it exists, such that the estimated $(1-\\rho ^{\\prime })$ -quantile of cost $\\kappa ^{\\prime } = J_{k, \\lceil (1-\\rho ^{\\prime } )N_k \\rceil } $ satisfies $\\kappa ^{\\prime } \\le \\gamma -\\varepsilon $ .",
"Then let $\\gamma = \\kappa ^{\\prime }$ and also let $\\rho = \\rho ^{\\prime }$ .",
"Continue to step 4).",
"However, if no such $\\rho ^{\\prime }$ exists, then there is no update in the threshold $\\gamma $ but the sample size is increased to $N_{k+1} = \\lceil (1+\\alpha ) N_k\\rceil $ .",
"Let $\\theta _{k+1} =\\theta _k$ , $k=k+1$ , and continue to step 2).",
"Parameter(Policy) update: We update parameters $\\theta _{k+1}$ for iteration $k+1$ .",
"First, we define a set $E = \\lbrace w \\mid J(x_0; w) \\le \\gamma , w \\in W_k, j=1,\\ldots , k \\rbrace $ of elite samples.",
"Note that the parameter update in $\\theta $ is to ensure a higher probability for elite samples.",
"To achieve that, for each elite sample $w\\in E $ , we associated a weight such that a higher weight is associated with a weight vector with a lower cost and a lower probability in the current distribution.",
"The next parameter $\\theta _{k+1}$ is selected to maximize the weighted sum of probabilities of elite samples.",
"To this end, we update the parameter as follows.",
"$\\theta _{k+1} ^\\ast \\\\=\\arg \\max _{\\theta \\in \\Theta }\\mathbb {E}_{\\theta _k}\\left[\\frac{S(J(x_0,w))^k}{p(w,\\theta _k)}I_{J(x_0, w)\\le \\gamma }\\ln p(w,\\theta )\\right]$ where $\\mathbb {E}_{\\theta }(\\nu )$ is the expected value of a random variable $\\nu $ given distribution $p(\\cdot , \\theta )$ , $S: \\mathbf {R}\\rightarrow \\mathbf {R}^+$ is a strictly decreasing and positive function Possible choices can be $S(x) = \\exp (-x)$ or $S(x)=\\frac{1}{x}$ if $x$ is strictly positive.. $S(J(x_0; w))^k/p(w,\\theta _k)$ is the weight for parameter $w$ .",
"If $k=1$ , we introduce a threshold $ \\gamma = \\kappa $ .",
"If $k \\ne 1$ , the following cases are further distinguished: $\\kappa \\le \\gamma - \\varepsilon $ , i.e., the estimated $(1-\\rho )$ -quantile of cost has been reduced by the amount of $\\varepsilon $ from the last iteration, then let $ \\gamma =\\kappa $ .",
"Let $N_{k+1}=N_k$ and continue to step 4).",
"Otherwise $\\kappa > \\gamma - \\varepsilon $ , we find the largest $ \\rho ^{\\prime } $ , if it exists, such that the estimated $(1-\\rho ^{\\prime })$ -quantile of cost $\\kappa ^{\\prime } = J_{k, \\lceil (1-\\rho ^{\\prime } )N_k \\rceil } $ satisfies $\\kappa ^{\\prime } \\le \\gamma -\\varepsilon $ .",
"Then let $\\gamma = \\kappa ^{\\prime }$ and also let $\\rho = \\rho ^{\\prime }$ .",
"Continue to step 4).",
"However, if no such $\\rho ^{\\prime }$ exists, then there is no update in the threshold $\\gamma $ but the sample size is increased to $N_{k+1} = \\lceil (1+\\alpha ) N_k\\rceil $ .",
"Let $\\theta _{k+1} =\\theta _k$ , $k=k+1$ , and continue to step 2).",
"$\\kappa \\le \\gamma - \\varepsilon $ , i.e., the estimated $(1-\\rho )$ -quantile of cost has been reduced by the amount of $\\varepsilon $ from the last iteration, then let $ \\gamma =\\kappa $ .",
"Let $N_{k+1}=N_k$ and continue to step 4).",
"Otherwise $\\kappa > \\gamma - \\varepsilon $ , we find the largest $ \\rho ^{\\prime } $ , if it exists, such that the estimated $(1-\\rho ^{\\prime })$ -quantile of cost $\\kappa ^{\\prime } = J_{k, \\lceil (1-\\rho ^{\\prime } )N_k \\rceil } $ satisfies $\\kappa ^{\\prime } \\le \\gamma -\\varepsilon $ .",
"Then let $\\gamma = \\kappa ^{\\prime }$ and also let $\\rho = \\rho ^{\\prime }$ .",
"Continue to step 4).",
"However, if no such $\\rho ^{\\prime }$ exists, then there is no update in the threshold $\\gamma $ but the sample size is increased to $N_{k+1} = \\lceil (1+\\alpha ) N_k\\rceil $ .",
"Let $\\theta _{k+1} =\\theta _k$ , $k=k+1$ , and continue to step 2).",
"Parameter(Policy) update: We update parameters $\\theta _{k+1}$ for iteration $k+1$ .",
"First, we define a set $E = \\lbrace w \\mid J(x_0; w) \\le \\gamma , w \\in W_k, j=1,\\ldots , k \\rbrace $ of elite samples.",
"Note that the parameter update in $\\theta $ is to ensure a higher probability for elite samples.",
"To achieve that, for each elite sample $w\\in E $ , we associated a weight such that a higher weight is associated with a weight vector with a lower cost and a lower probability in the current distribution.",
"The next parameter $\\theta _{k+1}$ is selected to maximize the weighted sum of probabilities of elite samples.",
"To this end, we update the parameter as follows.",
"$\\theta _{k+1} ^\\ast \\\\=\\arg \\max _{\\theta \\in \\Theta }\\mathbb {E}_{\\theta _k}\\left[\\frac{S(J(x_0,w))^k}{p(w,\\theta _k)}I_{J(x_0, w)\\le \\gamma }\\ln p(w,\\theta )\\right]$ where $\\mathbb {E}_{\\theta }(\\nu )$ is the expected value of a random variable $\\nu $ given distribution $p(\\cdot , \\theta )$ , $S: \\mathbf {R}\\rightarrow \\mathbf {R}^+$ is a strictly decreasing and positive function Possible choices can be $S(x) = \\exp (-x)$ or $S(x)=\\frac{1}{x}$ if $x$ is strictly positive.. $S(J(x_0; w))^k/p(w,\\theta _k)$ is the weight for parameter $w$ .",
"Assumption 3 The optimal parameter $\\theta ^\\ast $ is the interior point of $\\Theta $ for all $k$ .",
"Lemma 1 (based on Theorem 1 [9]) Assuming REF ,REF , and REF and the compactness of $W$ , with probability one, $\\lim _{k\\rightarrow \\infty }\\mu _k = w^\\ast , \\text{ and }\\lim _{k\\rightarrow \\infty }\\Sigma _k = 0_{N\\times N}.$ where $w^\\ast $ is the optimal weight vector and $0_{N \\times N}$ is an $N$ -by-$N$ zero matrix.",
"Note that since $\\Sigma _k$ converges in the limit a zero matrix, the stopping criterion is justified.",
"Building on the convergence result of mras, the proposed sampling-based planner ensures a convergence to a Dirac Delta function concentrating on the optimum.",
"In practice, the parameter update is performed using the expectation—maximization (EM) algorithm.",
"EM-based parameter update/policy improvement Since our choice of probability distribution is the multivariate Gaussian, the parameter $\\theta ^\\ast _{k+1} =(\\mathbf {\\mu },\\Sigma )$ is computed as follows $\\mathbf {\\mu }= \\frac{\\mathbb {E}_{\\theta _k} [ S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E} w}{\\mathbb {E}_{\\theta _k} [S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E}} \\\\\\approx \\frac{\\sum _{w\\in W_k}[S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E} w}{ \\sum _{w\\in W_k} [S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E}},$ and $\\Sigma = \\frac{\\mathbb {E}_{\\theta _k} [S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E} (w - \\mathbf {\\mu })(w - \\mathbf {\\mu })^\\intercal }{\\mathbb {E}_{\\theta _k}[S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E}}\\\\ \\approx \\frac{\\sum _{w\\in W_k} [S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E} (w -\\mathbf {\\mu })(w - \\mathbf {\\mu })^\\intercal }{ \\sum _{w\\in W_k}[S(J(x_0,w))^k/p(w,\\theta _k)]I_{w\\in E}},$ where we approximate $\\mathbb {E}_{\\theta _k}(h(\\mathbf {w})) $ with its estimate $ \\frac{1}{N_k} \\sum _{w\\in W_k}h(w)$ for $\\mathbf {w}\\sim p(\\cdot , \\theta _k)$ and the fraction $\\frac{1}{N_k}$ was canceled as the term is shared by the numerator and the denominator.",
"Smoothing: Due to limited sample size, a greedy maximization for parameter update can be premature if too few samples are used.",
"To ensure the convergence to the global optimal solution, a smoothing update is needed.",
"To this end, we select the parameter for the next iteration to be $ \\theta _{k+1} \\leftarrow \\lambda \\theta _k +(1-\\lambda )\\theta ^\\ast _{k+1}.$ where $\\lambda \\in [0,1)$ is the smoothing parameter.",
"Let $k=k+1$ .",
"We check if the iteration can be terminated based on a given stopping criterion.",
"If the stopping criterion is met, then we output the latest $\\theta _{k}$ .",
"Otherwise, we continue to update of $\\theta $ by moving to step 2).",
"Stopping criterion Given the probability distribution will converge to a degenerated one that concentrates on the optimal weight vector.",
"We stop the iteration if the covariance matrix $\\Sigma _k$ becomes near-singular given the convergence condition in Lemma REF .",
"To conclude, the proposed algorithm using mras is probabilistic complete and converges to the global optimal solution.",
"If the assumptions are not met, the algorithm converges to a local optimum.",
"Being able to directly search within continuous control policy space, one major advantage is that one can enforce stability condition such that the search is restricted to stable and robust policy space.",
"In this subsection, we consider contraction theory to compute conditions that need to be satisfied by weight vectors to ensure stability and robustness under bounded disturbances.",
"Definition 2 [15] Given the system equation for the closed-loop system $\\dot{x} =f(x,t)$ , a region of the state space is called a contraction region if the Jacobian $\\frac{\\partial f}{\\partial x}$ is uniformly negative definite in that region, that is, $\\exists \\beta >0, \\forall x, \\forall t>0, \\frac{1}{2}\\left(\\frac{\\partial f}{\\partial x}M+ \\dot{M}+ M \\frac{\\partial f}{\\partial x}^\\intercal \\right) \\preceq - \\beta M.$ where $M(t)$ is a positive definite matrix for all $t\\ge 0$ .",
"Theorem 1 [15] Given the system model $\\dot{x} = f(x, t)$ , any trajectory, which starts in a ball of constant radius with respect to the matrix $M$ , centered about a given trajectory and contained at all times in a contraction region with respect to the matrix $M$ , remains in that ball and converges exponentially to this trajectory.",
"Furthermore, global exponential convergence to the given trajectory is guaranteed if the whole state space is a contraction region.",
"Theorem REF provides a necessary and sufficient condition for exponential convergence of an autonomous system.",
"Under bounded disturbances, the key idea is to incorporate a contraction analysis in the planning algorithm such that it searches for a weight vector $w$ that is not only optimal in the nominal system but also ensures that the closed-loop actual system under the controller $u = w^\\intercal \\phi $ has contraction dynamics within a tube around the nominal trajectory.",
"Using a similar proof in [16], we can show that for systems with contracting dynamics, the actual trajectory under bounded disturbances will be ultimately uniformly bounded along the nominal trajectory.",
"Lemma 2 Consider a closed-loop system $\\dot{x} =f(x)+\\omega (t) $ where $\\omega (t)$ is a disturbance with $\\max _{t } \\Vert \\omega (t) \\Vert \\le \\rho _{\\max } $ , let a state trajectory $x(t)$ be in the contraction region $X_\\ell $ at all time $t\\ge t_0$ , then for any time $t\\ge t_0$ , the deviation between $x(t)$ and the nominal trajectory $\\bar{x}(t)$ , whose dynamic model is given by $\\dot{\\bar{x}} = f(\\bar{x})$ , satisfies $\\Vert x(t) - \\bar{x}(t) \\Vert _M^2 \\le \\frac{2\\ell \\rho _{\\max }}{\\beta }(1- e^{\\beta t}))$ In other words, the error is uniformly ultimately bounded with the ultimate bound $ \\frac{2\\ell \\rho _{\\max }}{\\beta }$ .",
"Let's pick the Lyapunov function $V= (x-\\bar{x})^T M(x- \\bar{x}),$ whose time derivative is $\\dot{V}& = (x -\\bar{x})^T M (f(x)+\\omega - f(\\bar{x})) \\\\& + (f(x)+\\omega - f(\\bar{x}))^T M(x- \\bar{x}) \\\\& = (x-\\bar{x})^T M(f(x)-f(\\bar{x})) +2 (x-\\bar{x})^TM \\omega \\\\& (M \\mbox{ is symmetric})\\\\& = (x - \\bar{x})^T ( \\frac{\\partial f}{\\partial x}^T\\mid _{\\tilde{x}} M+M\\frac{\\partial f}{\\partial x} \\mid _{\\tilde{x}} ) (x -\\bar{x}) \\\\& +2 (x-\\bar{x})^TM \\omega ,$ where the following property is used: $f(x)- f(\\bar{x}) =\\frac{\\partial f}{\\partial x}^T \\mid _{\\tilde{x}} (x-\\bar{x})$ for some $\\tilde{x} \\in [\\bar{x}, x]$ if $\\bar{x} \\preccurlyeq x$ or $\\tilde{x} \\in [x,\\bar{x}]$ otherwise.",
"Since the trajectories stays within the contraction region, the following condition holds $\\frac{\\partial f}{\\partial x}^T\\mid _{\\tilde{x}} M+M \\frac{\\partial f}{\\partial x} \\mid _{\\tilde{x}} \\le - 2\\beta M$ , and we have $\\dot{V} \\le -(x-\\bar{x})^T \\beta M (x-\\bar{x})+2(x-\\bar{x})^T M\\omega .$ Meanwhile, $\\Vert x(t) - \\bar{x}(t) \\Vert _M \\le \\ell $ as the trajectory $x(t)$ stays within the region of contraction, and also $M (x- \\bar{x}) \\le \\sqrt{(x-\\bar{x})^T M M (x-\\bar{x})} =\\sqrt{\\Vert x-\\bar{x} \\Vert _M}$ we conclude that as $\\omega \\le \\rho _{\\max }$ , $\\dot{V} \\le -(x-\\bar{x})^T (\\beta M) (x-\\bar{x}) + 2\\omega \\sqrt{\\Vert x - \\bar{x} \\Vert _M}\\\\ \\le -(x-\\bar{x})^T (\\beta M) (x-\\bar{x}) + 2\\rho _{\\max } \\ell ,$ Since $\\dot{V} \\le -\\beta V + 2\\rho _{\\max } \\ell $ and under the condition that $x(0 )= \\bar{x}(0)$ , we obtain $V(t) \\le \\frac{2\\ell }{\\beta } \\rho _{\\max }(1- e^{-\\beta t})$ , and therefore $\\Vert x-\\bar{x} \\Vert _M^2= \\frac{2\\ell }{\\beta } \\rho _{\\max }(1- e^{-\\beta t})$ Thus, to search for the optimal and robust policies, we modify the algorithm by introducing the following step.",
"Contraction verification step: Suppose the closed-loop system is subject to bounded disturbances, the objective is to ensure the trajectory is contracting within the time-varying tube $\\lbrace x \\mid \\Vert x-\\bar{x} \\Vert \\le \\ell \\rbrace $ , for all $t$ , where $\\bar{x}$ is the nominal state trajectory.",
"The following condition translates the contraction condition into verifiable condition for a closed-loop system: Choose positive constants $\\beta $ , a positive definite symmetric and constant matrix $M = [m_{ij}]_{i=1,\\ldots , n, j=1,\\ldots , n}$ , and verify whether, at each time step along the nominal trajectory $ \\bar{x}(t)$ in the closed-loop system under control $ u(t)=w^\\intercal \\phi ( x(t))$ , the following condition holds.",
"$\\begin{split}\\max _{x : \\Vert x- \\bar{x} \\Vert _M \\le \\ell } g_{ij}(x) \\le -\\beta m_{ij}, \\quad \\forall i=1,\\ldots , n,\\;\\forall j=1,\\ldots , n\\end{split}$ where $g_{ij}$ is the $(i,j)$ th component in the matrix $\\frac{\\partial f}{\\partial x}^\\intercal M+M \\frac{\\partial f}{\\partial x} $ .",
"We verify this condition numerically at discrete time steps instead of continous time.",
"Further, if the function $g_{ij}(x)$ is semi-continuous, according to the Extreme Value Theorem, this condition can be verified by evaluating $g_{ij}(x)$ at all critical points where $\\frac{dg_{ij} (x)}{dx}=0$ and the boundary of the set $\\lbrace x\\mid \\Vert x-\\bar{x} \\Vert _M \\le \\ell \\rbrace $ .",
"The modification to the planning algorithm is made in Step 3), if a controller $u = \\langle w, \\phi \\rangle $ of elite sample $w$ does not meet the condition, then $w$ is rejected from the set of elite samples.",
"Alternatively, one can do so implicitly by associating $w$ with a very large cost.",
"However, since the condition is sufficient but not necessary as we have the matrix $M$ , constant $\\beta $ and $\\ell $ pre-fixed and $M$ is chosen to be a constant matrix, the obtained robust controller may not necessary be optimal among all robust controllers in $\\Pi _\\phi $ .",
"A topic for future work is to extend joint planning and control policies with respect to adaptive bound $\\beta $ , $\\ell $ , and a uniformly positive definite and time-varying matrix $M(x,t)$ .",
"In this section, we use two examples to illustrate the correctness and efficiency of the proposed method.",
"The simulation experiments are implemented in MATLAB on a desktop with Intel Xeon E5 CPU and 16 GB of RAM.",
"To illustrate the correctness and sampling efficiency in the planning algorithm, we consider an optimal control of lti systems with non-quadratic cost.",
"For this class of optimal control problems, since there is no admissible heuristic, one cannot use any planning algorithm facilitated by the usage of a heuristic function.",
"Moreover, the optimal controller is nonlinear given the non-quadratic cost.",
"Consider a lti system $\\dot{x} = Ax + Bu$ where $A = \\begin{bmatrix} -1 & 1 \\\\ 0& 0\\end{bmatrix}$ and $B=\\begin{bmatrix}0\\\\1\\end{bmatrix}$ with $x\\in X =\\mathbf {R}^2$ and $u\\in \\mathbf {R}^1$ .",
"The initial state is $x_0 = [5,5]$ .",
"The cost functional is $J(x_0,u) =\\int _0^T( \\Vert x \\Vert ^2+ \\Vert u \\Vert ^2+ 0.5 \\Vert x \\Vert ^4+ 0.8\\Vert x \\Vert ^6 )dt+ \\Vert x(T) \\Vert ^2.",
"$ For a non-quadratic cost functional, the optimal controller is no longer linear and cannot be computed by LQR unless the running cost can be written in the sum-of-square form.",
"Thus, we consider an approximate feedback controller with basis vector $\\phi =[x_1,x_2,x_1^2,x_2^2, x_1^3,x_2^3]^\\intercal $ .",
"Suppose the magnitude of external disturbance is bounded by $\\rho _{\\max }= 0.5$ .",
"The following parameters are used in stability verification: $\\beta = 2$ , at any time $t$ , for all $x$ such that $\\Vert x(t)- \\bar{x}(t) \\Vert \\le \\ell $ , the controller ensures $\\Vert x(t) - \\bar{x}(t) \\Vert \\le \\frac{2\\ell \\rho _{\\max }}{\\beta }(1-e^{\\beta t})$ because $2\\frac{\\ell \\rho _{\\max }}{\\beta } = 0.5 \\ell \\le \\ell $ .",
"With this choice for stability analysis, the constraint $\\frac{\\partial f}{\\partial x} + \\frac{\\partial f}{\\partial x}^\\intercal & = \\begin{bmatrix}-2 & 3 w(5) x_1^2 + 2 w(3)x_1+ w(1 ) + 1\\\\\\mbox{Sym.}",
"& 6 w(6)x_2^2 + 4 w(4) x_2 + 2w(2) \\end{bmatrix}\\\\& \\le \\begin{bmatrix}-2 &0 \\\\0& -2\\\\\\end{bmatrix}$ In this case, if we select $w(6), w(4), w(5), w(3) $ nonpositive, $w(1)\\le -1$ and $w(2)\\le -1$ , then closed-loop system, which is a nonlinear polynomial system, will become globally contracting.",
"Figure: Convergence of the saop algorithm on the lti systemwith a nonquadratic cost functional.",
"(a) The mean of multivariate Gaussian as weight vectorover iterations.",
"(b) The state trajectory of the closed-loop systemunder bounded disturbance ρ max =0.5\\rho _{\\max }= 0.5 under feedbackcontroller computed with saop.Figures REF and REF show the convergence result with saop in one simulation in terms of cost and the mean of the multivariant Gaussian over iterations.",
"The following parameters are used: Initial sample size $N_1 =50$ , improvement parameter $\\epsilon =0.1$ , quantile percentage $\\rho =0.1$ , smoothing parameter $\\lambda = 0.5$ , sample increment parameter $\\alpha = 0.1$ .",
"The algorithm converges after 38 iterations with 3301 samples to the mean $\\bar{w}^\\ast = [ -1.0629 \\; -2.7517 \\; 0\\; -1.7939 \\; -0.0987\\;-2.1474 ]^\\intercal $ and the covariance matrix with a norm ${3.3401e-04}$ .",
"Each iteration took less than 10 seconds.",
"The approximate optimal cost under feedback controller $u =\\langle \\bar{w}^\\ast , \\phi \\rangle $ is $ 3863.3$ .",
"Figure REF shows the state trajectory for the closed-loop system with bounded disturbances.",
"With 25 independent runs of saop, the mean of $J(x_0; \\bar{w}_i^\\ast ), i=1,\\ldots , 25$ is $3903.3$ and the standard deviation is $ 104.1683 $ , $2.6\\% $ of the approximate optimal cost.",
"Note, if we only use linear feedback $u= Kx$ , the optimal cost is ${ 1.0943e+04} $ , which is about three times the optimal cost that can be achieved with a nonlinear controller.",
"Consider a Dubins car dynamics $\\dot{x} = u\\cos \\theta ,\\quad \\dot{y} = u \\sin \\theta \\; \\quad \\dot{\\theta }= v$ where $\\vec{x}= (x,y,\\theta )\\in \\mathbf {R}^2\\times \\mathbb {S}^1$ being the state (coordinates and turning angle with respect to $x$ -axis) and $u$ and $v$ are control variables including linear and angular velocities.",
"The system is kinematically constrained by its positive minimum turning radius $r$ which implies the following bound $\\vert v\\vert \\le \\frac{1}{r} $ .",
"Without loss of generality, we assume $\\vert v\\vert \\le 5$ and $\\vert u\\vert \\le 10$ are the input constraints.",
"The control objective is to reach the goal $x_f=20, y_f=20$ while avoiding static obstacles.",
"The cost function $J= \\int _{0}^T \\ell (x,u)dt + g(x,u)$ where $T=100$ , the running cost is $\\ell (x,u) = 0.1\\times (\\Vert x \\Vert + \\Vert u \\Vert )$ , and the terminal cost is $g(x(T),u(T)) =1000\\times \\Vert (x(T),y(T))-(x_f,y_f) \\Vert $ .",
"The initial state is $\\vec{x}_0 = \\mathbf {0}$ .",
"In simulation, we consider the robot reaches the target if $\\Vert (x,y)^{\\prime }-(x_f,y_f) \\Vert \\le \\varepsilon $ for $\\varepsilon \\in [0, 1]$ .",
"In simulation, $\\varepsilon =0.5$ .",
"We select rbf as basis functions and define $\\phi _{rbf} = [\\phi _1,\\ldots , \\phi _N]^\\intercal $ for $N$ center points.",
"In the experiment, the center points are includes [1)] uniform grids in $x-y$ coordinates with step sizes $\\delta x = 5$ , $\\delta y =5$ ; and vertices of the obstacle.",
"We also include linear basis functions $\\phi _{linear} = [(x-x_f), (y-y_f),\\theta ]$ .",
"The basis vector is $\\phi = [\\phi _{rbf}^\\intercal , \\phi _{linear}^\\intercal ]^\\intercal $ .",
"We consider a bounded domain $-5\\le x \\le 30$ and $-5\\le y\\le 30$ and $\\theta \\in [0, 2\\pi ]$ and thus the total number of basis functions is 80.",
"The control input $\\vec{u}= [u,v]^\\intercal $ where $u = w_u^\\intercal \\phi $ and $v =w_v^\\intercal \\phi $ .",
"The total number of weight parameters is twice the number of bases and in this case 160.",
"Figure: Convergence of the the planning algorithm algorithm on the Dubinscar.",
"(a) The planned trajectory under feedback policy〈μ,φ〉\\langle \\mu , \\phi \\rangle computed using the mean of multivariateGaussian over iterations (from the lightest to the darkest).",
"(b) The convergence of the covariancematrix.",
"(c) The total cost evaluated at the mean of the multivariateGaussian over iterations.The following parameters are used: Initial sample size $N_1= 100$ , improvement parameter $\\epsilon = 0.1 $ , smoothing parameter $\\lambda = 0.5$ , sample increment percentage $\\alpha =0.1$ , and $\\rho = 0.1$ .",
"In Fig.",
"REF we show the trajectory computed using the estimated mean of multivariate Gaussian distribution over iterations, from the lightest (1-th iteration) to the darkest (the last iteration when stopping criterion is met).",
"The optimal trajectory is the darkest line.",
"In Fig.",
"REF we show the cost computed using the mean of multivariate Gaussian over iterations.",
"saop converges after 22 iterations with 2200 samples and the optimal cost is $697.29$ .",
"Each iteration took about 20 to 30 seconds.",
"However, it generates a collision-free path only after 5 iterations.",
"Due to input saturation, the algorithm is only ensured to converge to a local optimum.",
"However, in 24 independent runs, all runs converges to a local optimum closer to the global one, as shown in the histogram in Fig.",
"REF .",
"Our current work is to implement trajectory-based contraction analysis using time-varying matrices $M(x,t)$ and adaptive bound $\\beta $ , which are needed for nonlinear Dubins car dynamics.",
"Figure: The frequency distribution of the optimal costs with 24independent runs.In this paper, an importance sampling-based approximate optimal planning and control method is developed.",
"In the control-theoretic formulation of optimal motion planning, the planning algorithm performs direct policy computation using simulation-based adaptive search for an optimal weight vector corresponding to an approximate optimal feedback policy.",
"Each iteration of the algorithm runs time linear in the number of samples and in the time horizon for simulated runs.",
"However, it is hard to quantify the number of iterations required for mras to converge.",
"One future work is to consider incorporate multiple-distribution importance sampling to achieve faster and better convergence results.",
"Based on contraction analysis of the closed-loop system, we show that by modifying the sampling-based policy evaluation step in the algorithm, the proposed planning algorithm can be used for joint planning and robust control for a class of nonlinear systems under bounded disturbances.",
"In future extension of this work, we are interested in extending this algorithm for stochastic optimal control."
]
] | 1612.05594 |
[
[
"Unsupervised Pixel-Level Domain Adaptation with Generative Adversarial\n Networks"
],
[
"Abstract Collecting well-annotated image datasets to train modern machine learning algorithms is prohibitively expensive for many tasks.",
"One appealing alternative is rendering synthetic data where ground-truth annotations are generated automatically.",
"Unfortunately, models trained purely on rendered images often fail to generalize to real images.",
"To address this shortcoming, prior work introduced unsupervised domain adaptation algorithms that attempt to map representations between the two domains or learn to extract features that are domain-invariant.",
"In this work, we present a new approach that learns, in an unsupervised manner, a transformation in the pixel space from one domain to the other.",
"Our generative adversarial network (GAN)-based method adapts source-domain images to appear as if drawn from the target domain.",
"Our approach not only produces plausible samples, but also outperforms the state-of-the-art on a number of unsupervised domain adaptation scenarios by large margins.",
"Finally, we demonstrate that the adaptation process generalizes to object classes unseen during training."
],
[
"Introduction",
"Large and well–annotated datasets such as ImageNet [9], COCO [29] and Pascal VOC [12] are considered crucial to advancing computer vision research.",
"However, creating such datasets is prohibitively expensive.",
"One alternative is the use of synthetic data for model training.",
"It has been a long-standing goal in computer vision to use game engines or renderers to produce virtually unlimited quantities of labeled data.",
"Indeed, certain areas of research, such as deep reinforcement learning for robotics tasks, effectively require that models be trained in synthetic domains as training in real–world environments can be excessively expensive [38], [43].",
"Consequently, there has been a renewed interest in training models in the synthetic domain and applying them in real–world settings [8], [48], [38], [43], [25], [32], [35], [37].",
"Unfortunately, models naively trained on synthetic data do not typically generalize to real images.",
"A solution to this problem is using unsupervised domain adaptation.",
"In this setting, we would like to transfer knowledge learned from a source domain, for which we have labeled data, to a target domain for which we have no labels.",
"Previous work either attempts to find a mapping from representations of the source domain to those of the target [41], or seeks to find domain-invariant representations that are shared between the two domains [14], [44], [31], [5].",
"While such approaches have shown good progress, they are still not on par with purely supervised approaches trained only on the target domain.",
"In this work, we train a model to change images from the source domain to appear as if they were sampled from the target domain while maintaining their original content.",
"We propose a novel Generative Adversarial Network (GAN)–based architecture that is able to learn such a transformation in an unsupervised manner, i.e.",
"without using corresponding pairs from the two domains.",
"Our unsupervised pixel-level domain adaptation method (PixelDA) offers a number of advantages over existing approaches:"
],
[
"Decoupling from the Task-Specific Architecture:",
"In most domain adaptation approaches, the process of domain adaptation and the task-specific architecture used for inference are tightly integrated.",
"One cannot switch a task–specific component of the model without having to re-train the entire domain adaptation process.",
"In contrast, because our PixelDA model maps one image to another at the pixel level, we can alter the task-specific architecture without having to re-train the domain adaptation component."
],
[
"Generalization Across Label Spaces:",
"Because previous models couple domain adaptation with a specific task, the label spaces in the source and target domain are constrained to match.",
"In contrast, our PixelDA model is able to handle cases where the target label space at test time differs from the label space at training time."
],
[
"Training Stability:",
"Domain adaptation approaches that rely on some form of adversarial training [5], [14] are sensitive to random initialization.",
"To address this, we incorporate a task–specific loss trained on both source and generated images and a pixel similarity regularization that allows us to avoid mode collapse [40] and stabilize training.",
"By using these tools, we are able to reduce variance of performance for the same hyperparameters across different random initializations of our model (see sec:experiments)."
],
[
"Data Augmentation:",
"Conventional domain adaptation approaches are limited to learning from a finite set of source and target data.",
"However, by conditioning on both source images and a stochastic noise vector, our model can be used to create virtually unlimited stochastic samples that appear similar to images from the target domain."
],
[
"Interpretability:",
"The output of PixelDA, a domain–adapted image, is much more easily interpreted than a domain adapted feature vector.",
"To demonstrate the efficacy of our strategy, we focus on the tasks of object classification and pose estimation, where the object of interest is in the foreground of a given image, for both source and target domains.",
"Our method outperforms the state-of-the-art unsupervised domain adaptation techniques on a range of datasets for object classification and pose estimation, while generating images that look very similar to the target domain (see fig:teaser)."
],
[
"Related Work",
"Learning to perform unsupervised domain adaptation is an open theoretical and practical problem.",
"While much prior work exists, our literature review focuses primarily on Convolutional Neural Network (CNN) methods due to their empirical superiority on the problem [14], [31], [41], [45]."
],
[
"Unsupervised Domain Adaptation:",
"Ganin [13], [14] and Ajakan [3] introduced the Domain–Adversarial Neural Network (DANN): an architecture trained to extract domain-invariant features.",
"Their model's first few layers are shared by two classifiers: the first predicts task-specific class labels when provided with source data while the second is trained to predict the domain of its inputs.",
"DANNs minimize the domain classification loss with respect to parameters specific to the domain classifier, while maximizing it with respect to the parameters that are common to both classifiers.",
"This minimax optimization becomes possible in a single step via the use of a gradient reversal layer.",
"While DANN's approach to domain adaptation is to make the features extracted from both domains similar, our approach is to adapt the source images to look as if they were drawn from the target domain.",
"Tzeng [45] and Long [31] proposed versions of DANNs where the maximization of the domain classification loss is replaced by the minimization of the Maximum Mean Discrepancy (MMD) metric [20], computed between features extracted from sets of samples from each domain.",
"Ghifary propose an alternative model in which the task loss for the source domain is combined with a reconstruction loss for the target domain, which results in learning domain-invariant features.",
"Bousmalis [5] introduce a model that explicitly separates the components that are private to each domain from those that are common to both domains.",
"They make use of a reconstruction loss for each domain, a similarity loss (eg.",
"DANN, MMD) which encourages domain invariance, and a difference loss which encourages the common and private representation components to be complementary.",
"Other related techniques involve learning a mapping from one domain to the other at a feature level.",
"In such a setup, the feature extraction pipeline is fixed during the domain adaptation optimization.",
"This has been applied in various non-CNN based approaches [17], [6], [19] as well as the more recent CNN-based Correlation Alignment (CORAL) [41] algorithm."
],
[
"Generative Adversarial Networks:",
"Our model uses GANs [18] conditioned on source images and noise vectors.",
"Other recent works have also attempted to use GANs conditioned on images.",
"Ledig [28] used an image-conditioned GAN for super-resolution.",
"Yoo [47] introduce the task of generating images of clothes from images of models wearing them, by training on corresponding pairs of the clothes worn by models and on a hanger.",
"In contrast to our work, neither method conditions on both images and noise vectors, and ours is also applied to an entirely different problem space.",
"The work perhaps most similar to ours is that of Liu and Tuzel [30] who introduce an architecture of a pair of coupled GANs, one for the source and one for the target domain, whose generators share their high-layer weights and whose discriminators share their low-layer weights.",
"In this manner, they are able to generate corresponding pairs of images which can be used for unsupervised domain adaptation.",
"on the ability to generate high quality samples from noise alone."
],
[
"Style Transfer:",
"The popular work of Gatys [15], [16] introduced a method of style transfer, in which the style of one image is transferred to another while holding the content fixed.",
"The process requires backpropagating back to the pixels.",
"Johnson [24] introduce a model for feed forward style transfer.",
"They train a network conditioned on an image to produce an output image whose activations on a pre-trained model are similar to both the input image (high-level content activations) and a single target image (low-level style activations).",
"However, both of these approaches are optimized to replicate the style of a single image as opposed to our work which seeks to replicate the style of an entire domain of images."
],
[
"Model",
"We begin by explaining our model for unsupervised pixel-level domain adaptation (PixelDA) in the context of image classification, though our method is not specific to this particular task.",
"Given a labeled dataset in a source domain and an unlabeled dataset in a target domain, our goal is to train a classifier on data from the source domain that generalizes to the target domain.",
"Previous work performs this task using a single network that performs both domain adaptation and image classification, making the domain adaptation process specific to the classifier architecture.",
"Our model decouples the process of domain adaptation from the process of task-specific classification, as its primary function is to adapt images from the source domain to make them appear as if they were sampled from the target domain.",
"Once adapted, any off-the-shelf classifier can be trained to perform the task at hand as if no domain adaptation were required.",
"Note that we assume that the differences between the domains are primarily low-level (due to noise, resolution, illumination, color) rather than high-level (types of objects, geometric variations, etc).",
"More formally, let ${\\mathbf {X}}^s = \\lbrace {\\mathbf {x}}_i^s, {\\mathbf {y}}_i^s\\rbrace _{i=0}^{N^s}$ represent a labeled dataset of $N^s$ samples from the source domain and let ${\\mathbf {X}}^t = \\lbrace {\\mathbf {x}}_i^t\\rbrace _{i=0}^{N^t}$ represent an unlabeled dataset of $N^t$ samples from the target domain.",
"Our pixel adaptation model consists of a generator function $G({\\mathbf {x}}^s, {\\mathbf {z}}; {\\mathbf {\\theta }}_G) \\rightarrow {\\mathbf {x}^f}$ , parameterized by ${\\mathbf {\\theta }}_G$ , that maps a source domain image ${\\mathbf {x}}^s \\in {\\mathbf {X}}^s$ and a noise vector ${\\mathbf {z}} \\sim p_z$ to an adapted, or fake, image ${\\mathbf {x}^f}$ .",
"Given the generator function $G$ , it is possible to create a new dataset ${\\mathbf {X}}^f = \\left\\lbrace G({\\mathbf {x}}^s, {\\mathbf {z}}), {\\mathbf {y}}^s\\right\\rbrace $ of any size.",
"Finally, given an adapted dataset ${\\mathbf {X}}^f$ , the task-specific classifier can be trained as if the training and test data were from the same distribution."
],
[
"Learning",
"To train our model, we employ a generative adversarial objective to encourage $G$ to produce images that are similar to the target domain images.",
"During training, our generator $G({\\mathbf {x}}^s, {\\mathbf {z}}; {\\mathbf {\\theta }}_G) \\rightarrow {\\mathbf {x}^f}$ maps a source image ${\\mathbf {x}}^s$ and a noise vector ${\\mathbf {z}}$ to an adapted image ${\\mathbf {x}^f}$ .",
"Furthermore, the model is augmented by a discriminator function $D({\\mathbf {x}};{\\mathbf {\\theta }}_D)$ that outputs the likelihood $d$ that a given image $\\mathbf {x}$ has been sampled from the target domain.",
"The discriminator tries to distinguish between `fake' images ${\\mathbf {X}}^f$ produced by the generator, and `real' images from the target domain ${\\mathbf {X}}^t$ .",
"Note that in contrast to the standard GAN formulation [18] in which the generator is conditioned only on a noise vector, our model's generator is conditioned on both a noise vector and an image from the source domain.",
"In addition to the discriminator, the model is also augmented with a classifier $T({\\mathbf {x}};{\\mathbf {\\theta }}_T) \\rightarrow {\\hat{\\mathbf {y}}}$ which assigns task-specific labels ${\\hat{\\mathbf {y}}}$ to images ${\\mathbf {x}} \\in \\lbrace {\\mathbf {X}}^f, {\\mathbf {X}}^t\\rbrace $ .",
"Our goal is to optimize the following minimax objective: $\\min _{{\\mathbf {\\theta }}_G, {\\mathbf {\\theta }}_T} \\max _{{\\mathbf {\\theta }}_D} \\; \\alpha \\, {\\cal L}_{d}(D,G) + \\beta {\\cal L}_{t}(G, T)$ where $\\alpha $ and $\\beta $ are weights that control the interaction of the losses.",
"${\\cal L}_{d}$ represents the domain loss: ${\\cal L}_{d}(D, G) = & \\; \\mathbb {E}_{{\\mathbf {x}}^t} [\\log D({\\mathbf {x}}^t; {\\mathbf {\\theta }}_D)] + \\nonumber \\\\ & \\; \\mathbb {E}_{{\\mathbf {x}}^s, {\\mathbf {z}} }[\\log (1 - D(G({\\mathbf {x}^s}, {\\mathbf {z}}; {\\mathbf {\\theta }}_G); {\\mathbf {\\theta }}_D))]$ ${\\cal L}_{t}$ is a task-specific loss, and in the case of classification we use a typical softmax cross–entropy loss: ${\\cal L}_{t}(G, T) \\; = \\; \\mathbb {E}_{{\\mathbf {x}}^s,{\\mathbf {y}}^s, {\\mathbf {z}}}\\big [&-\\mathbf {y^s}^\\top \\log T\\left(G({\\mathbf {x}^s},{\\mathbf {z}}; {\\mathbf {\\theta }}_G); {\\mathbf {\\theta }}_T\\right) \\nonumber \\\\&-\\mathbf {y^s}^\\top \\log {T({\\mathbf {x}}^s); {\\mathbf {\\theta }}_T}\\big ]$ where $\\mathbf {y}^s$ is the one-hot encoding of the class label for source input ${\\mathbf {x}}^s$ .",
"Notice that we train $T$ with both adapted and non-adapted source images.",
"When training $T$ only on adapted images, it's possible to achieve similar performance, but doing so may require many runs with different initializations due to the instability of the model.",
"Indeed, without training on source as well, the model is free to shift class assignments (e.g.",
"class 1 becomes 2, class 2 becomes 3 etc) while still being successful at optimizing the training objective.",
"We have found that training classifier $T$ on both source and adapted images avoids this scenario and greatly stabilizes training (See Table REF ).",
"might use a different label space (See Table REF ).",
"In our implementation, $G$ is a convolutional neural network with residual connections that maintains the resolution of the original image as illustrated in figure REF .",
"Our discriminator $D$ is also a convolutional neural network.",
"The minimax optimization of eq:objective1 is achieved by alternating between two steps.",
"During the first step, we update the discriminator and task-specific parameters ${\\mathbf {\\theta }}_D, {\\mathbf {\\theta }}_T$ , while keeping the generator parameters ${\\mathbf {\\theta }}_G$ fixed.",
"During the second step we fix ${\\mathbf {\\theta }}_D, {\\mathbf {\\theta }}_T$ and update ${\\mathbf {\\theta }}_G$ ."
],
[
"Content–similarity loss",
"In certain cases, we have prior knowledge regarding the low-level image adaptation process.",
"For example, we may expect the hues of the source and adapted images to be the same.",
"In our case, for some of our experiments, we render single objects on black backgrounds and consequently we expect images adapted from these renderings to have similar foregrounds and different backgrounds from the equivalent source images.",
"Renderers typically provide access to z-buffer masks that allow us to differentiate between foreground and background pixels.",
"This prior knowledge can be formalized via the use of an additional loss that penalizes large differences between source and generated images for foreground pixels only.",
"Such a similarity loss grounds the generation process to the original image and helps stabilize the minimax optimization, as shown in Sect.",
"REF and Table REF .",
"Our optimization objective then becomes: $\\min _{{\\mathbf {\\theta }}_G, {\\mathbf {\\theta }}_T} \\max _{{\\mathbf {\\theta }}_D} \\; \\alpha {\\cal L}_{d}(D,G) + \\beta {\\cal L}_{t}(T,G) + \\gamma {\\cal L}_{c}(G)$ where $\\alpha $ , $\\beta $ , and $\\gamma $ are weights that control the interaction of the losses, and ${\\cal L}_c$ is the content–similarity loss.",
"A number of losses could anchor the generated image to the original image in some meaningful way (e.g.",
"L1, or L2 loss, similarity in terms of the activations of a pretrained VGG network).",
"In our experiments for learning object instance classification from rendered images, we use a masked pairwise mean squared error, which is a variation of the pairwise mean squared error (PMSE) [11].",
"This loss penalizes differences between pairs of pixels rather than absolute differences between inputs and outputs.",
"Our masked version calculates the PMSE between the generated foreground and the source foreground.",
"Formally, given a binary mask ${\\mathbf {m}} \\in {\\mathbb {R}}^k$ , our masked-PMSE loss is: $\\mathcal {L}_{c}(G) &= \\mathbb {E}_{{\\mathbf {x}}^s, {\\mathbf {z}}} \\Big [\\frac{1}{k} \\left\\Vert \\left({\\mathbf {x}}^s - G({\\mathbf {x}^s}, {\\mathbf {z}}; {\\mathbf {\\theta }}_G) \\right) \\circ {\\mathbf {m}}\\right\\Vert _2^2 \\nonumber \\\\& - \\frac{1}{k^2}\\left(({\\mathbf {x}}^s - G({\\mathbf {x}^s}, {\\mathbf {z}}; {\\mathbf {\\theta }}_G) )^\\top {\\mathbf {m}}\\right)^2 \\Big ]$ where $k$ is the number of pixels in input $\\mathbf {x}$ , $\\Vert \\cdot \\Vert _2^2$ is the squared $L_2$ -norm, and $\\circ $ is the Hadamard product.",
"This loss allows the model to learn to reproduce the overall shape of the objects being modeled without wasting modeling power on the absolute color or intensity of the inputs, while allowing our adversarial training to change the object in a consistent way.",
"Note that the loss does not hinder the foreground from changing but rather encourages the foreground to change in a consistent way.",
"In this work, we apply a masked PMSE loss for a single foreground object because of the nature of our data, but one can trivially extend this to multiple foreground objects."
],
[
"Evaluation",
"We evaluate our method on object classification datasets used in previous workThe most commonly used dataset for visual domain adaptation in the context of object classification is Office [39].",
"However, we do not use it in this work as there are significant high–level variations due to label pollution.",
"For more information, see the relevant explanation in [5]., including MNIST, MNIST-M [14], and USPS [10] as well as a variation of the LineMod dataset [22], [46], a standard for object instance recognition and 3D pose estimation, for which we have synthetic and real data.",
"Our evaluation is composed of qualitative and quantitative components, using a number of unsupervised domain adaptation scenarios.",
"The qualitative evaluation involves the examination of the ability of our method to learn the underlying pixel adaptation process from the source to the target domain by visually inspecting the generated images.",
"The quantitative evaluation involves a comparison of the performance of our model to previous work and to “Source Only” and “Target Only” baselines that do not use any domain adaptation.",
"In the first case, we train models only on the unaltered source training data and evaluate on the target test data.",
"In the “Target Only” case we train task models on the target domain training set only and evaluate on the target domain test set.",
"The unsupervised domain adaptation scenarios we consider are listed below: MNIST to USPS: Images of the 10 digits (0-9) from the MNIST [27] dataset are used as the source domain and images of the same 10 digits from the USPS [10] dataset represent the target domain.",
"To ensure a fair comparison between the “Source–Only” and domain adaptation experiments, we train our models on a subset of 50,000 images from the original 60,000 MNIST training images.",
"The remaining 10,000 images are used as validation set for the “Source–Only” experiment.",
"The standard splits for USPS are used, comprising of 6,562 training, 729 validation, and 2,007 test images.",
"MNIST to MNIST-M: MNIST [27] digits represent the source domain and MNIST-M [14] digits represent the target domain.",
"MNIST-M is a variation on MNIST proposed for unsupervised domain adaptation.",
"Its images were created by using each MNIST digit as a binary mask and inverting with it the colors of a background image.",
"The background images are random crops uniformly sampled from the Berkeley Segmentation Data Set (BSDS500) [4].",
"All our experiments follow the experimental protocol by [14].",
"We use the labels for 1,000 out of the 59,001 MNIST-M training examples to find optimal hyperparameters.",
"Synthetic Cropped LineMod to Cropped LineMod: The LineMod dataset [22] is a dataset of small objects in cluttered indoor settings imaged in a variety of poses.",
"We use a cropped version of the dataset [46], where each image is cropped with one of 11 objects in the center.",
"The 11 objects used are `ape', `benchviseblue', `can', `cat', `driller', `duck', `holepuncher', `iron', `lamp', `phone', and `cam'.",
"A second component of the dataset consists of CAD models of these same 11 objects in a large variety of poses rendered on a black background, which we refer to as Synthetic Cropped LineMod.",
"We treat Synthetic Cropped LineMod as the source dataset and the real Cropped LineMod as the target dataset.",
"We train our model on 109,208 rendered source images and 9,673 real-world target images for domain adaptation, 1,000 for validation, and a target domain test set of 2,655 for testing.",
"Using this scenario, our task involves both classification and pose estimation.",
"Consequently, our task–specific network $T({\\mathbf {x}};{\\mathbf {\\theta }}_T) \\rightarrow \\lbrace {\\hat{\\mathbf {y}}}, \\hat{\\mathbf {q}}\\rbrace $ outputs both a class $\\hat{\\mathbf {y}}$ and a 3D pose estimate in the form of a positive unit quaternion vector $\\hat{\\mathbf {q}}$ .",
"The task loss becomes: ${\\cal L}_{t}&(G, T) \\; = \\; \\nonumber \\\\&\\mathbb {E}_{{\\mathbf {x}}^s,{\\mathbf {y}}^s, {\\mathbf {z}}}\\Big [-\\mathbf {y}^{s^\\top } \\log {\\hat{\\mathbf {y}}}^s -\\mathbf {y}^{s^\\top } \\log {\\hat{\\mathbf {y}}}^f+ \\nonumber \\\\&\\quad \\xi \\log \\left(1-\\left|{\\mathbf {q}^s}^\\top {\\hat{\\mathbf {q}}}^s\\right|\\right)+\\xi \\log \\left(1-\\left|{\\mathbf {q}^s}^\\top {\\hat{\\mathbf {q}}}^f\\right|\\right)\\Big ]$ where the first and second terms are the classification loss, similar to eq:taskloss, and the third and fourth terms are the log of a 3D rotation metric for quaternions [23].",
"$\\xi $ is the weight for the pose loss, ${\\mathbf {q}}^s$ represents the ground truth 3D pose of a sample, $\\lbrace {\\hat{\\mathbf {y}}}^s, \\hat{\\mathbf {q}}^s\\rbrace =T({\\mathbf {x}}^s;{\\mathbf {\\theta }}_T)$ , ${\\lbrace {\\hat{\\mathbf {y}}}^f, \\hat{\\mathbf {q}}^f\\rbrace =T(G({\\mathbf {x}}^s,{\\mathbf {z}};{\\mathbf {\\theta }}_G);{\\mathbf {\\theta }}_T)}$ .",
"tab:poseresults reports the mean angle the object would need to be rotated (on a fixed 3D axis) to move from predicted to ground truth pose [22].",
"Figure: Visualization of our model's ability to generate samples when trainedto adapt Synth Cropped Linemod to Cropped Linemod.",
"Top Row: Source RGB andDepth image pairs from Synth Cropped LineMod 𝐱 s {\\mathbf {x}}^s; Middle Row: Thesamples adapted with our model G(𝐱 s ,𝐳)G({\\mathbf {x}}^s, {\\mathbf {z}}) with random noise 𝐳{\\mathbf {z}};Bottom Row: The nearest neighbors between the generated samples in the middlerow and images from the target training set.",
"Differences between the generated andtarget images suggest that the model is not memorizing the target dataset."
],
[
"Implementation Details",
"All the models are implemented using TensorFlowOur code is available here: https://goo.gl/fAwCPw [1] and are trained with the Adam optimizer [26].",
"We optimize the objective in eq:objective1 for “MNIST to USPS” and “MNIST to MNIST-M” scenarios and the one in eq:objective2 for the “Synthetic Cropped Linemod to Cropped Linemod” scenario.",
"We use batches of 32 samples from each domain and the input images are zero-centered and rescaled to $[-1, 1]$ .",
"In our implementation, we let $G$ take the form of a convolutional residual neural network that maintains the resolution of the original image as shown in Figure REF .",
"$\\mathbf {z}$ is a vector of $N^z$ elements, each sampled from a uniform distribution $z_i\\sim {\\cal U}(-1,1)$ .",
"It is fed to a fully connected layer which transforms it to a channel of the same resolution as that of the image channels, and is subsequently concatenated to the input as an extra channel.",
"In all our experiments we use a $\\mathbf {z}$ with $N^z=10$ .",
"The discriminator $D$ is a convolutional neural network where the number of layers depends on the image resolution: the first layer is a stride 1x1 convolution (motivated by [33]), which is followed by repeatedly stacking stride 2x2 convolutions until we reduce the resolution to less or equal to 4x4.",
"The number of filters is 64 in all layers of $G$ , and is 64 in the first layer of $D$ and repeatedly doubled in subsequent layers.",
"The output of this pyramid is fed to a fully–connected layer with a single activation for the domain classification loss.",
"Our architecture details can be found in the supplementary material.",
"For all our experiments, the CNN topologies used for the task classifier $T$ are identical to the ones used in [14], [5] to be comparable to previous work in unsupervised domain adaptation."
],
[
"Quantitative Results",
"We have not found a universally applicable way to optimize hyperparameters for unsupervised domain adaptation.",
"Consequently, we follow the experimental protocol of [5] and use a small set ($\\sim $ 1,000) of labeled target domain data as a validation set for the hyperparameters of all the methods we compare.",
"We perform all experiments using the same protocol to ensure fair and meaningful comparison.",
"The performance on this validation set can serve as an upper bound of a satisfactory validation metric for unsupervised domain adaptation.",
"As we discuss in section REF , we also evaluate our model in a semi-supervised setting with 1,000 labeled examples in the target domain, to confirm that PixelDA is still able to improve upon the naive approach of training on this small set of target labeled examples.",
"We evaluate our model using the aforementioned combinations of source and target datasets, and compare the performance of our model's task architecture $T$ to that of other state-of-the-art unsupervised domain adaptation techniques based on the same task architecture $T$ .",
"As mentioned above, in order to evaluate the efficacy of our model, we first compare with the accuracy of models trained in a “Source Only” setting for each domain adaptation scenario.",
"This setting represents a lower bound on performance.",
"Next we compare models in a “Target Only” setting for each scenario.",
"This setting represents a weak upper bound on performance—as it is conceivable that a good unsupervised domain adaptation model might improve on these results, as we do in this work for “MNIST to MNIST-M”.",
"Quantitative results of these comparisons are presented in Tables REF and REF .",
"Our method is able to not just achieve better results than previous work on the “MNIST to MNIST-M” scenario, it is also able to outperform the “Target Only” performance we are able to get with the same task classifier.",
"Furthermore, we are also able to achieve state-of-the art results for the “MNIST to USPS” scenario.",
"Finally, PixelDA is able to reduce the mean angle error for the “Synth Cropped Linemod to Cropped Linemod” scenario to more than half compared to the previous state-of-the-art."
],
[
"Qualitative Results",
" The qualitative results of our model are illustrated in figures REF , REF , and REF .",
"In figures REF and REF one can see the visualization of the generation process, as well as the nearest neighbors of our generated samples in the target domain.",
"In both scenarios, it is clear that our method is able to learn the underlying transformation process that is required to adapt the original source images to images that look like they could belong in the target domain.",
"As a reminder, the MNIST-M digits have been generated by using MNIST digits as a binary mask to invert the colors of a background image.",
"It is clear from figure REF that in the “MNIST to MNIST-M” case, our model is able to not only generate backgrounds from different noise vectors $\\mathbf {z}$ , but it is also able to learn this inversion process.",
"This is clearly evident from e.g.",
"digits $\\mathbf {3}$ and $\\mathbf {6}$ in the figure.",
"In the “Synthetic Cropped Linemod to Cropped Linemod” case, our model is able to sample, in the RGB channels, realistic backgrounds and adjust the photometric properties of the foreground object.",
"In the depth channel it is able to learn a plausible noise model.",
"Table: Mean classification accuracy and pose error for the “Synth Cropped Linemod to Cropped Linemod” scenario."
],
[
"Model Analysis",
" We present a number of additional experiments that demonstrate how the model works and to explore potential limitations of the model.",
"Table: Mean classification accuracy and pose error when varying the background of images from the source domain.",
"For these experiments we used only the RGB portions of the images, as there is no trivial or typical way with which we could have added backgrounds to depth images.",
"For comparison, we display results with black backgrounds and Imagenet backgrounds (INet), with the “Source Only” setting and with our model for the RGB-only case."
],
[
"Sensitivity to Used Backgrounds",
"In both the “MNIST to MNIST-M” and “Synthetic-Cropped LineMod to Cropped LineMod” scenarios, the source domains are images of digits or objects on black backgrounds.",
"Our quantitative evaluation (Tables REF and REF ) illustrates the ability of our model to adapt the source images to the target domain style but raises two questions: Is it important that the backgrounds of the source images are black and how successful are data-augmentation strategies that use a randomly chosen background image instead?",
"To that effect we ran additional experiments where we substituted various backgrounds in place of the default black background for the Synthetic Cropped Linemod dataset.",
"The backgrounds are randomly selected crops of images from the ImageNet dataset.",
"In these experiments we used only the RGB portion of the images —for both source and target domains— since we don't have equivalent “backgrounds” for the depth channel.",
"As demonstrated in Table REF , PixelDA is able to improve upon training `Source-only' models on source images of objects on either black or random Imagenet backgrounds."
],
[
"Generalization of the Model",
"Two additional aspects of the model are relevant to understanding its performance.",
"Firstly, is the model actually learning a successful pixel-level data adaptation process, or is it simply memorizing the target images and replacing the source images with images from the target training set?",
"Secondly, is the model able to generalize about the two domains in a fashion not limited to the classes of objects seen during training?",
"To answer the first question, we first run our generator $G$ on images from the source images to create an adapted dataset.",
"Next, for each transferred image, we perform a pixel-space $L2$ nearest neighbor lookup in the target training images to determine whether the model is simply memorizing images from the target dataset or not.",
"Illustrations are shown in figures REF and REF , where the top rows are samples from ${\\mathbf {x}}^s$ , the middle rows are generated samples $G({\\mathbf {x}}^s, {\\mathbf {z}})$ , and the bottom rows are the nearest neighbors of the generated samples in the target training set.",
"It is clear from the figures that the model is not memorizing images from the target training set.",
"Next, we evaluate our model's ability to generalize to classes unseen during training.",
"To do so, we retrain our best model using a subset of images from the source and target domains which includes only half of the object classes for the “Synthetic Cropped Linemod” to “Cropped Linemod” scenario.",
"Specifically, the objects `ape', `benchviseblue', `can', `cat', `driller', and `duck' are observed during the training procedure, and the other objects are only used during testing.",
"Once $G$ is trained, we fix its weights and pass the full training set of the source domain to generate images used for training the task-classifier $T$ .",
"We then evaluate the performance of $T$ on the entire set of unobserved objects (6,060 samples), and the test set of the target domain for all objects for direct comparison with Table REF .",
"Table: Performance of our model trained on only 6 out of 11 Linemod objects.",
"The first row, `Unseen Classes,' displays the performance on all the samples of the remaining 5 Linemod objects not seen during training.",
"The second row, `Full test set,' displays the performance on the target domain test set for all 11 objects."
],
[
"Stability Study",
"We also evaluate the importance of the different components of our model.",
"We demonstrate that while the task and content losses do not improve the overall performance of the model, they dramatically stabilize training.",
"Training instability is a common characteristic of adversarial training, necessitating various strategies to deal with model divergence and mode collapse [40].",
"We measure the standard deviation of the performance of our models by running each model 10 times with different random parameter initialization but with the same hyperparameters.",
"Table REF illustrates that the use of the task and content–similarity losses reduces the level of variability across runs.",
"Table: The effect of using the task and content losses L t L_{t}, L c L_{c} on the standard deviation (std) of the performance of our model on the “Synth Cropped Linemod to Linemod” scenario.",
"L t source L_{t}^{source} means we use source data to train TT; L t adapted L_t^{adapted} means we use generated data to train TT; L c L_c means we use our content–similarity loss.",
"A lower std on the performance metrics means that the results are more easily reproducible."
],
[
"Semi-supervised Experiments",
"Finally, we evaluate the usefulness of our model in a semi–supervised setting, in which we assume we have a small number of labeled target training examples.",
"The semi-supervised version of our model simply uses these additional training samples as extra input to classifier $T$ during training.",
"We sample 1,000 examples from the Cropped Linemod not used in any previous experiment and use them as additional training data.",
"We evaluate the semi-supervised version of our model on the test set of the Cropped Linemod target domain against the 2 following baselines: (a) training a classifier only on these 1,000 target samples without any domain adaptation, a setting we refer to as `1,000-only'; and (b) training a classifier on these 1,000 target samples and the entire Synthetic Cropped Linemod training set with no domain adaptation, a setting we refer to as `Synth+1000'.",
"As one can see from Table REF our model is able to greatly improve upon the naive setting of incorporating a few target domain samples during training.",
"We also note that PixelDA leverages these samples to achieve an even better performance than in the fully unsupervised setting (Table REF ).",
"Table: Semi-supervised experiments for the “Synthetic Cropped Linemod to Cropped Linemod” scenario.",
"When a small set of 1,000 target data is available to our model, it is able to improve upon baselines trained on either just these 1,000 samples or the synthetic training set augmented with these labeled target samples."
],
[
"Conclusion",
" We present a state-of-the-art method for performing unsupervised domain adaptation.",
"Our PixelDA models outperform previous work on a set of unsupervised domain adaptation scenarios, and in the case of the challenging “Synthetic Cropped Linemod to Cropped Linemod” scenario, our model more than halves the error for pose estimation compared to the previous best result.",
"They are able to do so by using a GAN–based technique, stabilized by both a task-specific loss and a novel content–similarity loss.",
"Furthermore, our model decouples the process of domain adaptation from the task-specific architecture, and provides the added benefit of being easy to understand via the visualization of the adapted image outputs of the model."
],
[
"Acknowledgements",
"The authors would like to thank Luke Metz, Kevin Murphy, Augustus Odena, Ben Poole, Alex Toshev, and Vincent Vanhoucke for suggestions on early drafts of the paper.",
"Supplementary Material Additional Generated Images Figure: Linear interpolation between two random noise vectors demonstrates that the model is able to separate out style from content in the MNIST-M dataset.",
"Each row is generated from the same MNIST digit, and each column is generated with the same noise vector.Figure: Additional generation examples for the LineMod dataset.",
"The left 4 columns are generated images and depth channels, while the corresponding right 4 columns are L2 nearest neighbors.",
"Model Architectures and Parameters Figure: Task classifier (T) architectures for each dataset.",
"We use the same task classifiers as , to enable fair comparisons.",
"The MNIST-M classifier is used for USPS, MNIST, and MNIST-M classification.During training, the task classifier is applied to both the synthetic and generated images when L t source L_t^{source} is enabled (see Paper tab:stability).The 'Private' parameters are only trained on one of these sets of images, while the 'Shared' parameters are shared between the two.",
"At test time on the target domain images, the classifier is composed of the Shared parameters and the Private parameters which are part of the generated images classifier.",
"These first private layers allow the classifier to share high level layers even when the synthetic and generated images have different channels (such as 1 channel MNIST and RGB MNIST-M).We present the exact model architectures used for each experiment along with hyperparameters needed to reproduce results.",
"The general form for the generator, G, and discriminator, D, are depicted in fig:arch of the paper.",
"For G, we vary the number of filters and the number of residual blocks.",
"For D, we vary the amount of regularization and the number of layers.",
"Optimization consists of alternating optimization of the discriminator and task classifier parameters, referred to as the D step, with optimization of the generator parameters, referred to as the G step.",
"Unless otherwise specified, the following hyperparameters apply to all experiments: Batch size 32 Learning rate decayed by 0.95 every 20,000 steps All convolutions have a 3x3 filter kernel Inject noise drawn from a zero centered Gaussian with stddev 0.2 after every layer of discriminator Dropout every layer in discriminator with keep probability of 90% Input noise vector is 10 dimensional sampled from a uniform distribution ${\\cal U}(-1,1)$ We follow conventions from the DCGAN paper [36] for several aspects An L2 weight decay of $1e^{-5}$ is applied to all parameters Leaky ReLUs have a leakiness parameter of 0.2 Parameters initialized from zero centered Gaussian with stddev 0.02 We use the ADAM optimizer with $\\beta _1 = 0.5$ USPS Experiments The Generator and Discriminator are identical to the MNIST-M experiments.",
"Loss weights: Base learning rate is $2e^{-4}$ The discriminator loss weight is 1.0 The generator loss weight is 1.0 The task classifier loss weight in G step is 1.0 There is no similarity loss between the synthetic and generated images MNIST-M Experiments (Paper tab:results) Generator: The generator has 6 residual blocks with 64 filters each Discriminator: The discriminator has 4 convolutions with 64, 128, 256, and 512 filters respectively.",
"It has the same overall structure as paper fig:arch Loss weights: Base learning rate is $1e^{-3}$ The discriminator loss weight is 0.13 The generator loss weight is 0.011 The task classifier loss weight in G step is 0.01 There is no similarity loss between the synthetic and generated images LineMod Experiments All experiments are run on a cluster of 10 TensorFlow workers.",
"We benchmarked the inference time for the domain transfer on a single K80 GPU as 30 ms for a single example (averaged over 1000 runs) for the LineMod dataset.",
"Generator: The generator has 4 residual blocks with 64 filters each Discriminator: The discriminator matches the depiction in paper fig:arch.",
"The dropout keep probability is set to 35%.",
"Parameters without masked loss (Paper tab:poseresults): Base learning rate is $2.2e^{-4}$ , decayed by 0.75 every 95,000 steps The discriminator loss weight is 0.004 The generator loss weight is 0.011 The task classification loss weight is 1.0 The task pose loss weight is 0.2 The task classifier loss weight in G step is 0 The task classifier is not trained on synthetic images There is no similarity loss between the synthetic and generated images Parameters with masked loss (Paper tab:stability): Base learning rate is $2.6e^{-4}$ , decayed by 0.75 every 95,000 steps.",
"The discriminator loss weight is 0.0088 The generator loss weight is 0.011 The task classification loss weight is 1.0 The task pose loss weight is 0.29 The task classifier loss weight in G step is 0 The task classifier is not trained on synthetic images The MPSE loss weight is 22.9 InfoGAN Connection In the case that $T$ is a classifier, we can show that optimizing the task loss in the way described in the main text amounts to a variational approach to maximizing mutual information [2], akin to the InfoGAN model [7], between the predicted class and both the generated and the equivalent source images.",
"The classification loss could be re-written, using conventions from [7] as: ${\\cal L}_{t} \\; &= \\; -\\mathbb {E}_{{\\mathbf {x}}^s \\sim {\\cal D}^s} [ \\mathbb {E}_{{y}^{\\prime } \\sim p({y}|{\\mathbf {x}}^s)}\\log q({ y}^{\\prime }|{\\mathbf {x}}^s)] \\nonumber \\\\&\\quad -\\mathbb {E}_{{\\mathbf {x}}^f \\sim G({\\mathbf {x}}^s, {\\mathbf {z}})} [ \\mathbb {E}_{{y}^{\\prime } \\sim p({y}|{\\mathbf {x}}^f)}\\log q({ y}^{\\prime }|{\\mathbf {x}}^f)]\\\\&\\ge -I(y^{\\prime }, {\\mathbf {x}}^s) - I(y^{\\prime }, {\\mathbf {x}}^f)+ 2H(y),$ where $I$ represents mutual information, $H$ represents entropy, $H(y)$ is assumed to be constant as in [7], $y^{\\prime }$ is the random variable representing the class, and $q({y}^{\\prime }|.",
")$ is an approximation of the posterior distribution $p({y}^{\\prime }|.",
")$ and is expressed in our model with the classifier $T$ .",
"Again, notice that we maximize the mutual information of $y^{\\prime }$ and the equivalent source and generated samples.",
"By doing so, we are effectively regularizing the adaptation process to produce images that look similar for each class to the classifier $T$ .",
"This helps maintain the original content of the source image and avoids, for example, transforming all objects belonging to one class to look like objects belonging to another.",
"Deep Reconstruction-Classification Networks Ghifary report a result of 91.80% accuracy on the MNIST $\\rightarrow $ USPS domain pair, versus our result of 95.9%.",
"We attempted to reproduce these results using their published code and our own implementation, but we were unable to achieve comparable performance."
],
[
"Model Architectures and Parameters",
"We present the exact model architectures used for each experiment along with hyperparameters needed to reproduce results.",
"The general form for the generator, G, and discriminator, D, are depicted in fig:arch of the paper.",
"For G, we vary the number of filters and the number of residual blocks.",
"For D, we vary the amount of regularization and the number of layers.",
"Optimization consists of alternating optimization of the discriminator and task classifier parameters, referred to as the D step, with optimization of the generator parameters, referred to as the G step.",
"Unless otherwise specified, the following hyperparameters apply to all experiments: Batch size 32 Learning rate decayed by 0.95 every 20,000 steps All convolutions have a 3x3 filter kernel Inject noise drawn from a zero centered Gaussian with stddev 0.2 after every layer of discriminator Dropout every layer in discriminator with keep probability of 90% Input noise vector is 10 dimensional sampled from a uniform distribution ${\\cal U}(-1,1)$ We follow conventions from the DCGAN paper [36] for several aspects An L2 weight decay of $1e^{-5}$ is applied to all parameters Leaky ReLUs have a leakiness parameter of 0.2 Parameters initialized from zero centered Gaussian with stddev 0.02 We use the ADAM optimizer with $\\beta _1 = 0.5$"
],
[
"USPS Experiments",
"The Generator and Discriminator are identical to the MNIST-M experiments."
],
[
"Loss weights:",
" Base learning rate is $2e^{-4}$ The discriminator loss weight is 1.0 The generator loss weight is 1.0 The task classifier loss weight in G step is 1.0 There is no similarity loss between the synthetic and generated images"
],
[
"Generator:",
"The generator has 6 residual blocks with 64 filters each"
],
[
"Discriminator:",
"The discriminator has 4 convolutions with 64, 128, 256, and 512 filters respectively.",
"It has the same overall structure as paper fig:arch"
],
[
"Loss weights:",
" Base learning rate is $1e^{-3}$ The discriminator loss weight is 0.13 The generator loss weight is 0.011 The task classifier loss weight in G step is 0.01 There is no similarity loss between the synthetic and generated images"
],
[
"LineMod Experiments",
"All experiments are run on a cluster of 10 TensorFlow workers.",
"We benchmarked the inference time for the domain transfer on a single K80 GPU as 30 ms for a single example (averaged over 1000 runs) for the LineMod dataset."
],
[
"Generator:",
"The generator has 4 residual blocks with 64 filters each"
],
[
"Discriminator:",
"The discriminator matches the depiction in paper fig:arch.",
"The dropout keep probability is set to 35%."
],
[
"Parameters without masked loss (Paper tab:poseresults):",
" Base learning rate is $2.2e^{-4}$ , decayed by 0.75 every 95,000 steps The discriminator loss weight is 0.004 The generator loss weight is 0.011 The task classification loss weight is 1.0 The task pose loss weight is 0.2 The task classifier loss weight in G step is 0 The task classifier is not trained on synthetic images There is no similarity loss between the synthetic and generated images"
],
[
"Parameters with masked loss (Paper tab:stability):",
" Base learning rate is $2.6e^{-4}$ , decayed by 0.75 every 95,000 steps.",
"The discriminator loss weight is 0.0088 The generator loss weight is 0.011 The task classification loss weight is 1.0 The task pose loss weight is 0.29 The task classifier loss weight in G step is 0 The task classifier is not trained on synthetic images The MPSE loss weight is 22.9"
],
[
"InfoGAN Connection",
"In the case that $T$ is a classifier, we can show that optimizing the task loss in the way described in the main text amounts to a variational approach to maximizing mutual information [2], akin to the InfoGAN model [7], between the predicted class and both the generated and the equivalent source images.",
"The classification loss could be re-written, using conventions from [7] as: ${\\cal L}_{t} \\; &= \\; -\\mathbb {E}_{{\\mathbf {x}}^s \\sim {\\cal D}^s} [ \\mathbb {E}_{{y}^{\\prime } \\sim p({y}|{\\mathbf {x}}^s)}\\log q({ y}^{\\prime }|{\\mathbf {x}}^s)] \\nonumber \\\\&\\quad -\\mathbb {E}_{{\\mathbf {x}}^f \\sim G({\\mathbf {x}}^s, {\\mathbf {z}})} [ \\mathbb {E}_{{y}^{\\prime } \\sim p({y}|{\\mathbf {x}}^f)}\\log q({ y}^{\\prime }|{\\mathbf {x}}^f)]\\\\&\\ge -I(y^{\\prime }, {\\mathbf {x}}^s) - I(y^{\\prime }, {\\mathbf {x}}^f)+ 2H(y),$ where $I$ represents mutual information, $H$ represents entropy, $H(y)$ is assumed to be constant as in [7], $y^{\\prime }$ is the random variable representing the class, and $q({y}^{\\prime }|.",
")$ is an approximation of the posterior distribution $p({y}^{\\prime }|.",
")$ and is expressed in our model with the classifier $T$ .",
"Again, notice that we maximize the mutual information of $y^{\\prime }$ and the equivalent source and generated samples.",
"By doing so, we are effectively regularizing the adaptation process to produce images that look similar for each class to the classifier $T$ .",
"This helps maintain the original content of the source image and avoids, for example, transforming all objects belonging to one class to look like objects belonging to another."
],
[
"Deep Reconstruction-Classification Networks",
"Ghifary report a result of 91.80% accuracy on the MNIST $\\rightarrow $ USPS domain pair, versus our result of 95.9%.",
"We attempted to reproduce these results using their published code and our own implementation, but we were unable to achieve comparable performance."
]
] | 1612.05424 |
[
[
"Beyond Volkov: Solving the Second-Order Klein-Gordon Equation"
],
[
"Abstract Whether monochromatic, pulsed, or even constant and crossed, the field used to describe the interaction of charged fermions with an intense laser beam is mainly assumed to be of plane-wave form.",
"We consider a simple extension to plane-wave fields and consider a scalar particle in a non-lightlike, univariate and transverse propagating electromagnetic wave.",
"The existence of some known exact solutions in this case allows us to analyse various proposed approximations in the literature as well as the plane wave model.",
"The results also describe some of the quantum dynamics of a scalar particle in a standing wave background."
],
[
"Introduction",
"We wish to calculate the behaviour of an electron in a realistic external electromagnetic (EM) field.",
"There are few classes of external field, for which the Dirac equation has been solved analytically.",
"One example is the solution to the Dirac equation in a plane wave background, the so-called “Volkov solution” [1].",
"This solution is central to the plane wave model of laser-based strong-field quantum electrodynamics (SFQED) and has dominated calculations for the last few decades (more detail on SFQED can be found in reviews [2], [3], [4], [5], [6], [7]).",
"However, to acquire the high field intensities in experiment, a laser beam has structure in both space, via focussing, and time, via pulse compression, so is clearly not a plane wave.",
"To assess whether the plane wave model is a good approximation for SFQED processes in highly intense laser pulses, one should be able to find the plane wave limit from more complicated backgrounds and investigate its domain of applicability.",
"The standard argument of the plane wave model is the following [2].",
"QED is a covariant theory, so the probability of any process $P_{\\tiny {\\textrm {QED}}}$ can be written entirely in terms of relativistic invariants.",
"The natural field scale is given by the “Schwinger” field $E_{\\tiny {\\textrm {cr}}} = m^{2}c^{3}/e\\hbar $ , where $m$ and $e$ are the mass and charge of a positron (from here on, we set $c$ , the speed of light in vacuo and $\\hbar $ , Planck's constant to the values $c=\\hbar =1$ ).",
"Then the relevant field invariants are the usual EM invariants, scaled by the Schwinger field $\\mathcal {F} = -e^{2}F^{\\mu \\nu }F_{\\mu \\nu }/4 m^{4}$ and $\\mathcal {G} = -e^{2}F^{\\mu \\nu }F^{\\ast }_{\\mu \\nu }/4 m^{4}$ , where $F_{\\mu \\nu }$ and $F^{\\ast }_{\\mu \\nu }$ are the Faraday tensor and its dual [8].",
"Two further invariants can be defined.",
"The intensity parameter $\\xi = e \\sqrt{\\langle p\\cdot T(\\varphi )\\cdot p \\rangle _{\\varphi }} / m\\,(k\\cdot p)$ [9] where $T^{\\mu \\nu }=(F^{2})^{\\mu \\nu } - \\eta ^{\\mu \\nu }\\, F^{2} / 4$ is the energy-momentum tensor [8], $\\langle \\cdot \\rangle _{\\varphi }$ indicates a cycle-average over $\\varphi $ , $p$ is the electron momentum and $k$ is the external-field wavevector, quantifies the work done by the external field over a Compton wavelength in units of a the field's photon energy.",
"The energy parameter $\\eta = k\\cdot p/m^{2}$ quantifies the seed particle energy.",
"Most discussions of the plane wave model choose the quantum parameter $\\chi = \\eta \\xi $ instead of $\\eta $ , so we choose that here as well.",
"Then probabilities in QED can be written $P_{\\tiny {\\textrm {QED}}} = P_{\\tiny {\\textrm {QED}}}(\\xi ,\\eta ,\\mathcal {F},\\mathcal {G})$ .",
"Typical laser intensities (to the best of our knowledge the highest recorded is $2\\times 10^{22}\\,\\textrm {Wcm}^{-2}$ at the HERCULES laser [10]), are much less than the equivalent Schwinger intensity for a linearly-polarised pulse: $I_{\\tiny {\\textrm {cr}}} = 4.6\\times 10^{29}\\,\\textrm {Wcm}^{-2}$ , and so one can assume $\\mathcal {F}$ , $\\mathcal {G}$ are the smallest parameters.",
"This allows one to Taylor-expand probabilities in these parameters and assuming they enter only perturbatively, or that their non-perturbative contribution is vanishingly small, $P_{\\tiny {\\textrm {QED}}}(\\xi ,\\xi \\,\\eta ,\\mathcal {F},\\mathcal {G})\\approx P_{\\tiny {\\textrm {QED}}}(\\xi ,\\xi \\,\\eta ,0,0)$ .",
"This implies that when $\\mathcal {F},\\mathcal {G}\\ll \\xi , \\xi \\,\\eta $ and $\\mathcal {F},\\mathcal {G} \\ll 1$ , it is a good approximation to assume an arbitrary background is crossed (electric and magnetic fields perpendicular and equal in magnitude).",
"A focussed laser pulse is then taken to be a perturbation around the plane wave background which is an example of a propagating crossed field.",
"The plane wave model is central to numerical codes that wish to include QED effects in laser-plasma interactions [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"This is because they rely upon the locally constant field approximation.",
"This has the premise that when $\\xi \\gg 1$ , formation regions of QED processes become much smaller than the pulse wavelength and so the field can be approximated as locally constant [20].",
"Therefore a better understanding of when the accuracy of the plane wave model is questionable will also have an impact on experimental design and analysis, which invariably invokes numerical simulation.",
"The Proceeding is organised as follows: we begin in section 2 with a recap of non-lightlike fields, followed by an analysis of solutions to the Klein Gordon equation for $k^{2}<0$ in section 3, detailing scalar charged particle dynamics for the case of over-the-barrier, under-the-barrier and periodic background scattering.",
"In sections 4 and 5 we analyse the solution and various approximations by calculating the scalar particle quasimomentum and the field-theory current.",
"In section 6 we conclude the presentation of results."
],
[
"Non-lightlike fields",
"The Faraday tensor of a plane wave can be written $F^{\\mu \\nu } = F^{\\mu \\nu }(\\varphi )$ where $\\varphi = k\\cdot x$ and $k^{2}=0$ .",
"We choose to relax one of these conditions and study non-lightlike fields, for which $F = F(\\varphi )$ but $k^{2} \\ne 0$ .",
"If $k^{2}>0$ , then one can perform a Lorentz transformation to a frame in which $k = \\omega (1,0,0,0)$ and the wave is entirely timelike.",
"This would correspond to a homogeneous but time-dependent electric field.",
"This case has been studied in various works [21], [22], [23], [24], [25], [26], [27], [28].",
"If $k^{2}<0$ then one can perform a Lorentz transformation to a frame in which $k = \\omega (0,0,0,1)$ and the wave is entirely spacelike.",
"This would correspond to an inhomogeneous but constant magnetic field.",
"This has also been studied in various works [21], [29].",
"Some motivation for studying SFQED in non-lightlike backgrounds also comes from experiments using energetic particle beams with oriented crystals [30], suggested experiments using laboratory plasmas [31] and observations of strongly-magnetised astrophysical systems [32].",
"If one takes the magnetic case $k = \\omega (0,0,0,1)$ , then after performing a Lorentz transformation in the $z$ -direction, the wavevector becomes $k = \\omega \\gamma \\beta (1,0,0,1/\\beta )$ .",
"As $\\beta \\rightarrow 1$ , the wavevector tends to that of a plane wave background.",
"But however relativistic the transformation, $k^{2} \\ne 0$ , as it is invariant.",
"We highlight that non-lightlike fields are relativistically inequivalent to plane wave fields.",
"In particular, charges in undulators and charges in lasers are not equivalent [33].",
"Combinations of plane-wave fields can be both of magnetic and electric character.",
"Take a standing wave made from two, counterpropagating, circularly-polarised plane waves, which has a vector potential: $A(\\varphi _{1},\\varphi _{2}) = C\\left\\lbrace \\varepsilon _{1}\\cos \\varphi _{1}+\\varepsilon _{2}\\sin \\varphi _{1} + \\varepsilon _{1}\\cos \\varphi _{2}+\\varepsilon _{2}\\sin \\varphi _{2}\\right\\rbrace ,$ where $\\lbrace k_{1}, \\varepsilon _{1}, \\varepsilon _{2}\\rbrace $ and $\\lbrace k_{2}, \\varepsilon _{1}, \\varepsilon _{2}\\rbrace $ are two dreibeins and we pick $k_{1} = \\omega (1,\\vec{n})$ , $k_{2} = \\omega (1,-\\vec{n})$ , with $\\vec{n}\\cdot \\vec{n} = 1$ .",
"Ponderomotive terms in a plane wave are related to the square of the vector potential.",
"We see: $A^{2}(\\varphi _{\\Delta }) = -2C^{2}(1+ \\cos \\varphi _{\\Delta }); \\qquad \\varphi _{\\Delta } = k_{\\Delta }\\cdot x, \\quad k_{\\Delta } = k_{1}-k_{2},$ and $k_{\\Delta }^{2} < 0$ .",
"Alternatively, at a magnetic node of this standing wave, for example if $\\vec{n} = (0,0,1)$ , in the $z=0$ plane, we see $\\varphi _{1}=\\varphi _{2} = \\omega (1,0,0,0)\\cdot x= \\bar{k}\\cdot x$ where $\\bar{k} = \\omega (1,0,0,0)$ , for which $\\bar{k}^{2} >0$ .",
"Classical and quantum electron dynamics for these two cases have recently been studied in [34], in the following, we concentrate on a single plane wave with magnetic character $k^{2}<0$ ."
],
[
"The second-order Klein-Gordon equation",
"In recent publications [34], [35], the classical and scalar quantum dynamics in some example non-lightlike backgrounds have been analysed.",
"Here we concentrate on the solution of the Klein-Gordon equation: $\\left[D^{2}+m^{2}\\right]\\Phi = 0; \\qquad D = \\partial + ia; \\qquad a = eA.$ One can proceed with the usual ansatz that is employed to acquire the plane wave solution $\\Phi = F(\\varphi )\\exp (i p\\cdot x)$ , to give: $k^{2}\\,F^{\\prime \\prime } - 2i\\,k\\cdot p\\,F^{\\prime } + (2\\,a\\cdot p - a^{2})F = 0.$ If the plane-wave case is taken and $k^{2}\\rightarrow 0$ (not via a boost, but via a co-ordinate rotation [36], [37], [38], [39]) set, we see that the solution can be solved immediately by exponentiation and we recover: $F = \\exp \\left[-i u_{\\tiny \\textrm {pw}}(\\varphi )\\right]; \\qquad u_{\\tiny \\textrm {pw}}(\\varphi ) = \\int ^{\\varphi } \\frac{2 a(\\phi )\\cdot p - a^{2}(\\phi )}{2k\\cdot p}d\\phi ,$ where $u_{\\tiny \\textrm {pw}}$ is the plane wave or “Volkov” exponent [1].",
"However, if $k^{2}\\ne 0$ , the Klein-Gordon (KG) equation is of second-order.",
"We immediately identify that a perturbative approach of neglecting the $k^{2}$ term is problematic: a condition for perturbation theory to apply is that the number of solutions to an equation should not change in the limit of of zero perturbation [40].",
"Still, one of the solutions will turn out to be unphysical, so there is some hope the plane wave limit may still be a good approximation.",
"Making the alternative ansatz $\\Phi = G(\\varphi )\\exp (i \\tilde{p}\\cdot x)$ where $\\tilde{p} = p - (k\\cdot p/k^{2})k$ , we acquire a second-order equation for $G$ in normal form: $k^{2} G^{\\prime \\prime } + 2(a\\cdot \\tilde{p} - a^{2})G = \\left(\\tilde{p}^{2}-m^{2}\\right)G. $ Since we are interested in the plane-wave limit and perturbations around this, $k^{2}$ can be the smallest invariant in the problem.",
"This motivates us to map Eq.",
"(REF ) onto the nonlinear Schrödinger equation: $-\\frac{\\hbar ^{2}}{2}G^{\\prime \\prime } + V(\\varphi ) G = \\mathcal {E} G, $ for potential $V$ and energy $\\mathcal {E}$ .",
"We distinguish three cases: i) over the barrier; ii) under the barrier scattering and iii) a periodic background."
],
[
"Over the barrier / head-on scattering",
"Let us consider $k^{2}<0$ and $a = m\\xi \\hat{f}_{\\mu }$ where $\\hat{f}_{\\mu }$ is of order unity.",
"$k^{2}$ is the smallest parameter, and to set the scattering to be over the barrier, we pick $k\\cdot p/\\sqrt{-k^{2}}\\gg p_{\\perp }$ , where $p_{\\perp }$ is the electron's momentum perpendicular to the external field wavevector.",
"We also assume $p_{\\perp } \\ll m \\xi $ (the transverse momentum acquired by an electron in a plane wave field is of the order $p_{\\perp } \\approx m \\xi $ [9]).",
"Normalising Eq.",
"(REF ) by $(m\\xi )^{2}$ , we identify: $V \\sim \\hat{f}^{2} \\sim 1; \\qquad -\\frac{\\hbar ^{2}}{2} = \\frac{k^{2}}{(m\\xi )^{2}} \\ll 1.$ For this to be over-the-barrier scattering, we require the particle energy to be much larger than the potential.",
"In other words, we require $\\mathcal {E}\\sim (\\gamma /m\\xi )^{2} \\gg 1$ .",
"This “high-energy” set-up then ensures the smallest parameter is multiplying the highest derivative, which is the typical situation where one can employ the WKB method to acquire a semiclassical solution [40]: $G(\\varphi ) \\sim \\left[\\frac{1}{\\mathcal {E}-V(\\varphi )}\\right]^{1/4}\\exp \\left[\\pm \\frac{i}{\\hbar } \\int ^{\\varphi } \\sqrt{2(\\mathcal {E}-V)}\\right] $ (the $\\pm $ sign will be fixed by the $\\xi \\rightarrow 0$ limit or the asymptotic sign of $k\\cdot p$ ).",
"This approach is known in SFQED [41], [42] and has recently been further developed to study high-energy particle-laser collisions [43], [44], [45]."
],
[
"Under the barrier / wide-angle scattering",
"In contrast to the high-energy, head-on scattering case, this type of dynamics is characterised by lower energies and large transverse momenta.",
"It is known that quantum effects can lead to a transverse spreading of an electron beam that is distinct from classical or beam-shape effects [46].",
"To demonstrate barrier effects, we take the background field to be of the form: $a_{\\mu } = m \\xi l_{\\mu } \\textrm {sech}(\\varphi ) = m \\xi l_{\\mu } g(\\varphi )$ , which produces a $\\textrm {sech}$ -like localised potential maximum.",
"In this case we have: $-\\frac{\\hbar ^{2}}{2} = \\frac{k^{2}}{3m^{2}\\xi ^{2}}; \\quad V = \\frac{g^{2} + 2g}{3}; \\quad \\mathcal {E} = \\frac{\\tilde{p}^{2}-m^{2}}{3m^{2}\\xi ^{2}},$ and we have chosen the electron's perpendicular momentum to be of the order of that acquired in a plane-wave background $p_{\\perp } = -m\\xi l_{\\perp }$ .",
"We also take $\\tilde{p}^{2} \\approx m^{2}$ .",
"We then pick $\\mathcal {E} = 0.995$ for a potential of height $V=1$ , to demonstrate barrier transmission.",
"In Fig.",
"REF the reflection and transmission of the electron wavefunction becomes evident.",
"Also plotted is the perturbative, plane-wave-like result, which is effectively blind to the barrier.",
"Figure: Comparison of the numerical solution of the Schrödinger equationin the sech-type potential with the perturbative approximation of neglecting the k 2 k^{2} term in the KG equation.",
"The particle energy ℰ=0.995\\mathcal {E}=0.995(horizontal dashed line) is chosen to be just below the peak of the potential (black dashed line).",
"The initial conditionsinitial conditions G(-5)=1G(-5)=1, G ' (-5)=0G^{\\prime }(-5)=0 have been used."
],
[
"Periodic fields",
"If the KG equation is written for monochromatic, circularly-polarised vector potential, it can be cast in the form: $\\frac{d^{2}G}{dy^{2}} - 2Q\\cos (2y)\\,G = -A\\, G; \\qquad A = \\frac{4}{k^{2}}\\left(\\frac{(k\\cdot p)^{2}}{k^{2}} + m^{2}\\xi ^{2}\\right); \\quad Q = -\\frac{4m\\xi |p_{\\perp }|}{k^{2}},$ and $y = \\varphi /2$ , which is a recognised form of the Mathieu Equation [40] (the case of linear polarisation leads to a Hill equation, recently studied in [24], [25], [26]).",
"Solutions to the Mathieu equation can be written in the form $G = \\phi (\\varphi ) \\exp [i\\nu (A,Q)\\varphi ]$ , where $\\phi $ is a periodic function and $\\nu (A,Q)$ is referred to as the “Mathieu characteristic exponent” [40] or “Floquet exponent” [47].",
"Certain regions of $A$ -$Q$ parameter-space lead to an imaginary Floquet exponent and since $\\varphi $ can be arbitrarily large and of positive or negative sign, this would indicate an infinitely large wavefunction normalisation constant.",
"These regions are referred to as “gaps” due to the vanishing probability of an electron possessing these parameters for an arbitrarily-long time.",
"Regions where the imaginary part of the Floquet exponent are zero are referred to as “bands”.",
"(A recent discussion of the connection of this band structure with resurgence can be found in [48] and references therein.)",
"The position of these structures is indicated in Fig.",
"REF .",
"Figure: A plot of the Floquet exponent.",
"Forbidden parameter regions or “gaps”, in which the imaginary part of the Floquet exponent is non-zero, are indicated by linear hatching.",
"Permissible regions or “bands”, in which the imaginary part of the Floquet exponent is zero “bands” are indicated by solid colours.",
"The shaded area within the dotted line is in principle accessible to an electron in a circularly-polarised monochromatic plane wave for which k 2 <0k^{2}<0.Also the Mathieu equation can be mapped onto the Schrödinger equation in Eq.",
"(REF ) using the following assignment: $\\frac{\\hbar ^{2}}{2} = \\frac{2}{Q}, \\quad V = \\cos \\varphi , \\quad \\mathcal {E} = \\frac{A}{2Q}.$ The cosine potential describes an infinite number of degenerate local minima.",
"The large-$Q$ limit corresponds to an electron being captured in a minimum which locally resembles a harmonic oscillator with high and steep walls.",
"In the small-$Q$ limit, one expects tunneling between neighbouring minima to become more probable."
],
[
"Quasimomentum",
"To compare various approximations to the periodic background solution, one can study the quasimomentum $q$ in each approach.",
"The quasimomentum is that quantity which occurs in global energy-momentum conserving delta-functions.",
"For the $k^{2}<0$ periodic background case, the exact quasimomentum can be written as [21], [29] $q_{\\mu } = \\tilde{p}_{\\mu } -\\textrm {sign}(k\\cdot p)\\,\\nu (A,Q)\\,k_{\\mu }/2$ .",
"The exponent in the WKB case Eq.",
"(REF ) contains an incomplete elliptic integral of the second kind, $\\textrm {E}(\\cdot |\\cdot )$ , leading to a quasimomentum similar in form to the exact quasimomentum, but with $\\nu (A,Q)\\rightarrow \\sqrt{A-2Q}\\,\\textrm {E}[\\pi /2,-4Q/(A-2Q)]$ when the cycle-average is performed.",
"If $Q$ is small, the exact quasimomentum can be written using the small-$Q$ expansion of the exponent $\\nu (A,Q) \\approx \\sqrt{A}$ [47].",
"The accuracy of these approximations as well as the classical solution for the longitudinal component of the quasimomentum, are displayed in Fig.",
"REF .",
"Figure: Comparison of the longitudinal quasimomentum component for various approximations.",
"Left: Q=1/10Q=1/10 small-QQ example.",
"Right: Q=10Q=10 large-QQ example.",
"The imaginary parts of the exact solution are shown with orange dashed lines, whereas the real part is shown with a solid green line.",
"Left of the vertical line delineates the under-the-barrier, classically-forbidden region.",
"Horizontal lines in the exact solution indicate forbidden regions.",
"For A<2QA < 2Q, under the barrier, theband and gap structure is clearly visible.",
"However, the approximations are blind to the band/gap structure, which also occurs at values over the barrier."
],
[
"Current conservation",
"One way to understand the discrepancy between the perturbative, plane-wave-like result and the correct electron wavefunction is to calculate the field theory current: $J_{\\mu }(x) = \\Phi ^{\\dagger }\\partial _{\\mu }\\Phi - \\partial _{\\mu }\\!\\left(\\Phi ^{\\dagger }\\right)\\Phi +2 a_{\\mu } \\Phi ^{\\dagger }\\Phi .$ From current conservation, we know $\\partial ^{\\mu }J_{\\mu } = (d/d\\varphi ) k\\cdot J = 0$ .",
"Then we find: $J_{\\mu }^{\\tiny \\textrm {pw}} = p_{\\mu } -a_{\\mu }(\\varphi ) + u_{\\tiny \\textrm {pw}}(\\varphi )\\,k_{\\mu }; \\qquad J_{\\mu }^{\\tiny \\textrm {wkb}} = \\frac{\\pi _{\\mu }(\\varphi )}{s(\\varphi )}$ where we have used the definition $\\pi _{\\mu } = p_{\\mu } - a_{\\mu }(\\varphi ) + \\frac{k\\cdot p}{k^{2}}\\left[s(\\varphi )-1\\right]\\,k_{\\mu }\\qquad s(\\varphi ) = \\sqrt{1+\\frac{2k^{2}}{k\\cdot p}\\,u_{\\tiny \\textrm {pw}}}.$ Since $\\varphi = k\\cdot p$ , we see that for WKB, and hence classically, the current is conserved as $k\\cdot J^{\\tiny \\textrm {wkb}}/ k\\cdot p = 1$ .",
"However, for the plane wave model we have: $k\\cdot J^{\\tiny \\textrm {pw}} = 1 + 2\\varepsilon u(\\varphi )$ , for $\\varepsilon = k^{2}/2k\\cdot p \\ll 1$ .",
"Therefore, the plane wave model violates current conservation to the order $O(\\varepsilon )$ , which is particularly relevant for cases of nontrivial barrier transmission/reflection, as already shown in Fig.",
"REF ."
],
[
"Conclusions",
"The plane wave model has been the focus of laser-based SFQED calculations for several decades.",
"Due to a linear relationship between the electron phase and and the proper time, the classical dynamics in a plane wave is integrable (there exist three conserved quantities in addition to the mass-shell condition).",
"The quantum case is essentially the WKB solution, which is exact.",
"However, when the null condition $k^{2}=0$ is relaxed, since $d\\varphi (\\tau )/d\\tau = s[\\varphi (\\tau )]\\, k\\cdot p/m$ the relation between phase and proper time becomes implicit and an integration is required to obtain the explicit relationship.",
"We have presented some solutions to the Klein Gordon equation for non-null ($k^{2}<0$ ), univariate transverse fields.",
"The WKB solution ceases to be exact in this case, making the exact solution non-Volkov in character.",
"Approximations based on WKB seem to work well, but can miss some of the non-perturbative structure e.g.",
"the band-gap structure in parameter space for an electron in a periodic background field.",
"Non-null fields were mentioned to occur at the magnetic node of a standing wave, for which $k^{2}>0$ .",
"This is particularly relevant to simulations of electromagnetic cascades, which often invoke the (constant) plane wave model in a homogeneous, time-dependent electric field [49], [11], [12], [13].",
"If a perturbative solution is invoked, in which the $k^{2}$ term is neglected, or included to first order by a “reduction-of-order approach” [35], for high-energy, over-the-barrier problems, the approximation appears promising (this depends on the field set-up: for example, see the problem at the magnetic node of a standing wave detailed in [34]).",
"The advantage with such an approximation is that it is independent of the form of the background.",
"However, when barrier reflection or tunneling becomes relevant, important dynamical details are missed by all such first-order approaches."
],
[
"Acknowledgments",
"B. K. acknowledges fruitful and productive work with A. Ilderton, T. Heinzl and H. Hu.",
"B. K. is thankful for the support from the Royal Society International Exchanges Scheme, CARDC (Chinese Aerodynamics Research and Development Center) and of SWUST (Southwest University of Science and Technology), Sichuan, China."
]
] | 1612.05436 |
[
[
"Association of Plages With Sunspots: A multi wavelength Study Using\n Kodaikanal Ca $\\scriptsize{{\\textrm{II}}}$ K and Greenwich sunspot area Data"
],
[
"Abstract Plages are the magnetically active chromospheric structures prominently visible in Ca $\\scriptsize{{\\textrm{II}}}$ K line (3933.67 \\r{A}).",
"A plage may or may not be associated with a sunspot which is a magnetic structure visible in the solar photosphere.",
"In this study we explore this aspect of association of plages with sunspots using the newly digitized Kodaikanal Ca $\\scriptsize{{\\textrm{II}}}$ K plage data and the Greenwich sunspot area data.",
"Instead of using the plage index or fractional plage area and their comparison with the sunspot number, we use, to our knowledge for the first time, the individual plage areas and compared it with the sunspot area time series.",
"Our analysis shows that these two structures formed at two different layers are highly correlated with each other on a time scale comparable to the solar cycle.",
"The area and the latitudinal distributions of plages are also similar to that of the sunspots.",
"Different area thresholdings on the `Butterfly diagram' reveal that plages with area $\\geq$4 arcmin$^2$ are mostly associated with a sunspot in the photosphere.",
"Apart from this, we found that the cyclic properties change when different sized plages are considered separately.",
"These results may help us to better understand the generation and the evolution of the magnetic structures in different layers of the solar atmosphere."
],
[
"Introduction",
"Sun is a magnetically active star with a dynamic atmosphere which varies on a time scale from seconds to years.",
"Different solar features are basically the manifestations of the solar magnetic concentraions in different layers of the Sun.",
"Plages are the chromospheric features which appear as bright patches on the solar disc when seen through Ca $\\scriptsize {{\\textrm {II}}}$ K line (3933.67 Å) images, whereas sunspots are the dark photospheric features prominently visible in white light images.",
"Sunspots and plages both vary periodically in a 11 year time scale known as `solar cycle'.",
"Apart from that plages are found to be highly correlated with the location of the magnetic field concentrations [13], [6], very similar to sunspots.",
"Satellite measurements over the past few decade revealed that the changes in solar irradiance happen over various time scales and it has a strong dependence on the various solar features present on the solar disc [8], [7].",
"It has been found that the total solar irradiance is highly correlated with the fractional plage area or the plage index [3].",
"Recently, [2] found a strong correlation between the sunspot number and the Ca $\\scriptsize {{\\textrm {II}}}$ K emission index.",
"Thus the study of long term plage data is not only of importance in connection to the solar irradiation variation study but also with the evolution of the solar magnetic fields and its cyclic changes.",
"Different observatories around the globe has been taking routine observations of the plages in Ca $\\scriptsize {{\\textrm {II}}}$ K line.",
"Mount Wilson data series is one of such plage index time series in the world [3].",
"Kodaikanal observatory in India has obtained daily photoheliograms of the Sun since 1904 to till 2007.",
"This century long data has been recently digitized [12].",
"[6] used this digitized data to identify the plages using an automated algorithm and generated a plage area time series.",
"In this paper, we use this data to find an association of the plages with the sunspots.",
"We also use this data to find different distributions in the plage sizes and their latitudinal locations."
],
[
"Data Description ",
"In this study we have used the plage area time series obtained from the newly digitized Kodaikanal Ca $\\scriptsize {{\\textrm {II}}}$ K data.",
"The complete time series covers more than 100 years of data (1904 – 2007).",
"Due to issues with the conditions of the photographic plates and also large number of missing days in the later half of the data [see Fig.",
"1b and Fig.",
"4 of [6]], we chose to limit our analysis between the period of 1907 to 1965 which covers from cycle 14 (descending phase only) to cycle 19.",
"For every detected plage form the daily Ca $\\scriptsize {{\\textrm {II}}}$ K images, we have the heliographic latitude, longitude (in degrees) and the area (in arcmin$^2$ ) information.",
"For the sunspot area data we have used the Greenwich daily sunspot record, for the same duration, available in the website https://solarscience.msfc.nasa.gov/greenwch.shtml."
],
[
"Yearly Averaged data and The Hemispheric Asymmetry",
"We generate the yearly averaged data from the daily plage observations and plotted it (black solid line) in panel (a) of Figure REF along with the sunspot area data (red solid line).",
"Since we are interested in the association of the two structures, we plot the normalized values of the yearly averaged data.",
"From the figure we readily see that the two time series show a good match with each other.",
"Every feature, including the double peaks seen in the sunspot data are also present in the plage area time series.",
"To estimate this association quantitatively we plot the scatter diagram as shown in panel (b) of Figure REF .",
"A correlation coefficient of 0.97 again confirms the close association of these two solar features which have formed in two different layers in the solar atmosphere.",
"Though we must emphasize that this high correlation in the yearly data does not imply the same for the smaller time scales (in months or days).",
"This is because when a sunspot decays away, the remnant small magnetic field may still continue to show itself as a plage in the higher atmosphere.",
"Also the small scale magnetic fields, which lives in the order of days or less, do not always develop as a sunspot.",
"Figure: Panel (a) shows the comparison plot of the yearly averaged sunspot and the plage area data.",
"Scatter plot between the two area data is shown in panel (b).",
"Panels (c - d) show the North (South) hemispheric yearly averaged plage area as marked.Now the hemispheric asymmetry in the sunspot area is a well known phenomena.",
"We investigate the same from the plage area time series by computing the yearly data separately for the two hemispheres.",
"Panels (c-d) in Figure REF show the yearly averaged plage area data in the northern and southern hemispheres (plotted in red and green solid line) respectively.",
"The sunspot area data, for the corresponding hemispheres, is also overplotted in the panels as shown in the black dotted lines.",
"Hemispheric plage area series shows a very good match with the sunspot area data.",
"Similar to the sunspots, in this case too we find that the double peaks near the cycle maximum is not a persistent feature in both of the hemispheres.",
"For an example, cycle 16th is a double peaked cycle (see Figure REF a) but from Figure REF (c-d) we see that in this case only northern hemisphere shows a double peak signature.",
"In the case of 19th cycle, there is no such double peak seen in the overall case but both the hemispheres have prominent double peak signatures.",
"Figure: Normalized plage area asymmetry is plotted in panel (a).",
"Comparison of the same with the sunspot area asymmetry is shown in panel (b).",
"Correlation value between these two are printed in the panel.Next we compute the normalized asymmetry coefficient, defined as (A$_{pn}$ -A$_{ps}$ )/(A$_{pn}$ +A$_{ps}$ ), as a measure of the hemispheric asymmetry in the plage area (A$_{pn}$ ,A$_{ps}$ are the yearly averaged plage area values in the northern and the southern hemispheres).",
"The evolution of this asymmetry coefficient is plotted in panel (a) of Figure REF .",
"There are quite a few distinct features readily noticeable from the plot.",
"During the minima of cycles 14, 15 and 16, we see that the northern hemisphere dominates whereas for the later cycles, i.e cycles 17, 18 and 19, the southern hemisphere dominates over the north.",
"This behavior is highlighted in the plot (Figure REF a) by using red arrows.",
"It is also interesting to note that during the progression of a cycle the asymmetry coefficient changes its sign quite a few times and this does not show any meaningful correlation with the cycle amplitude or any other properties of that particular cycle.",
"We also notice that on an average the northern hemisphere dominates over the south for the six cycles analyzed here.",
"We again revisited this northern hemispheric dominance in the following section.",
"Since we are interested in the plage-sunspot association, we look for the same in the asymmetry coefficient also.",
"In panel (b) in Figure REF we plot the scatter diagram between the asymmetry coefficients obtained from hemispheric plage area and the same computed for sunspot area.",
"We find a very good match between these two coefficients with a correlation value of 0.90.",
"Though we found some outliers also which probably have occurred during the cycle minima during which the asymmetry coefficient is prone to large departures.",
"Individual plage area values are considered for the size distribution analysis.",
"In panel (a) of Figure REF we show the plage area (sizes) distribution for the whole time period i.e from 1907 to 1965.",
"The distribution pattern look similar to an exponentially decaying function.",
"We fit the histogram with a decaying exponential function of the form, Y=A$_0\\exp ^{-(\\frac{X}{B})}$ , as shown by the red dashed line in Figure REF a.",
"From the fit we notice that though the initial part of the histogram gets a good fit but the fitted function drops very rapidly in the wing and leaves a large deviation around that region.",
"To get a better fit, we consider the lognormal function next.",
"This is inspired from that fact the sunspots are known to have a lognormal size distribution [4], [1], [11].",
"Thus a lognormal function of the form, $\\mathrm {y}=\\frac{1}{\\sqrt{2\\pi }\\sigma x}\\exp {-\\frac{[\\ln (x)-\\mu ]^2}{2\\sigma ^2}}$ is considered and fitted the histogram as shown with the black dotted curve in Figure REF a.",
"In this case we notice that the full histogram along with the tail gets a very good fit.",
"Thus the individual size distribution of the plages also follow the same `lognormal distribution' as we find for the sunspots.",
"Here we must highlight the fact that the good match of the two functions (exponential and the lognormal) at the core of the histogram (near to the origin, 1 arcmin$^{2}$ ) is due to the use of a rigid cutoff of 1 arcmin$^{2}$ as the minimum detectable plage area.",
"Thus the initial increment of the lognormal distribution got suppressed and the function decays exponentially thereafter.",
"Figure: Panels (a) and (b) show the area and the latitudinal distributions of the plages.",
"Corresponding fits to these distributions are also overplotted in these panels.",
"Panels (c) and (d) show (with solid black and red circles for the northern and southern hemisphere) the evolution of the fitted Gaussian parameters (C and σ\\sigma ) with the cycle number.",
"We also overplot the same for the sunspots in open circles."
],
[
"Latitudinal Distribution",
"In the beginning of a cycle, plages occur at higher latitudes and progressively move towards the equator as the cycle progresses.",
"We plot the distribution of the number of plages with their latitudes, for the full time span (1907-1695), in panel (b) of Figure REF (we also analyzed same for the individual cycles also).",
"It shows two bell-shaped distributions corresponding to the two hemispheres which are then fitted with two separate Gaussian functions.",
"From the plot (panel b) we notice that the peak height of the northern hemisphere is greater than that of southern hemisphere.",
"This compliments our findings in the section 3.1, where we obtained a similar result from the area asymmetry analysis.",
"To make a comparison, we repeat the same procedure for the Greenwich sunspots area data.",
"From every Gaussian fit we note down two parameters: center (C) and the sigma ($\\sigma $ ) values.",
"In panels (c-d) in Figure REF , we plot the evolution of these parameters for the plages and the sunspots simultaneously (solid red and black circles represent the southern and northern hemispheric values for the plages whereas the open circles corresponds to sunspots).",
"In the case of the center (C) plot (panel c) we see that the the centers of the plage distributions, for the two hemispheres, is always higher than that of the sunspots, although the trends remain the same.",
"At the same time we notice that the differences is more for the southern hemisphere.",
"We also observe that the center of the southern hemisphere for the 16th cycle is higher than that of the 19th cycle though the cycle amplitude of the 19th cycle is much higher than that of the 16th cycle.",
"In panel (d) we plot the evolution of the sigma parameters.",
"Overall the evolution of this parameter for both the indices i.e for sunspots and the plages follow the same pattern.",
"Again we find that there is a noticeable difference in the southern hemisphere.",
"The maximum sigma value occurs for the 17th cycle whereas for the sunspot it occurs at 16th cycle.",
"Currently there is a very little theoretical understanding on the relation of these parameters in connection to the solar dynamo.",
"Recently [5] found a connection of the parameter $\\sigma $ with the diffusivity ($\\eta $ ) parameter used in the dynamo theory."
],
[
"Correspondence Between Plage and Sunspot Locations",
"Plages may be or may not be always associated with a sunspot due to the reasons mentioned in section 3.1.",
"To probe this into depth, we divide the plages into four classes according to their individual sizes (A$_{p}$ ) and make use of the `Butterfly diagram' for our further analysis.",
"In different panels in Figure REF we plot the time-latitude diagram for the individually detected plages with black circles whereas the green circles represent the locations of the sunspots.",
"In the first size class (where A$_{p}\\ge $ 1 arcmin$^{2}$ ), we notice that there are substantially large number of plages which do not have any sunspots associated with them.",
"Also we notice that at the end of every cycle (or may be from the next cycle due to the overlapping period) large number of plages appear at high latitudes ($\\approx $ 60$^\\circ $ ) and this is much more prominent in the southern hemisphere.",
"Figure: [Top to bottom]: Comparison of the `butterfly diagram' for the plages (black circles) and the sunspots (green circles).",
"The longitudinal scatter diagram between the two are shown in the side panels.",
"Different plage sizes (A p _{p}) are mentioned in the title of the panels.",
"An animated GIF of the above figure is available at ftp://ftp.iiap.res.in/dipu/plage_sunspot.gif.Now as we progressively go towards higher size thresholds (A$_{p}\\ge $ 2 arcmin$^{2}$ , A$_{p}\\ge $ 3 arcmin$^{2}$ , A$_{p}\\ge $ 4 arcmin$^{2}$ ) we see that the two butterfly diagrams (one for the sunspots and the other for the plages) show progressively better match with each other.",
"Thus, we conclude that the latitudinal locations of the plages with an area $\\ge $ 4 arcmin$^{2}$ is showing a very good match with the sunspot latitudes.",
"A latitudinal match does not necessarily imply an one-to-one correspondence between the two.",
"This is because these features can be at same latitudes but in different longitudes implying the two of them not connected at all.",
"To better establish the association of plage sizes with the sunspot locations, we plot the longitudinal scatter diagrams for every plage size range as shown in the side panels in Figure REF .",
"We record the area weighted average longitudes for both the plages and the sunspots respectively during the simultaneous observing days.",
"A careful analysis reveals that the scatter plots between the plage and the sunspot longitudes become progressively linear as we consider higher plage areas.",
"Thus, in combination with our previous results from the `butterfly diagram' analysis, we conclude that plages with area $\\ge $ 4arcmin$^{2}$ have better one to one correspondence with the sunspot locations.",
"We must again remind the reader that the decay of a sunspot does not necessarily mean the disappearance of the associated plage."
],
[
"Plage Sizes and The 11-Year Cycle",
"Different long-term and short-term properties of the sunspot cycle show a strong size dependence [10].",
"Inspired by the findings of the plage size dependence on the plage-sunspot association in the previous section, we look for different signatures in the cycle properties when the plages are considered according to their sizes.",
"In panel (a) of Figure REF we plot the `no thresholding' case whereas in panels (b-d) we plot the yearly averaged plage area variations for different size thresholds.",
"For the smallest sized plages (1arcmin$^{2}\\le $ A$_{p}$$<$ 2 arcmin$^{2}$ ) we do not see much differences from the overall cycle characteristics (as found in panel REF a), except for the double peak behavior for certain cycles.",
"For an example, the double peaks of cycle 16 and cycle 18 become less prominent or weaker.",
"Also the overall strength of the cycle do not change much for the smallest plage size range i.e the 19th cycle is still the strongest during the analyzed time span.",
"The scenario almost remains the same for the medium plage size range (panel c).",
"As we move towards the biggest plage sizes (panel d, we notice that the double peaks near the cycle maxima appear almost for every cycle.",
"This is consistent with the behavior of biggest sized sunspots found by [10].",
"Apart from that we also note an interesting behavior i.e the presence of a weaker peak near the cycle minima.",
"This is highlighted in the panel by red arrows.",
"The location of this peak from the cycle maxima seems to move inwards as we move from cycle 14 to cycle 19.",
"This trend is though not prominently visible for the 19th cycle as we see a double peak late after the cycle maxima.",
"Next, we investigate the `odd-even rule' or the `G-O rule' which states that the odd numbered cycles are stronger than the preceding even numbered one [9].",
"This is well established for the sunspot area and the sunspot number data.",
"From Figure REF we notice that this rule is also valid for plage area time series for all the size ranges.",
"Still it should me mentioned here that the relative heights of the cycles change slightly when one considers different size ranges (panels (b-d) of Figure REF ).",
"Figure: Different panels showing the yearly averaged plage area time series as obtained using different plage size criteria.",
"Individual size ranges are printedon every panel.Long term multi wavelength study of different solar features help us to better understand the magnetic field evolution in different layers of the solar atmosphere in different time scales.",
"Sunspot and plages, though formed at different heights in the solar atmospheres, show a good correlation with each other at the time scales comparable to the solar cycle.",
"In shorter time scales of months or days, we find certain differences though.",
"Such differences can be explained by considering the complex evolution of the magnetic fields associated with sunspots.",
"When a sunspot decay the fields get fragmented and this progress results in disappearance of the same from the white light images.",
"The left over small scale fields still live for quite a few days and continue to appear as plages in the subsequent Ca $\\scriptsize {{\\textrm {II}}}$ K images.",
"Analyzing the individual plages area we find that they follow a lognormal distribution similar to the sunspots.",
"We also obtain a Gaussian distribution, in each of the hemispheres, of their latitudinal appearances.",
"Some of the properties of the fitted Gaussian parameters show different evolution with the solar cycle as compared to the sunspots.",
"Though not well understood but this hints towards a small scale component of the solar dynamo which is responsible for the evolution of the small scale fields.",
"This aspect is explored further by implementing different size criteria on the individual plages and considering their time evolution.",
"We find that different properties of the cycle change with the plage sizes which again points towards a complex dynamo operating into the Sun.",
"Finally we use the `butterfly diagram' along with longitudinal scatter plots to show that the plages with sizes $\\ge $ 4 arcmin$^{2}$ are always associated with a sunspot.",
"To conclude, we have analyzed the newly digitized Ca $\\scriptsize {{\\textrm {II}}}$ K data from Kodaikanal observatory to investigate the association of the plages, a chromospheric structure, with the sunspots which are the photospheric structures.",
"To our knowledge, for the first time, individual plage sizes are considered and compared with the sunspot area data.",
"Our analysis shows that the two layers (chromosphere and the photosphere) are magnetically coupled and the dynamo responsible for the magnetic fields in the Sun may have a complex action (generation of large scale and small scale fields)."
],
[
"Acknowledgment",
"We would like to thank the Kodaikanal facility of Indian Institute of Astrophysics, Bangalore, India for providing the data.",
"This data is now available for public use at http://kso.iiap.",
"res.in/data."
]
] | 1612.05711 |
[
[
"Holomorphic motions and complex geometry"
],
[
"Abstract We show that the graph of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C n ."
],
[
"Introduction and Main Result",
"Let $B$ be a connected complex $(n-1)$ -manifold with a basepoint $z_0 \\in B$ .",
"A holomorphic motion of the unit disc $\\mathbb {C}$ parametrized by $B$ is a continuous map $f: B \\times \\mathbb {C} \\mathbb {P}^1 = \\mathbb {C}\\cup \\lbrace \\infty \\rbrace $ satisfying the following conditions: (1) $f(z_0,w)=w$ for all $w \\in ,$ (2) the map $f(z,.",
"): \\mathbb {C} \\mathbb {P}^1$ is injective for each $z \\in B$ , (3) the map $f(.,w): B \\rightarrow \\mathbb {C} \\mathbb {P}^1$ is holomorphic for each $w \\in .",
"\\vspace{5.69046pt}$ Holomorphic motions were introduced by R. Mãne, P. Sad and D. Sullivan [11] and have been intensively studied since then (see, for instance, [13], [4], [5], [1]).",
"In this note we study the complex-analytic structure of the graph $D$ of $f$ : $D = \\lbrace (z,f(z,w)), \\ z \\in B, w \\in \\subset B \\times \\mathbb {C}\\mathbb {P}^1.$ Our main result is the following Theorem 1.1 The graph $D$ of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in $\\mathbb {C}^n$ .",
"Denoting the unit ball in $\\mathbb {C}^n$ by $\\mathbb {B}^n$ , Theorem REF will be a consequence of the following Theorem 1.2 Let $S \\subset \\mathbb {C}$ be a bounded domain.",
"If $A({\\mathbb {C}}^n)$ denotes the set of complex affine $(n-1)$ -dimensional subspaces of $\\mathbb {C}^n$ , then there does not exist a map $f: S \\rightarrow A({\\mathbb {C}}^n)$ satisfying the following conditions: (1) For $t \\in S$ , if $W_t = f(t) \\cap \\mathbb {B}^n$ , then $\\mathbb {B}^n= \\cup _{t \\in S} W_t$ .",
"(2) Either $W_t \\cap W_s = \\phi $ or $W_s=W_t$ for $s,t \\in S$ .",
"(3) There is a holomorphic map $\\pi : \\mathbb {B}^n \\rightarrow \\mathbb {B}^{n-1}$ such that $\\pi : W_t \\rightarrow \\mathbb {B}^{n-1}$ is bijective for all $t \\in S$ .",
"To derive Theorem REF from Theorem REF , we use a rescaling argument based on a recent result of K. T. Kim and L. Zhang [8].",
"To put these results in context, we recall the following result of K. Liu [10] and V. Koziarz-N. Mok [9] : Theorem 1.3 ([10], [9]) Let $n > m \\ge 1$ and let $\\Gamma _1 \\subset SU(n,1), \\ \\Gamma _2 \\subset SU(m,1)$ be torsion-free cocompact lattices.",
"Then there does not exist a holomorphic submersion from ${\\mathbb {B}^n}/{\\Gamma _1}$ to ${\\mathbb {B}^m}/{\\Gamma _2}$ .",
"This was proved for $n=2, m=1$ by K. Liu [10] and for all $n > m \\ge 1$ by V. Koziarz and N. Mok [9].",
"This result is natural from various points of view.",
"In particular, it is related to the following well-known question in the study of negatively curved Riemannian manifolds: Let $f: M^n \\rightarrow N^m$ be a smooth fibre bundle where $M$ and $N$ are smooth compact manifolds of dimensions $n > m \\ge 2$ .",
"Can $M$ admit a Riemannian metric with negative sectional curvature ?",
"If the bundle above is trivial then Preissman's theorem implies that the answer to the above question is in the negative.",
"Also, it is essential that $m \\ge 2$ : a theorem of W. Thurston states that certain 3-manifolds fibering over a circle admit hyperbolic metrics.",
"Theorem REF implies the Liu-Koziarz-Mok result when $n=m+1$ by the Bers-Griffiths uniformization theorem as explained later in the paper.",
"The compactness of the manifolds is essential in the question above and the result of Liu-Koziarz-Mok.",
"In other words, cocompact group actions on universal covers are needed.",
"Our point of view is that given the Bers-Griffiths theorem, the cocompact actions are not necessary.",
"The proof we present involves some elementary facts about the Kobayashi metric and Riemannian geometric techniques.",
"Note that an equivalent formulation of Theorem REF is that the graph of a holomorphic motion cannot admit a complete Kähler metric with constant negative holomorphic sectional curvature.",
"Hence the following question is natural: Can the graph of a holomorphic motion of the unit disc admit a complete Kähler metric with variable negative sectional curvature?",
"A related question, mentioned to the authors by Benoit Claudon and Pierre Py, is: Can the graph of a holomorphic motion of the unit disc be Gromov hyperbolic with respect to the Kobayashi metric ?",
"The method in this paper appears to hold some promise for tackling these questions.",
"In this connection, it is important to point out that metrics with weaker negative curvature conditions can exist on such domains: a result of S. K. Yeung [15] asserts that the universal cover of a Kodaira fibration, which is necessarily the graph of a holomorphic motion by the Bers-Griffiths theorem, admits complete Kähler metrics with negative holomorphic bisectional curvature."
],
[
"The Kobayashi metric on $D$",
" Let $\\bullet $ $B$ be a connected complex $(n-1)$ -manifold with a basepoint $z_0 \\in B$ , $\\bullet $ $f: B \\times \\mathbb {C} ~\\mathbb {P}^1$ a holomorphic motion, $\\bullet $ $D = \\lbrace (z,f(z,w)) \\ : \\ z \\in B, w \\in \\subset B \\times ,$$\\bullet $ $F: B \\times D$ be defined by $F(z,w)=(z,f(z,w))$ , $\\bullet $ $\\pi : D \\rightarrow B$ denote the first projection, $\\bullet $ for $p \\in D$ , let $S_p= \\pi ^{-1}(\\pi (p))$ , $\\bullet $ for $w \\in , let $$F_w: B \\rightarrow D \\ \\ {\\rm be} \\ \\ F_w(z)=f(z,w) \\ \\ \\ \\ {\\rm and} \\ \\ \\ \\ \\Sigma _w= F_w(B).$$\\vspace{5.69054pt}$ Note that $\\pi ^{-1}(z) =F(\\lbrace z\\rbrace \\times $ for every $z \\in B$ .",
"Lemma 2.1 For every $w \\in , the map $ F:B {w} D$ is a holomorphic embedding which is totally geodesic for the Kobayashi metrics on $ B$ and $ D$.",
"\\vspace{5.69054pt}$ Proof: Since $ d_{^K(z_1,z_2) \\ge d_D^K(F(z_1,w),F(z_2,w))\\ge d_{^K(\\pi \\circ F(z_1,w), \\pi \\circ F(z_2,w))= d_{^K(z_1,z_2)by the distance decreasing property of the Kobayashi metric the result follows.",
"}}\\subsection {Proof of Theorem \\ref {ball}}Let S \\subset \\mathbb {C} be a bounded domain and f: S \\rightarrow A({\\mathbb {C}}^n) be a map, where A({\\mathbb {C}}^n) is the set of complex affine (n-1)-dimensional subspaces of \\mathbb {C}^n.",
"Suppose that we have \\vspace{5.69054pt}}(1) a partition $ Bn= t S Wt$ where $ Wt = f(t) Bn$ and \\vspace{5.69054pt}$ (2) a holomorphic map $\\pi : \\mathbb {B}^n \\rightarrow \\mathbb {B}^{n-1}$ such that $\\pi \\vert _{W_t}: W_t \\rightarrow \\mathbb {B}^{n-1}$ is bijective.",
"Before stating the next lemma, we recall that the Kobayashi metric on $\\mathbb {B}^n$ coincides with the Bergman metric and is, in particular, a $C^2$ Riemannian metric.",
"Lemma 2.2 (1) For every $t \\in S$ and $z \\in \\mathbb {B}^{n-1},$ $W_t \\cap \\pi ^{-1}(z)$ consists of a single point and the intersection is orthogonal.",
"(2) The fibers of $\\pi $ are equidistant, i.e., for any $z_1,z_2 \\in \\mathbb {B}^{n-1}$ and $p \\in \\pi ^{-1}(z_1)$ we have $d_D^K(p, \\ \\pi ^{-1}(z_2)) =d_D^K(\\pi ^{-1}(z_1), \\ \\pi ^{-1}(z_2))$ Proof: Let $t \\in S$ and $z_1 \\in \\mathbb {B}^{n-1}$ .",
"It is clear that $W_t \\cap \\pi ^{-1}(z_1)$ is a singleton since $\\pi \\vert _{W_t}: W_t \\rightarrow \\mathbb {B}^{n-1}$ is bijective.",
"Fix $p \\in \\pi ^{-1}(z_1)$ .",
"Since $\\mathbb {B}^n= \\cup _{t \\in S} W_t$ , $p \\in W_t$ for some $t \\in S$ .",
"Let $\\gamma :[0,1] \\rightarrow D$ be a unit-speed geodesic with $\\gamma (0) =p$ and $\\gamma ^{\\prime }(0) \\in T_p W_t$ .",
"By Lemma REF we can assume that $\\gamma ([0,1]) \\subset W_t$ .",
"Let $\\gamma (1)= q \\in \\pi ^{-1}(z) \\cap W_t$ .",
"We claim that $\\gamma $ is the shortest geodesic between $\\pi ^{-1}(z_1)$ and $\\pi ^{-1}(z)$ .",
"This is because $ d_D^K(p, q)) \\ge d_{\\mathbb {B}^{n-1}}^K(z_0,z)= l(\\gamma ) \\ge d_D^K(p,q)$ for any $w_1,w_2 \\in .",
"The equality above comes from the assumption that $ Wt: Wt Bn-1$ is bijective and hence an isometry.",
"Since $ k$ is a Riemannian metric, the first variation for arc-length implies that $$ meets $ -1(z1)$ and$ -1(z)$ orthogonally.",
"Hence $ '(0)$ is orthogonal to $ Tp -1(z1)$.",
"\\hfill $$\\vspace{8.53581pt}$ In what follows we use the following notation: for any $p \\in D$ $S_p :=\\pi ^{-1}(\\pi (p)).$ Corollary 2.3 Let $\\gamma :[0,L] \\rightarrow D$ be a geodesic with $\\gamma (0)=p, \\ \\gamma ^{\\prime }(0) \\in (T_p S_p) ^{\\perp }$ .",
"If $P_s$ denotes the parallel transport of $T_p S_p$ along $\\gamma $ , then $P_s = T_{\\gamma (s)} S_{\\gamma (s)}.$ Proof: Let $\\lbrace e_1,e_2,...,e_{2n}\\rbrace $ be an orthonormal basis of $T_pD$ with $e_1, \\ e_2 \\in T_p S_p$ and let $E_i(s)$ be the parallel translate of $e_i$ , $i=1,...,2n$ , along $\\gamma $ .",
"By Lemma REF for $i \\ge 2$ , $\\gamma $ lies in $W_t$ for some $t \\in S$ .",
"By Lemma (1) of REF , $e_i \\in T_pW_t$ for $3 \\le i \\le 2n$ .",
"Since $W_t$ is totally geodesic, $E_i(s)$ is tangent to $W_t$ for all $s \\in [0,L]$ and $3 \\le i \\le 2n$ .",
"Hence $E_1(s), E_2(s) \\in (T_{\\gamma (s)} W_t)^\\perp = T_{\\gamma (s)}S_{\\gamma (s)}$ .",
"$\\square $ ."
],
[
"Distance between complex submanifolds",
"Fix $z_0,z_1 \\in \\mathbb {B}^{n-1}$ .",
"Let $p_0 \\in \\pi ^{-1}(z_0), p_1 \\in \\pi ^{-1}(z_1)$ be points satisfying $d^K_D(p_0,p_1) =d_D^K(\\pi ^{-1}(z_0), \\ \\pi ^{-1}(z_1)).$ Let $\\gamma :[0,L] \\rightarrow D$ be the unit speed geodesic with $\\gamma (0)=p_0, \\ \\gamma (L)=p_1$ which realizes the distance between $\\pi ^{-1}(z_0)$ and $\\pi ^{-1}(z_1)$ .",
"Since $ \\gamma ^{\\prime }(0)$ is orthogonal to $T_p \\pi ^{-1}(z_0)$ , Lemma REF implies that $\\gamma ^{\\prime }(0) \\in T_pW_t$ and $\\gamma ([0,\\infty )) \\subset W_t$ for some $t \\in S$ .",
"By considering the normal exponential map to $\\pi ^{-1}(z_0)$ , we can find a unit normal vector field $X$ to $\\pi ^{-1}(z_0)$ in a neighbourhood $U$ (in $\\pi ^{-1}(z_0)$ ) of $p_0$ such that the holds: for any $q \\in U$ , the geodesic $\\gamma _q: [0,L] \\rightarrow D$ with $\\gamma _q(0)=q, \\gamma ^{\\prime }_q(0)=X_q$ satisfies $\\gamma _q(L) \\in \\pi ^{-1}(z_1)$ .",
"Note that $X_{p_0}= \\gamma ^{\\prime }(0)$ .",
"Let $u \\in T_{p_0} \\pi ^{-1}(z_0)$ and $\\sigma :[-a,a] \\rightarrow \\pi ^{-1}(z_0) \\subset D$ a curve with $\\sigma (0) = p_0$ and $\\sigma ^{\\prime }(0)=u$ .",
"Define a geodesic variation $H:[0,L] \\times [-a,a] \\rightarrow D$ of $\\gamma $ by $H(s,t) \\ = \\ Exp_{\\sigma (t)} (sX_{\\sigma (t)}).$ Let $Y(s,t)= \\frac{\\partial H}{\\partial t} (s,t)$ be the variation vector field and, for each $t \\in [-a,a]$ , let $\\gamma _t$ be the geodesic given by $\\gamma _t(s)= H(s,t)$ .",
"Let $T(s,t)= \\gamma _t^{\\prime }(s) = \\frac{\\partial H}{\\partial s} (s,t)$ .",
"Lemma 2.4 For any $(s,t) \\in [0,L] \\times [-a,a]$ we have (i) $Y(s,t) \\in T_{\\gamma _t(s)}S_{\\gamma _t(s)}$ .",
"(ii) $Y^{\\prime }(s,t) := \\nabla _T Y (s,t) \\in T_{\\gamma _t(s)} S_{\\gamma _t(s)}$ .",
"Proof: (i) This follows if we can show that for each $s \\in [0,L]$ the curve $t \\mapsto H(s,t)$ lies in a fiber of $\\pi $ .",
"To see this consider the curves, for $t_1, t_2\\in (-a,a)$ , $s \\mapsto (\\pi \\circ \\gamma _{t_0})(s)$ and $s \\mapsto (\\pi \\circ \\gamma _{t_1})(s)$ .",
"These curves are unit-speed geodesics in $ \\mathbb {B}^{n-1}$ connecting $\\pi (z_0)$ and $\\pi (z_1)$ by Lemma REF .",
"Since $ \\mathbb {B}^{n-1}$ has negative curvature, uniqueness of geodesics forces $ \\pi (H(s,t_0)) = \\pi (H(s,t_1))$ .",
"(ii) We show this for $t=0$ for notational simplicity.",
"Let $\\lbrace e_1, \\ e_2 \\rbrace $ be an orthonormal basis of $T_{p_0} S_{p_0}$ .",
"Let $E_1(s), \\ E_2(s)$ be the parallel vector fields along $\\gamma $ with $E_i(0)=e_i$ .",
"We then have, by (i) and Corollary REF , $Y(s,0)= f_1(s)E_1(s) +f_2(s)E_2(s)$ for some functions $f_1,f_2:[0,L] \\rightarrow \\mathbb {R}$ .",
"Hence $Y^{\\prime }(s,0)= f_1^{\\prime }(s)E_1(s) +f_2^{\\prime }(s)E_2(s) \\ \\in \\ T_{\\gamma (s)} S_{\\gamma (s)}.",
"\\ \\ \\ \\ \\square $ We continue to denote $p=F(z_0,w_0)$ in what follows.",
"We recall the notation and constructions above: $\\bullet $ $X$ denotes a local unit normal vector field on $S_p$ such that the geodesics starting in the direction $X$ pass through the same fibers subsequently, $\\bullet $ $\\gamma :[0, \\infty ) \\rightarrow D$ is a geodesic with $\\gamma (0)=p, \\gamma ^{\\prime }(0) =X_p$ , $\\bullet $ for $u \\in T_p \\ S_p$ , $H_u(s,t)=\\gamma ^u_t(s)$ denotes a geodesic variation of $\\gamma $ such that (a) $\\frac{\\partial H_u}{\\partial s} (0,t) = X_{H(0,t)}$ (b) the variation vector field $Y_u(s,t)=\\frac{\\partial H_u}{\\partial t}(s,t)$ satisfies $Y_u(0,0)=u$ , $\\bullet $ $T_u(s,t)=\\frac{\\partial H_u}{\\partial s} (s,t)$ denotes the tangent vector $(\\gamma _t^u)^{\\prime }(s)$ to the variation geodesic $\\gamma _t^u(s)$ .",
"Note that $T_u(s,0)= \\gamma ^{\\prime }(s)$ for all $s$ and $T_u(0,t) = X_{H_u(0,t)}$ for all $t$ .",
"Lemma 2.5 For any $u \\in T_pS_p$ and $s \\in [0,\\infty )$ , we have (i) $ \\langle \\nabla _{Y_u}Y_u (s,0), \\gamma ^{\\prime }(s) \\rangle = -\\frac{1}{2} ( \\vert Y_u \\vert ^2)^{\\prime }(s,0).$ (ii) $\\nabla _uX \\ \\in \\ T_pS_p.$ Proof: $\\langle \\nabla _{Y_u}Y_u, \\gamma ^{\\prime } \\rangle \\ &= \\ Y_u \\langle Y_u, \\gamma ^{\\prime } \\rangle - \\langle Y_u, \\nabla _{Y_u} \\gamma ^{\\prime } \\rangle \\\\ &= \\ - \\langle Y_u, \\nabla _{\\gamma ^{\\prime }} Y_u \\rangle \\\\ &= \\ -\\frac{1}{2} ( \\vert Y_u \\vert ^2)^{\\prime } $ where we have used $\\langle T_u, Y_u \\rangle =0$ and $\\nabla _{Y_u}T_u=\\nabla _{T_u}Y_u$ .",
"This proves (i).",
"For (ii), $[Y_u,T_u]=0$ and (ii) of Lemma REF imply that $\\nabla _uX =\\nabla _{\\gamma ^{\\prime }(0)}Y_u \\ \\in \\ T_pS_p.$ $\\square $ By (ii) above, the shape operator $L: T_p S_p \\rightarrow T_p S_p$ of $S_p$ along the normal vector field $X$ is given by $L(v)= \\nabla _v X.$ As $L$ is a symmetric operator we can find an orthonormal basis $\\lbrace e_1,e_2\\rbrace $ of $T_p S_p$ consisting of eigenvectors of $L$ .",
"Since $S_p$ is a minimal submanifold (being a complex subvariety), the corresponding eigenvalues are given by $\\alpha , -\\alpha $ for some $\\alpha \\ge 0$ .",
"As before we denote the parallel transports of $e_1,e_2$ along $\\gamma $ by $E_1(s),E_2(s)$ .",
"Next we observe that if $Y_1 (s):=Y_{e_1}(s,0), \\ Y_2(s):=Y_{e_2}(s,0)$ are Jacobi fields constructed as earlier with $Y_1(0)=e_1, \\ Y_2(0)=e_2$ then $Y_1^{\\prime }(0) = \\nabla _{\\gamma ^{\\prime }(0)} Y_1 = \\nabla _{e_1}X = L(e_1) = \\alpha e_1.$ Similarly $Y_2^{\\prime }(0) = -\\alpha e_2.$ Hence if $K_1:[0,L] \\rightarrow \\mathbb {R}$ denotes the function $ K_1(s) = R(E_1(s), \\gamma ^{\\prime }(s), \\gamma ^{\\prime }(s), E_1(s))$ and $f_1: [0,L] \\rightarrow $ is the solution to $ y^{\\prime \\prime }+K_1y=0, \\ \\ \\ y(0)=1, \\ \\ y^{\\prime }(0)=\\alpha $ then $Y_1 =f_1E_1.$ Similarly $Y_2=f_2E_2$ where $f_2$ satisfies $y^{\\prime \\prime }+K_2y=0$ , $y(0)=1$ , $y^{\\prime }(0)=-\\alpha $ with $K_2(s) =R(E_2(s), \\gamma ^{\\prime }(s), \\gamma ^{\\prime }(s), E_2(s))$ ."
],
[
"The case of $\\mathbb {C}{\\mathbb {H}}^n$",
"In case $D$ is biholomorphic to the unit ball in $\\mathbb {C}^n$ , the Kobayashi metric on $D$ has constant holomorphic sectional curvature $-1$ and the curvature tensor has the property that $\\langle R(X,Y)Y,X \\rangle = - \\frac{1}{4}$ whenever $\\lbrace X,Y\\rbrace $ is an orthonormal pair spanning a totally real 2-plane, i.e., whenever $\\langle X,Y \\rangle =\\langle X, JY \\rangle =0$ .",
"Hence $f_1$ and $f_2$ satisfy $y^{\\prime \\prime } - \\frac{y}{4}=0.$ It follows that $f_1(s)= \\cosh (\\frac{s}{2}) + 2\\alpha \\sinh (\\frac{s}{2}), \\ \\ \\ \\ f_2(s)= \\cosh (\\frac{s}{2}) - 2 \\alpha \\sinh (\\frac{s}{2}).$ Case 1: $\\alpha \\ne \\frac{1}{2}$ .",
"In this case, (i) of Lemma REF implies that $ \\langle \\nabla _{Y_i}Y_i, \\gamma ^{\\prime } \\rangle (s) = - \\frac{1}{2} (f^2_i)^{\\prime }(s) <0 $ for $i=1,2$ and $s$ large enough.",
"On the other hand, Lemma REF and the fact that $S_{\\gamma (s)}$ is a minimal submanifold implies that $0= \\sum _{i=1}^2 \\langle \\nabla _{E_i}E_i, \\gamma ^{\\prime } \\rangle (s)=\\sum _{i=1}^2 f_i^{-2} \\langle \\nabla _{Y_i}Y_i, \\gamma ^{\\prime } \\rangle (s).$ This contradiction completes the proof.",
"Case 2: $\\alpha = \\frac{1}{2}$ for all $p \\in D$ , all $q \\in S_p$ and all $T_0 \\in (T_q S_p)^\\perp $ .",
"In this case, one can check that the second fundamental form of $S_p$ in every normal direction is parallel.",
"O'Neill's formula [12] (Page 465, 2. of Corollary 1) for the curvature of a Riemannian submersion then shows that the sectional curvature of the 2-plane $\\lbrace u,v\\rbrace $ is zero where $u \\in T_pS_p$ and $v \\in (T_pS_p)^\\perp $ ."
],
[
"Holomorphic motions and Bers-Griffiths uniformization",
"The following fundamental theorem allows us to deduce (when $n=m+1$ ) Theorem REF from Theorem REF : Theorem 2.6 ([2], [6]) Let $M$ and $N$ be compact complex manifolds and $\\phi :M \\rightarrow N$ a holomorphic submersion.",
"Suppose that $dim(M)=dim(N)+1$ and the fibers of $\\phi $ are compact Riemann surfaces of genus $\\ge 2$ .",
"Then the universal cover of $M$ is biholomorphic to the graph of a holomorphic motion over $\\tilde{N}$ .",
"For a detailed account of holomorphic motions and uniformization we refer the reader to [4]."
],
[
"Proof of Theorem ",
"We assume, to get a contradiction, that the graph $D$ of some holomorphic motion is biholomorphic to some bounded strictly pseudoconvex domain $\\Omega $ .",
"We denote by $\\Phi $ a biholomorphism from $D$ to $\\Omega $ .",
"For every $\\nu \\ge 1$ , let $F_\\nu : B \\rightarrow D$ be the totally geodesic holomorphic embedding defined by $F_{\\nu }(z) := F(z,1-\\frac{1}{\\nu })$ .",
"For every $\\nu $ , the map $\\Phi \\circ F_{\\nu }$ is holomorphic from $B$ to $\\Omega $ .",
"Let $z_0 \\in B$ .",
"We may assume, taking a subsequence if necessary, that $\\lim _{\\nu \\rightarrow \\infty } z^{\\nu }:=\\Phi \\circ F_{\\nu }(z_0) = p \\in \\partial \\Omega $ .",
"According to [8] it holds $\\lim _{z \\rightarrow p}\\sigma _{\\Omega }(z) = 1$ , where $\\sigma _{\\Omega }$ is the squeezing function of $\\Omega $ (see Definition in [8]).",
"This means that for every $\\nu \\ge 1$ there exists a biholomorphism $\\varphi _{\\nu }$ from $\\Omega $ to some strongly pseudoconvex domain $\\Omega _{\\nu }$ and there exists a sequence $(r_{\\nu })_{\\nu }$ with $\\lim _{\\nu \\rightarrow \\infty }r_{\\nu } = 1$ such that for every $\\nu \\ge 1$ : $\\varphi _{\\nu }(z^{\\nu }) = 0 \\ {\\rm and} \\ B(0,r_{\\nu }) \\subset \\Omega _{\\nu } \\subset \\mathbb {B}^n.$ Here $B(0,r_{\\nu })$ denotes the ball in $\\mathbb {C}^n$ centered at the origin with radius $r_{\\nu }$ .",
"For every $\\nu \\ge 1$ , let $\\Sigma _{\\nu }:=\\lbrace F_{\\nu }(z),\\ z \\in B\\rbrace $ and let $\\tilde{\\Sigma }_0^{\\nu }:=(\\varphi _{\\nu } \\circ \\Phi )(\\Sigma _{\\nu })$ .",
"Lemma 3.1 For every $\\nu \\ge 1$ , the set $\\tilde{\\Sigma }_0^{\\nu }$ is a totally geodesic complex submanifold of $\\Omega _{\\nu }$ .",
"Proof of Lemma REF .",
"This follows from Lemma REF and the fact that biholomorphisms are isometries for the Kobayashi metric.",
"$\\square $ Moreover, we get: Lemma 3.2 The sequence $(\\tilde{\\Sigma }_0^{\\nu })_{\\nu }$ converges, for the local Hausdorff convergence of sets, to some totally geodesic complex submanifold of $\\mathbb {B}^n$ .",
"Proof of Lemma REF .",
"Let, for every $\\nu \\ge 1$ , $\\Psi _{\\nu }:= \\varphi _{\\nu } \\circ \\Phi \\circ F_{\\nu }$ .",
"Then $\\Psi _{\\nu }$ is a holomorphic isometric embedding of $(B,d^K_B)$ into $(\\Omega _{\\nu },d^K_{\\Omega _{\\nu }})$ satisfying $\\Psi _{\\nu }(z_0) = 0$ .",
"Since for every $\\nu \\ge 1$ we have the inclusion $\\Omega _\\nu \\subset \\mathbb {B}^n$ , the sequence $(\\Psi _{\\nu })_{\\nu }$ is normal and extracting a subsequence if necessary, we may assume that $(\\Psi _{\\nu })_{\\nu }$ converges, uniformly on compact subsets of $B$ , to some holomorphic map $\\Psi _{\\infty }:B \\rightarrow \\mathbb {B}^n$ satisfying $\\Psi _{\\infty }(z_0) = 0$ .",
"Finally, let $ 0 \\in L \\subset \\subset \\mathbb {B}^n$ .",
"Since $\\Psi _{\\nu }$ is an isometry for the Kobayashi distances, there exists $K \\subset \\subset B$ such that for every $\\nu \\ge 1$ we get: $L \\cap \\tilde{\\Sigma }_0^{\\nu } \\subset \\Psi _{\\nu }(K)$ .",
"Now the uniform cnvergence of $(\\Psi _{\\nu })_{\\nu }$ on $K$ implies that the sets $\\tilde{\\Sigma }_0^{\\nu }$ converge to $\\tilde{\\Sigma }_0^{\\infty }$ for the Hausdorff convergence on $L$ .",
"Let $\\tilde{\\Sigma }_0^{\\infty }:=\\Psi _{\\infty }(B)$ and let $z,z^{\\prime } \\in B$ .",
"There exist $q_{\\nu }, q^{\\prime }_{\\nu } \\in \\tilde{\\Sigma }_0^{\\nu }$ , converging respectively to $\\Psi _{\\infty }(z)$ and $\\Psi _{\\infty }(z^{\\prime })$ and we have by Lemma REF : $\\begin{array}{lllll}d^K_{\\tilde{\\Sigma }_0^{\\infty }}(\\Psi _{\\infty }(z),\\Psi _{\\infty }(z^{\\prime })) & = & \\lim _{\\nu \\rightarrow \\infty }d^K_{\\tilde{\\Sigma }_0^{\\nu }}(\\Psi _{\\nu }(z),\\Psi _{\\nu }(z^{\\prime })) & = &\\lim _{\\nu \\rightarrow \\infty }d^K_{\\Omega _{\\nu }}(\\Psi _{\\nu }(z),\\Psi _{\\nu }(z^{\\prime }))\\\\& & & = & d^K_{\\mathbb {B}^n}(\\Psi _{\\infty }(z),\\Psi _{\\infty }(z^{\\prime })).\\end{array}$ $\\Box $ In particular, since totally geodesic complex submanifolds of $\\mathbb {B}^n$ , of complex dimension $(n-1)$ , are intersections of $\\mathbb {B}^n$ with complex affine subspaces of complex dimension $(n-1)$ , we may assume that $\\tilde{\\Sigma }_0^{\\infty }=\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ .",
"Let $\\zeta \\in \\Delta $ and let $q:=(0,\\zeta ) \\in \\mathbb {B}^n$ .",
"Then $q \\in \\Omega _{\\nu }$ for sufficiently large $\\nu $ and there exists $(b_{\\nu },\\zeta _{\\nu }) \\in B \\times \\Delta $ such that $q=\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }}(b_{\\nu })$ .",
"We set $\\tilde{\\Sigma }^{\\nu }_q:= \\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }}(B)$ .",
"We prove, exactly as for $\\tilde{\\Sigma }^{\\nu }_0$ , that $\\tilde{\\Sigma }^{\\nu }_q$ is a totally geodesic complex submanifold of $\\Omega _{\\nu }$ .",
"Lemma 3.3 The sequence $(b_{\\nu })_{\\nu }$ is relatively compact in $B$ .",
"Proof of Lemma REF .",
"Since $D$ is complete hyperbolic by assumption, it follows from Lemma REF that $B$ is complete hyperbolic.",
"Assume to get a contradiction that $(b_{\\nu })_{\\nu }$ is not relatively compact in $B$ .",
"We recall that $\\pi :D \\rightarrow B$ is holomorphic.",
"Hence we get for every sufficiently large $\\nu $ : $d^K_D\\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right), F(b_{\\nu },\\zeta _{\\nu })\\right) \\ge \\lim _{\\nu \\rightarrow \\infty }d^K_B(z_0,b_{\\nu }).$ Consequently, extracting a subsequence if necessary, we may assume that: $\\lim _{\\nu \\rightarrow \\infty }d^K_D\\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right), F\\left(b_{\\nu },\\zeta _{\\nu }\\right)\\right)=\\infty .$ Hence $\\lim _{\\nu \\rightarrow \\infty }d^K_{\\Omega _{\\nu }}\\left(\\varphi _{\\nu } \\circ \\Phi \\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right)\\right), \\varphi _{\\nu } \\circ \\Phi (F(b_{\\nu },\\zeta _{\\nu }))\\right)=\\infty .$ However, since $\\varphi _{\\nu } \\circ \\Phi \\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right)\\right)=0$ and $\\varphi _{\\nu } \\circ \\Phi (F(b_{\\nu },\\zeta _{\\nu }))=q$ for every $\\nu $ we get: $\\lim _{\\nu \\rightarrow \\infty }d^K_{\\Omega _{\\nu }}\\left(\\varphi _{\\nu } \\circ \\Phi \\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right)\\right), \\varphi _{\\nu } \\circ \\Phi (F(b_{\\nu },\\zeta _{\\nu }))\\right)=d^K_{\\mathbb {B}^n}(0,q) < \\infty .$ This contradicts Condition (REF ).",
"$\\Box $ It follows now from Lemma REF that we may extract from $(b_{\\nu })_{\\nu })$ a subsequence, still denoted $(b_{\\nu })_{\\nu }$ , that converges to some point $b_{\\infty } \\in B$ .",
"Hence, extracting a subsequence if necessary, we may assume that the sequence $(\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }})_{\\nu }$ converges uniformly on compact subsets of $B$ to a holomorphic map $\\Psi ^q_{\\infty }:B \\rightarrow \\mathbb {B}^n$ satisfying $\\Psi ^q_{\\infty }(b_{\\infty })= q$ .",
"This implies that $\\tilde{\\Sigma }^{\\nu }_q=\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }}(B)$ converges to $\\tilde{\\Sigma }^{\\infty }_q:=\\Psi ^q_{\\infty }(B)$ and $\\tilde{\\Sigma }^{\\infty }_q$ is a totally geodesic complex submanifold of $\\mathbb {B}^n$ .",
"We finally prove Proposition 3.4 For every $q \\in \\mathbb {B}^n \\cap (\\lbrace 0^{\\prime }\\rbrace \\times $ there exists a totally geodesic complex submanifold $\\tilde{\\Sigma }^{\\infty }_q$ of $\\mathbb {B}^n$ passing through $q$ .",
"For every $\\nu $ , let $\\pi _{\\nu } : D \\rightarrow \\Sigma _{\\nu }$ be given by $\\forall (z,\\zeta ) \\in B \\times \\Delta ,\\ \\pi _{\\nu }(F(z,\\zeta )) = F\\left(z,1-\\frac{1}{\\nu }\\right)$ and let $\\begin{array}{ccccc}\\tilde{\\pi }_{\\nu } & : & \\Omega _{\\nu } & \\rightarrow & \\Sigma _0^{\\nu }\\\\& & z & \\mapsto & \\varphi _{\\nu } \\circ \\Phi \\circ \\pi _{\\nu } \\circ \\Phi ^{-1} \\circ \\varphi _{\\nu }^{-1}\\end{array}.$ Since $\\varphi _{\\nu } \\circ \\Phi (F(z_0,1-\\frac{1}{\\nu })) = 0$ according to (REF ), we have $\\tilde{\\pi }_{\\nu }(0) = 0$ for every $\\nu $ .",
"Hence we may extract from $(\\tilde{\\pi }_{\\nu })_{\\nu }$ a subsequence, still denoted $(\\tilde{\\pi }_{\\nu })_{\\nu }$ , that converges to a holomorphic map $\\tilde{\\pi }_{\\infty } : \\mathbb {B}^n \\rightarrow \\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ .",
"Moreover we have: Proposition 3.5 For every $q \\in \\lbrace 0\\rbrace \\times \\Delta $ , the restriction of $\\tilde{\\pi }_{\\infty }$ to $\\tilde{\\Sigma }^{\\infty }_q$ is a biholomorphism from $\\tilde{\\Sigma }^{\\infty }_q$ to $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ .",
"Proof of Proposition REF .",
"By the very definition of $\\tilde{\\pi }_{\\nu }$ , the restriction of $\\tilde{\\pi }_{\\nu }$ to $\\tilde{\\Sigma }_q^{\\nu }$ is a biholomorphism from $\\tilde{\\Sigma }_q^{\\nu }$ to $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ for every $\\nu $ .",
"Moreover, $\\tilde{\\Sigma }_q^{\\nu }$ converges to $\\tilde{\\Sigma }_q^{\\infty }$ for the Hausdorff distance.",
"Finally, we have for every $\\nu \\ge 1$ : $\\pi _{\\nu } \\circ \\Phi ^{-1} \\circ \\varphi _{\\nu }^{-1}(q) = \\left(b_{\\nu },1-\\frac{1}{\\nu }\\right) = F_{\\nu }(b_{\\nu }).$ Since $\\lim _{\\nu \\rightarrow \\infty }b_{\\nu } = b_{\\infty } \\in B$ and since the sequence $(\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\nu })_{\\nu }$ converges, uniformly on compact subsets of $B$ to $\\Psi _{\\infty }$ (see the proof of Lemma REF ) we obtain that: $\\tilde{\\pi }_{\\infty }(q) = \\lim _{\\nu \\rightarrow \\infty }\\varphi _{\\nu } \\circ \\Phi \\circ \\pi _{\\nu } \\circ \\Phi ^{-1} \\circ \\varphi _{\\nu }^{-1}(q) = \\Psi _{\\infty }(b_{\\infty }) \\in \\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace .$ Hence if $g_{\\nu }$ denotes the inverse of the restriction of $\\tilde{\\pi }_{\\nu }$ to $\\tilde{\\Sigma }^{\\infty }_q$ then $g_{\\nu }$ is defined on $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ and the sequence ($g_{\\nu })_{\\nu }$ converges, uniformly on compact subsets of $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ , to some holomorphic map $g_{\\infty } : \\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace \\rightarrow \\tilde{\\Sigma }^{\\infty }_q$ such that $g_{\\infty } \\circ \\tilde{\\pi }_{\\infty } = id_{\\tilde{\\Sigma }^{\\infty }_q}$ and $\\tilde{\\pi }_{\\infty } \\circ g_{\\infty }= id_{|\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace }$ .",
"$\\square $ Finally, let $q \\ne q^{\\prime }$ be two points in $\\mathbb {B}^n$ .",
"By construction, for every $\\nu \\ge 1$ , the intersection between the totally geodesics submanifolds $\\tilde{\\Sigma }_q^\\nu $ and $\\tilde{\\Sigma }_{q^{\\prime }}^\\nu $ is empty.",
"Since $(\\tilde{\\Sigma }_q^\\nu )_{\\nu }$ converges to $\\tilde{\\Sigma }_q^{\\infty }$ and $\\tilde{\\Sigma }_{q^{\\prime }}^\\nu $ converges to $\\tilde{\\Sigma }_{q^{\\prime }}^{\\infty }$ , it follows from the positivity of intersection that: $\\tilde{\\Sigma }_q^{\\infty } \\cap \\tilde{\\Sigma }_{q^{\\prime }}^{\\infty } = \\emptyset .$ Now Proposition REF and Proposition REF give a contradiction, according to Theorem REF .",
"$\\Box $ We end the note by studying some metric properties of $D$ .",
"We assume that $B$ is complete hyperbolic and that $B$ admits an exhaustion $(B_k)_{k \\in \\mathbb {N}}$ : $B_{k} \\subset \\subset B_{k+1}$ for every $k$ and $B=\\sup _{k \\in \\mathbb {N}} B_k$ , such that $B_k$ is complete (Kobayashi) hyperbolic for every $k$ .",
"Proposition 3.6 The domain $D$ is complete (Kobayashi) hyperbolic.",
"Proof of Proposition REF .",
"It is proved in [3] that for every $k$ , $D_k:=F(B_k \\times \\Delta )$ is complete hyperbolic.",
"Let $Z^0=(z_0,w_0), \\ Z^{\\nu }=(z_{\\nu },w_{\\nu }) \\in B \\times \\Delta $ be such that $\\lim _{\\nu \\rightarrow \\infty }d^K_B(z_0,z_{\\nu }) = \\infty $ .",
"Then: $\\forall \\nu \\ge 1,\\ d^K_D(F(Z^0),F(Z^{\\nu })) = d^K_B(z_0,z_{\\nu }) \\longrightarrow _{\\nu \\rightarrow \\infty } \\infty .$ Hence, to prove that $D$ is complete hyperbolic, it is sufficient to prove that if $(z_{\\nu })_{\\nu } \\subset \\subset B_{k_0}$ for some $k_0 \\in \\mathbb {N}$ and $|w_{\\nu }| \\longrightarrow _{\\nu \\rightarrow \\infty }1$ , then $d^K_D(F(Z^0),F(Z^{\\nu })) \\longrightarrow _{\\nu \\rightarrow \\infty } \\infty $ .",
"Assume, to get a contradiction, that there exists $c > 0$ such that $d^K_D(F(Z^0),F(Z^{\\nu })) \\le c$ for every $k \\in \\mathbb {N}$ (extracting a subsequence if necessary).",
"There exists $k_1 \\ge k_0$ such that the set $\\lbrace y \\in D /\\ d^K_D(y,F(Z^0)) < c +1\\rbrace $ is contained in $D_{k_1}$ according to (REF ).",
"Moreover, it follows from Lamma 5.1 in [7] that: $d^K_{D_{k_1}}(F(Z^0),F(Z^{\\nu })) \\le \\frac{1}{\\tanh (1)}d^K_D(F(Z^0),F(Z^{\\nu })).$ This contradicts the fact that $D_{k_1}$ is complete hyperbolic.",
"$\\Box $ We assume, to get a contradiction, that the graph $D$ of some holomorphic motion is biholomorphic to some bounded strictly pseudoconvex domain $\\Omega $ .",
"We denote by $\\Phi $ a biholomorphism from $D$ to $\\Omega $ .",
"For every $\\nu \\ge 1$ , let $F_\\nu : B \\rightarrow D$ be the totally geodesic holomorphic embedding defined by $F_{\\nu }(z) := F(z,1-\\frac{1}{\\nu })$ .",
"For every $\\nu $ , the map $\\Phi \\circ F_{\\nu }$ is holomorphic from $B$ to $\\Omega $ .",
"Let $z_0 \\in B$ .",
"We may assume, taking a subsequence if necessary, that $\\lim _{\\nu \\rightarrow \\infty } z^{\\nu }:=\\Phi \\circ F_{\\nu }(z_0) = p \\in \\partial \\Omega $ .",
"According to [8] it holds $\\lim _{z \\rightarrow p}\\sigma _{\\Omega }(z) = 1$ , where $\\sigma _{\\Omega }$ is the squeezing function of $\\Omega $ (see Definition in [8]).",
"This means that for every $\\nu \\ge 1$ there exists a biholomorphism $\\varphi _{\\nu }$ from $\\Omega $ to some strongly pseudoconvex domain $\\Omega _{\\nu }$ and there exists a sequence $(r_{\\nu })_{\\nu }$ with $\\lim _{\\nu \\rightarrow \\infty }r_{\\nu } = 1$ such that for every $\\nu \\ge 1$ : $\\varphi _{\\nu }(z^{\\nu }) = 0 \\ {\\rm and} \\ B(0,r_{\\nu }) \\subset \\Omega _{\\nu } \\subset \\mathbb {B}^n.$ Here $B(0,r_{\\nu })$ denotes the ball in $\\mathbb {C}^n$ centered at the origin with radius $r_{\\nu }$ .",
"For every $\\nu \\ge 1$ , let $\\Sigma _{\\nu }:=\\lbrace F_{\\nu }(z),\\ z \\in B\\rbrace $ and let $\\tilde{\\Sigma }_0^{\\nu }:=(\\varphi _{\\nu } \\circ \\Phi )(\\Sigma _{\\nu })$ .",
"Lemma 3.1 For every $\\nu \\ge 1$ , the set $\\tilde{\\Sigma }_0^{\\nu }$ is a totally geodesic complex submanifold of $\\Omega _{\\nu }$ .",
"Proof of Lemma REF .",
"This follows from Lemma REF and the fact that biholomorphisms are isometries for the Kobayashi metric.",
"$\\square $ Moreover, we get: Lemma 3.2 The sequence $(\\tilde{\\Sigma }_0^{\\nu })_{\\nu }$ converges, for the local Hausdorff convergence of sets, to some totally geodesic complex submanifold of $\\mathbb {B}^n$ .",
"Proof of Lemma REF .",
"Let, for every $\\nu \\ge 1$ , $\\Psi _{\\nu }:= \\varphi _{\\nu } \\circ \\Phi \\circ F_{\\nu }$ .",
"Then $\\Psi _{\\nu }$ is a holomorphic isometric embedding of $(B,d^K_B)$ into $(\\Omega _{\\nu },d^K_{\\Omega _{\\nu }})$ satisfying $\\Psi _{\\nu }(z_0) = 0$ .",
"Since for every $\\nu \\ge 1$ we have the inclusion $\\Omega _\\nu \\subset \\mathbb {B}^n$ , the sequence $(\\Psi _{\\nu })_{\\nu }$ is normal and extracting a subsequence if necessary, we may assume that $(\\Psi _{\\nu })_{\\nu }$ converges, uniformly on compact subsets of $B$ , to some holomorphic map $\\Psi _{\\infty }:B \\rightarrow \\mathbb {B}^n$ satisfying $\\Psi _{\\infty }(z_0) = 0$ .",
"Finally, let $ 0 \\in L \\subset \\subset \\mathbb {B}^n$ .",
"Since $\\Psi _{\\nu }$ is an isometry for the Kobayashi distances, there exists $K \\subset \\subset B$ such that for every $\\nu \\ge 1$ we get: $L \\cap \\tilde{\\Sigma }_0^{\\nu } \\subset \\Psi _{\\nu }(K)$ .",
"Now the uniform cnvergence of $(\\Psi _{\\nu })_{\\nu }$ on $K$ implies that the sets $\\tilde{\\Sigma }_0^{\\nu }$ converge to $\\tilde{\\Sigma }_0^{\\infty }$ for the Hausdorff convergence on $L$ .",
"Let $\\tilde{\\Sigma }_0^{\\infty }:=\\Psi _{\\infty }(B)$ and let $z,z^{\\prime } \\in B$ .",
"There exist $q_{\\nu }, q^{\\prime }_{\\nu } \\in \\tilde{\\Sigma }_0^{\\nu }$ , converging respectively to $\\Psi _{\\infty }(z)$ and $\\Psi _{\\infty }(z^{\\prime })$ and we have by Lemma REF : $\\begin{array}{lllll}d^K_{\\tilde{\\Sigma }_0^{\\infty }}(\\Psi _{\\infty }(z),\\Psi _{\\infty }(z^{\\prime })) & = & \\lim _{\\nu \\rightarrow \\infty }d^K_{\\tilde{\\Sigma }_0^{\\nu }}(\\Psi _{\\nu }(z),\\Psi _{\\nu }(z^{\\prime })) & = &\\lim _{\\nu \\rightarrow \\infty }d^K_{\\Omega _{\\nu }}(\\Psi _{\\nu }(z),\\Psi _{\\nu }(z^{\\prime }))\\\\& & & = & d^K_{\\mathbb {B}^n}(\\Psi _{\\infty }(z),\\Psi _{\\infty }(z^{\\prime })).\\end{array}$ $\\Box $ In particular, since totally geodesic complex submanifolds of $\\mathbb {B}^n$ , of complex dimension $(n-1)$ , are intersections of $\\mathbb {B}^n$ with complex affine subspaces of complex dimension $(n-1)$ , we may assume that $\\tilde{\\Sigma }_0^{\\infty }=\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ .",
"Let $\\zeta \\in \\Delta $ and let $q:=(0,\\zeta ) \\in \\mathbb {B}^n$ .",
"Then $q \\in \\Omega _{\\nu }$ for sufficiently large $\\nu $ and there exists $(b_{\\nu },\\zeta _{\\nu }) \\in B \\times \\Delta $ such that $q=\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }}(b_{\\nu })$ .",
"We set $\\tilde{\\Sigma }^{\\nu }_q:= \\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }}(B)$ .",
"We prove, exactly as for $\\tilde{\\Sigma }^{\\nu }_0$ , that $\\tilde{\\Sigma }^{\\nu }_q$ is a totally geodesic complex submanifold of $\\Omega _{\\nu }$ .",
"Lemma 3.3 The sequence $(b_{\\nu })_{\\nu }$ is relatively compact in $B$ .",
"Proof of Lemma REF .",
"Since $D$ is complete hyperbolic by assumption, it follows from Lemma REF that $B$ is complete hyperbolic.",
"Assume to get a contradiction that $(b_{\\nu })_{\\nu }$ is not relatively compact in $B$ .",
"We recall that $\\pi :D \\rightarrow B$ is holomorphic.",
"Hence we get for every sufficiently large $\\nu $ : $d^K_D\\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right), F(b_{\\nu },\\zeta _{\\nu })\\right) \\ge \\lim _{\\nu \\rightarrow \\infty }d^K_B(z_0,b_{\\nu }).$ Consequently, extracting a subsequence if necessary, we may assume that: $\\lim _{\\nu \\rightarrow \\infty }d^K_D\\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right), F\\left(b_{\\nu },\\zeta _{\\nu }\\right)\\right)=\\infty .$ Hence $\\lim _{\\nu \\rightarrow \\infty }d^K_{\\Omega _{\\nu }}\\left(\\varphi _{\\nu } \\circ \\Phi \\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right)\\right), \\varphi _{\\nu } \\circ \\Phi (F(b_{\\nu },\\zeta _{\\nu }))\\right)=\\infty .$ However, since $\\varphi _{\\nu } \\circ \\Phi \\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right)\\right)=0$ and $\\varphi _{\\nu } \\circ \\Phi (F(b_{\\nu },\\zeta _{\\nu }))=q$ for every $\\nu $ we get: $\\lim _{\\nu \\rightarrow \\infty }d^K_{\\Omega _{\\nu }}\\left(\\varphi _{\\nu } \\circ \\Phi \\left(F\\left(z_0,1-\\frac{1}{\\nu }\\right)\\right), \\varphi _{\\nu } \\circ \\Phi (F(b_{\\nu },\\zeta _{\\nu }))\\right)=d^K_{\\mathbb {B}^n}(0,q) < \\infty .$ This contradicts Condition (REF ).",
"$\\Box $ It follows now from Lemma REF that we may extract from $(b_{\\nu })_{\\nu })$ a subsequence, still denoted $(b_{\\nu })_{\\nu }$ , that converges to some point $b_{\\infty } \\in B$ .",
"Hence, extracting a subsequence if necessary, we may assume that the sequence $(\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }})_{\\nu }$ converges uniformly on compact subsets of $B$ to a holomorphic map $\\Psi ^q_{\\infty }:B \\rightarrow \\mathbb {B}^n$ satisfying $\\Psi ^q_{\\infty }(b_{\\infty })= q$ .",
"This implies that $\\tilde{\\Sigma }^{\\nu }_q=\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\zeta _{\\nu }}(B)$ converges to $\\tilde{\\Sigma }^{\\infty }_q:=\\Psi ^q_{\\infty }(B)$ and $\\tilde{\\Sigma }^{\\infty }_q$ is a totally geodesic complex submanifold of $\\mathbb {B}^n$ .",
"We finally prove Proposition 3.4 For every $q \\in \\mathbb {B}^n \\cap (\\lbrace 0^{\\prime }\\rbrace \\times $ there exists a totally geodesic complex submanifold $\\tilde{\\Sigma }^{\\infty }_q$ of $\\mathbb {B}^n$ passing through $q$ .",
"For every $\\nu $ , let $\\pi _{\\nu } : D \\rightarrow \\Sigma _{\\nu }$ be given by $\\forall (z,\\zeta ) \\in B \\times \\Delta ,\\ \\pi _{\\nu }(F(z,\\zeta )) = F\\left(z,1-\\frac{1}{\\nu }\\right)$ and let $\\begin{array}{ccccc}\\tilde{\\pi }_{\\nu } & : & \\Omega _{\\nu } & \\rightarrow & \\Sigma _0^{\\nu }\\\\& & z & \\mapsto & \\varphi _{\\nu } \\circ \\Phi \\circ \\pi _{\\nu } \\circ \\Phi ^{-1} \\circ \\varphi _{\\nu }^{-1}\\end{array}.$ Since $\\varphi _{\\nu } \\circ \\Phi (F(z_0,1-\\frac{1}{\\nu })) = 0$ according to (REF ), we have $\\tilde{\\pi }_{\\nu }(0) = 0$ for every $\\nu $ .",
"Hence we may extract from $(\\tilde{\\pi }_{\\nu })_{\\nu }$ a subsequence, still denoted $(\\tilde{\\pi }_{\\nu })_{\\nu }$ , that converges to a holomorphic map $\\tilde{\\pi }_{\\infty } : \\mathbb {B}^n \\rightarrow \\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ .",
"Moreover we have: Proposition 3.5 For every $q \\in \\lbrace 0\\rbrace \\times \\Delta $ , the restriction of $\\tilde{\\pi }_{\\infty }$ to $\\tilde{\\Sigma }^{\\infty }_q$ is a biholomorphism from $\\tilde{\\Sigma }^{\\infty }_q$ to $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ .",
"Proof of Proposition REF .",
"By the very definition of $\\tilde{\\pi }_{\\nu }$ , the restriction of $\\tilde{\\pi }_{\\nu }$ to $\\tilde{\\Sigma }_q^{\\nu }$ is a biholomorphism from $\\tilde{\\Sigma }_q^{\\nu }$ to $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ for every $\\nu $ .",
"Moreover, $\\tilde{\\Sigma }_q^{\\nu }$ converges to $\\tilde{\\Sigma }_q^{\\infty }$ for the Hausdorff distance.",
"Finally, we have for every $\\nu \\ge 1$ : $\\pi _{\\nu } \\circ \\Phi ^{-1} \\circ \\varphi _{\\nu }^{-1}(q) = \\left(b_{\\nu },1-\\frac{1}{\\nu }\\right) = F_{\\nu }(b_{\\nu }).$ Since $\\lim _{\\nu \\rightarrow \\infty }b_{\\nu } = b_{\\infty } \\in B$ and since the sequence $(\\varphi _{\\nu } \\circ \\Phi \\circ F_{\\nu })_{\\nu }$ converges, uniformly on compact subsets of $B$ to $\\Psi _{\\infty }$ (see the proof of Lemma REF ) we obtain that: $\\tilde{\\pi }_{\\infty }(q) = \\lim _{\\nu \\rightarrow \\infty }\\varphi _{\\nu } \\circ \\Phi \\circ \\pi _{\\nu } \\circ \\Phi ^{-1} \\circ \\varphi _{\\nu }^{-1}(q) = \\Psi _{\\infty }(b_{\\infty }) \\in \\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace .$ Hence if $g_{\\nu }$ denotes the inverse of the restriction of $\\tilde{\\pi }_{\\nu }$ to $\\tilde{\\Sigma }^{\\infty }_q$ then $g_{\\nu }$ is defined on $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ and the sequence ($g_{\\nu })_{\\nu }$ converges, uniformly on compact subsets of $\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace $ , to some holomorphic map $g_{\\infty } : \\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace \\rightarrow \\tilde{\\Sigma }^{\\infty }_q$ such that $g_{\\infty } \\circ \\tilde{\\pi }_{\\infty } = id_{\\tilde{\\Sigma }^{\\infty }_q}$ and $\\tilde{\\pi }_{\\infty } \\circ g_{\\infty }= id_{|\\mathbb {B}^{n-1} \\times \\lbrace 0\\rbrace }$ .",
"$\\square $ Finally, let $q \\ne q^{\\prime }$ be two points in $\\mathbb {B}^n$ .",
"By construction, for every $\\nu \\ge 1$ , the intersection between the totally geodesics submanifolds $\\tilde{\\Sigma }_q^\\nu $ and $\\tilde{\\Sigma }_{q^{\\prime }}^\\nu $ is empty.",
"Since $(\\tilde{\\Sigma }_q^\\nu )_{\\nu }$ converges to $\\tilde{\\Sigma }_q^{\\infty }$ and $\\tilde{\\Sigma }_{q^{\\prime }}^\\nu $ converges to $\\tilde{\\Sigma }_{q^{\\prime }}^{\\infty }$ , it follows from the positivity of intersection that: $\\tilde{\\Sigma }_q^{\\infty } \\cap \\tilde{\\Sigma }_{q^{\\prime }}^{\\infty } = \\emptyset .$ Now Proposition REF and Proposition REF give a contradiction, according to Theorem REF .",
"$\\Box $ We end the note by studying some metric properties of $D$ .",
"We assume that $B$ is complete hyperbolic and that $B$ admits an exhaustion $(B_k)_{k \\in \\mathbb {N}}$ : $B_{k} \\subset \\subset B_{k+1}$ for every $k$ and $B=\\sup _{k \\in \\mathbb {N}} B_k$ , such that $B_k$ is complete (Kobayashi) hyperbolic for every $k$ .",
"Proposition 3.6 The domain $D$ is complete (Kobayashi) hyperbolic.",
"Proof of Proposition REF .",
"It is proved in [3] that for every $k$ , $D_k:=F(B_k \\times \\Delta )$ is complete hyperbolic.",
"Let $Z^0=(z_0,w_0), \\ Z^{\\nu }=(z_{\\nu },w_{\\nu }) \\in B \\times \\Delta $ be such that $\\lim _{\\nu \\rightarrow \\infty }d^K_B(z_0,z_{\\nu }) = \\infty $ .",
"Then: $\\forall \\nu \\ge 1,\\ d^K_D(F(Z^0),F(Z^{\\nu })) = d^K_B(z_0,z_{\\nu }) \\longrightarrow _{\\nu \\rightarrow \\infty } \\infty .$ Hence, to prove that $D$ is complete hyperbolic, it is sufficient to prove that if $(z_{\\nu })_{\\nu } \\subset \\subset B_{k_0}$ for some $k_0 \\in \\mathbb {N}$ and $|w_{\\nu }| \\longrightarrow _{\\nu \\rightarrow \\infty }1$ , then $d^K_D(F(Z^0),F(Z^{\\nu })) \\longrightarrow _{\\nu \\rightarrow \\infty } \\infty $ .",
"Assume, to get a contradiction, that there exists $c > 0$ such that $d^K_D(F(Z^0),F(Z^{\\nu })) \\le c$ for every $k \\in \\mathbb {N}$ (extracting a subsequence if necessary).",
"There exists $k_1 \\ge k_0$ such that the set $\\lbrace y \\in D /\\ d^K_D(y,F(Z^0)) < c +1\\rbrace $ is contained in $D_{k_1}$ according to (REF ).",
"Moreover, it follows from Lamma 5.1 in [7] that: $d^K_{D_{k_1}}(F(Z^0),F(Z^{\\nu })) \\le \\frac{1}{\\tanh (1)}d^K_D(F(Z^0),F(Z^{\\nu })).$ This contradicts the fact that $D_{k_1}$ is complete hyperbolic.",
"$\\Box $"
]
] | 1612.05765 |
[
[
"The Bayesian analysis of contingency table data using the bayesloglin R\n package"
],
[
"Abstract For log-linear analysis, the hyper Dirichlet conjugate prior is available to work in the Bayesian paradigm.",
"With this prior, the MC3 algorithm allows for exploration of the space of models to try to find those with the highest posterior probability.",
"Once top models have been identified, a block Gibbs sampler can be constructed to sample from the posterior distribution and to estimate parameters of interest.",
"Our aim in this paper, is to introduce the bayesloglin R package \\citep{R} which contains functions to carry out these tasks."
],
[
"Introduction",
"Data in the form of a contingency table arise when individuals are cross classified according to a finite number of criteria.",
"Log-linear modeling (see e.g., [1], [4], or [3]) is a popular and effective methodology for analyzing such data enabling the practitioner to make inferences about dependencies between the various criteria.",
"For hierarchical log-linear models, the interactions between the criteria can be represented in the form of a graph; the vertices represent the criteria and the presence or absence of an edge between two criteria indicates whether or not the two are conditionally independent [11].",
"This kind of graphical summary greatly facilitates the interpretation of a given model.",
"For log-linear analysis, we can use the conjugate prior of [14] to work in the Bayesian paradigm.",
"With this prior, the MC3 algorithm of [13] allows for exploration of the space of models to try to find those with the highest posterior probability.",
"Once top models have been identified, a block Gibbs sampler can be constructed to sample from the posterior distribution and to estimate parameters of interest.",
"Our aim in this paper, is to introduce the bayesloglin R package [17] to carry out these tasks.",
"The outline of this paper is as follows: In section 2, we develop the notation for hierarchical log-linear models which is based on [12].",
"In section 3, we give the conjugate prior for hierarchical models under Poisson sampling.",
"In section 4, we describe the MC3 algorithm for searching the spaces of hierarchical, graphical, and decomposable models.",
"Section 5 deals with Gibbs sampling and section 6 gives some exact results for the normalizing constant and the mean and variance of the log-linear parameters for decomposable models.",
"In section 7 we illustrate the use of the bayesloglin package for analyzing the often studied Czech autoworkers data from [9]."
],
[
"Preliminaries",
"The notation in this section is adapted from [12] with some minor changes to move from multinomial to Poisson sampling.",
"Let $V$ be a finite set of indices representing $|V|$ criteria.",
"We assume that the criterion labeled by $v\\in V$ can take values in a finite set $\\mathcal {I}_{v}$ .",
"The resulting counts are gathered in a contingency table such that $\\mathit {\\mathcal {I}=}\\prod _{v\\in V}\\mathcal {I}_{v}$ is the set of cells $i=\\left(i_{v},v\\in V\\right)$ .",
"The vector of cell counts is denoted $n=\\left(n(i),i\\in \\mathcal {I}\\right)$ with corresponding mean $m(i)=E(n)=\\left(m(i),i\\in \\mathcal {I}\\right)$ .",
"For $D\\subset V,$ $\\mathcal {I}_{D}=\\prod _{v\\in D}\\mathcal {I}_{v}$ is the set of cells $i_{D}=(i_{v},v\\in D)$ in the $D$ -marginal table.",
"The marginal counts are $n(i_{D})=\\sum _{j:j_{D}=i_{D}}n(j)$ with $m(i_{D})=E\\left(n(i_{D})\\right)$ .",
"Let $\\mathcal {D}$ be a family of subsets of $V$ such that $D\\in \\mathcal {D}$ and $D_{1}\\subset D$ implies that $D_{1}\\in \\mathcal {D}$ .",
"We will assume that $\\cup _{D\\in \\mathcal {D}}D=V$ .",
"The hierarchical log-linear model generated by $\\mathcal {D}$ is $\\log m(i)=\\sum _{D\\in \\mathcal {D}}\\lambda _{D}(i)$ where $m(i)$ is assumed positive and $\\lambda _{D}(i)$ is a real valued function that depends on $i$ only through $i_{D}$ .",
"We now select a select a special element in each $\\mathcal {I}_{v}$ .",
"For convenience, we denote it 0.",
"We also denote 0 in $i$ the cell with all its components equal to 0.",
"The choice of special element 0 in each $\\mathcal {I}_{v}$ is arbitrary.",
"If $i\\in \\mathcal {I}$ , the support of $i$ is the subset of $V$ defined as $S(i)=\\left\\lbrace v\\in V,i_{v}\\ne 0\\right\\rbrace $ .",
"We let $J=\\left\\lbrace j\\in \\mathcal {I},S(j)\\in \\mathcal {D}\\right\\rbrace $ and define the notation $j\\triangleleft i$ for $i\\in \\mathcal {I}$ and $j\\in J$ to mean that $j_{S(j)}=i_{S(j)}$ .",
"By convention, we say that $0\\triangleleft i$ for any $i\\in \\mathcal {I}$ .",
"For any $a\\in \\mathcal {D}$ , we also define the sub-model $J_{a}=\\left\\lbrace j\\in J:S(j)\\subseteq a\\right\\rbrace $ .",
"Let $\\left(e_{j},j\\in J\\right)$ be the canonical basis of $R^{J}.$ For all $i\\in \\mathcal {I},$ we define $f_{i}\\in R^{J}$ by $f_{i}=\\sum _{j\\in J,j\\triangleleft i}e_{j}.$ The baseline constrained hierarchical log-linear model generated by $\\mathcal {D}$ has the unique representation $\\log m(i)=\\sum _{j\\in J:j\\triangleleft i}\\theta _{j}=\\left\\langle f_{i},\\theta \\right\\rangle $ for $i\\in \\mathcal {I}$ and $\\theta =\\left(\\theta _{j},j\\in J\\right)\\in R^{J}$ .",
"In matrix notation, we have $\\log m=X\\theta $ where $X$ is an $\\mathcal {I}\\times J$ design matrix of full column rank with rows $\\left\\lbrace f_{i},i\\in \\mathcal {I}\\right\\rbrace $ .",
"It is worth noting that $X$ is a binary 0/1 matrix with a first column that is all 1's."
],
[
"Prior distribution under Poisson sampling",
"We assume that the components of $n$ are independent and follow a Poisson distribution.",
"The sufficient statistic $t=X^{T}n$ has a probability distribution in the natural exponential family $f(t)=\\exp \\left(\\left\\langle \\theta ,t\\right\\rangle -\\sum _{i\\in \\mathcal {I}}\\exp \\left(\\left\\langle f_{i},\\theta \\right\\rangle \\right)\\right)\\nu \\left(dt\\right)$ with respect to a discrete measure $\\nu $ that has convex support $C^{p}=\\left\\lbrace \\sum _{i\\in \\mathcal {I}}y(i)f_{i},y(i)\\ge 0,i\\in \\mathcal {I}\\right\\rbrace =\\mathrm {cone}\\left\\lbrace f_{i},i\\in \\mathcal {I}\\right\\rbrace $ i.e.",
"the convex cone generated by the rows of the design matrix $X$ .",
"The Diaconis and Ylvisaker [7] conjugate prior with respect to the Lebesgue measure for the log-linear parameters is $f(\\theta )=I(r,\\alpha )^{-1}\\exp \\left(\\alpha \\left\\langle r,\\theta \\right\\rangle -\\alpha \\sum _{i\\in \\mathcal {I}}\\exp \\left\\langle f_{i},\\theta \\right\\rangle \\right)$ where $I(r,\\alpha )=_{\\theta \\in R^{J}}\\exp \\left(\\alpha \\left\\langle r,\\theta \\right\\rangle -\\alpha \\sum _{i\\in \\mathcal {I}}\\exp \\left\\langle f_{i},\\theta \\right\\rangle \\right)d\\theta $ and is proper when $\\alpha >0$ and $r=X^{T}y$ for some $y>0$ i.e.",
"$r$ is in the relative interior of $C^{p}$ .",
"The Bayes factor for comparing two models $J_{1}$ and $J_{2}$ is $B_{12}=\\frac{P(t|J_{1})}{P(t|J_{2})}=\\frac{I_{1}\\left(\\frac{t+\\alpha r}{1+\\alpha },1+\\alpha \\right)/I_{1}(r,\\alpha )}{I_{2}\\left(\\frac{t+\\alpha r}{1+\\alpha },1+\\alpha \\right)/I_{2}\\left(r,\\alpha \\right)}$"
],
[
"The MC3 algorithm for model selection",
"The Bayesian paradigm to model selection involves choosing models with high posterior probability from a set $\\mathcal {M}$ of competing models.",
"We associate with each model $J\\in \\mathcal {M}$ a neighbourhood $\\mathrm {nbd}\\left(J\\right)\\subset \\mathcal {M}$ .",
"The $MC^{3}$ algorithm proposed by [13] constructs an irreducible Markov chain with state space $\\mathcal {M}$ and and equillibrium distribution $\\left\\lbrace p(J|n):J\\in \\mathcal {M}\\right\\rbrace $ where $P(J|t)$ is the posterior probability of $J$ .",
"We assume that all models are apriori equally likely; hence $P(J|t)$ is proportional to the marginal likelihood $P(t|J)=I\\left(t+r,\\alpha +1\\right)/I(r,\\alpha )$ .",
"If the chain is in state $J$ at we draw a candidate model $J^{\\prime }$ from a uniform distribution on $\\mathrm {nbd}(J)$ .",
"The chain moves to $J^{\\prime }$ with probability $\\mathrm {min}\\left\\lbrace 1,\\frac{P\\left(t|J\\right)/\\mathrm {\\#nbd}(J)}{P(t|J^{\\prime })/\\mathrm {\\#nbd}(J^{\\prime })}\\right\\rbrace $ where $\\#\\mathrm {nbd}(J)$ denotes the number of neighbours of $J$ .",
"Otherwise the chain does not move.",
"The evaluation of the marginal likelihoods and the specification of model neighbourhoods is done with respect to the particular properties of the set of candidate models considered.",
"Hierarchical log-linear models.",
"We calculate the marginal likelihood through the Laplace approximation to the normalizing constants for the prior and posterior distribution of the log-linear model parameters.",
"The neighbourhood of a hierarchical model $J$ consists of the hierarchical models obtained from $J$ by adding one of its dual generators (i.e.",
"minimal terms not present in the model) or deleting one of its generators (i.e.",
"maximal terms present in the model).",
"For details see [9] and [6].",
"Graphical log-linear models.",
"We evaluate the marginal likelihood using the Laplace approximation to the normalizing constants as we do in the hierarchical case.",
"The neighbourhood of a graphical model with corresponding graph $G$ consists of those models whose independence graphs are obtained from $G$ by adding or removing one edge.",
"Decomposable log-linear models.",
"In this case, the marginal likelihood can be obtained explicitly.",
"See Section 6 for the formula.",
"The neighbourhood of a decomposable model with corresponding graph $G$ consists of those models whose independence graphs are decomposable and are obtained by adding or deleting one edge from $G$ ."
],
[
"Gibbs sampling",
"Our aim in this section is to develop a blocked Gibbs sampler to sample from the posterior distribution and to estimate parameters of interest.",
"We begin by partitioning the cells and the prior into blocks.",
"For $a\\in \\mathcal {D}$ we define the sets $B_{i_{a}}=\\left\\lbrace j\\in \\mathcal {I}:j_{a}=i_{a}\\right\\rbrace $ for $i_{a}\\in \\mathcal {I}_{a}$ .",
"These sets are disjoint and partition $\\mathcal {I}.$ Define the vectors $\\chi _{i_{a}},i_{a}\\in \\mathcal {I}_{a}$ with $\\mathcal {\\chi }_{i_{a}}(i)={\\left\\lbrace \\begin{array}{ll}1 & i\\in B_{i_{a}}\\\\0 & \\mathrm {otherwise}\\end{array}\\right.",
"}$ and the matrix $\\chi $ with columns $x_{i_{a}}(i)$ .",
"We can then write: $f_{i}=f_{(i_{a},i_{a}^{c})}=f_{(i_{a},0)}+f_{(0,i_{a}^{c})}$ and $\\theta =\\left(\\theta _{a},\\theta _{a^{c}}\\right)$ where $\\theta _{a}=\\left(\\theta _{j}:S(j)\\subseteq a\\right)$ and $\\theta _{a^{c}}=\\left(\\theta _{j}:S(j)\\lnot \\subset a\\right)$ .",
"The marginal counts $m(i_{a}),i_{a}\\in \\mathcal {I}_{a}$ follow a log-linear model with $m\\left(i_{a}\\right) & = & \\sum _{i\\in B_{i_{a}}}\\exp \\left(\\left\\langle f_{i},\\theta \\right\\rangle \\right)\\\\& = & \\exp \\left(\\left\\langle f_{(i_{a},0)},\\theta \\right\\rangle \\right)\\sum _{i\\in B_{i_{a}}}\\exp \\left(\\left\\langle f_{(0,i_{a^{c}})},\\theta \\right\\rangle \\right)$ and, taking logs, $\\log m\\left(i_{a}\\right) & = & \\sum _{i\\in B_{i_{a}}}\\left\\langle f_{(i_{a},0)},\\theta \\right\\rangle +\\log \\left(\\sum _{i\\in B_{i_{a}}}\\exp \\left(\\left\\langle f_{(0,i_{a^{c}})},\\theta \\right\\rangle \\right)\\right)$ Let $m_{a}=\\left(m(i_{a}),i_{a}\\in \\mathcal {I}_{a}\\right)$ and partition the matrix $X$ such that $X=\\left[X_{a},X_{\\bar{a}}\\right]$ where $X_{a}$ is a matrix made up of those columns of $X$ corresponding to $j$ such that $S(j)\\subseteq a$ and $X_{\\bar{a}}$ is a matrix with all the other columns.",
"Then, in matrix notation, $\\log m_{a}=\\left(\\frac{\\chi ^{T}X_{a}}{|\\mathcal {I}\\mathcal {\\backslash I}_{a}|}\\right)\\theta _{a}+\\log \\left(\\chi ^{T}\\exp \\left(X_{\\bar{a}}\\theta _{\\bar{a}}\\right)\\right)$ Returning to the prior, parametrized temporarily in terms of $m$ , we can partition $f$ as $f(m|J) & \\propto & \\exp \\left(\\alpha \\left\\langle y,\\log m\\right\\rangle -\\alpha \\sum _{i\\in \\mathcal {I}}m(i)\\right)\\\\& = & \\left\\lbrace \\prod _{i_{a}\\in \\mathcal {I}_{a}}\\prod _{i\\in B_{i_{a}}}\\left(\\frac{m(i)}{m\\left(i_{a}\\right)}\\right)^{\\alpha y(i)}\\right\\rbrace \\left\\lbrace \\prod _{i_{a}\\in \\mathcal {I}_{a}}m\\left(i_{a}\\right)^{\\alpha y\\left(i_{a}\\right)}\\exp \\left(-\\alpha m\\left(i_{a}\\right)\\right)\\right\\rbrace \\\\& = & f\\left(m_{\\bar{a}}\\right)f\\left(m_{a}|m_{\\bar{a}}\\right)$ and we see that $f\\left(m_{a}|m_{\\bar{a}}\\right)$ is the product of independent $\\mathrm {Gamma}\\left(1+\\alpha y\\left(i_{a}\\right),1/\\alpha \\right),i_{a}\\in \\mathcal {I}_{a}$ distributions.",
"Since it is easy to generate from $f\\left(m_{a}|m_{\\bar{a}}\\right)$ for each $a\\in \\mathcal {D}$ , a blocked Gibbs sampler [10] is feasible to sample from $f(\\theta )$ .",
"Following [8], we begin by choosing an arbitrary initial value of $\\theta ^{(0)}$ .",
"For a given value of $\\theta ^{(k)}$ , we update as follows: Generate independent $m\\left(i_{a}\\right)\\sim \\mathrm {Gamma}\\left(\\alpha y\\left(i_{a}\\right),1/\\alpha \\right)$ random variables for all $a\\in \\mathcal {D}$ and $i_{a}\\in \\mathcal {I}_{a}$ .",
"For each $a\\in \\mathcal {D}$ , in any arbitrary order set, $\\theta _{a}^{(k)}=\\left(\\frac{\\chi ^{T}X_{a}}{|\\mathcal {I}\\mathcal {\\backslash I}_{a}|}\\right)^{-1}\\left(\\log \\left(m_{a}\\right)-\\log \\left(x^{T}\\exp \\left(X_{\\bar{a}}\\theta _{\\bar{a}}\\right)\\right)\\right)$ using the most recent value of $\\theta _{\\bar{a}}$ available.",
"After a suitable burn-in, the resulting samples come from $f(\\theta )$ .",
"We note that the above Gibbs sampler is also known as the Bayesian Iterative Proportional Fitting algorithm.",
"See [8],[2],[15], and[16] for more details."
],
[
"Some exact results for decomposable models",
"For decomposable models, some exact results exist for the normalizing constant and the mean and variance of the log-linear parameters.",
"Let us reconsider the prior defined in section 3 as $f(\\theta )=I(r,\\alpha )^{-1}\\exp \\left(\\alpha \\left\\langle r,\\theta \\right\\rangle -\\alpha \\sum _{i\\in \\mathcal {I}}\\exp \\left\\langle f_{i},\\theta \\right\\rangle \\right)$ with $I(r,\\alpha )=_{\\theta \\in R^{J}}\\exp \\left(\\alpha \\left\\langle r,\\theta \\right\\rangle -\\alpha \\sum _{i\\in \\mathcal {I}}\\exp \\left\\langle f_{i},\\theta \\right\\rangle \\right)d\\theta $ where $\\alpha >0$ and $r=\\left(r_{j},j\\in J\\right)\\in \\mathrm {ri\\left(C_{p}\\right)}$ .",
"Then $\\mathrm {E}\\left(\\alpha \\theta \\right)=\\frac{\\partial \\log I(r,\\alpha )}{\\partial r}$ and $\\mathrm {Cov}(\\alpha \\theta )=\\frac{\\partial ^{2}\\log I(r,\\alpha )}{\\partial r^{2}}$ In the case of log-linear models where $m$ is Markov with respect to a decomposable graph $G=(V,E)$ , with vertex set $V$ and edge set $E$ , an explicit formula exists for $I(r,\\alpha )$ .",
"Let $C$ denote the set of cliques and $S$ the set of minimal vertex separators.",
"For a given $s\\in S$ , let $V_{1},V_{2},...,V_{p}$ be the connected components of the subgraph $G_{V\\backslash s}$ and $q$ be the number of $j=1,2,...,p$ such that $s$ is not a clique of $S\\cup V_{j}$ .",
"Then $\\nu (s)=q-1$ is called the multiplicity of $s$ and $\\sum _{s\\in S}\\nu (s)=|C|-1$ [11].",
"Based on proposition 4.2 of [14], adapted to Poisson sampling, we have $I(r,\\alpha )=\\alpha ^{-\\alpha \\sum _{i\\in \\mathcal {I}}y(i)}\\frac{\\prod _{c\\in C}\\prod _{i_{c}\\in \\mathcal {I}_{c}}\\Gamma \\left(\\alpha y(i_{c})\\right)}{\\prod _{s\\in S}\\prod _{i_{s}\\in \\mathcal {I}_{s}}\\left\\lbrace \\Gamma \\left(\\alpha y(i_{s})\\right)\\right\\rbrace ^{\\nu (s)}}$ Taking logs and differentiating with respect to $r$ gives $E(\\theta )=-\\frac{d\\sum _{i\\in \\mathcal {I}}y(i)}{dr}\\log \\alpha +\\sum _{c\\in C}\\sum _{i_{c}\\in \\mathcal {I}_{c}}\\psi \\left(\\alpha y(i_{c})\\right)\\frac{dy\\left(i_{c}\\right)}{dr}-\\sum _{s\\in C}\\sum _{i_{s}\\in \\mathcal {I}_{s}}\\nu (s)\\psi \\left(\\alpha y(i_{s})\\right)\\frac{dy\\left(i_{s}\\right)}{dr}$ where $\\psi $ is the digamma function.",
"Note that the derivatives in the right hand side of $\\mathrm {E}(\\theta )$ are vectors.",
"In particular, $d\\sum _{i\\in \\mathcal {I}}y(i)/dr=\\left(1,0,...,0\\right)^{T}$ since $r_{0}=\\sum _{i\\in \\mathcal {I}}y(i)$ .",
"Differentiating once more we have $\\mathrm {Cov}\\left(\\theta \\right)=\\sum _{c\\in C}\\sum _{i_{c}\\in \\mathcal {I}_{c}}\\psi _{1}\\left(\\alpha y(i_{c})\\right)\\frac{dy\\left(i_{c}\\right)}{dr}\\frac{dy\\left(i_{c}\\right)}{dr}^{T}-\\sum _{s\\in S}\\sum _{i_{s}\\in \\mathcal {I}_{s}}\\nu (s)\\psi _{1}\\left(\\alpha y(i_{s})\\right)\\frac{dy\\left(i_{s}\\right)}{dr}\\frac{dy\\left(i_{s}\\right)}{dr}^{T}$ with $\\psi _{1}$ being the trigamma function.",
"We note that for decomposable models, the subgraphs $G_{c},c\\in C$ and $G_{s},s\\in S$ are all complete and we have a saturated model on those subgraphs.",
"For $a\\in C\\cup S$ , it is easy to find $d\\left(y(i_{a})\\right)/dr,i_{a}\\in \\mathcal {I}_{a}$ by inverting the design matrix for the model $J_{a}$ ."
],
[
"The bayesloglin R package.",
"The bayesloglin package includes the $2^{6}$ Czech autoworkers data from [9].",
"This cross-classification of 1841 men gives six potential risk-factors for coronary thrombosis: (a) smoking, (b) strenuous mental work, (c) strenuous physical work, (d) systolic blood pressure, (e) ratio of beta and alpha lipoproteins and (f) family anamnesis of coronary heart disease.",
"Currently, bayesloglin only allows choice of the hyperparameter $\\alpha $ and sets $y(i)=1/|\\mathcal {I}|$ for each $i\\in \\mathcal {I}$ .",
"Consequently, $r_{0}=\\sum _{i\\in \\mathcal {I}}y(i)=1$ .",
"The required R code to search for the top decomposable, graphical, and hierarchical log-linear models is: > data(czech) > s1 <- MC3 (init = NULL, alpha = 1, iterations = 5000, replicates = 1, data = czech, mode = \"Decomposable\") > s2 <- MC3 (init = NULL, alpha = 1, iterations = 5000, replicates = 1, data = czech, mode = \"Graphical\") > s3 <- MC3 (init = NULL, alpha = 1, iterations = 5000, replicates = 1, data = czech, mode = \"Hierarchical\") The top models in terms of posterior probability are > head(s1, n = 5) formula logPostProb 1 [a,c,e][b,c][d,e][f] 5271.975 2 [a,c,e][a,d,e][b,c][f] 5271.103 3 [a,c,e][a,d][b,c][f] 5271.077 4 [a,c][b,c][b,e][d,e][f] 5270.549 5 [a,c,e][b,c][b,f][d,e] 5270.394 > head(s2, n = 5) formula logPostProb\t\t 1 [a,c][a,d,e][b,c][b,e][f] 7122.398 2 [a,c][a,e][b,c][b,e][d,e][f] 7121.580 3 [a,c][a,d,e][b,c][b,e][b,f] 7121.374 4 [a,c][a,d][a,e][b,c][b,e][f] 7120.683 5 [a,c][a,e][b,c][b,e][b,f][d,e] 7120.556 > head(s3, n = 4) formula logPostProb\t\t 1 [a,c][a,d][a,e][b,c][c,e][d,e][f] 7125.171 2 [a,c][a,d][a,e][b,c][b,e][d,e][f] 7124.704 3 [a,c][a,d][a,e][b,c][b,e][c,e][d,e][f] 7124.229 \t 4 [a,c][a,d][a,e][b,c][b,f][c,e][d,e] 7124.147 These results match those obtained by the same methods in [14].",
"Consider the top hierarchical model $[a,c][a,d][a,e][b,c][c,e][d,e][f]$ .",
"We can use the function $\\mathrm {\\mathtt {gibbsSampler}}$ to sample from the posterior and obtain estimates of the mean and variances of the log-linear parameters.",
"We use a burn-in of 5000 iterations.",
"> formula <- freq ~ a*c + a*d + a*e + b*c + c*e + d*e + f > s <- gibbsSampler (formula, alpha = 1, data = czech, nSamples = 15000, verbose = T) > postMean <- colSums(s[5000:15000,]) / 10000 > postCov <- cov(s[5000:15000,]) > postVar <- diag(postCov) The values of $\\mathrm {\\mathtt {postMean}}$ and $\\mathtt {\\mathrm {\\mathtt {postVar}}}$ are > postMean (Intercept) a1 c1 b1 d1 e1 3.0915633 -0.4150080 1.0199107 0.9010453 -0.2877865 -0.4890538 f1 a1:c1 b1:c1 a1:d1 a1:e1 c1:e1 -1.8057132 0.5409632 -2.8017859 -0.3542662 0.4871123 -0.4479492 d1:e1 0.3784125 > postVar (Intercept) a1 c1 b1 d1 e1 0.006921940 0.008033988 0.008498167 0.005310040 0.005564232 0.008184625 f1 a1:c1 b1:c1 a1:d1 a1:e1 c1:e1 0.004433024 0.009185728 0.015035403 0.009219168 0.009280780 0.009133959 d1:e1 0.009298324 We now consider the decomposable model $[a,c,e][b,c][d,e][f]$ .",
"The $\\mathrm {\\mathtt {findPostMean}}$ and $\\mbox{$\\mathtt {\\mathtt {findPostCov}}$}$ functions can compute the posterior mean and covariance matrix, which for decomposable models, is available in closed form.",
"In R we have > formula <- freq ~ a*c*e + b*c + d*e + f > postMean <- findPostMean (formula, alpha = 1, data = czech) > postCov <- findPostCov(formula, alpha = 1, data = czech) > postVar <- diag(postCov) > postMean (Intercept) b1 c1 a1 e1 d1 3.1561271 0.9002899 1.0149757 -0.5565110 -0.4621862 -0.4387784 f1 b1:c1 a1:c1 a1:e1 c1:e1 d1:e1 -1.8051306 -2.8012942 0.5494842 0.4645452 -0.4380842 0.3412027 a1:c1:e1 -0.0194745 > postVar (Intercept) b1 c1 a1 e1 d1 0.006563014 0.005252849 0.009530313 0.008807288 0.009375078 0.003956279 f1 b1:c1 a1:c1 a1:e1 c1:e1 d1:e1 0.004478660 0.014932109 0.015834157 0.018016838 0.018531263 0.009099995 a1:c1:e1 0.037264994 The reader can verify that the Gibbs sampler gives a close approximation to the exact values for this model.",
"Acknowledgements.",
"The author is grateful to the creators of the $\\mathtt {igraph}$ R package [5] which was used extensively for manipulating graphs.",
"Special thanks also to Adrian Dobra, whose C++ code for representing hierarchical log-linear models, was adapted for use in R."
]
] | 1612.05501 |
[
[
"Microscopic Muscle Image Enhancement"
],
[
"Abstract We propose a robust image enhancement algorithm dedicated for muscle fiber specimen images captured by optical microscopes.",
"Blur or out of focus problems are prevalent in muscle images during the image acquisition stage.",
"Traditional image deconvolution methods do not work since they assume the blur kernels are known and also produce ring artifacts.",
"We provide a compact framework which involves a novel spatially non-uniform blind deblurring approach specialized to muscle images which automatically detects and alleviates degraded regions.",
"Ring artifacts problems are addressed and a kernel propagation strategy is proposed to speedup the algorithm and deals with the high non-uniformity of the blur kernels on muscle images.",
"Experiments show that the proposed framework performs well on muscle images taken with modern advanced optical microscopes.",
"Our framework is free of laborious parameter settings and is computationally efficient."
],
[
"Introduction",
"Skeletal muscle is an extremely adaptive tissue that is able to change size depending upon external stimuli.",
"Increases in muscle mass in response to resistance training and losses in mass caused by disuse or associated with chronic diseases such as cancer and HIV are primarily due to hypertrophy or atrophy of individual muscle fibers, respectively, rather than addition or loss of fibers [1].",
"Thus, The ability to accurately and efficiently quantify the morphological characteristics of muscle cells, such as cross-sectional areas (CSAs), is vital for assessing muscle function, since muscle mass is the primary determinant of muscle strength [2], [3].",
"One example is the Idiopathic Inflammatory Myophathy (IIM) detection, which is characterized by weakness and inflammation of skeletal muscles [4].",
"A very important prerequisite for accurate CSA quantification is the high-quality microscopic image acquisition, after the muscle specimen is stained with H&E or fluorescence.",
"Unfortunately, the non-flat surface of muscle specimens, which involves massive tissue protuberances and concavities, impedes the users holding the desired observatory regions to uniformly fall into the focus plane of the microscope.",
"Nevertheless, modern advanced optical microscopes for fluorescence or H&E specimens often involve a lot of laborious manual configurations of confusing parameters.",
"It is usually a prerequisite for the users to be well trained and extensively practiced in order to properly adjust those settings.",
"Thus, improper or even erroneous settings of parameters are inevitable.",
"Blur/defocus is consequently produced occasionally at some regions, or the entire scope of the captured muscle microscopic images.",
"Vagueness and ambiguity are observed on these regions not only degrading the visual quality but also hindering the success of the subsequent computer aided diagnosis operations potentially included in an automatic diagnosis system such as segmentation of cells and the detection of a certain disease.",
"In this work, we propose a compact framework to enhance the visual quality of the poorly captured muscle images.",
"The proposed framework involves the following steps: 1) Local blur/defocus kernel estimation.",
"A group of spatially variant blur kernels for part of the muscle image are estimated based on the gradient information under carefully chosen constraints and priors.",
"2) Kernel propagation.",
"The estimated kernels are propagated to their neighborhood and speedup their kernel estimation.",
"3) Non-blind image enhancement.",
"An efficient image deconvolution algorithm involving the most recent findings in image statistics is performed locally based on the estimated kernels.",
"Fast convergence speed is achieved.",
"The proposed method has an automatic design that reduces blur and out of focus in a one-click style.",
"The users are thus liberated from laborious parameter tweaks.",
"Figure: An example of (a) a blur/defocus region observed on a microscopic muscle images and (b) enhanced by the proposed framework.",
"(c) The gradient map of (a) used in the proposed method.",
"Darker color corresponds to lower values.",
"(d) The estimated blur kernel.We do not confine the type of the degradation as specific defocus caused by the limited or misplaced depth of field (DoF) produced by the optical system of the microscope.",
"Rather, we put it into a more general blur model that captures the local linear transforms represented as a (or a group of) freely shaped kernel to cope with the degradation caused by sophisticated blur such as lens aberration [5].",
"The proposed muscle image enhancement is characterized as blind, non-uniform blur kernel estimation and deconvolution with the purposes of both computational efficiency and automation.",
"Conventional methods, such as as Weiner filters [6] and Richard-Lucy deconvolution [7], are parametric since they assume the precise blur kernels are known.",
"In addition, the enhanced images provided by them are often less constrained and thus have strong artifacts.",
"The natural image defocus removal methods such as [8] usually rely on a robust map estimation of Gaussian parameter and the existence of similar patch pairs between the blurry and the good regions.",
"In contrast, the proposed method utilizes the sparsity of the gradient as clue to guide the algorithm automatically produce good quality results.",
"No heuristic computations are involved and the noise is also addressed with proper models.",
"The enhancement framework is thus both robust and automatic.",
"The blur kernel estimation step involved in the proposed framework differs from the existing non-uniform methods in literature significantly.",
"Those works are dedicated to model the spatial motion of a hand-held camera.",
"3D translations, in-plane and/or out-of-plane rotations are involved and are estimated during the energy optimization stage.",
"However, rotations do not happen on a microscopic muscle image.",
"What is more important is that the blur happened in microscopic muscle images is much different from typical natural image blur, exhibiting much more severe spatial variances but less (still exists though) directional blur.",
"Although part of the blur can be understood as out of focus and is proper to be modeled using an exponential family function (such as Gaussian or Poisson), putting such rigid constraints does not perform well for lack of flexibility, especially on muscle images with strong spatial variance.",
"We address this problem by proposing a new spatially variant kernel estimation algorithm.",
"Specifically, a group of spatially invariant kernels are estimated for some randomly chosen local image regions.",
"These estimations are propagated to their neighboring regions serving as initiations of their optimization arguments.",
"This step not only preserves the coherence of the inter-region image structures but also accelerates the computational efficiency drastically.",
"Although we also encourage the estimated kernels to be Gaussian-shaped, the weight assigned to this prior is gradually deducted during the iteration of the algorithm.",
"This brings the benefit that the blur is treated as defocus problem at the first few iterations to produce a reasonable intermediate result.",
"The most recent findings in image statistics suggesting the sparsity constraint on gradient of the image are utilized during the kernel estimation.",
"An efficient deconvolution method is used to recover the blurred muscle images after all kernels are estimated.",
"Although the proposed method can effectively remove the blur and defocus, the recovery of some regions that are severely blurred or defocused cannot rely on only local neighbored pixel information that typical deblur methods utilize since the original image information is mostly lost.",
"We propose to use a patch synthesis algorithm to address this problem.",
"These severely blurred regions are detected via a simple gradient computation after the muscle image is processed via the proposed spatially non-uniform deblur algorithm.",
"Then, a fast non-local similar patch search is used to find out candidates of synthesis for the detected regions.",
"Candidates with both high similarity and good patch-wise coherence preservation are selected and replace the highly degraded patches.",
"Finally, a smoothing algorithm is performed to remove the artifacts produced by during the synthesis step.",
"Extensive experiments are conducted on muscle images captured with advanced optical microscopes where both mild and severe blur happen.",
"We use an Olympus VS 120 advanced optical microscope dedicated to medical specimen observation and capture 44 muscle images.",
"Most images more or less contain some regions showing blur or defocus problems.",
"One example of the enhancement provided by our framework is shown in Figure REF .",
"The users are free of empirically choosing parameters for our algorithm since the framework is completely automatic.",
"The processing time for each three color channel muscle image ranges from two minutes to eleven minutes, depending on the resolution which is normally more than 10m pixels.",
"Results show that the proposed framework is both effective and computationally efficient.",
"The remaining of this paper is organized as follows: We briefly survey relevant works in the current literature for the proposed framework in section .",
"The proposed muscle image enhancement framework is introduced in details in section .",
"We present the experimental results in section .",
"We conclude our paper and give a discussion on the future work in section ."
],
[
"Related Work",
"The most relevant works to the proposed muscle image enhancement framework can be categorized as the single image blind deconvolution.",
"The problem is typically modeled as a blur kernel (or point spread function (PSF)) convolves with the latent, distortion free image.",
"The blur kernel can be a Gaussian-shaped distribution to model out of focuse (defocus), or a motion trajectory to model the blur caused by spatial shifts of a camera during a long time exposure.",
"Blind image deconvolution is considered as extremely difficult due to its severe ill-posedness since both the latent image and the blur kernel are assumed unknown.",
"Traditional methods such as Weiner filters [6] and Richard-Lucy deconvolution [7] are popular due to their simplicity and efficiency.",
"However, these methods cannot work on blind image deconvolution since they assume the blur kernel to the latent image is known.",
"In addition, these works produce large amount of undesired ring-like artifacts during the latent image recovery, even if we assume the kernel is available (hand-designed model with super-tweaked parameters).",
"In the past decade, there has been a significant progress on the blind image deconvolution involving both the phases of blur kernel estimation and latent image recovery.",
"Various constraints are designed on both the latent image and the blur kernel with much more improved results being obtained.",
"The reason of their success is twofold: 1) The constraints made on the kernel and the latent image alleviate the ill-posedness of the blind image deconvolution problem.",
"2) Well designed constraints can effectively suppress the artifacts on the latent image and exclude those unnatural latent images from the solution space.",
"Noise in the estimated blur kernel can be also reduced with a proper constraint on it.",
"The former produces more visually pleasant result while the latter usually leads to more robust kernel estimations.",
"Shan et al.",
"apply the recent findings derived from image statistics that in spatial domain, the natural image gradient should follow a heavy-tailed distribution [9] and make such constraint on their latent image in gradient domain [10].",
"Goldstein et al.",
"use the power law of natural image in frequency domain $|\\hat{I}(\\omega )^2|\\propto ||\\omega ||^{-\\beta },$ where $\\hat{I}$ is the Fourier transform of a natural image, $\\omega $ is the frequency variable and $\\beta $ is assumed to be roughly two [11].",
"Yue et al.",
"further exploit the power spectrum of the natural images and propose a new model for the blur kernel in frequency domain [12].",
"However, the image statistics of both spatial and spectrum domain may not hold for muscle images since they are significantly different from the natural images and often contain more special shape and texture pattern.",
"Weak edges, subtle curves or corners included in muscle images usually convey important pathological meanings and diagnostic clues.",
"The sparsity on the gradient map for the latent image is a more general and robust regularization promising for muscle images.",
"In this context, Whyte et al.",
"adopt $l_1$ sparsity to constrain the gradient [13] of the latent image.",
"Krishnan et al.",
"further improve this term by normalizing the $l_1$ norm with the $l_2$ norm to better express the nature of the blur [14].",
"Xu et.",
"al adopt non-linear penalties to $l_1$ norm [15] and sophisticated $l_0$ norm [16] [17] of the gradient of the latent image.",
"The $l_1$ norm [14] [18] or the $l_2$ norm [19] are typically used to regularize the estimated kernel.",
"Without proper constraints, ring artifacts are usually produced due to outliers/noise of the data or the saturated pixels.",
"Shan et.",
"al propose a heuristic method for ring effect reduction by localizing the smooth regions of the latent image and suppress their gradients [10].",
"Whyte et al.",
"examine the reasons why ring artifacts emerge and propose that it is because of the nonlinear mapping caused by saturated pixels and the lack of constrains when solving on the near-zero domain of the transform made by the kernel [13].",
"They propose to use a mask to filter out those saturated pixels to avoid wrong estimations and also use some regularizations to further forbid those errors from propagating to good estimations.",
"Cho et al.",
"propose a method in a non-blind deconvolution with similar rationale but the mask and the outliers are explicitly modeled and optimized, leading to a better result [20].",
"The context of aforementioned deblur algorithms is spatially uniform which assumes a constant kernel across the entire image.",
"Despite their amazingly good performance [21], this is not true in either natural images taken with a camera or the microscopic muscle images in this paper.",
"Several works [22] [23] [17] extend their kernel models to 3D or 6D degree of freedom (DoM)Suppose the 2D image locates on the plane determined by the $x$ and $z$ axises, the 3D modeling is either rotations along the three axises $\\lbrace \\theta _x,\\theta _y,\\theta _z\\rbrace $ , or the translations on $x$ and $z$ axises and the rotation on visual axis $z$ $\\lbrace \\tau _x,\\tau _y,\\theta _y\\rbrace $ .",
"The 6D modeling is to use both.",
"See [21] for more details., describing a complete motion domain caused by in-plane translations, rotations, and out-plane rotations.",
"However, the performance of the deblur does improve significantly [21].",
"We show in this paper that uniform assumption is good for local regions of the muscle image while non-uniformity must be considered in the global domain."
],
[
"Muscle Image Enhancement",
"In this section, we introduce the proposed spatially non-uniform deblur framework involving blur kernel estimation and muscle image deconvolution.",
"We do not assume the blur kernel for muscle images is known and explicitly estimate it.",
"The proposed kernel estimation algorithm is overall spatially variant but can be separated into a group of spatially invariant estimations.",
"These estimations are made on a few random subregions of the muscle image.",
"Estimated kernels are treated as seeds and are propagated to their neighboring regions as the initializations for kernel estimation.",
"We show in the experimental results that the proposed scheme has fast convergence maintains good coherence of the image structure.",
"An overall flowchart of the proposed method is shown in Fig.",
"REF .",
"Figure: The flowchart of the proposed method."
],
[
"Kernel Estimation",
"In the spatially invariant blur kernel estimation context, one effective Bayesian inference approach, namely, the MAP model, is frequently used.",
"Suppose the latent image (or a subregion) $X$ is blurred with a kernel $k$ with some additive noise $\\epsilon $ that follows a specific statistical distribution.",
"The degradation of blur and defocus on a latent image can be both modeled as convolution computation written as: $I=X\\otimes k+\\epsilon ,$ where $I$ denotes the degraded image, $X$ is the latent distortion-free image, and $\\otimes $ denotes convolution operation.",
"Note that there is an option to reorganize $k$ or $X$ to write the convolution as one matrix multiplies the other as in Eq.",
"REF : $I = AX + \\epsilon ,$ where $A$ is the Toeplitz form of the kernel $k$Each row of $A$ corresponds to a vectorized version of $k$ with a phase shifted by one element (and reversed for convolution)..",
"However, this is not necessary for our algorithm and all terms are written as their original version.",
"The blur kernel $k$ , or PSF, can be either a motion trajectory to model the physical motion blur, or a Gaussian distribution to model defocus, or both (in which the kernel usually shows a blurred version of a motion trajectory).",
"In MAP-estimation framework, deblur algorithms typically assume the pixels of the image are independent and attempt to maximize the posteriori $p(X,k|I)$ : $\\max _{X,k} p(X,k|I)\\propto \\prod ^{N}_{i=1} p(I|x_i,k)P(k)p(X),$ where $p(I|x_i,k)$ is the likelihood of the degraded image and $X=\\lbrace x_i|i=1...N\\rbrace $ where $N$ denotes the number of pixels of the latent image.",
"$p(k)$ , $p(x)$ are the prior probability of the kernel and the latent image, respectively.",
"This form can be interpreted as an energy minimization problem by defining $E(X,k)=-log*(p(X,k|k))$ and write Eq.",
"REF as: $\\min _{X,k} E(X,k) \\propto \\sum ^N_{i=1} E(I|x_i,k)+E(k)+E(X),$ where $\\sum ^N_{i=1}E(I|x_i,k) $ is referred to as the data fidelity term.",
"$E(k)$ and $E(X)$ are the priors of the blur kernel and the latent image, respectively.",
"In the following sections, we introduce the choices of the MAP terms in the proposed muscle image enhancement framework and provide the optimization solution.",
"The purpose of the data fidelity term is to constraint the estimated latent image not to deviate too much from its blurred version.",
"Recent findings show that gradient plays an essential role during the kernel estimation [10].",
"In fact, $X$ is not necessarily a “real\" image but can be either gradient map $(\\partial _xx_i,\\partial _yx_i)$ [10] [14] or some “unnatural\" representations of the image [19] [17].",
"Due to the severe ill-posed nature of this problem, these options usually provide more helpful information of the blur kernel than the natural image estimated during the optimization or the initial blurred image.",
"After the MAP optimization reaches the convergence, only the estimated kernel is used and the final output image is computed using some non-blind deconvolution approaches with image priors to suppress artifacts.",
"In this paper, we propose to use gradient map as the input to the kernel estimation problem due to its simplicity and efficiency.",
"The blurry muscle image is converted to gray scale and two derivative kernels are used to obtain the gradient maps at both the vertical and horizontal directions: $\\nabla X=\\lbrace \\partial _x X,\\partial _y X\\rbrace =\\lbrace X \\otimes \\partial _x,X \\otimes \\partial _y\\rbrace ,$ where $\\otimes $ denotes the convolution operation and $\\partial _x = \\begin{bmatrix}-1&1\\\\0&0\\end{bmatrix},\\partial _y = \\begin{bmatrix}-1&0\\\\1&0\\end{bmatrix}.$ There are several options to model noise in muscle images such as Gaussain distribution, Poisson distribution [7] [24], and impulse distribution [25].",
"For the convenience of computation, we assume the noise follows white Gaussian distribution with a uniform variance $\\sigma $ .",
"The data fidelity term becomes: $\\sum ^N_{i=1} E(I|x_i,k)=||\\nabla X\\otimes k-\\nabla I||_F^2 \\sim \\mathcal {N}(0,\\sigma ),$ where $\\nabla X=\\lbrace \\partial _x x_i, \\partial _y x_i\\rbrace $ and $\\mathcal {N}(0,\\sigma )$ denotes a white Gaussian model.",
"Although noise is explicitly modeled and addressed, strong noise hinders the accurate estimation of the blur kernel and blur removal.",
"We propose to use an automatic denoising framework proposed in [26] to suppress noise without the prior knowledge of the noise parameter."
],
[
"Image Prior",
"Image priors are used as regularizations during the optimization process to alleviate the ill-posedness problem and suppress artifacts such as rings.",
"There are large number of image priors proposed in image deblur literature.",
"Whyte et al.",
"apply a simple $l_1$ norm to the Richard-Lucy deconvolution algorithm [13].",
"Shan et al.",
"adopt the fact in image statistics that the image gradient should have a heavy-tailed distribution [10].",
"Rudin et al propose to minimize the total variation ($l_2$ norm) of the gradient map.",
"Krishnan et al.",
"propose a normalized $l_1$ [14] to constraint the high frequency components of the latent image.",
"More sophisticated but computationally tractable $l_0$ norm is proposed in [17].",
"We use $l_1$ norm on the gradient of the latent image, modeling its sparsity nature: $E(X) = ||\\nabla X||_F^1.$ The sparsity of the gradient image is increased in blurry/defocus microscopic muscle images compared to its latent distortion free version.",
"Thus, the $l_1$ norm can be adopted in a energy minimization framework to constrain the solutions and alleviate the ill-posedness problem in REF .",
"Empirically, this simple $l_1$ norm is not only effective in practice but also provides significant convenience in computation since the objective function is inherently convex.",
"Gradient descent-like algorithms such as (fast) iterative shrinkage-thresholding algorithm (ISTA or FISTA) [27] can be used to find its solution efficiently, providing we adopt Bayesian inference and assume the noise follows the white Gaussian noise model in Eq.",
"REF .",
"Other priors, such as the prior in [10] and the sophisticated $l_0$ norm [17], are not used because they is computational efficient despite their promising performance.",
"The total variation of the gradient tends to over-smooth the latent image.",
"The normalized $l_1$ norm in [14] is highly non-convex and has many local minima, which hinders the successful estimation.",
"works well but it is usually slow to optimize."
],
[
"Kernel Prior",
"Unlike typical motion blur problems in which the blur kernel can be modeled as a motion trajectory in 3D space, the degradation of the microscopic muscle image is largely the defocus problem with high variances in 2D space of the entire slide.",
"Although we explicitly confine the blur kernel to be Gaussian at the initialization stage of the optimization, the weight assigned to this constrain is quickly decreased and diminished.",
"We adopt $l_2$ norm to suppress the noise.",
"$l_1$ norm tends to over-suppress the kernel in our case though it is a good regularization in common natural image deblurring problem.",
"The kernel prior used in our method is written as: $\\begin{aligned}E(k) &= \\eta p(k|\\sigma )+\\nu ||k||_F^2,\\\\p(k|\\sigma ) &= ||k(a,b)-\\frac{1}{2\\pi \\sigma ^2}exp\\lbrace -\\frac{a^2+b^2}{2\\sigma ^2}\\rbrace ||_F^2,\\end{aligned}$ where $p(k|\\sigma )$ denotes the likelihood of the kernel to be a circular Gaussian with the parameter $\\sigma $ which takes $(a,b)$ as variables.",
"$\\eta $ and $\\nu $ are weights of controlling the importance of both terms.",
"$\\eta $ is decayed exponentially and is set to 0 after a few iterations."
],
[
"Optimization",
"Combining Eq.",
"REF and Eq.",
"REF to Eq.",
"REF , the overall optimization of the proposed deblur is written as: $\\min _{\\nabla X,k} ||\\nabla X\\otimes k-\\nabla I||_F^2 + \\lambda ||\\nabla X||_F^1 + \\eta p(k|\\sigma )+\\nu ||k||_F^2,$ where $\\lambda $ , $\\eta $ and $\\nu $ are weight terms.",
"We discuss the detailed configurations of these parameters in the experiment section and the user does not have to .",
"Eq.",
"REF is solved using Alternating Minimization (AM) and obtain the estimated blur kernel and the gradient of the latent image (which is an “unnatural\" representation of the latent image and is not used after the kernel is estimated): $\\min _{\\nabla X} ||\\nabla X\\otimes k-\\nabla I||_F^2 + \\lambda ||\\nabla X||_F^1,$ and $\\min _k ||\\nabla X\\otimes k-\\nabla I||_F^2 + \\eta p(k|\\sigma )+\\nu ||k||_F^2.$ To optimize Eq.",
"REF , we use ISTA algorithm by replacing matrix multiplications with convolutions as shown in algorithm REF and find the solution of the latent muscle image in gradient domain.",
"Note that $\\Phi _{\\pi }(\\cdot )$ denotes a rotation of $180^\\circ $ operation on a matrixThis can be implemented via the “rot90(k, 2)\" command in MATLAB..",
"This rotation operation replaces the original matrix transpose when 1) typical ISTA algorithm or 2) the Toeplitz form for convolution as in Eq.",
"REF is used.",
"$sign(\\cdot )$ is the element-wise sign function for a matrix and $\\odot $ denotes the matrix element-wise multiplication.",
"Note that we obtain the updates for $\\partial _x X$ and $\\partial X_y$ individually using Eq.",
"REF .",
"In Eq.",
"REF , the two updated gradient maps are input jointly as one term $\\nabla X$ .",
"Gradient map update [1] Task: Obtain the solution of the latent image in gradient domain $\\nabla \\tilde{X}$ under $l_1$ norm regularization; Initialization: Observed gradient image $\\nabla I$ in current resolution level, initial latent image in gradient domain $\\nabla X^0\\leftarrow \\nabla I$ , initial kernel $k^0$ at coarsest resolution level or the optimization result $k=\\tilde{k}$ from a coarser resolution level, regularization parameter $\\lambda $ , ISTA threshold $\\zeta $ , maximum inner iteration number $T$ , current iteration $t \\leftarrow 1$ ; Set $\\nabla Y \\leftarrow \\nabla X^t-\\zeta \\Phi _\\pi (k) \\otimes (k \\otimes \\nabla X^t - \\nabla I)$ .",
"Set $\\nabla X^{t+1} \\leftarrow sign(\\nabla Y) \\odot \\max (|\\nabla X^t|-\\lambda \\zeta ,0)$ .",
"Compute the energy cost in Eq.",
"REF .",
"Set $t\\leftarrow t+1$ .",
"$t>T$ or energy cost convergence reached.",
"Set $\\nabla \\tilde{X} \\leftarrow \\nabla X^t$ .",
"Kernel update [1] Task: Obtain the solution of the blur kernel $\\tilde{k}$ under both $l_2$ norm and Gaussian similarity regularization; Initialization: Observed gradient image $\\nabla I$ at current resolution level, regularization parameter $\\eta $ and $\\nu $ decayed by $\\Delta \\eta $ at each iteration; Compute the least square estimation of the kernel $\\hat{k}$ in Eq.",
"REF .",
"$\\eta \\ne 0$ Compute the maximum likelihood solution of the approximate Gaussian function $\\mathcal {N}(0,\\tilde{\\sigma }) \\simeq \\hat{k}$ in Eq.",
"REF .",
"Search for the solution $\\tilde{k}\\sim \\mathcal {N}(0,\\sigma +\\nabla \\sigma )$ in Eq.",
"REF .",
"Decrease the Gaussian likelihood weight $\\eta \\leftarrow \\max ((\\eta -\\Delta \\eta ),0)$ .",
"Set $\\tilde{k} \\leftarrow \\hat{k}$ .",
"Normalize the estimated kernel $\\tilde{k} \\leftarrow \\tilde{k}/ ||\\tilde{k}||_F^1$ .",
"We present the details of solving the optimization problem for kernel $k$ in Eq.",
"REF .",
"We first solve $k$ with $l_2$ regulation in a least square minimization: $\\begin{aligned}\\hat{k}&=\\operatornamewithlimits{argmin}_k E_k,\\\\&=\\operatornamewithlimits{argmin}_k ||\\nabla X\\otimes k-\\nabla I||_F^2 + \\nu ||k||_F^2,\\end{aligned}$ whose solution, according to the Parseval's theoremThe sum of the square of a function is equal to the sum of the square of its Fourier transform., is efficiently obtained in closed-form by Fast Fourier Transform (FFT) and setting derivative $\\frac{\\partial \\mathcal {F}(E_k)}{\\partial \\mathcal {F}(k)}=0$ [19]: $\\begin{aligned}\\hat{k} &= \\mathcal {F}^{-1} \\Big ( \\frac{\\overline{\\mathcal {F}(\\nabla X)}\\odot \\mathcal {F}(\\nabla I)}{\\overline{\\mathcal {F}(\\nabla X)}\\odot \\mathcal {F}(\\nabla X)+\\nu \\mathbf {1}} \\Big )\\\\&= \\mathcal {F}^{-1} \\Big ( \\frac{\\overline{\\mathcal {F}(\\partial _x X)}\\odot \\mathcal {F}(\\partial _x I)+\\overline{\\mathcal {F}(\\partial _y X)}\\odot \\mathcal {F}(\\partial _y I)}{\\overline{\\mathcal {F}(\\partial _x X)}\\odot \\mathcal {F}(\\partial _x X) + \\overline{\\mathcal {F}(\\partial _y X)}\\odot \\mathcal {F}(\\partial _y X) + \\nu \\mathbf {1}} \\Big ),\\end{aligned}$ where $\\mathcal {F}$ and $\\mathcal {F}^{-1}$ is the Fourier/inverse Fourier transform pairThe Fourier/inverse Fourier transform pair is implemented with fast (inverse) Fourier transform (FFT and iFFT).",
"In MATLAB, we use “fft2\" to compute the coefficients in frequency domain for gradient maps and use “otf2psf\" to compute the inverse Fourier transform to convert the estimated blur kernel to spatial domain.",
"This is because “fft2\" centers the input image at $(1,1)$ while “otf2psf\" centers at the geometric center.. $\\mathbf {1}$ denotes a matrix whose elements are uniformly 1.",
"Then, we find a Gaussian blur kernel $\\mathcal {N}(0,\\hat{\\sigma })$ closest to $\\hat{k}$ with maximum likelihood inference: $\\begin{aligned}\\tilde{k}&\\sim \\mathcal {N}(0,\\tilde{\\sigma }),\\\\\\tilde{\\sigma }&=\\operatornamewithlimits{argmax}_{\\sigma \\in \\Theta }\\hat{\\ell }(\\sigma ;\\hat{k},\\mathcal {N}(0,\\tilde{\\sigma })),\\end{aligned}$ where $\\hat{\\sigma }$ is the maximum likelihood solution of the parameter and $\\Theta $ is its value domain.",
"Finally, we search in a small window for this parameter: $\\tilde{\\sigma } \\pm \\Delta \\sigma ,$ to find a local minima for the cost function in Eq.",
"REF .",
"We use brute force here and search for the result for each possible value of $\\tilde{\\sigma } \\pm \\Delta \\sigma _i$ , where $\\sigma _i$ is some preset step sizes.",
"Eq.",
"REF and Eq.",
"REF are served as regularizations to control the overall shape of the estimated kernel and to suppress noise.",
"We empirically find out that they are helpful only in the first few iterations.",
"Thus, we gradually decrease the value of the weight $\\nu $ to zero after a few iterations.",
"The brute force search in Eq.",
"REF looks ad-hoc but it is fast in practice.",
"In fact, we are not the first to propose this approach.",
"Hu et al.",
"assume their estimated kernel to be plate-shaped and use brutal force to search for its axis parameters [18].",
"They claim that the computational cost is not a problem.",
"In addition, we normalize the estimated kernel to keep its $l_1$ norm to be 1 in order to preserve its energy at each iteration of the optimization.",
"The kernel update steps are summarized in Algorithm REF .",
"We follow the popular coarse-to-fine strategy using pyramid representation of the muscle image as Fig.",
"REF depicts.",
"We conduct our blur kernel estimation algorithm from low to high resolution levels.",
"Each resolution level provides an estimation of the kernel (with an increasing size) to the next finer level.",
"At each resolution level, we run Algorithm REF and Algorithm REF several but not too many iterations for refinement.",
"The overall kernel estimation algorithm is summarized in Algorithm REF .",
"Figure: Pyramid representation of the muscle image.",
"k ˜ ω \\tilde{k}_\\omega represents the initial/estimated blur kernels.",
"[t] Overall blur kernel estimation Task: Obtain the solution of the blur kernel $\\tilde{k}$ ; Initialization: Observed image $I$ , set the initial blur kernel as $k^0_w \\leftarrow \\delta $ (Dirac function) or as an estimated kernel from the neighboring region $k^0_w\\leftarrow \\tilde{k}^{\\prime }$ , regularization parameter $\\lambda $ , $\\eta $ and $\\nu $ , total outer iteration number $N$ and the current iteration $n$ , total pyramid levels $\\Omega $ and the current processing resolution level $\\omega _{ini}\\leftarrow (k^0_w=\\delta )?1:\\Omega $ , where $\\omega =1$ relates to the coarsest resolution level of the image; Compute the image pyramid $\\lbrace I_\\omega \\rbrace $ with $\\Omega $ levels for $I$ .",
"$\\omega = \\omega _{ini}$ to $\\Omega $ Initialize the gradient map at the current resolution level: $\\nabla X^0_\\omega \\leftarrow (\\omega = 1)?",
"\\nabla I_\\omega :\\nabla \\tilde{X}_{\\omega -1}$ .",
"Set $\\nu $ as its initial value.",
"$n \\leftarrow 1$ to $N$ Update the blur kernel $\\tilde{k}^n_\\omega $ for $\\nabla I_\\omega $ at the current resolution level with Algorithm REF by inputting $\\nabla X = \\nabla X^n_\\omega $ .",
"Update the gradient map $\\nabla X^n_\\omega $ of the latent image at the current resolution level with Algorithm REF by inputting $k=k^n_\\omega $ .",
"$\\omega \\ne \\Omega $ Upscale $\\tilde{k}_\\omega $ and $\\nabla \\tilde{X}_\\omega $ to the next finer resolution level."
],
[
"Latent Image Recovery",
"After the kernel for the finest resolution level is estimated, we deconvolve the muscle image with it and obtain the estimation of the latent image.",
"Note that we only obtain the latent image in gradient domain in previous steps.",
"Numerous works for non-blind (with knowledge of the kernel) deconvolution have been proposed in literature, ranging from the traditional inverse filtering and Weiner filtering [6] to the modern sophisticated approaches focusing on artifact suppression, e.g., Laplacian regularization [28], $l_1$ regularization on Richard-Lucy algorithm [13] [23], expectation maximization (EM) scheme with outliers (saturated pixels and noise) handling [20], or a hybrid method combining the advantages from the previous two [18].",
"We adopt the Laplacian regularization based approach in [29] to deconvolve the degraded muscle images with estimated blur kernels due to its good performance and fast convergence: $\\tilde{X} = \\operatornamewithlimits{argmin}_{X}\\beta ||X\\otimes \\tilde{k}-I||^2_F+||\\nabla X||_{\\alpha },$ where $\\tilde{X}$ is the enhanced image, $\\tilde{k}$ is the estimated blur kernel from Eq.",
"REF .",
"We set parameters such as $\\beta =3000$ and $\\alpha =0.8$ exactly the same as suggested in [14]."
],
[
"Local Kernel Propagation",
"The proposed deblur algorithm until now assumes the blur kernel is constant across the image.",
"This is not true on typical microscopic muscle images where blur and defocus happen in a very non-uniform style.",
"See Fig.",
"REF for examples.",
"Rather than on the entire scope, we perform our deblur algorithms on muscle sub-image (200 by 200 pixels in this paper) and optimize the local gradient sparsity in Eq.",
"REF of the patch and obtain a group of spatially variant blur kernels.",
"The entire image is thus recovered via the non-blind deconvolution on each of the sub-images using Eq.",
"REF .",
"The recovered sub-images are finally stitched.",
"We extract sub-images with small marginal overlaps.",
"The recovered sub-image are simply averaged on the overlapped parts to preserve inter sub-image coherence of the image structure.",
"Although this method is simple, it benefits the blur removal of the microscopic muscle image in several ways: The strong non-uniform nature of the blur kernel of the muscle image is addressed.",
"In the experiment section, we show that the proposed method outperforms the existing non-uniform deblur algorithms significantly due to this flexibility.",
"Fast deblur computations is enabled by feeding the cpu/gpu with image patches in parallel.",
"The deblur algorithm is typically very slow on the entire, large muscle image.",
"Figure: Examples of the non-uniform blur on the muscle images.To further speedup the algorithm, we randomly randomly pick up $n$ sub-image centersWe set $n$ as the number of the CPUs of our experiment computer.",
"away from the boundary (to avoid picking sub-images that have too few number of neighbors).",
"We run Algorithm REF and estimate their blur kernels individually.",
"The estimated kernels are treated as seeds and are propagated to their neighboring sub-images as initializations during their kernel estimation.",
"The rationale behind this method is that most neighboring sub-image share similar blur degree and thus should have similar blur kernels (Globally, they still display high non-uniformity).",
"The sub-images having their kernels propagated by seeds are not represented with pyramid structure during the kernel estimationSee the different initialization of of $\\omega _{ini}$ for seeds and non-seeds in Algorithm REF .",
"Thus, the computation is accelerated by skipping the kernel estimation on coarser resolution levels.",
"Fig.",
"REF shows an example of kernel propagation made by to estimated kernels.",
"Figure: Local kernel propagation.",
"The estimated blur kernels k ˜\\tilde{k} and k ˜ ' \\tilde{k}^{\\prime } are propagated to their neighboring subregions to accelerate their kernel estimations."
],
[
"Experiment Settings",
"We obtain 44 microscopic muscle images using an Olympus VS 120 advanced optical microscope, each of which has very large size.",
"Thus, the muscle images used in our experiments are cropped with large size, assuming each channel of the muscle image data is normalized to range in $[0,1]$ .",
"We empirically set $\\lambda =80$ , $\\eta =15$ , $\\nu =6$ and decay term $\\Delta \\nu =2$ and do not make any further tunings.",
"We assume the kernel size to be 15 pixelThis size is the maximum size while in practice most estimated blur kernels are smaller than it..",
"The outer iteration number $N$ and the inner iteration number $T$ are set as 5 and 3, respectively.",
"The search window in Eq.",
"REF is set as $\\Delta \\sigma = \\lbrace 0.1/255,0.2/255,...0.5/255\\rbrace $ .",
"The number of resolution levels of the image pyramid $\\Omega $ is set as 5.",
"We test the proposed framework on a desktop computer with Intel Xeon E5-1650 3.50 GHz (12 threads) CPU and 128GB memory."
],
[
"Synthesis Blur",
"In this experiment, we manually crop 20 muscle sub-images (round 2000 by 1000)The actual size of the that we consider as sharp ones without visually obvious blur or defocus.",
"We synthesize a number of Gaussian blur on them to simulate the spatially variant blur and defocus.",
"We denote $b_i=\\lbrace (a_i,b_i),\\sigma _s,\\sigma _l\\rbrace $ as a blur of the latent muscle image centered at the coordinate $(a_i,b_i)$ and compute the blurred image as follows: $I(m,n) = \\sum _{(i,j)\\in \\Phi (m,n)} \\frac{1}{W_k}\\cdot X(i,j) \\cdot k(i,j),$ where $I(m,n)$ is the pixel value of the blurred muscle image, $(i,j)$ denotes the arbitrary location in the neighborhood of the central pixel $\\Phi (m,n)$ , $X$ is the sharp muscle image and $k$ is the synthesis Gaussian blur kernel with its normalization term $W_k$ defined as: $\\begin{split}k &\\sim \\mathcal {N}(0,\\rho (m,n)),\\\\W_k&=\\sum _{(x,y)\\in k} k(x,y),\\end{split}$ where $\\rho (m,n)$ is the standard deviation of the blur kernel.",
"It is determined by the distance between $(m,n)$ and the blur center $(a,b)$ : $\\rho (m,n) = \\sigma _s \\cdot exp\\lbrace \\frac{-|(a,b)-(m,n)|^2}{2\\sigma _l}\\rbrace ,$ where $\\sigma _s$ is the blur strength of the kernel $b_i$ and is decayed as the location deviates from the blur center and is controlled by the parameter $\\sigma _l$ .",
"We randomly set the center of the blur kernel and set $\\sigma _s\\in (0.5/255,4.5/255]$ , $\\sigma _l=100$ .",
"Figure REF shows a distortion free muscle image corrupted by the defined synthesis blur with $\\sigma _s=5/255$ .",
"We uniformly split the clean images into four distortion levels degraded by different number of synthesis Gaussian kernels and different $\\sigma _s$ .",
"The settings of the distortion levels are described in Table REF .",
"Table: Settings of different levels of synthesis distortion.At each level, we randomly choose the value of $\\sigma _s$ uniformly distributed at the interval described in REF , whose value is for pixel value range $[0,255]$ and is dived by 255 in our experiment.",
"Figure: An example of the muscle image corrupted with the synthesis blur.",
"Blur circle locates roughly the blurry area.We test the proposed deblur algorithm of our framework and set parameters as mentioned before without further tweaking.",
"We compare our the visual quality of our results with the state-of-art spatially non-uniform deblur algorithm by L. Xu et al.",
"[17] using peak signal to noise ratio (PSNR) as the image quality metric.",
"The Table: Comparison of the enhancement results for synthesis blur.",
"Average PSNR value is compared at each distortion level.Table: Comparison of the enhancement results for real blur using blind IQA score."
],
[
"Real Blur",
"We test our algorithm on 12 cropped muscle images (around 2000 by 1000) with significant noticeable blur.",
"To quantitatively evaluate the performance, we use a popular blind image quality assessment (blind IQA) [30] to compare the results with [17].",
"The blind IQA in [30] is a image quality evaluation scoring system which returns a score regarding to the image's sharpness, noise, transmission loss, etc.",
"without the necessity of the availability of the distortion-free version of the image.",
"We normalize the score to the range [0,1] for convenience.",
"Table REF summarizes the blind IQA scores for the original blur images, enhanced by the proposed algorithm, and those by [17].",
"It is observed that the proposed algorithm outperforms the [17] on 9 out of 12 blur muscle images, although [17] exhibits a little more robustness on certain images.",
"The parameters for both compared methods are set with consistent configurations we empirically choose and defaulted by the authors, respectively.",
"The proposed algorithm is also computationally efficient due to its simple design.",
"The average computation time for one cropped muscle image of its CPU implementation on Matlab, compared with [17], is presented in figure REF .",
"The proposed algorithm is significantly efficient compared with the algorithm in [17], since the latter is designed for natural image deblur which is typically not so large as muscle images are.",
"Figure: Examples of enhanced muscle images by the proposed algorithm.",
"The figures in the left column are the original blur image and the enhanced images are in the right column.",
"Best reviewed in enlarged resolution.Figure: Average execution time per one cropped muscle image for the proposed algorithm and ."
],
[
"Conclusion",
"We present a novel image enhance system in which a simple but effective deblur algorithm specialized for muscle image is involved.",
"The system is designed with one-click-style interface which frees the users from laborious parameter tunings.",
"The effectiveness of the algorithm is verified on both synthesis and practical blur quantified by an objective image quality evaluation metric."
],
[
"Acknowledgements",
"The authors would like to thank to XX projects, fundings, and institutions."
]
] | 1612.05719 |
[
[
"Correlations generated from high-temperature states: nonequilibrium\n dynamics in the Fermi-Hubbard model"
],
[
"Abstract We study interaction quenches of the Fermi-Hubbard model initiated from various high-temperature and high-energy states, motivated by cold atom experiments, which currently operate above the ordering temperature(s).",
"We analytically calculate the dynamics for quenches from these initial states, which are often strongly-interacting, to the non-interacting limit.",
"Even for high-temperature uncorrelated initial states, transient connected correlations develop.",
"These correlations share many features for all considered initial states.",
"We observe light-cone spreading of intertwined spin and density correlations.",
"The character of these correlations is quite different from their low-temperature equilibrium counterparts: for example, the spin correlations can be ferromagnetic.",
"We also show that an initially localized hole defect affects spin correlations near the hole, suppressing their magnitude and changing their sign."
],
[
"Introduction",
"The development of correlations out of equilibrium is the topic of much recent research in AMO and condensed matter systems.",
"Major areas of interest include the relaxation dynamics of a system driven out of equilibrium [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and the possibility of relaxation to nonthermal steady states [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] that have unusual properties [25], [16], [26], [27], [28], [29], [30], [31], [32], [33], [34].",
"An emerging direction involves inducing nonequilibrium correlations at temperatures above those required for equilibrium order.",
"This has been demonstrated in some solid state systems in the presence of continuous driving [35].",
"However, the dynamics after quenches has been less studied, and numerous questions exist in all cases: What conditions are required for correlations to develop?",
"What timescales are involved?",
"What will the character of these correlations be?",
"In this paper we study quenches of the Fermi-Hubbard model from finite (and in some cases very high) initial temperatures to noninteracting final Hamiltonians.",
"This is a useful complement to studies that consider dynamics from low temperature initial conditions [6], [36], [37], [4].",
"Besides its intrinsic interest, this regime is important to ongoing experiments.",
"This is because despite much recent progress towards realizing low temperature equilibrium states experimentally, the regime well below the ordering temperatures (e.g.",
"the Néel temperature for the antiferromagnet) remains elusive due to the very low temperatures and entropies required [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58].",
"We find that, even when initiated from high temperature initial states that are above the superexchange or even tunneling energy scales, such quenches generate transient particle number and spin correlations between two sites; after the quench a light cone of connected correlations between increasingly distant sites develops over time.",
"A wide range of initial product states exhibit qualitatively similar correlation dynamics.",
"In particular we calculate the dynamics for high temperature Mott insulators in one and two dimensions, a strongly interacting metal, a partially spin-polarized Mott insulator, and a perfect product state antiferromagnet.",
"The transient correlations can be qualitatively different from the correlations of the equilibrium low temperature states of the same initial Hamiltonian.",
"For example, we observe the generation of ferromagnetic spin correlations from a Hamiltonian with initially repulsive on-site interactions, in contrast to the antiferromagnetic spin correlations that occur in equilibrium for the repulsive Hubbard model.",
"Going forward, our results will help one understand quenches with finite interactions after the quench.",
"On the one hand, when phenomena persist with interactions our results provide a foundation for understanding them.",
"On the other hand, when interactions lead to phenomena that are absent in our results, it signals that the physics is intrinsically interacting.",
"Given how surprising out-of-equilibrium dynamics can be, it is crucial to sort out which surprises result from the interactions and which arise from the inherent nonequilibrium nature of the problem (independent of interactions).",
"An example drives this home.",
"Imagine that one found - perhaps in a strongly interacting system - that spin and density correlations were evolving dynamically with exactly the same magnitude.",
"This intriguing behavior is reminiscent of “intertwined\" spin and density order in equilibrium strongly correlated systems [59].",
"Although a natural instinct is to imagine this observation is similarly non-trivial, one of our results is to show that such dynamics occurs even for non-interacting quenches.",
"Thus, remarkably intricate phenomena can occur even in the non-interacting dynamics.",
"Comparing to this important baseline allows one to assess how dramatic a given observation in a strongly-interacting system really is.",
"Another interesting example that we study in this context is the transport of a hole defect after a quench.",
"We show that as the hole propagates it affects the development of correlations around it.",
"Superficially, it appears that the hole is dressed with a cloud of spin correlations.",
"This is another example where, if this were observed in a strongly interacting system one might leap to the conclusion that the physics was highly non-trivial, but in fact the richness here appears already in the noninteracting dynamics.",
"This paper is organized as follows: Section describes how we calculate the dynamics of observables quenched from initial spatial product states to non-interacting Hamiltonians.",
"Section applies the theory to calculate the connected correlations in a one-dimensional Mott insulator.",
"Section shows that qualitatively similar phenomena persist for multiple initial conditions.",
"Section describes how an initially localized hole defect modifies the dynamics of spin correlations.",
"Section presents conclusions and outlook."
],
[
"Quench dynamics in the non-interacting limit",
"We consider interaction quenches from initial product states to the noninteracting limit for ultracold fermions in an optical lattice, illustrated in Fig.",
"REF .",
"The initial state is $\\rho =\\bigotimes _{i}\\rho _{i}^{(1)}.$ where $\\rho _i^{(1)}$ is an arbitrary density matrix for site $i$ (in general a mixed state).",
"The system is described by the Hubbard Hamiltonian $H=-J\\underset{\\left\\langle ij\\right\\rangle ,\\sigma }{\\sum }c_{i\\sigma }^{\\dagger }c_{j\\sigma }+U\\underset{i}{\\sum }n_{i\\uparrow }n_{i\\downarrow }$ where $\\left\\langle ij\\right\\rangle $ indicates a nearest neighbor pair of sites, $\\sigma \\in \\left\\lbrace \\uparrow \\,, \\downarrow \\right\\rbrace $ , $c_{i\\sigma }$ is the fermionic annihilation operator at site $i$ with spin $\\sigma $ , and $n_{i\\sigma }=c^\\dagger _{i\\sigma }c_{i\\sigma }$ is the corresponding number operator.",
"This describes fermions in a deep lattice with a nearest-neighbor tunneling amplitude $J>0$ and on-site interaction energy $U$ [60].",
"Many of our initial states arise as high-temperature ($T\\gg J$ ) equilibrium states of Eq.",
"(REF ), and the post-quench dynamics is governed by its $U=0$ limit.",
"Figure REF illustrates our quench protocol, in which the system starts in equilibrium at some value of $U$ and the interaction is turned off at $t=0$ : $U(t)=U_{0}\\left[1-\\Theta (t)\\right]$ where $\\Theta $ is the Heaviside step function and $\\left| U_{0}\\right| \\gg J$ .",
"When the temperature $T$ before the quench is large compared to $J_{\\text{init}}$ (the tunneling before the quench) – i.e.",
"$T\\gg J_{\\text{init}} $ – the initial state takes the form of Eq.",
"(REF ).",
"(We will consider a few alternative product states later.)",
"Experimentally, the interaction can be dynamically controlled by using a Feshbach resonance or changing the lattice depth.",
"We note that our calculations actually describe a variety of more general quenches of the Fermi-Hubbard model.",
"The only required conditions are that the initial temperature $T$ satisfies $T\\gg J_{\\text{init}}$ and $U=0$ after the quench.",
"So, for example, one could suddenly change both $U$ and $J$ at time $t=0$ as long as these conditions are met.",
"Figure: (a) Quench protocol for the dynamics in this paper (arbitrary units).",
"(b) Pre-quench, the system is in a product of single-site states.",
"An important class of states of this form that we consider arise from the J≪T≪UJ\\ll T \\ll U equilibrium state of the Fermi-Hubbard Hamiltonian.",
"(c) Post-quench, the system evolves in the noninteracting limit of the Fermi-Hubbard Hamiltonian, with conserved momentum occupation numbers.Our goal is to calculate the density and spin expectation values and two-site correlation functions for $t>0$ .",
"We define the total density operator $n_i=\\sum _\\sigma n_{i\\sigma }$ and the spin operators $ \\vec{S}_i = \\frac{1}{2}\\sum _{\\alpha \\beta }c_{i\\alpha }^\\dagger \\vec{\\sigma }_{\\alpha \\beta } c_{i\\beta }^{\\phantom{\\dagger }}$ , where $\\vec{\\sigma }$ is the vector of Pauli matrices.",
"We focus on these observables as the most basic correlations that characterize equilibrium systems, and because they can be measured in experiments.",
"Note that the correlation functions can be expressed as $\\langle n_{i} n_{j}\\rangle &=& \\sum _{\\alpha \\beta } \\langle c^\\dagger _{i\\alpha }c_{i\\alpha }^{\\phantom{\\dagger }}c^\\dagger _{j\\beta }c_{j\\beta }^{\\phantom{\\dagger }} \\rangle \\\\\\langle S_{i}^a S_{j}^b \\rangle &=& \\frac{1}{4}\\sum _{\\alpha \\beta \\gamma \\delta }\\sigma ^{a}_{\\alpha \\beta }\\sigma ^{b}_{\\gamma \\delta }\\langle c^\\dagger _{i\\alpha }c_{i\\beta }^{\\phantom{\\dagger }}c^\\dagger _{j\\gamma }c_{j\\delta }^{\\phantom{\\dagger }} \\rangle $ where $a \\,, b \\in \\left\\lbrace x \\,, y \\,, z \\right\\rbrace $ .",
"Therefore we turn to calculating the dynamics of a general two-site correlation $\\left\\langle c_{i\\alpha }^\\dagger c_{i\\beta }^{\\phantom{\\dagger }} c^\\dagger _{j\\gamma } c^{\\phantom{\\dagger }}_{j\\delta }\\right\\rangle $ , from which we can obtain the density and spin correlations.",
"For compactness, we define $C^{nn}_{ij} &=& \\mathinner {\\langle {n_i n_j}\\rangle }- \\mathinner {\\langle {n_i}\\rangle }\\mathinner {\\langle {n_j}\\rangle } \\\\C^{ab}_{ij} &=& \\mathinner {\\langle {S_i^a S_j^b}\\rangle }- \\mathinner {\\langle {S_i^a}\\rangle }\\mathinner {\\langle {S_j^b}\\rangle } $ where $a,b \\in \\lbrace x,y,z\\rbrace $ .",
"Because the Hamiltonian after the quench is non-interacting, one can analytically express the time-evolution of the annihilation operator as $c_{j\\alpha } (t)=\\underset{l}{\\sum }A_{jl}(t)c_{l\\alpha }$ where $A_{jl}(t)$ is the propagator from site $l$ to site $j$ of a single particle on the lattice.",
"Eq.",
"(REF ) follows because our Hamiltonian can be written $H=\\sum _{k\\alpha } {\\mathcal {E}}_k b_{k\\alpha }^\\dagger b_{k\\alpha }$ for some set of annihilation operators $b_{k\\alpha }$ .",
"The time evolution of these operators is $b_{k\\alpha }(t)=e^{-i{\\mathcal {E}}_k t} b_{k\\alpha }$ .",
"(If no time argument is provided, the operator is evaluated at $t=0$ , and we set $\\hbar =1$ throughout.)",
"The annihilation operators $c_{j\\alpha }$ can be expressed $c_{j\\alpha } = \\sum _k S_{j k} b_{k\\sigma }$ for some $S_{j k}$ .",
"Conversely, $b_{k\\alpha } = \\sum _{j} (S^{-1})_{k j} c_{j\\alpha }$ .",
"Hence at time $t$ , $c_{j\\alpha }(t) = \\sum _k S_{j k} b_{k\\alpha }(t) = \\sum _{k} S_{j k} e^{-i {\\mathcal {E}}_k t} b_{k\\alpha } = \\sum _{k l} e^{-i {\\mathcal {E}}_k t} S_{j k} (S^{-1})_{k l} c_{l\\alpha }$ .",
"In one dimension the single particle eigenstates $k$ can be identified with quasi-momentum states in the first Brillouin zone, for which $\\mathcal {E}_{k} = -2 J\\cos \\left(ka\\right)$ and $S_{jk}=\\exp {\\left(ijka\\right)}/\\sqrt{N}$ .",
"Taking $N \\rightarrow \\infty $ we see that Eq.",
"(REF ) holds with $A_{jl}(t)=\\left(-i\\right)^{\\left|j-l\\right|}\\mathcal {J}_{\\left|j-l\\right|}\\left(2Jt\\right)$ where $\\mathcal {J}_{m}\\left(z\\right)$ is a Bessel function of the first kind.",
"The expectation value of the general two-site correlator that determines the density and spin correlations at time $t$ is given in terms of initial expectation values by $\\hspace{-14.22636pt}\\left\\langle c_{i\\alpha }^{\\dagger }(t)c_{i\\beta }(t)c_{j\\gamma }^{\\dagger }(t)c_{j\\delta }(t)\\right\\rangle ={}\\\\ \\hspace{28.45274pt}\\underset{p,q,r,s}{\\sum }A_{ip}^{*}(t)A_{iq}(t)A_{jr}^{*}(t)A_{js}(t)\\left\\langle c_{p\\alpha }^{\\dagger }c_{q\\beta }c_{r\\gamma }^{\\dagger }c_{s\\delta }\\right\\rangle _0 $ using Eq.",
"(REF ), where $\\left\\langle \\cdots \\right\\rangle _0$ indicates the expectation value at time $t=0$ .",
"We compute these initial expectation values by taking advantage of the product state nature of Eq.",
"(REF ).",
"In this state, expectation values of operators factor by site: $\\left\\langle P_{i}Q_{j}\\right\\rangle _{0}=\\left\\langle P_{i}\\right\\rangle _{0}\\left\\langle Q_{j}\\right\\rangle _{0}$ if $i \\ne j$ for operators $P_i$ and $Q_j$ supported on single sites.",
"Then Eq.",
"(REF ) factors into a sum of three types of non-vanishing terms: (i) $p=q=r=s$ , (ii) $p=q$ and $r=s$ with $p\\ne r$ , and (iii) and $p=s$ and $r=q$ with $p\\ne r$ .",
"Writing the expectation in terms of these sums (and renaming summation indices) we have $\\left\\langle c_{i\\alpha }^{\\dagger }c_{i\\beta }^{\\phantom{\\dagger }}c_{j\\gamma }^{\\dagger }c_{j\\delta }^{\\phantom{\\dagger }}\\right\\rangle (t) & = & \\sum _{p}\\left|A_{ip}(t)\\right|^{2}\\left|A_{jp} (t )\\right|^{2}\\left\\langle c_{p\\alpha }^{\\dagger }c_{p\\beta }^{\\phantom{\\dagger }}c_{p\\gamma }^{\\dagger }c_{p\\delta }^{\\phantom{\\dagger }}\\right\\rangle _{0}\\nonumber \\\\& & \\hspace{-57.81621pt}{}+\\sum _{p\\ne q}\\left|A_{ip}(t)\\right|^{2}\\left\\langle c_{p\\alpha }^{\\dagger }c_{p\\beta }^{\\phantom{\\dagger }}\\right\\rangle _{0}\\left|A_{jq}(t)\\right|^{2}\\left\\langle c_{q\\gamma }^{\\dagger }c_{q\\delta }^{\\phantom{\\dagger }}\\right\\rangle _{0}+\\sum _{p\\ne q}A_{ip}^{*}(t)A_{jp}(t)\\left\\langle c_{p\\alpha }^{\\dagger }c_{p\\delta }^{\\phantom{\\dagger }}\\right\\rangle _{0}A_{iq}(t)A_{jq}^{*}(t)\\left\\langle c_{q\\beta }^{\\phantom{\\dagger }}c_{q\\gamma }^{\\dagger }\\right\\rangle _{0}\\!.$ Although the last two terms are double sums over $p$ and $q$ with $p\\ne q$ , the summand factors.",
"The sums can be written as products of single sums because in general $\\sum _{p\\ne q} P_p Q_q = \\sum _{p, q} P_p Q_q- \\sum _p P_p Q_p$ .",
"Using this, and using $\\left\\langle c^{\\phantom{\\dagger }}_{p\\alpha }c_{p\\beta }^{\\dagger }\\right\\rangle _{0}=\\delta _{\\alpha \\beta }-\\left\\langle c_{p\\beta }^{\\dagger }c^{\\phantom{\\dagger }}_{p\\alpha }\\right\\rangle _{0}$ to write each expectation value in a structurally similar form allows us to rewrite Eq.",
"(REF ) as $\\left\\langle c_{i\\alpha }^{\\dagger }c_{i\\beta }^{\\phantom{\\dagger }}c_{j\\gamma }^{\\dagger }c_{j\\delta }^{\\phantom{\\dagger }}\\right\\rangle (t)& = & \\sum _{p}\\left|A_{ip}(t)\\right|^{2}\\left|A_{jp}(t)\\right|^{2}\\left[g _{\\alpha \\beta \\gamma \\delta }^{p}-f _{\\alpha \\beta }^{p}f _{\\gamma \\delta }^{p}-f _{\\alpha \\delta }^{p}\\left(\\delta _{\\beta \\gamma }-f _{\\gamma \\beta }^{p}\\right)\\right]\\nonumber \\\\& & \\hspace{-57.81621pt}{}+\\left[\\sum _{p}\\left|A_{ip}(t)\\right|^{2}f _{\\alpha \\beta }^{p}\\right]\\left[\\sum _{q}\\left|A_{jq}(t)\\right|^{2}f_{\\gamma \\delta }^{q}\\right]+\\left[\\sum _{p}A_{ip}^{*}(t)A_{jp}(t)f _{\\alpha \\delta }^{p}\\right]\\left[\\sum _{q}A_{iq}(t)A_{jq}^{*}(t)\\left(\\delta _{\\beta \\gamma }-f _{\\gamma \\beta }^{q}\\right)\\right]\\!.$ We have defined $f_{\\alpha \\beta }^{i} &=& \\left\\langle c_{i\\alpha }^{\\dagger }c_{i\\beta }^{\\phantom{\\dagger }}\\right\\rangle _{0} \\\\g_{\\alpha \\beta \\gamma \\delta }^{i} &=& \\left\\langle c_{i\\alpha }^{\\dagger }c_{i\\beta }^{\\phantom{\\dagger }}c_{i\\gamma }^{\\dagger }c_{i\\delta }^{\\phantom{\\dagger }}\\right\\rangle _{0} $ to simplify notation.",
"With this rearrangement, double sums are eliminated (they factor) and only single sums remain.",
"In combination with Eqs.",
"(REF ) and () this allows the time evolution of the density-density and spin-spin correlators to be calculated efficiently.",
"We note that our calculations are similar to those by Gluza et al.",
"in [61] who also study noninteracting lattice fermions.",
"We consider a concrete, physically relevant case and focus on interesting phenomena that occur during the transient dynamics."
],
[
"Quench from $T \\gg J$ Mott insulator",
"We now apply the results of Section to a Mott insulating initial state (i.e., no spin polarization and unit filling).",
"In particular we consider $T \\gg J$ and (on-average) unit filling enforced by choosing the chemical potential to be $\\mu =U/2$ .",
"Define $H_{i} = U n_{i\\uparrow }n_{i\\downarrow } - \\mu n_{i} $ In this limit, the density matrix is given by $\\rho &=&Z^{-1}\\exp \\left(-\\beta H\\right) \\nonumber \\\\&=&Z^{-1}\\exp \\left(-\\beta \\sum _i H_i + O (J/T)\\right)\\nonumber \\\\&\\approx &Z^{-1}\\bigotimes _{i}\\exp \\left(-\\beta H_{i}\\right)$ where $Z$ is a constant enforcing $\\operatorname{Tr}\\rho = 1$ , $\\beta =1/T$ is the inverse temperature and we set $k_B=1$ throughout.",
"In what follows, we will associate with any energy $A$ a dimensionless ratio $\\tilde{A}=\\beta A$ .",
"Then the expectation values in the initial state are $f_{\\alpha \\beta }^{i} &=& \\frac{1}{2}\\delta _{\\alpha \\beta } \\\\g_{\\alpha \\beta \\gamma \\delta }^{i} &=& {\\left\\lbrace \\begin{array}{ll}\\frac{1}{2} &\\alpha =\\beta =\\gamma =\\delta \\\\\\frac{1}{2}\\frac{1}{1+e^{\\frac{1}{2} \\tilde{U}}} &\\alpha =\\beta \\ne \\gamma =\\delta \\\\\\frac{1}{2}\\left(1-\\frac{1}{1+e^{\\frac{1}{2} \\tilde{U}}}\\right) &\\alpha =\\delta \\ne \\beta =\\gamma \\\\0 &\\rm {otherwise}\\\\\\end{array}\\right.}",
"$ Figure REF shows the post-quench correlation dynamics of this $T \\gg J$ Mott insulating initial state obtained by Eq.",
"(REF ) using $f^i_{\\alpha \\beta }$ and $g^i_{\\alpha \\beta \\gamma \\delta }$ given in Eq. ().",
"For a fixed distance, transient connected correlations develop after the quench.",
"Connected correlations of both spin [Fig.",
"REF (a,c)] and density [Fig.",
"REF (b,d)] develop as a function of time in the shape of a light cone: correlations develop inside, and at the edge of, a region in space whose size grows as $vt$ for some velocity $v$ .",
"We observe that connected correlations spread at a velocity $v \\approx 4 J a$ , which is twice the maximum group velocity of a single particle with dispersion relation $\\mathcal {E}_k = -2 J \\cos \\left( k a \\right)$ .",
"The correlations can spread with twice the velocity of a single particle since two lattice sites can be mutually influenced by signals from a source halfway between them.",
"This is consistent with previous work describing the spread of correlations after a quench [62], [63], [64], [65], [66].",
"Figure: Connected correlations of a J≪T≪UJ\\ll T \\ll U 1D Mott insulator quenched to a noninteracting Hamiltonian.",
"(a) Spin-spin correlations C ij xx =〈σ i x σ j x 〉-〈σ i x 〉〈σ j x 〉C^{xx}_{ij} = \\langle \\sigma ^x_i \\sigma ^x_j\\rangle - \\langle \\sigma ^x_i \\rangle \\langle \\sigma ^x_j\\rangle and (b) density-density correlations C ij nn =〈n i n j 〉-〈n i 〉〈n j 〉C^{nn}_{ij} = \\langle n_i n_j\\rangle - \\langle n_i \\rangle \\langle n_j\\rangle between sites with an offset of one (solid lines), two (dashed lines), or three (dotted lines).",
"(c) Spin-spin and (d) density-density correlations as a function of time and site offset.In contrast to the low-temperature equilibrium state, the correlations are ferromagnetic rather than antiferromagnetic.",
"Furthermore, the spin and density correlations are intertwined, suggesting an emergent symmetry.",
"Specifically, the system develops positive spin-spin connected correlations that are independent of the spin orientation (i.e.",
"$C^{xx}_{ij}=C^{yy}_{ij}=C^{zz}_{ij}$ ) and negative density-density connected correlations $C^{nn}_{ij}$ of equal magnitude.",
"The intertwined spin and density correlations stem from the fact that in the noninteracting dynamics, there is only one energy scale, which is set by the tunneling $J$ .",
"Thus the spin and density correlations are controlled by the same energy scale.",
"To qualitatively understand the correlation dynamics, it is useful to consider the dynamics of a two-site model, which is shown schematically in Fig.",
"REF .",
"Let $\\mathinner {|{p\\,q}\\rangle }$ denote a state with $p$ and $q$ referring to the left and right sites respectively, and taking on the values 0 (empty), $\\uparrow $ (one atom with spin up), $\\downarrow $ (one atom with spin down), and $d$ (two atoms).",
"The state $\\mathinner {|{\\uparrow \\, \\uparrow }\\rangle }$ does not evolve since Pauli blocking prevents it from coupling to any other states, while the state $\\mathinner {|{\\uparrow \\, \\downarrow }\\rangle }$ evolves in the Schrödinger picture as $\\mathinner {|{\\uparrow \\, \\downarrow }\\rangle }(t)=\\cos ^{2}(t)\\mathinner {|{\\uparrow \\, \\downarrow }\\rangle }_{0}+\\sin ^{2}(t)\\mathinner {|{\\downarrow \\, \\uparrow }\\rangle }_{0}\\\\{}-i\\cos (t)\\sin (t)\\left(\\mathinner {|{d \\, 0}\\rangle }_{0}+\\mathinner {|{0 \\, d}\\rangle }_{0}\\right)$ as shown in Figs.",
"REF (a) and REF (b) for $J t = \\frac{\\pi }{4}$ .",
"In Fig.",
"REF (c), the ferromagnetic character of the dynamic spin correlations becomes apparent by observing that although the initial density matrix has equal weight on aligned (e.g.",
"$\\mathinner {|{\\uparrow \\, \\uparrow }\\rangle }$ ) and anti-aligned (e.g.",
"$\\mathinner {|{\\uparrow \\, \\downarrow }\\rangle }$ ) spin configurations, the time-evolved matrix has more weight on the aligned states.",
"This is because the aligned states stay frozen in time, while the anti-aligned states can partially convert to states with doublons and holes, reducing their spin correlations.",
"The density correlations can be explained similarly: the initial density matrix has no weight on doubly-occupied states, but the time-evolved matrix does.",
"The double-occupancy next to a vacant site represents a negative two-site density correlation, or equivalently, a (short-ranged) density wave correlation.",
"Figure: Schematic diagram of the time evolution of a two-site model at unit filling, initially with J≪T≪UJ\\ll T \\ll U, and then quenched to U=0U=0.",
"(a) Aligned initial states do not evolve.",
"(b) Anti-aligned initial states (the two-site equivalent of AFM initial states) evolve into superpositions with weight on doubly-occupied states at later times.",
"(c) The Mott insulator-like initial density matrix on two sites transfers some weight from its matrix elements for anti-aligned states to doubly-occupied states at later times, while aligned states' matrix elements do not change.",
"The matrix elements' magnitudes are indicated by color, from white (zero) to dark blue (maximal).Local observables approach constant values at large times.",
"Although the noninteracting system is clearly integrable and thus not expected to thermalize, the expectation values of the local observables as $t \\rightarrow \\infty $ are consistent with those of a thermal equilibrium state, in particular one at $T = \\infty $ .",
"This occurs because the initial state is a product state in the site basis.",
"Thermalization is not expected for other, more general initial states."
],
[
"Quenches from more general initial states: doped and spin-imbalanced systems, 2-dimensional Mott insulators, and antiferromagnets",
"The light-cone spreading of correlations from an uncorrelated initial state is not restricted to a 1D Mott insulator, but also occurs for a variety of initial conditions, as shown in Fig.",
"REF .",
"We demonstrate this for a $T\\gg J$ metal (with $n<1$ ), a spin imbalanced $T\\gg J$ Mott insulator, a product state antiferromagnet, and a 2D $T\\gg J$ Mott insulator.",
"We note that the metal can be alternatively viewed as a doped Mott insulator when $U\\gg J$ .",
"Figure: Spreading of correlations is generic, demonstrated by four additional classes of initial conditions.",
"(a) A 1D product state antiferromagnet aligned along the zz-axis at half-filling, (b) a J≪T≪UJ\\ll T \\ll U 1D hole-doped system, with 〈n〉≈0.85 \\langle n \\rangle \\approx 0.85, (c) a J≪T≪UJ\\ll T \\ll U spin-imbalanced 1D system with 〈σ z 〉≈0.25 \\langle \\sigma ^z \\rangle \\approx 0.25, and (d) a J≪T≪UJ\\ll T \\ll U 2D Mott insulator.",
"Site offsets to the right are along the (1,0) direction, and those to the left are along (1,1).Although both the spin imbalanced and hole-doped 1D initial states show correlations developing in light cones as in the Mott insulator, the magnitude of the correlations is reduced, as shown in Fig.",
"REF ; this follows from Eq.",
"(REF ) with the $f$ and $g$ in Eq.",
"() evaluated in these limits (see below).",
"One can induce a partially spin-polarized initial state by adding a term $B S^z_i$ to Eq.",
"(REF ), and likewise induce a number density other than one per site by taking the chemical potential to be $\\mu =U/2+\\Delta $ with $\\Delta \\ne 0$ .",
"For a partially spin-polarized system at unit filling, one finds $\\hspace{-28.45274pt}f_{\\alpha \\beta }^{i} &=&\\frac{1}{\\mathcal {N}_{1}}\\delta _{\\alpha \\beta }\\left(1+e^{\\frac{1}{2}\\tilde{U}+\\frac{1}{2}\\sigma ^{z}_{\\alpha \\beta }\\tilde{B}}\\right) \\\\\\hspace{-28.45274pt}g_{\\alpha \\beta \\gamma \\delta }^{i} &=& {\\left\\lbrace \\begin{array}{ll}f_{\\alpha \\alpha }^{i} &\\alpha =\\beta =\\gamma =\\delta \\\\\\frac{1}{\\mathcal {N}_{1}} &\\alpha =\\beta \\ne \\gamma =\\delta \\\\\\frac{1}{\\mathcal {N}_{1}}\\left(1+e^{\\frac{1}{2}\\tilde{U}+\\frac{1}{2}\\sigma ^{z}_{\\alpha \\delta }\\tilde{B}}\\right)&\\alpha =\\delta \\ne \\beta =\\gamma \\\\0 &\\rm {otherwise}\\\\\\end{array}\\right.}",
"$ with the normalization factor $\\mathcal {N}_{1}=2+2\\exp \\left(\\frac{1}{2}\\tilde{U}\\right)\\cosh \\left(\\frac{1}{2}\\tilde{B}\\right).$ For a system doped away from unit filling, $f_{\\alpha \\beta }^{i} &=&\\frac{1}{\\mathcal {N}_{2}}\\delta _{\\alpha \\beta }\\left(e^{\\tilde{\\Delta }}+e^{\\frac{1}{2}\\tilde{U}}\\right) \\\\g_{\\alpha \\beta \\gamma \\delta }^{i} &=& {\\left\\lbrace \\begin{array}{ll}f_{\\alpha \\alpha }^{i} &\\alpha =\\beta =\\gamma =\\delta \\\\\\frac{1}{\\mathcal {N}_{2}}e^{\\tilde{\\Delta }} &\\alpha =\\beta \\ne \\gamma =\\delta \\\\\\frac{1}{\\mathcal {N}_{2}}e^{\\frac{1}{2}\\tilde{U}} &\\alpha =\\delta \\ne \\beta =\\gamma \\\\0 &\\rm {otherwise}\\\\\\end{array}\\right.}",
"$ with $\\mathcal {N}_{2}=2\\left[\\cosh \\left(\\tilde{\\Delta }\\right)+\\exp \\left(\\frac{1}{2}\\tilde{U}\\right)\\right].$ Finally, for dynamics initiated from a 1D antiferromagnetic product state given by $ \\rho =\\bigotimes _{i}{\\left\\lbrace \\begin{array}{ll}\\mathinner {|{\\uparrow }\\rangle }_{i}\\mathinner {\\langle {\\uparrow }|}_{i} & i \\quad \\rm {even}\\\\ \\mathinner {|{\\downarrow }\\rangle }_{i}\\mathinner {\\langle {\\downarrow }|}_{i} & i \\quad \\rm {odd}\\end{array}\\right.",
"}$ one finds $f_{\\alpha \\beta }^{i} &=& {\\left\\lbrace \\begin{array}{ll}\\delta _{\\alpha \\beta }\\delta _{\\alpha \\uparrow }& \\quad i \\quad \\rm { even}\\\\\\delta _{\\alpha \\beta }\\delta _{\\alpha \\downarrow }& \\quad i \\quad \\rm {odd}\\\\\\end{array}\\right.",
"}\\\\g_{\\alpha \\beta \\gamma \\delta }^{i} &=& f_{\\alpha \\delta }^{i}\\delta _{\\beta \\gamma } $ The antiferromagnet-initiated dynamics displays a distinctive feature: anisotropy in the spin correlations.",
"As shown in Fig.",
"REF , the $C^{xx}$ and $C^{yy}$ connected correlations remain positive and equal in magnitude, as they were in previous cases, but the $C^{zz}$ and $C^{nn}$ connected correlations are negative.",
"They are, however, still equal in magnitude.",
"The magnitude of the correlations is larger than those of the 1D Mott insulator.",
"The anisotropy manifests despite the SU(2) symmetry of the Hamiltonian due to the broken symmetry of the initial state.",
"Figure: Spreading of correlations in a 1D product state antiferromagnet aligned along the zz-axis.",
"(a) Spin-spin correlations in the xx-direction C ij xx C^{xx}_{ij}.",
"The yy-yy correlations are identical: C ij yy =C ij xx C^{yy}_{ij}=C^{xx}_{ij} (b) Spin-spin correlations in the zz-direction C ij zz C^{zz}_{ij}.",
"(c) Density-density correlations C ij nn C^{nn}_{ij}.The light-cone spreading of correlations is not restricted to one-dimensional systems.",
"As seen in Fig.",
"REF (d) a two-dimensional Mott insulator on a square lattice shows qualitatively similar dynamics, but develops weaker transient correlations than a 1D Mott insulator with the same post-quench tunneling amplitude $J$ .",
"In a two-dimensional Mott insulator, the initial expectation values do not differ from the 1D case, but the propagators take a different form.",
"For a square lattice, they are $A_{\\mathbf {p}\\mathbf {q}}(t)=A_{p_{x}q_{x}}(t)A_{p_{y}q_{y}}(t)$ where $\\mathbf {p}$ and $\\mathbf {q}$ are integer vectors indicating sites on the square lattice.",
"Note that the 2D propagators factor into 1D components in this way due to the properties of the square lattice.",
"It could be interesting to explore the effects of other geometries, where interference between different paths can give propagators with structures other than Eq.",
"(REF )."
],
[
"Hole transport dynamics",
"Now we consider a system in which a single hole is added, localized to a single site, to the $T\\gg J$ Mott insulator state discussed in Section .",
"The behavior of hole defects in fermionic systems underpins the physics of many strongly correlated materials, where doping Mott insulators leads to a panoply of intriguing phenomena, most famously high-temperature superconductivity.",
"The topic has long been of interest [67], [68], and thermodynamics, spectral properties, and dynamics have been investigated [69], [70], [71], [72].",
"It has recently been shown that a hole defect in a Mott insulating system disperses neither purely ballistically nor diffusively: the hole in fact leaves a trace in the background as it travels, preventing the quantum interference of some trajectories [73].",
"In light of this interesting result it is useful to consider the noninteracting analog.",
"The initial density matrix for a hole initially at site $j_0$ in a $T\\gg J$ Mott insulator is $\\rho =Z^{-1}\\bigotimes _{i}{\\left\\lbrace \\begin{array}{ll}\\mathinner {|{0}\\rangle }_{i}\\mathinner {\\langle {0}|}_{i} & i = j_0\\\\ \\exp \\left(-\\beta H_{i}\\right) & \\rm {otherwise}\\end{array}\\right.",
"}$ which is identical to the Mott insulator density matrix except at $j_0$ .",
"Likewise $f^{j_0}$ and $g^{j_0}$ are zero, with $f$ and $g$ otherwise identical to those of the Mott insulator.",
"We find that, as expected for single particle ballistic motion, the hole disperses outwards according to the distribution $1-\\left\\langle n_{j}\\right\\rangle (t)=\\left|\\mathcal {J}_{\\left|j-j_{0}\\right|}\\left(2Jt\\right)\\right|^{2}$ as shown in Fig.",
"REF (a).",
"Figure REF (b) shows that as the hole disperses, it modifies spin correlations between pairs of nearby sites.",
"In particular, the correlations obtain contributions of the opposite sign, suppressing the correlations and even at some points reversing their sign, compared to their values in the absence of the hole.",
"This gives the impression that the spreading hole is dressed with a cloud of spin correlations.",
"Such a phenomenon might be thought to be unique to an interacting system, and is certainly of interest there.",
"Our results show that apparently similar phenomena occur even without interactions, although they arise from different causes."
],
[
"Conclusions",
"We have shown that interaction quenches of the Fermi-Hubbard model from initial product states to the non-interacting limit produce transient connected two-site correlations.",
"The correlations develop despite the initial states being at high temperature or, in the case of the product state antiferromagnet, high energy.",
"Even when the temperature is much greater than the initial tunneling, and very much greater than the superexchange energy scale, significant correlations exist in the dynamics.",
"This finding contrasts with the natural idea that at high temperature or high entropy correlations should be absent.",
"For example, in the context of previous work that has observed correlations out of equilibrium from a high-entropy initial state of ultracold molecules [74], [75], [76] it has sometimes been argued that long-range interactions are crucial.",
"Our present work shows that in contrast, correlations are ubiquitous out of equilibrium, even when one starts from high entropy states.",
"We generally observe that the correlations grow in a light cone and then fade away.",
"In the process, interesting structures emerge, such as correlations with a sign opposite that of the equilibrium system and intertwined density and spin correlations of equal magnitude.",
"It is noteworthy that these and other phenomenon, such as the appearance of a spin correlation cloud around a hole, that look intriguing and might typically be associated with strong interactions, can occur quite generally in out-of-equilibrium systems, even in the absence of interactions.",
"In this light, our work provides a useful comparison for future work with interacting quenches.",
"We note that the peak magnitudes of the connected correlations are up to about $\\sim 0.15$ .",
"While the precise values depend strongly on the initial conditions, these values are comparable in magnitude to recent equilibrium observations of correlations in 1D [48] and 2D [52], and 3D [77], [78], [39].",
"This indicates that the correlations generated dynamically from uncorrelated initial states are large enough to be experimentally measured.",
"One interesting future direction involves understanding how the features of integrability of this system manifest for finite temperature initial states.",
"This is much less explored than quenches from low temperature initial states.",
"It is expected to be fruitful to start by understanding the simplest integrable systems – non-interacting ones.",
"In our system, even though one expects the steady state to be non-thermal, i.e.",
"to prethermalize, due to our initial conditions the prethermalization coincides with a $T=\\infty $ thermal equilibrium state as measured by spin, density, and correlation operators.",
"This is because the initial product state has equal overlap with all the states in the noninteracting band (i.e., the eigenstates of the final Hamiltonian), leading to a final state with occupation numbers independent of energy.",
"We expect that perturbing the initial state away from a perfect product state will lead to a detectable difference from a thermal steady state.",
"Likewise, including weak interactions in the post-quench Hamiltonian should allow for the investigation of integrability breaking and prethermalization.",
"We thank Rafael Poliseli-Teles for conversations.",
"K.R.A.H.",
"thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, for its hospitality while part of this work was carried out, and the Welch foundation, Grant No.",
"C-1872.",
"R.G.H.",
"thanks the National Science Foundation (Grant No.",
"PHY-1408309), the Welch Foundation (Grant No.",
"C-1133), an ARO-MURI (Grant No.",
"W911NF-14-1-0003), and the ONR."
]
] | 1612.05671 |
[
[
"New R Coronae Borealis and DY Persei Candidates in the SMC"
],
[
"Abstract We report 3 new R Coronae Borealis and 63 new DY Persei candidates in the Small Magellanic Cloud.",
"Our analysis, based on data published by the OGLE team, consisted in a search for the characteristic drops in brightness that define these classes.",
"All candidates had been previously classified as semi-regular or Mira variables.",
"We briefly remark upon the possible existence of a \"borderline\" DY Per-like star and a \"transitional\" DY Per/RCB star.",
"Follow-up observations are needed to conclusively establish the nature of our candidates."
],
[
"[twoside, epsf]article COMMISSIONS 27 AND 42 OF THE IAU INFORMATION BULLETIN ON VARIABLE STARS Number 5xxx Konkoly ObservatoryBudapest 00 Month 200xHU ISSN 0374 – 0676 NEW R CORONAE BOREALIS AND DY PERSEI CANDIDATES IN THE SMC NIKZAT, F.$^{1,2}$ ; CATELAN, M.$^{1,2}$ $^{1}$ Instituto de Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, Av.",
"Vicuña Mackenna 4860, 782-0436 Macul, Santiago, Chile; e-mail: [email protected], [email protected] $^{2}$ Millennium Institute of Astrophysics, Santiago, Chile R Coronae Borealis (RCB) stars are C-rich, H-deficient red supergiants that undergo dramatic dimming episodes at irregular intervals.",
"The dimming episodes are caused by self-obscuration by dust, occurring as a consequence of mass loss events (e.g., Lambert & Rao 1994; Clayton 1996, 2012; Catelan & Smith 2015, and references therein).",
"The purpose of this contribution is to report on a new search for RCB stars in the Small Magellanic Cloud (SMC), carried out using $VI$ light curves from the OGLE project (Soszyński et al.",
"2011).",
"To detect candidates, the $VI$ light curves of all SMC red variable stars were visually inspected, and compared against templates from the literature.",
"New RCB candidates were detected in the process, which had previously been classified as semi-regular or Mira variables.",
"Additionally, DY Persei candidates were also identified.",
"Compared to their RCB counterparts, the DY Per stars tend to be cooler, have slower decline rates, and more symmetrical declines (e.g., Začs et al.",
"2007; Catelan & Smith 2015, and references therein).",
"If confirmed, these detections would lead to a significant increase in the number of known RCB + DY Per stars in the SMC.",
"The RCB stars have traditionally been classified as eruptive variables, due to their massive ejection episodes.",
"However, they may also be classified as self-eclipsing variable stars, because of the self-obscuration due to formation of carbon dust around the star during mass loss events.",
"Consequently, the RCB stars show a deep drop in their light curves which is a distinguishing characteristic of this class of variables.",
"Since the dust forms in discrete clouds, the declines only occur when dust condenses along our line of sight.",
"The evolutionary origin of the RCB stars is not understood yet.",
"Two scenarios have been proposed for their formation (Iben et al.",
"1996; Saio & Jeffery 2002): a merger of two white dwarfs or a final He-shell flash of the central object of a planetary nebula-hosting post-asymptotic giant branch (AGB) star.",
"In the latter case, they would represent so-called “born-again stars,” to the extent that they would constitute (pre-) white dwarf stars that have been brought back to giant dimensions (Renzini 1990); in the former, they would be low-mass analogs of the same process that is believed to result in type Ia supernovae.",
"RCB stars are rare, with only about a hundred currently known (Tisserand et al.",
"2016), of which roughly one quarter are found in the Magellanic Clouds (Tisserand et al.",
"2013).",
"To properly understand their evolutionary origin, more RCB stars in different environments with different metallicities are required.",
"Furthermore, AGB stars are known as one of the main producers of dust to the interstellar medium (ISM), and likewise RCB stars may also significantly contribute to the dust enrichment of the ISM.",
"As dust has different behavior in different environments, building significant samples of low-metallicity RCB stars can provide useful constraints on the role such stars may have played in the course of cosmic history.",
"In this note, we present new RCB candidates found in the relatively low-metallicity environment of the SMC, based on an analysis of the morphology of the light curves of red variables published by the OGLE team.",
"Their names and coordinates are provided in Table 1.",
"To date, only three RCB and three DY Per stars have been confirmed in the SMC (Tisserand et al.",
"2009), with an additional two RCB plus three DY Per candidates also having been reported in the literature (Kraemer et al.",
"2005; Tisserand et al.",
"2009).",
"Most recently, a new RCB candidate, Gaia16aau, was discovered using Gaia data (Tisserand et al.",
"2016).",
"The catalog data for red variable stars in the SMC are available online from the OGLE project.ftp://ftp.astrouw.edu.pl/ogle/ogle3/OIII-CVS/smc/lpv/ The data were taken with the 1.3-meter Warsaw telescope at Las Campanas Observatory, northern Chile, in the course of the OGLE-III campaigns (Soszyński et al.",
"2011).",
"All $VI$ light curves were visually inspected, in a search for dramatic, non-periodic drops in brightness that might be indicative of RCB-like behavior.",
"Our results are summarized in Table 1 and Figures 1 through 14.",
"In total, we present two new RCB (Fig.",
"1) and 63 new DY Per (Figs.",
"2-14) candidates.",
"A third RCB candidate was also identified, and will be discussed later in this note.",
"For completeness, previously confirmed and candidate RCB and DY Per stars in the SMC are also listed in Table 2.",
"Among these, three confirmed DY Per stars (OGLE-SMC-LPV-03068, OGLE-SMC-LPV-04633, OGLE-SMC-LPV-11903) and three DY Per candidates (OGLE-SMC-LPV-05023, OGLE-SMC-LPV-06616, OGLE-SMC-LPV-12291) from the EROS2 project (Tisserand et al.",
"2009) were detected in the OGLE data.",
"Their corresponding light curves are shown in Figures 15 and 16 for the confirmed and candidate DY Per stars, respectively.",
"Note that we include OGLE-SMC-LPV-05007 among the DY Per candidates in this paper, even though it was rejected by Tisserand et al.",
"(2009) due to the presence of strong TiO bands.",
"However, the light curve morphology bears some resemblance to those of other C-rich stars, and indeed the star has been classified as a C-star in the OGLE-III catalog.",
"This star may thus be an interesting example of what might perhaps be called a “borderline” DY Per-like star, not clearly conforming to the canonical DY Per classification scheme.",
"Its coordinates are given in the bottom row of Table 2.",
"We emphasize, in this context, that OGLE-SMC-LPV-11903Note that the OGLE-2 ID for this star is missing in the OGLE-III catalog.",
"(EROS2-SMC-DYPer-3), which has previously been classified as a DY Per star based on spectroscopic data, presents a light curve morphology that is strongly reminiscent of an RCB star (Fig. 15).",
"This may also hint at the possibility of a “transitional” DY Per/RCB status.",
"The latter might be consistent with the presence of an evolutionary sequence among hydrogen-deficient carbon stars, as suggested by Saio & Jeffery (2002) and supported by De Marco et al.",
"(2002) and Schaefer (2016).",
"We also note that, while OGLE-SMC-LPV-03068 (EROS2-SMC-DYPer-2) is classified as an O-type LPV in the OGLE-III catalog, it has already been spectroscopically confirmed to be a DY Per C-star (see Tisserand et al.",
"2004, 2009).",
"The star's light curve, as shown in Figure 15, is indeed consistent with that expected for a C-star.",
"We were able to match the spectroscopic RCB candidate MSX-SMC-014 (Kraemer et al.",
"2005; Tisserand et al.",
"2009) to OGLE-SMC-LPV-05719; the light curve is shown in Figure 17.",
"We point out that this light curve bears some resemblance to that of OGLE-SMC-LPV-17611; in both cases, we see several photometric declines during the OGLE-III monitoring, and the time interval between adjacent minima/maxima is roughly similar (Fig. 17).",
"We accordingly propose OGLE-SMC-LPV-17611 as an additional candidate RCB star in the SMC.",
"To close, we note that, in our work, we have only used light curve morphology as indicative of potential RCB/DY Per status.",
"Follow-up observations, both photometric and spectroscopic, are required in order to conclusively establish the nature of our candidates.",
"Acknowledgments: We warmly thank the referee, E. J. Montiel, for his thoughtful report.",
"Support for this project is provided by the Ministry for the Economy, Development, and Tourism's Millennium Science Initiative through grant IC 120009, awarded to the Millennium Institute of Astrophysics (MAS); by Proyecto Basal PFB-06/2007; by CONICYT's PCI program through grant DPI20140066; and by FONDECYT grants #1141141.",
"FN is grateful for financial support by Proyecto Gemini CONICYT grants #32130013 and #32140036.",
"References: Catelan, M., Smith, H. A., 2015, Pulsating Stars (Wiley-VCH) Clayton, G. C. 1996, PASP, 108, 225 Clayton, G. C., 2012, JAVSO, 40, 539 De Marco, O., Clayton, G. C., Herwig, F., et al.",
"2002, AJ, 123, 3387 Kraemer, K. E., Sloan, G. C., Wood, P. R., et al., 2005, ApJ, 631, L147 Lambert, D. L., & Rao, N. K. 1994, JApA, 15, 47 Renzini, A., 1990, ASPC, 11, 549 Saio, H., & Jeffery, C. S. 2002, MNRAS, 333, 121 Schaefer, B. E. 2016, MNRAS, 460, 1233 Soszyński, I., Udalski, A., Szymański, M. K., et al., 2011, AcA, 61, 217 Tisserand, P., Clayton, G. C., Welch, D. L., et al., 2013, A&A, 551, A77 Tisserand, P., Marquette, J.",
"B., Beaulieu, J. P., et al., 2004, A&A, 424, 245 Tisserand, P., Wood, P. R., Marquette, J.",
"B., et al., 2009, A&A, 501, 985 Tisserand, P., Wyrzykowski, L., Clayton, G., et al., 2016, ATel, 8681, 1T Začs, L., Mondal, S., Chen, W. P., et al.",
"2007, A&A, 472, 247 Table 1.",
"New RCB and DY Per Candidates in the SMC Table: NO_CAPTION Table 1.",
"New RCB and DY Per Stars in the SMC (cont.)",
"Table: NO_CAPTION Table 2.",
"Known RCB and DY Per Stars in the SMC Table: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION Figure: NO_CAPTION"
]
] | 1612.05546 |
[
[
"Chiral 2d Theories from N=4 SYM with Varying Coupling"
],
[
"Abstract We study 2d chiral theories arising from 4d N=4 Super-Yang Mills (SYM) with varying coupling tau.",
"The 2d theory is obtained by dimensional reduction of N=4 SYM on a complex curve with a partial topological twist that accounts for the non-constant tau.",
"The resulting 2d theories can preserve (0,n) with n = 2, 4, 6, 8 chiral supersymmetry, and have a natural realization in terms of strings from wrapped D3-branes in F-theory.",
"We determine the twisted dimensional reduction, as well as the spectrum and anomaly polynomials of the resulting strings in various dimensions.",
"We complement this by considering the dual M-theory configurations, which can either be realized in terms of M5-branes wrapped on complex surfaces, or M2-branes on curves that result in 1d supersymmetric quantum mechanics."
],
[
"Introduction",
"Two-dimensional chiral supersymmetric and superconformal theories, or strings, have enjoyed revived interest in recent years.",
"A useful starting point for generating new theories of this type are compactifications of higher-dimensional superconformal theories in a suitable background.",
"For instance 4d $N=2$ or $N=4$ SYM on a complex curve with a partial topological twist have been studied in [1], [2], [3].",
"In this paper we provide a new class of such 2d theories obtained from $N=4$ SYM by compactification along a curve where the complexified coupling $\\tau $ varies.",
"The resulting 2d theories have chiral supersymmetry, and in the limit of the curve volume going to zero size, they give rise to new superconformal theories.",
"The goal of this paper is to present a setup where such theories can be naturally studied and classified, and to derive the dimensional reductions including the spectra.",
"There are several topological twists of $N=4$ SYM with constant coupling, known as Vafa-Witten [4], Geometric-Langlands [5] and half-twist [6].",
"Allowing the coupling to vary requires changing the background for the $N=4$ SYM theory.",
"This can be realized, whilst preserving supersymmetry, by the so-called topological duality twist, introduced in [7] for the abelian case and generalized in a non-abelian manner in [8].",
"This twist combines both an R-symmetry background field, and one for the so-called bonus-symmetry $U(1)_D$ [9], [10], [5] of the abelian $N=4$ SYM theory.",
"As often, such non-trivial backgrounds for supersymmetric gauge theories have a brane realization.",
"As we consider $N=4$ SYM with varying coupling, the natural setting are D3-branes in IIB backgrounds where the axio-dilaton, which is identified with $\\tau $ on the brane world-volume, varies.",
"Such backgrounds go by the name F-theory [11], [12], [13].",
"In fact we will show that F-theory on elliptically fibered Calabi-Yau spaces provides supersymmetric backgrounds for the $N=4$ SYM theory which preserve chiral supersymmetry in 2d.",
"There are several possibilities for twisting the $N=4$ SYM theory including the $U(1)_D$ bonus-symmetry, each of which will have an interpretation in terms of an F-theory background.",
"The compactification space of F-theory is an elliptic Calabi-Yau manifold, with the complex structure of the elliptic fiber representing the axio-dilaton of IIB string theory.",
"The gauge sector is realized in terms of 7-branes.",
"In addition, D3-branes can be mutually supersymmetric and provide a new and interesting sector of the theory.",
"The $SL(2,\\mathbb {Z})$ symmetry of Type IIB theory is identified with the Montonen-Olive duality of the $N=4$ SYM theory on the worldvolume of the D3-brane, acting on the complexified coupling constant $\\tau $ .",
"In this paper we will study D3-branes which are wrapped on complex curves $C$ , and in the low energy limit thus correspond to strings in $d$ dimensions.",
"As the curve is part of the base of the elliptic fibration, the D3-coupling $\\tau $ indeed varies over $C$ , as long as $C$ has transversal intersections with the 7-branes, which source the $\\tau $ -monodromy.",
"Such strings not only result in new 2d chirally supersymmetric theories, which become conformal as the volume of $C$ vanishes, but also form important additional sectors in F-theory compactifications.",
"Specifically in compactifications to 2d, the D3-brane sector is crucial for the consistency of the F-theory vacuum.",
"Indeed, recently 2d string compactifications have attracted renewed interest [14], [15], [16], [17], [18], [19], [20].",
"In the F-theory realization of 2d $(0,2)$ string vacua on Calabi-Yau five-folds [16], [18], consistency of the theory forces the introduction of space-filling D3-branes wrapping a curve in the compactification manifold.",
"Thus understanding the D3 sector is paramount to fully characterizing such chiral 2d string vacua.",
"In higher dimensions, strings usually play the role of additional sectors of the theory which pinpoint loci in the moduli space with interesting physics.",
"The prime example for this are strings in 6d $N=(1,0)$ theories, which have recently seen a resurgence of interest, following the seminal paper [21].",
"Supersymmetric self-dual strings are an important subsector of these theories [22], [23], [24], [25]: In their tensionless limit, corresponding to wrapped curves of zero volume, they are indicators of superconformal invariance of the underlying 6d theories.",
"Multiple examples of strings in 6d have been investigated from various points of view, including [26], [27], [28], [29], [30], [31], [32] and references therein.",
"Strings in 4d $N=1$ compactifications have been comparatively less explored.",
"In 4d, strings are dual to instantons.",
"Again when they become tensionless, interesting physics can be expected in the 4d theories.",
"However, unlike in 6d, these are not BPS objects since the string vacuum only preserves four real supercharges.",
"Thus their tension is a priori not protected and may receive, albeit possibly small, quantum corrections [33].",
"Nevertheless they are an inevitable part of the rich landscape of 4d string compactifications, which deserve further study.",
"The strings we consider in this work preserve $(0,p)$ supersymmetry along their worldvolume, with the value of $p$ depending on the dimensionality of the transverse space as summarized in table REF .",
"We will determine the string action, as well as the BPS equations and zero-mode spectrum in each dimension and utilize these to study anomaly cancellation on the strings.",
"For strings from single D3-branes, as studied in this paper, the variation of $\\tau $ has the interesting effect of breaking the $U(1)$ gauge symmetry in the 2d effective field theory along the string.",
"This is in fact counter to naive expectations from the weakly coupled Type IIB limit, which we describe in more detail in section REF .",
"The breaking of the abelian gauge group is owed to non-perturbative dynamics localised in the vicinity of the orientifold plane.",
"Such effects induce a quantum Higgsing of the perturbative $U(1)$ in F-theory.",
"As we will explain, the order parameter of the quantum Higgsing corresponds to the distance between the two mutually non-local 7-branes into which the orientifold plane splits in F-theory.",
"Another interesting aspect of the duality twist which we will find is that it considerably modifies the BPS equations, and hence the Hitchin system, along the D3-brane, already in the abelian case.",
"The general solutions of the gauge field components along $C$ can be viewed as duality-twisted flat connections on $C$ .",
"It would be interesting to study the properties of this duality-twisted Hitchin system further, including the non-abelian formulation.",
"The duality twist as defined in [7] has the shortcoming that it only strictly applies to the abelian $N=4$ SYM theory, as the $U(1)_D$ does not survive in the non-abelian generalization [9], [10].",
"Nevertheless, the expectation is that a stack of D3-branes wrapped on the base of an F-theory compactification should be a consistent setup and thus an analogue of the topological twist should exist also in the non-abelian theory.",
"It was subsequently pointed out in [8] and shown that the duality twist in fact, once mapped to M-theory via M/F duality, becomes a standard (geometric) topological twist of the M5-brane theory and a non-abelian generalization of this was proposed.",
"The D3-brane wrapped on a cycle in the base of the Calabi-Yau maps to an M5-brane wrapping in addition the elliptic fiber of the Calabi-Yau, and the $U(1)_D$ becomes the symmetry associated to the elliptic fibration.",
"This topological twist has a clear generalization to the non-abelian case and for the D3-branes wrapping four-cycle in the base of the Calabi-Yau this was presented in [8].",
"For strings the M-theory dual description also provides a way to first of all test the duality twist, in addition to providing a generalization to non-abelian strings.",
"There are two options how to dualize the D3-brane, either to an M5 or an M2.",
"We will utilize both points of view and show agreement between the resulting theories.",
"The M2-brane approach leads to a 1d supersymmetric Quantum Mechanics (SQM) theory, which is the dimensional reduction of the string on a circle.",
"These theories have so far received comparatively little attention, but recently for M2-branes on curves in K3 they have been studied in [34], [35].",
"The BPS spectrum which we find on M2 branes wrapping a curve $C$ in the base of a general elliptic fibration is matched with the spectrum on the dual D3-brane setup.",
"As we will explain, this makes use of an interesting speciality of supersymmetric theories in one dimension known as automorphic duality [36], [37].",
"Again, similar to the M5-brane approach, this point of view allows for a generalization to the BLG theory [38], [39], and thus a non-abelian version of the dimensionally reduced string.",
"So far we have only discussed the D3-brane in terms of the $N=4$ SYM degrees of freedom.",
"However, the key players of a generic F-theory compactification, the 7-branes, intersect the D3-brane world-volume in complex codimension-one loci.",
"From the point of view of the $N=4$ SYM coupling $\\tau $ , the 7-branes are the loci where the coupling diverges.",
"For definiteness let us assume the F-theory fibration has a section, and thus a presentation in terms of a Weierstrass model $y^2 = x^3 + fx + g$ .",
"The so-defined variety is singular along $\\Delta =0$ in the base of the fibration, where $\\Delta = 4 f^3 + 27 g^2$ is the discriminant.",
"The discriminant characterizes precisely the loci where the 7-branes wrap the base of the fibration.",
"The intersection of the D3-branes with $\\Delta =0$ are, from the point of view of the $N=4$ SYM theory, loci where $\\tau $ undergoes monodromy in $SL(2,\\mathbb {Z})$ .",
"In fact the loci are so-called duality defects, which have been studied in [7], [8] for the D3-brane with duality twist wrapping a surface in the base of the elliptic fibration.",
"In that case the duality defects are 2d with chiral 3–7 strings localized on the defect, and as observed in [8] these duality defects themselves intersect along points.",
"In the present case where the D3-brane wraps a curve $C$ in the base of the Calabi-Yau, the duality defects are point-like in $C$ , but again there are degrees of freedom localized on them, which correspond to 3–7 strings.",
"These give rise to a number of left-moving degrees of freedom, which we show is universally given in terms of $n_{37}= 8 \\, \\text{deg}(\\mathcal {L}_D) = 8\\, c_1(B_{n-1}) \\cdot C \\,,$ where $\\mathcal {L}_D$ is the line bundle associated to the duality $U(1)_D$ .",
"This contribution can be corroborated in terms of the dual M5-brane picture.",
"Directly counting these strings, however, is in general a formidable task, which we address only in passing, but the technology for determining the spectra is readily available, see for instance [40], [41], [42], [43], [44].",
"For Type IIB orientifolds we provide a discussion of this sector in section REF .",
"The general anomaly polynomial, which we analyse in section , places further interesting constraints on the spectrum of 3–7 strings.",
"Specifically, demanding that the anomaly polynomial arising via inflow from the bulk in which the string propagates is cancelled by the spectrum on the string requires that there are additional states contributing as in (REF ).",
"An important off-spring of the present work is a full characterization of the gravitational anomalies along the string theories in various dimensions.",
"As noted already, in the 2d setting of [16], [18] the addition of extra D3-branes is in fact imperative due to tadpole constraints as the strings are spacetime-filling.",
"Tadpole cancellation in turn guarantees that the complete spectrum, consisting of the 7-brane sector, the 2d $(0,2)$ supergravity sector and the D3-brane sector, is anomaly free.",
"For the gauge anomalies this has already been investigated and related to M-theory Chern-Simons terms in [16], but in absence of a complete understanding of the D3-brane sector no such verification was possible.",
"In [45] we combine this article's results on the D3-brane sector in 2d $(0,2)$ theories together with a derivation of the supergravity spectrum to show cancellation of all gravitational anomalies in F-theory compactifications on Calabi-Yau five-folds.",
"Finally, it is of considerable interest to explore the theory on the strings from wrapped D3-branes as 2d super-conformal field theories by themselves.",
"The central charges can be computed for each given base and would yield interesting new holographic setups in IIB with varying axio-dilaton.",
"Other computations, which naturally have a starting point with the results in this paper are the computation of elliptic genera for these theories as already started in [31] for the 6d case, but looking ahead for the $N=(0,2)$ theories, the starting points would be the general expression in [46].",
"For ease of locating the results the various spectra can be found in the following locations in the paper.",
"An overview of the general setup for strings from D3, M5, and M2-branes is given in section .",
"The D3-branes with topological duality twist along $C$ are discussed in section .",
"The spectra in 6d, 4d and 2d are summarized in the tables REF , REF and REF , respectively.",
"The 8d case is special in that the D3-brane wraps the full base of the elliptic K3, and can be found in appendix .",
"The remainder of the paper then discusses the M5 and M2-duals in section .",
"The anomalies of the strings in all dimensions can be found in section .",
"Appendices providing shelter for conventions and details of the string actions as well as cohomological computations can be found at the end of the paper."
],
[
"Duality Twists of $N=4$ SYM and Brane Realizations",
"In this section, we study 4d $N=4$ SYM on a curve $C$ along which the coupling constant $\\tau $ varies, and determine the possible duality twists which retain supersymmetry in the non-compact two dimensions.",
"We then present three brane-setups related to this, in terms of D3-branes, M5-branes and M2-branes, respectively."
],
[
"Duality Twisted $N=4$ SYM",
"We determine all possible duality twisted theories and the supersymmetry that they preserve in the non-compact two dimensions.",
"For constant coupling the analog of this analysis has been performed in [2].",
"For $N=4$ SYM with varying coupling, the twists have to include the structure group $U(1)_C$ along the curve, an R-symmetry factor $U(1)_R \\subset SU(4)_R$ and the duality (or `bonus symmetry') $U(1)_D$ of the abelian $N=4$ SYM theory [9], [10].",
"Indeed, the variation of $\\tau $ can be understood in terms of a non-trivial line bundle whose structure group $U(1)_D$ forms a bonus symmetry of the underlying abelian $N = 4$ SYM theory.",
"If under the $SL(2,\\mathbb {Z})$ transformation the complexified gauge coupling $\\tau = \\frac{\\theta }{2\\pi } + \\frac{4 \\pi i }{g^2} = \\tau _1 + i \\tau _2$ is mapped to $\\tau \\ \\rightarrow \\ \\frac{a \\tau + b}{c \\tau + d} \\,,\\qquad ad-bc= 1\\,,\\qquad a,b,c,d \\in \\mathbb {Z} \\,,$ then a field of $U(1)_D$ charge $q$ transforms as [5] $ \\Phi \\rightarrow e^{i q \\alpha } \\, \\Phi \\qquad {\\rm with} \\qquad e^{i\\alpha } = \\frac{c \\tau + d}{|c \\tau + d|} \\,.$ We shall briefly recap the spectrum of $N=4$ SYM theory in four dimensions, including the charges of the fields under the $U(1)_D$ symmetry.",
"The field content consists of a vector multiplet containing a gauge field $A_\\mu $ , six scalars $\\phi _i$ , and fermions $\\Psi _\\alpha ^I$ and $\\widetilde{\\Psi }_{\\dot{\\alpha }I}$ .",
"The indices transform in various different representations of the symmetry group $SO(1,3)_L \\times SU(4)_R$ : $\\mu $ in the vector of $SO(1,3)_L$ , $\\alpha $ ($\\dot{\\alpha }$ ) in the fundamental of the left (right) $SU(2)$ of $SO(1,3)_L$ , $i$ in the ${\\bf 6}$ of $SU(4)_R$ , and finally $I$ in the ${\\bf 4}$ or ${\\bf \\overline{4}}$ of $SU(4)_R$ .",
"The field content transforms under the total symmetry group $G_{\\rm total} = SO(1,3)_L \\times SU(4)_R \\times U(1)_D \\,,$ as A : (2,2,1)* i : (1,1,6)0 I : (2,1,4)1 I : (1,2,4)-1 , where the subscript indicates the $U(1)_D$ representation.",
"The gauge field does not itself form a $U(1)_D$ eigenstate and thus has no well-defined $U(1)_D$ charge, but the self-dual and anti-self-dual components of the field strength have definite $U(1)_D$ charges $\\begin{aligned}q_D=+2:\\qquad \\sqrt{\\tau _2} F^+ &= - \\frac{i}{2\\sqrt{\\tau _2}} (F_D - \\bar{\\tau }F) \\cr q_D=-2:\\qquad \\sqrt{\\tau _2} F^- &= \\frac{i}{2\\sqrt{\\tau _2}} (F_D - {\\tau }F) \\,,\\end{aligned}$ where $F_D$ is the duality transformed field strength $F_D = \\tau _1 F + i \\tau _2 \\star F \\,.$ The sixteen supersymmetries $Q_{\\alpha I}$ and $\\widetilde{Q}_{\\dot{\\alpha }}^I$ transform under $G_{\\rm total}$ as [5] QI : (2,1,4)1 QI : (1,2,4)-1 , and, equivalently, the supersymmetry transformation parameters are given by I : (2,1,4)-1 I : (1,2,4)1 .",
"Associated to the $U(1)_D$ symmetry is the complex line bundle $\\mathcal {L}_D$ defined over the subspace of spacetime, $C$ , where $\\tau $ varies.",
"A field of $U(1)_D$ charge $q$ transforms as a section of the $q$ -th power of this $U(1)_D$ bundle $\\mathcal {L}_D$ .",
"The one-form connection on the bundle is given by $ {\\cal A} = \\frac{d \\tau _1}{2 \\tau _2} \\,.$ The connection can be decomposed into $(0,1)$ and $(1,0)$ forms on $C$ as $\\mathcal {A} = \\mathcal {A}^{(0,1)} + \\mathcal {A}^{(1,0)} \\,,$ where these forms defineWe use an identical notation in $\\mathcal {L}_D$ for both the complex and holomorphic line bundles merely to prevent a proliferation of notation.",
"Where not clarified the particular line bundle of interest at any point should be unambiguous by context.",
"respectively a holomorphic line bundle, $\\mathcal {L}_D$ , and an anti-holomorphic line bundle, $\\bar{\\mathcal {L}}_D$ .",
"To write expressions in an appropriately covariant way we shall also need to introduce covariant derivatives, on $C$ , with respect to the connections on these $U(1)_D$ bundles.",
"Given a field with $U(1)_D$ charge $q_D$ these are defined via $d_{\\cal A} = d + i q_D^{\\rm twist} {\\cal A} =\\frac{1}{2}(\\partial + i q_D {\\cal A}^{(1,0)}) +\\frac{1}{2}(\\bar{\\partial }+ i q_D {\\cal A}^{(0,1)})= \\frac{1}{2} (\\partial _{\\cal A} + \\bar{\\partial }_{\\cal A}) \\,,$ where $\\partial $ , $\\bar{\\partial }$ are the standard holomorphic and anti-holomorphic derivatives on $C$ .",
"Since the 4d spacetime of the $N=4$ SYM theory is the product space $\\mathbb {R}^{1,1}\\times C$ , the $SO(1,3)_L$ Lorentz symmetry is broken to $SO(1,1) \\times U(1)_C$ .",
"The relevant representations decompose as SO(1,3)L SO(1,1) U(1)C (2,2) 12,0 1-2,0 10,2 10,-2 (2,1) 11,1 1-1,-1 (1,2) 11,-1 1-1,1 .",
"The topological twist requires to turn on a $U(1)$ R-symmetry background gauge field.",
"We are therefore interested in decompositions of the R-symmetry $SU(4)_R$ of $N=4$ SYM which contain a $U(1)$ factor.",
"These are given by the following decompositions: .",
"$SU(4)_R &\\ \\rightarrow \\ SO(4)_T \\times \\underline{U(1)_R} \\cr {\\bf 4} &\\ \\rightarrow \\ {\\bf (2,1)}_1 \\oplus {\\bf (1,2)}_{-1} \\cr {\\bf 6} &\\ \\rightarrow \\ {\\bf (1,1)}_2 \\oplus {\\bf (1,1)}_{-2} \\oplus {\\bf (2,2)}_0$ CY$_3$ Duality-Twist .",
"$SU(4)_R &\\ \\rightarrow \\ SU(2)_R \\times \\underline{U(1)_R} \\times SO(2)_T \\cr {\\bf 4} &\\ \\rightarrow \\ {\\bf 2}_{0,1} \\oplus {\\bf 1}_{1,-1} \\oplus {\\bf 1}_{-1,-1} \\cr {\\bf 6} &\\ \\rightarrow \\ {\\bf 1}_{0,2} \\oplus {\\bf 1}_{0,-2} \\oplus {\\bf 2}_{1,0} \\oplus {\\bf 2}_{-1,0}$ CY$_4$ Duality-Twist .",
"$SU(4)_R &\\ \\rightarrow \\ SU(3)_R \\times \\underline{U(1)_R} \\cr {\\bf 4} &\\ \\rightarrow \\ {\\bf 1}_3 \\oplus {\\bf 3}_{-1} \\cr {\\bf 6} &\\ \\rightarrow \\ {\\bf 3}_2 \\oplus {\\bf 3}_{-2}$ CY$_5$ Duality-Twist .",
"As we will show later on, these particular decompositions are those induced when the curve $C$ is a complex curve in the base of a Calabi-Yau $n$ -fold in F-theory.",
"The case where the R-symmetry remains unbroken $SU(4)_R$ corresponds to compactification on CY$_2$ , i.e.",
"K3, and is discussed in appendix .",
"There is an additional case where the $SU(4)_R$ breaks to $U(1)^3$ , with decomposition .",
"$SU(4)_R &\\ \\rightarrow \\ U(1)_1 \\times U(1)_2 \\times U(1)_3 \\cr {\\bf 4} &\\ \\rightarrow \\ {\\bf 1}_{1,1,1,} \\oplus {\\bf 1}_{-1,1,1}\\oplus {\\bf 1}_{1,-1,1}\\oplus {\\bf 1}_{1,1,-1}\\cr {\\bf 6} &\\ \\rightarrow \\ {\\bf 1}_{\\pm 2, 0, 0}\\oplus {\\bf 1}_{0, \\pm 2, 0}\\oplus {\\bf 1}_{0, 0, \\pm 2}$ CY$_5$ ' Duality-Twist The standard topological twist combines the R-symmetry $U(1)_R$ in one of the decompositions in (REF ) - (REF ) with the internal Lorentz symmetry $U(1)_C$ in (REF ).",
"As the supercharges transform non-trivially under the $U(1)_D$ symmetry it is necessary to combine this standard topological twist of $U(1)_C$ with $U(1)_R$ with an additional one including $U(1)_D$ , as follows from (REF ).",
"This is the duality twist, discussed in [7] for Euclidean D3-branes wrapping a Kähler surface on a base $B_3$ and generalized to non-abelian theories in [8].",
"Throughout we shall twist $U(1)_D$ with the same $U(1)_R$ , as we have twisted $U(1)_C$ with.",
"The resulting theories then preserve chiral supersymmetry in 2d as summarized in table REF .",
"The twist in (REF ) with the R-symmetry taken as the diagonal $U(1)_{\\rm diag} = U(1)_R$ results also in a $(0,2)$ theory.",
"Finally, we should note that the decomposition in (REF ) can in addition give rise to an $N= (0,6)$ in 2d, which will be discussed in detail in section REF .",
"As pointed out above, the 4d vector field $A_\\mu $ , which is insensitive to the R-symmetry and so will have the same decomposition regardless of the dimension of the compactification space, does not have a single well-defined $U(1)_D$ charge.",
"We shall write the decompostion of this vector under the 4d Lorentz and duality group as $\\begin{aligned}SO(1,3)_L \\times U(1)_D&\\quad \\rightarrow \\quad SO(1,1)_L \\times U(1)_C \\times U(1)_D \\cr {\\bf (2,2)}_* &\\quad \\rightarrow \\quad {\\bf 1}_{2,0,*} \\oplus {\\bf 1}_{-2,0,*} \\oplus {\\bf 1}_{0,2,*} \\oplus {\\bf 1}_{0,-2,*} \\,,\\end{aligned}$ and label the four resulting fields respectively as $v_+$ , $v_-$ , $\\bar{a}$ , and $a$ .",
"We shall identify these fields in terms of the components of $A_\\mu $ by determining which combinations of components have the same charges as listed.",
"Further we will find that the twisted supersymmetry variations, which relate bosonic fields of unspecified $U(1)_D$ charge to fermionic fields of known $U(1)_D$ charge, require that the objects $\\sqrt{\\tau _2} a \\quad \\text{ and } \\quad \\sqrt{\\tau _2}\\bar{a} \\,,$ do indeed have a precise $U(1)_D$ charge.",
"The $v_\\pm $ fields do not have this feature, however, they do not give rise to any massless fields in the compactification to two dimensions, and so the ambiguity in the charge of $A_\\mu $ does not translate into an ambiguity in the field content of the low energy theory.",
"Table: The number of supersymmetries preserved by 4d N=4N=4 SYM on C×ℝ 1,1 C\\times \\mathbb {R}^{1,1} with duality twist as defined in () – ().The brane realization of this setup corresponds to D3-branes wrapping a holomorphic curve CC inside a Calabi-Yau nn-fold in F-theory, giving rise to strings in d=12-2nd= 12-2n dimensions.Equivalently, this is the supersymmetry of the strings fromM5-branes wrapped on the elliptic surface C ^×ℝ 1,1 \\widehat{C} \\times \\mathbb {R}^{1,1}.",
"We also list the supersymmetriesof the 1d SQM arising from an M2-brane wrapping C×ℝC\\times \\mathbb {R}."
],
[
"Brane Realizations",
"Naturally, the setup so far has a realization in terms of D3-branes in F-theory, which will be briefly discussed now.",
"We consider F-theory on an elliptically fibered Calabi-Yau $n$ -fold $Y_n$ , which gives rise to a $(12-2n)$ -dimensional theory.",
"The D3-brane worldvolume is $\\mathbb {R}^{1,1} \\times C$ , where $C$ is a holomorphic curve in the base $B_{n-1}$ , and the 2d theory obtained by reduction along $C$ represents a string in the bulk non-compact directions.",
"We only consider situations where the curve $C$ contains transversal intersection points with the 7-brane locus $\\Delta $ in the base F-theory $B_{n-1}$ .",
"As one encircles the 7-brane loci, the coupling $\\tau $ of the underlying $N=4$ $U(1)$ SYM theory on the D3-brane undergoes a monodromy.",
"In the F-theory description, the complex line bundle $\\mathcal {L}_D$ is nothing other than the anti-canonical bundle of the base of the fibration [47], [7] $ \\mathcal {L}_D \\cong K_{B_{n-1}}^{-1}|_C \\,.$ Indeed the anti-canonical bundle $K_{B_{n-1}}^{-1}$ fundamentally defines the Calabi-Yau elliptic fibration $Y_n$ over $B_{n-1}$ via the associated Weierstrass model $y^2 = x^3 + f x z^4 + g z^6 \\,,$ where $f$ and $g$ transform according to $f \\in H^0(B, K_{B_{n-1}}^{-4})$ and $g \\in H^0(B, K_{B_{n-1}}^{-6})$ .",
"Similarly the restriction of the elliptic fibration to $C$ describes itself a (generically non-Calabi-Yau) elliptic fibration $\\widehat{C}$ , defined precisely by the anti-canonical bundle restricted to $C$ , i.e.",
"$\\mathcal {L}_D$ .",
"The existence of the elliptic fibration $\\widehat{C}$ guarantees that $\\mathcal {L}_D$ is ample, and further $\\mathcal {L}_D$ is trivial if and only if $C$ does not intersect the discriminant locus of $Y_n$ .",
"Appendix REF explains such attributes of the elliptic surfaces $\\widehat{C}$ .",
"In view of the D3-brane realizations, the decomposition of the $SU(4)_R$ -symmetry underlying the twists (REF ) - (REF ) corresponds to remnant rotation symmetries of the string in the transverse directions.",
"If we were to consider flat space, the R-symmetry of the $N=4$ SYM theory living on the D3-brane would decompose as $SU(4)_R \\simeq SO(6)_R \\rightarrow SO(10 - 2n)_T \\times SO(2n-4)_R .$ The first factor in the decomposition is the rotation group of the non-compact spacetime directions transverse to the D3-brane, and the second factor comes from the structure of the normal bundle of $C$ in $B_{n-1}$ .",
"The Kähler nature of $B_{n-1}$ further reduces the structure from $SO(2n-4)_R\\quad \\rightarrow \\quad U(n-2)_R \\simeq SU(n-2)_R \\times U(1)_R \\,.$ In this way we identify the twists in (REF ) – (REF ) with the embeddings into Calabi-Yau elliptic fibrations.",
"The additional case of the CY$_5$ ' twist in (REF ) can be thought of as an F-theory compactification on an elliptic Calabi-Yau five-fold where the base is the sum of three line-bundles over $C$ .",
"One can dualise this D3-brane to an M5-brane wrapping the elliptic surface $\\widehat{C}$ formed by restricting the fibration to $C$ in $Y_n$ : A T-duality transverse to the D3-brane first leads to a D4-brane, which is then uplifted to an M5-brane in M-theory.",
"Alternatively one can T-dualise along one of the extended directions of the D3-brane, giving rise to a D2-brane which is uplifted to an M2-brane in M-theory.",
"The interrelations between the three different viewpoints are shown in figure REF .",
"The $\\tau $ -monodromies in the D3-brane F-theoretic setup are geometrised in the M5-brane picture as the M5-brane also wraps the elliptic fiber above $C$ , which documents the variation of the coupling as already observed in [8].",
"As such the only topological twist necessary is that mandated by the curvature of the elliptic surface $\\widehat{C}$ , and there is no additional duality twist.",
"The only instance when the M5-brane point of view is not applicable is the case of strings in 2d, where the low energy effective theory of M-theory on the Calabi-Yau five-fold compactification is a 1d Super Quantum Mechanics (SQM).",
"In this instance, the only window into the worldvolume theory in M-theory arises from the point of view of the M2-brane.",
"Generally, the M2-brane point of view gives rise to a 1d SQM which is the circle compactification of the 2d theory living on the worldvolume of the string from the D3-brane.",
"Such SQM theories have had some appearance in the literature, see [34], [35].",
"By analysing the structure of the supersymmetric multiplets in the 1d theory and how they arise from the 2d theory before the $S^1$ reduction one can match the content of the two theories.",
"The number of supersymmetries preserved in the 1d SQM arising from an M2-brane wrapping $\\mathbb {R} \\times C$ for $C$ a holomorphic curve inside a Calabi-Yau $n$ -fold is known for $n \\le 5$ [48] and is listed in table REF .",
"We shall consider only situations with strictly this much supersymmetry preserved, and ignore special cases where the supersymmetry is enhanced further due to non-generic choices of $C$ or $Y_n$ .",
"By the highlighted dualities the same number of supersymmetries should be preserved when a D3-brane wraps $\\mathbb {R}^{1,1} \\times C$ with $C \\subset B_{n-1}$ in F-theory.",
"Indeed one can observe that it is necessary to consider a non-trivial $\\tau $ -profile to preserve exactly the same number of supersymmetries as in table REF ; without 7-brane insertions the elliptic fibration is trivial, $Y_n = B_{n-1} \\times T^2$ , and thus the base $B_{n-1}$ is also Calabi-Yau.",
"In such a situation the number of supersymmetries preserved after the partial twist is known [49] to be precisely double the number shown in table REF realized in a non-chiral fashionWhen $n=2$ the Calabi-Yau $Y_2$ is a K3-surface, which is a special case in this analysis and is described in appendix ..",
"Figure: Overview of setup and duality maps between D3-branes and their M-theory dual description in terms of M2- and M5-branes wrapping the curve CC or the elliptic surface with base CC, respectively."
],
[
"M5-branes on Elliptic Surfaces",
"Duality with M-theory maps a D3-brane wrapped on $C \\subset B_{n-1}$ to the theory of a single M5-brane on $\\mathbb {R}^{1,1} \\times \\widehat{C}$ , where $\\widehat{C}$ is a non-trivial elliptically fibered Kähler surface obtained by restricting the F-theory elliptic Calabi-Yau $n$ -fold $Y_n$ to the fiber over a curve $C$ inside the base.",
"The theory living on a single M5-brane is known as the 6d abelian tensor multiplet theory which preserves $N=(0,2)$ supersymmetry.",
"It has an $Sp(4)$ R-symmetry in addition to the $SO(1,5)$ Lorentz symmetry.",
"The abelian tensor multiplet contains a self-dual two-form field strength, $B_{\\mu \\nu }$ , scalars, $\\Phi ^{ij}$ , and symplectic Majorana-Weyl fermions $\\rho ^i$ .",
"Each of these fields transforms in the following representations of the $SO(1,5)\\times Sp(4)$ symmetry group: B : (15, 1) ij : (1,5) i : (4, 4) .",
"The field $B_{\\mu \\nu }$ is self-dual so there are not, in flat space, fifteen independent degrees of freedom contained in the field.",
"Often it shall be clearer to consider the field strength $H = dB$ , which by self-duality, $H = \\star _6 H \\,,$ transforms as $H \\,:\\quad ({\\bf \\overline{10}, 1}) \\,,$ under the symmetry groupIn our conventions a self-dual (resp.",
"anti-self-dual) three-form in six dimensions transforms in the ${\\bf \\overline{10}}$ (resp.",
"${\\bf 10}$ ) representation of $SO(1,5)$ ..",
"The supercharges of the theory transform in the same representation as the fermions, $Q^i \\,:\\quad ({\\bf \\overline{4}, 4}) \\,.$ Once we consider the theory on the geometry $\\mathbb {R}^{1,1} \\times \\widehat{C}$ and further given that $\\widehat{C}$ is Kähler, the 6d Lorentz group $SO(1,5)_L$ is broken to the subgroup $\\begin{aligned}SO(1,5) &\\quad \\rightarrow \\quad SU(2)_l \\times SO(1,1)_L \\times U(1)_l \\cr {\\bf 4} &\\quad \\rightarrow \\quad {\\bf 2}_{1,0} \\oplus {\\bf 1}_{-1,1} \\oplus {\\bf 1}_{-1,-1} \\cr {\\bf \\overline{10}} &\\quad \\rightarrow \\quad {\\bf 3}_{-2,0} \\oplus {\\bf 1}_{2, \\pm 2} \\oplus {\\bf 1}_{2, 0} \\oplus {\\bf 2}_{0, \\pm 1} \\cr {\\bf 15} &\\quad \\rightarrow \\quad {\\bf 1}_{0,0} \\oplus {\\bf 3}_{0,0} \\oplus {\\bf 1}_{0,\\pm 2}\\oplus {\\bf 1}_{0,0} \\oplus {\\bf 2}_{\\pm 2,\\pm 1} \\,,\\end{aligned}$ where all combinations of signs must be summed over.",
"Here $SU(2)_l \\times U(1)_l$ is the holonomy of the Kähler surface $\\widehat{C}$ and $SO(1,1)_L$ is the Lorentz rotations of $\\mathbb {R}^{1,1}$ .",
"As the two-form potential $B_{\\mu \\nu }$ is unconcerned with the particular decomposition of the R-symmetry it is also insensitive to the dimension $n$ of the compactification space $Y_n$ .",
"Instead of imposing the self-duality condition on $B_{\\mu \\nu }$ post-decomposition we can consider the decomposition of the $({\\bf \\overline{10}}, {\\bf 1})$ as in (REF ) to understand the field strengths in evidence when the M5-brane is placed on the product of $\\mathbb {R}^{1,1} \\times \\widehat{C}$ .",
"Indeed by studying the relationship between the decomposition of the $({\\bf 15, 1})$ and the $({\\bf \\overline{10}, 1})$ one can see that the resulting fields are ${\\bf 3}_{0,0} \\,,\\quad {\\bf 1}_{0,\\pm 2} \\,,\\quad {\\bf 1}_{0,0} \\,,\\quad {\\bf 2}_{{\\pm 2}, \\pm 1} \\,,$ at least on the level of the zero-modes of such fields as the derivative does not change the representation content of the internal symmetry groups.",
"The decomposition of the R-symmetry $Sp(4)$ will depend on the dimension of the Calabi-Yau $n$ -fold containing $\\widehat{C}$ .",
"Similarly to the decomposition for the D3-brane, ${ Sp}(4) \\rightarrow SO(9 - 2n)_T \\times U(n-2)_R \\,,$ where $SO(9 - 2n)_T$ is the group of rotations in the non-compact directions transverse to both the M5-brane and the Calabi-Yau, a $U(n-2)_R$ is the holonomy group associated with the directions internal to the Calabi-Yau that are transverse to the M5-brane.",
"It is clear from this discussion that it will not be possible to study the 2d theory arising from the compactification on a Calabi-Yau five-fold, since there are not two non-compact directions transverse to the five-fold for the M5-brane to fill – in this case we shall instead compare with the M2-brane theory expounded on in section REF .",
"The M5-brane theory has the feature that the $SL(2, \\mathbb {Z})$ self-duality of the D3-brane theory is manifest – this means that we do not have to include the bonus symmetry $U(1)_D$ explicitly.",
"The appearance of the D3-brane line bundle $\\mathcal {L}_D$ can be seen as follows.",
"To the elliptic surface $\\widehat{C}$ is associated a line bundle as part of its Weierstrass data (see appendix REF for more details) and this line bundle is exactly the restriction of the anti-canonical bundle of $B_{n-1}$ to $C$ , which is precisely $\\mathcal {L}_D$ .",
"In this way one can see that the duality becomes encoded in the elliptic surface $\\widehat{C}$ through its defining Weierstrass line bundle – as expected from the geometrization of the $SL(2,\\mathbb {Z})$ symmetry when uplifting from $N =4$ super-Yang-Mills to the 6d $(0,2)$ theory [50], [51]."
],
[
"M2-branes on Curves and Super-QM",
"Our final description of the effective string theories we consider is in terms of an M2-brane wrapping a curve $C$ in the base of an elliptically fibered Calabi-Yau $Y_n$ .",
"The resulting supersymmetric Quantum Mechanics (SQM) is related to the 2d effective field theory on a D3-brane wrapping the same curve $C$ by a circle reduction.",
"Since the compactified theory turns out to be quite subtle, let us first recall the well-known theory on a single M2 brane extended along ${\\mathbb {R}}^{1,2} \\subset \\mathbb {R}^{1,10}$ .",
"The effective theory on an M2-brane in flat space is a 3d $N=8$ superconformal theory with an $SO(8)_R$ symmetry group corresponding to the rotational group of the transverse space.",
"The supersymmetry parameters $\\epsilon $ and fields $\\Phi _i$ and $\\rho $ of the theory transform as follows: $\\begin{array}{c|c}& SO(3)_L \\times SO(8)_R \\cr \\hline \\epsilon & ({\\bf 2}, {\\bf 8}_s)\\cr \\Phi _i & ({\\bf 1}, {\\bf 8}_v) \\cr \\rho & ({\\bf 2}, {\\bf 8}_c) \\cr \\end{array}$ As is well-known, this field content can be derived from the field content of a Type IIA D2-brane along ${\\mathbb {R}}^{1,2} \\subset \\mathbb {R}^{1,9}$ : The transverse fluctuations of the D2-brane within $\\mathbb {R}^{1,9}$ give rise to seven real scalar fields; in addition the D2-brane carries a $U(1)$ gauge field potential, which in $\\mathbb {R}^{1,2}$ carries one dynamical degree of freedom and which can be dualised into a single real dynamical scalar field.",
"From the M2-perspective, this gauge-scalar is interpreted as the fluctuation scalar of the M2-brane in the direction of the M-theory circle and thus completes the set of scalars to transform in the $({\\bf 1}, {\\bf 8}_v)$ of $SO(8)_R$ .",
"In our setup the M2-brane worldvolume takes the form $C\\times \\mathbb {R}$ , giving rise to an $N=4$ or $N=2$ SQM.",
"Compactification on the holomorphic curve $C$ breaks the $SO(1,2)_L$ symmetry to the structure group $U(1)_L$ of the tangent bundle to $C$ , and also breaks the conformal symmetry for finite volume of the curve $C$ .",
"The ${\\bf 2}$ of $SO(1,2)_L$ clearly decomposes as ${\\bf 2} \\rightarrow {\\bf 1}_{1} \\oplus {\\bf 1}_{-1}$ , while the decomposition of the R-symmetry group depends on the dimensionality of $Y_n$ and will be discussed for the various situations in subsequent sections.",
"It is our desire to compare the 2d $(0,2)$ or $(0,4)$ theories from the worldvolumes of the D3- and M5-branes to the 1d $N = 2$ or $N = 4$ SQM from the M2-brane.",
"It is known, and is explained in appendix REF , that under the circle reduction the multiplets transform according to the rule $\\begin{aligned}& (0,2) \\text{ chiral multiplet} &\\rightarrow & \\quad N = 2B \\,\\, (2,2,0) \\text{ multiplet } \\cr &(0,2) \\text{ Fermi multiplet } &\\rightarrow &\\quad N = 2B \\,\\, (0,2,2) \\text{ multiplet } \\,,\\end{aligned}$ where the three entries in the $N=2B$ multiplet specify, respectively, the number of real scalars, fermions, and auxiliary fields.",
"A similar decomposition holds for the $(0,4)$ multiplets following the construction of $(0,4)$ multiplets in terms of $(0,2)$ multiplets in appendix .",
"However, in order to match the circle reduction of the 2d (0,2) and (0,4) theories to the M2 SQM, it is important to make use of a special property of SQM called automorphic duality.",
"Automorphic duality, as described in more detail in appendix REF , allows one to replace physical scalar fields in a 1d supersymmetric sigma model that are not a dependency of the moduli space metric with auxiliary fields.",
"In effect this transforms e.g.",
"a $(2,2,0)$ multiplet into a $(0,2,2)$ multiplet.In fact, this duality is also required to match the 2d $(0,2)$ supergravity spectrum from F-theory compactified on a Calabi-Yau five-fold [45] to the dual M-theory $N=2$ SQM of [52].",
"For the 2d theory from the D3-brane compactification we shall always observe that there exist Wilson line scalar fields $a$ , $\\bar{a}$ from the reduction of the 4d gauge field $A_\\mu $ along $C$ .",
"These scalars thus inherit a shift symmetry, and the moduli space metric cannot depend on them.",
"The same is true for the resulting $(2,2,0)$ multiplets obtained by circle reduction to SQM.",
"Hence for the $(2,2,0)$ multiplets associated with the shift-symmetric Wilson line scalars we shall always be able to invoke automorphic duality to transform the $(2,2,0)$ multiplet into a $(0,2,2)$ multiplet.",
"Such a situation also holds for the $N = 4B$ SQM from the compactification of the 2d $(0,4)$ theory on $S^1$ , where now we can exchange the $(4,4,0)$ multiplet containing $a$ and $\\bar{a}$ (along with two more real scalar fields) with a $(2,4,2)$ multiplet by dualising the Wilson line degrees of freedom only.",
"As we shall see the zero-modes of $a$ and $\\bar{a}$ are always counted by the same cohomology group $h^0(C, K_C \\otimes \\mathcal {L}_D) = g - 1 + \\text{deg}(\\mathcal {L}_D) \\,,$ as they come from the 4d Lorentz sector and do not couple to the R-symmetry.",
"We shall thus always be able to transform this many $(2,2,0)$ or $(4,4,0)$ multiplets in the compactification on $S^1$ to $(0,2,2)$ or $(2,4,2)$ multiplets respectively.",
"Indeed agreement between the spectra from the D3-brane theory compactified on $S^1$ and the M2-brane is reached only due to this possible reinterpretation.",
"Furthermore we expect there to be states analogous to the 3–7 strings in the F-theory picture, which have to be considered in addition on the M2-brane side.",
"While it would be interesting to understand the microscopic origin of these states explicitly in the M2-brane picture, it is clear by duality that we must add the usual number of $8 c_1(B_{n-1}) \\cdot C$ complex fermions to account for these modes."
],
[
"Strings from Duality-Twisted $N=4$ SYM",
"We now determine the dimensional reduction of the D3-branes wrapped on a complex curve in the base of an F-theory elliptic Calabi-Yau compactification.",
"Our analysis makes crucial use of the duality twist as introduced in [7] and applied recently in [31], [8].",
"This not only allows us to work out the 2d action for the abelian theory, but also the spectrum, multiplet structure, and the BPS equations.",
"The latter give rise to generalized Hitchin equations including a $\\tau $ -dependence.",
"The following sections discuss the strings in 6, 4, and 2 dimensions, respectively.",
"Each dimension is somewhat different and requires its own study, as the transverse symmetry groups depend on the dimension."
],
[
"Strings in 6d $N=(1,0)$ Theories",
"First we consider a 6d $N=(1,0)$ F-theory compactification on an elliptic Calabi-Yau three-fold $Y_3$ with a D3-brane along $\\mathbb {R}^{1,1} \\times C$ .",
"Here $C$ is a curve inside the base $B_2$ of $Y_3$ .",
"The resulting string, which propagates in the non-compact six dimensions transverse to $Y_3$ , has $(0,4)$ supersymmetry, and can be described in terms of a duality-twisted $N=4$ SYM theory, which we now derive."
],
[
"Duality Twist for 2d $N=(0,4)$",
"To determine the duality twist, we first recall from section REF that the background geometry breaks the $SU(4)_R$ -symmetry of the $N=4$ SYM theory on the D3-brane according to $\\begin{aligned}SU(4)_R &\\quad \\rightarrow \\quad SO(4)_T \\times U(1)_R \\cr {\\bf 4} &\\quad \\rightarrow \\quad {\\bf (2,1)}_1 \\oplus {\\bf (1,2)}_{-1} \\cr {\\bf 6} &\\quad \\rightarrow \\quad {\\bf (1,1)}_2 \\oplus {\\bf (1,1)}_{-2} \\oplus {\\bf (2,2)}_0\\, .\\end{aligned}$ Here $SO(4)_T$ and $U(1)_R$ represent the rotation groups in the four external directions normal to the D3-brane and, respectively, the two directions normal to $C$ inside $B_2$ .",
"The Lorentz symmetry decomposes as in (REF ).",
"Under the reduced symmetry group, the supercharges of the D3-theory transform as $\\begin{aligned}G_{\\rm total} &\\quad \\rightarrow \\quad SO(4)_T \\times SO(1,1)_L \\times U(1)_R \\times U(1)_C \\times U(1)_{D} \\cr ({\\bf 2}, {\\bf 1}, \\overline{\\bf 4})_1 &\\quad \\rightarrow \\quad ({\\bf 2}, {\\bf 1})_{1;-1,1,1} \\oplus ({\\bf 2}, {\\bf 1})_{-1;-1,-1,1} \\oplus ({\\bf 1}, {\\bf 2})_{1;1,1,1} \\oplus ({\\bf 1}, {\\bf 2})_{-1;1,-1,1}\\\\({\\bf 1}, {\\bf 2}, {\\bf 4})_{-1}&\\quad \\rightarrow \\quad ({\\bf 2}, {\\bf 1})_{1;1,-1,-1} \\oplus ({\\bf 2}, {\\bf 1})_{-1;1,1,-1} \\oplus ({\\bf 1}, {\\bf 2})_{1;-1,-1,-1} \\oplus ({\\bf 1}, {\\bf 2})_{-1;-1,1,-1} \\,.\\end{aligned}$ According to the discussion in section REF , a D3-brane wrapping a curve in the base of a Calabi-Yau three-fold is expected to preserve $(0,4)$ supersymmetry in 2d.",
"This is accomplished by the topological twist TCtwist = 12(TC + TR) TDtwist = 12(TD + TR) , where $T_C$ , $T_D$ , and $T_R$ are, respectively, the generators of $U(1)_C$ , $U(1)_D$ , and $U(1)_R$ .",
"The decomposition of the supercharges with respect to these twisted symmetry groups is $\\begin{aligned}G_{\\rm total}&\\quad \\rightarrow \\quad SO(4)_T \\times SO(1,1)_L \\times U(1)_C^{\\rm twist} \\times U(1)_{D}^{\\rm twist} \\cr ({\\bf 2}, {\\bf 1}, \\overline{\\bf 4})_1 &\\quad \\rightarrow \\quad ({\\bf 2}, {\\bf 1})_{1; 0,0} \\oplus ({\\bf 2}, {\\bf 1})_{-1; -1, 0} \\oplus ({\\bf 1}, {\\bf 2})_{1;1,1} \\oplus ({\\bf 1}, {\\bf 2})_{-1; 0,1}\\cr ({\\bf 1}, {\\bf 2}, {\\bf 4})_{-1}&\\quad \\rightarrow \\quad ({\\bf 2}, {\\bf 1})_{1; 0,0} \\oplus ({\\bf 2}, {\\bf 1})_{-1; 1,0} \\oplus ({\\bf 1}, {\\bf 2})_{1;-1, -1 } \\oplus ({\\bf 1}, {\\bf 2})_{-1; 0,-1}\\,.\\end{aligned}$ After the twist one thus ends up with four positive chirality scalar supercharges in 2d, QB+ : (2,1)1;0,0 QB+ : (2,1)1;0,0 , where $B$ is an index in the left $SU(2)$ inside $SO(4)_T$ .",
"A similar decomposition of the supersymmetry parameters (REF ) identifies these as B- : (2,1)-1;0,0 B- : (2,1)-1;0,0 .",
"From the field content (REF ) of the original $N=4$ SYM theory on the D3-brane we determine the spectrum of fields of the topologically half-twisted theory on $\\mathbb {R}^{1,1} \\times C$ : $\\begin{aligned}&\\qquad SO(4)_T \\times SO(1,1)_L \\times U(1)_C^{\\text{twist}} \\times U(1)_D^{\\text{twist}} \\times U(1)_R \\cr A \\,&:\\, \\quad ({\\bf 1,1})_{2,0,*,0} \\oplus ({\\bf 1,1})_{-2,0,*,0} \\oplus ({\\bf 1,1})_{0,1,*,0} \\oplus ({\\bf 1,1})_{0,-1,*,0} \\cr \\, & \\, \\qquad = v_+ \\oplus v_- \\oplus \\bar{a}_{\\bar{z}} \\oplus a_{z} \\cr \\Phi \\,&:\\, \\quad ({\\bf 1,1})_{0,1,1,2} \\oplus ({\\bf 1,1})_{0,-1,-1,-2} \\oplus ({\\bf 2,2})_{0,0,0,0} \\cr & \\, \\qquad = \\bar{\\sigma }_{\\bar{z}} \\oplus \\sigma _{ z} \\oplus \\varphi \\cr \\Psi \\,&:\\, \\quad ({\\bf 2,1})_{1,1,1,1} \\oplus ({\\bf 1,2})_{1,0,0,-1}\\oplus ({\\bf 2,1})_{-1,0,1,1} \\oplus ({\\bf 1,2})_{-1,-1,0,-1} \\cr & \\, \\qquad = \\psi _{+,\\bar{z}} \\oplus \\mu _+ \\oplus \\lambda _- \\oplus \\rho _{-,z} \\cr \\widetilde{\\Psi } \\,&:\\, \\quad ({\\bf 2,1})_{1,-1,-1,-1} \\oplus ({\\bf 1,2})_{1,0,0,1} \\oplus ({\\bf 2,1})_{-1,0,-1,-1} \\oplus ({\\bf 1,2})_{-1,1,0,1} \\cr & \\, \\qquad = \\tilde{\\psi }_{+,z} \\oplus \\tilde{\\mu }_+ \\oplus \\tilde{\\lambda }_- \\oplus \\tilde{\\rho }_{-,\\bar{z}} \\,.\\end{aligned}$ Table: The field content of the (0,4)(0,4) theory on a D3-brane wrapping a curve CC inside a Kähler surface B 2 B_{2} which is the base of an elliptic CY3 in F-theory.",
"The spectrum is obtained by the partially twisted reduction of 4d N=4N=4 SYM.",
"The representations in the second and third double-column refer to SO(4) T ×SO(1,1) L SO(4)_T \\times SO(1,1)_L, and fields are labeled by their SO(1,1) L SO(1,1)_L spin as well as their form-type along CC.At this stage all fields depend both on the coordiates $x^0, x^1$ along $\\mathbb {R}^{1,1}$ and on the local (anti-)holomorphic coordinates $z, \\bar{z}$ on $C$ .",
"The superscripts $\\pm $ of the fermionic fields denote the $SO(1,1)_L$ chirality.",
"This field content is summarized in table REF .",
"As a result of the topological twist, the fields assembled in table REF transform as differential forms on $C$ .",
"Their bidegree is fixed by the $U(1)^{\\rm twist}_C$ charge, which determines the transformation behaviour of the field with respect to the (twisted) structure group of $C$ .",
"The relation between the topological twist charge and the cotangent bundle of $C$ is $\\begin{aligned}&q^{\\rm twist}_C = +1 &\\qquad \\longleftrightarrow \\qquad &\\Omega ^{0,1}(C) \\cr &q^{\\rm twist}_C = -1 &\\qquad \\longleftrightarrow \\qquad &\\Omega ^{1,0}(C) \\, .\\end{aligned}$ As discussed in section REF , fields carrying in addition duality twist charge transform as form-valued sections of suitable powers of the duality bundle, $\\mathcal {L}_D$ , viewed as a complex line bundle on $C$ with connection (REF ).",
"To characterize the relevant sections we need the kinetic operators acting on the fields.",
"To determine these it is necessary to work out the topologically twisted action in detail.",
"The sections that the fields transform as listed in the rightmost column of table REF are the result of this analysis.",
"This way we will also be able to determine the massless spectrum in the effective 2d theory along the string."
],
[
"Action and BPS Equations",
"The string action and the supersymmetry variations of the fields follow by the decomposition of the 4d $N = 4$ SYM theory, or, equivalently, via dimensional reduction of the 10d $N = 1$ SYM theory with action $S_{10d} = \\int _{\\mathbb {R}^{1,9}} - \\frac{1}{4g^2} {\\rm Tr} \\hat{F}_{MN} \\hat{F}^{MN} - \\frac{i}{2 g^2} \\overline{\\hat{\\Psi }} \\Gamma ^M D_M \\hat{\\Psi }\\,,$ and supersymmetry variations $ \\delta \\hat{A}_M = -i \\overline{\\hat{\\epsilon }} \\Gamma _M \\hat{\\Psi }, \\qquad \\quad \\delta \\hat{\\Psi }= \\frac{1}{2} \\Gamma ^{M N} \\hat{F}_{M N} \\hat{\\epsilon }\\,.$ Here $\\hat{\\Psi }$ and $\\hat{\\epsilon }$ denote the 10d Majorana-Weyl spinor and SUSY parameter.",
"In addition after the dimensional reduction to four dimensions one adds explicitly a topological term $\\frac{i\\theta }{2\\pi } \\text{Tr}F \\wedge F \\,,$ where $F$ is the 4d reduction of the 10d field strength $\\hat{F}$ .",
"The 4d abelian ${N}=4$ SYM action for the fermions and scalars is $S_{\\rm fermions} + S_{\\rm scalars }= {i\\over 2 \\pi } \\int _{\\mathbb {R}^{1,1}\\times C} \\sqrt{-g} d^4x \\mathop {\\mathrm {Tr}}\\nolimits \\tilde{\\Psi }\\bar{\\sigma }^\\mu \\partial _\\mu \\Psi + {1\\over 4\\pi } \\int _{\\mathbb {R}^{1,1} \\times C} \\mathop {\\mathrm {Tr}}\\nolimits \\star _4 d\\Phi \\wedge d\\Phi \\,.$ The gauge fields in the abelian ${N}=4$ SYM theory have the action $S_{F } = {1\\over 4\\pi } \\int _{\\mathbb {R}^{1,1} \\times C} \\tau _2 \\mathop {\\mathrm {Tr}}\\nolimits F \\wedge \\star _4 F + {1\\over 4\\pi i} \\int _{\\mathbb {R}^{1,1} \\times C} \\tau _1\\mathop {\\mathrm {Tr}}\\nolimits F\\wedge F \\,.$ Via the decomposition (REF ) and including the twist we can determine the theory in terms of fields along $C$ and $\\mathbb {R}^{1,1}$ , and subsequently integrate out the internal degrees of freedom.",
"Let us introduce coordinates $x^\\pm = x^0 \\pm x^1$ along $\\mathbb {R}^{1,1}$ , with derivatives $\\partial _\\pm := \\partial _0 \\pm \\partial _1$ , as well as (anti-)holomorphic derivatives along $C$ , $\\partial \\equiv \\partial _z$ , $\\bar{\\partial }\\equiv \\partial _{\\bar{z}}$ .",
"The internal derivatives of fields with definite $U(1)_D$ transformation properties will be written in terms of the duality covariant derivatives $\\partial _{\\cal A}$ and $\\bar{\\partial }_{\\cal A}$ , which were defined in (REF ).",
"The action for the full theory after the decomposition and twist is $\\begin{aligned}S_{\\rm string} &= \\int _{\\mathbb {R}^{1,1} \\times C} d^4 x \\sqrt{|g|}\\mathcal {L} = \\int d^4 x \\sqrt{|g|} \\left(\\tau _2 \\mathcal {L}_\\text{gauge} +\\frac{2}{3}\\tau _1\\mathcal {L}_\\text{top} + 2i\\mathcal {L}_\\text{fermion} +\\mathcal {L}_\\text{scalar} \\right)\\cr & = \\int _{\\mathbb {R}^{1,1} \\times C} d^4 x \\sqrt{|g|} \\left(\\tau _2 F_{\\mu \\nu } F^{\\mu \\nu } + \\frac{2}{3}\\tau _1 \\left(\\epsilon ^{\\mu \\nu \\alpha \\beta } F_{\\mu \\nu }F_{\\alpha \\beta } \\right)+ 2i \\left(\\bar{\\Psi }\\Gamma ^\\mu \\partial _\\mu \\Psi \\right)+ \\sum _i \\partial _\\mu \\phi _i \\partial ^\\mu \\phi ^i \\right)\\,.\\end{aligned}$ The gauge field part takes the same universal form for the 2d actions obtained from D3-branes on any Calabi-Yau $n$ -fold, $\\begin{aligned}{\\cal L}_\\text{gauge} =&- \\frac{1}{2} (\\partial \\bar{a} - \\bar{\\partial }a)^2- \\frac{1}{2} (\\partial _- v_+ - \\partial _+ v_-)^2- \\partial _- a \\partial _+ \\bar{a}- \\partial _- \\bar{a} \\partial _+ a- \\partial v_+ \\bar{\\partial }v_- \\cr &- \\partial v_- \\bar{\\partial }v_++ \\bar{\\partial }v_+ \\partial _- a+ \\partial v_+ \\partial _- \\bar{a}+ \\partial v_- \\partial _+ \\bar{a}+ \\bar{\\partial }v_- \\partial _+ a \\\\{2\\over 3i} {\\cal L}_{\\rm top} =&(\\partial v_+\\bar{\\partial }v_-- \\partial v_- \\bar{\\partial }v_++ \\partial _- \\bar{a} \\partial _+ a- \\partial _- a \\partial _+ \\bar{a} ) \\cr &+ ( \\bar{\\partial }v_+ \\partial _- a- \\partial v_+ \\partial _- \\bar{a}+ \\partial \\bar{a} \\partial _- v_+- \\bar{\\partial }a \\partial _- v_+ ) \\cr &+ ( -\\bar{\\partial }v_- \\partial _+ a+ \\partial v_- \\partial _+ \\bar{a}- \\partial \\bar{a} \\partial _+ v_-+ \\bar{\\partial }a \\partial _+ v_- ) \\,.\\end{aligned}$ The part that is twist specific is the action for the fermions and scalars.",
"It depends on the embedding of the curve $C$ and consequently on the dimension of the compactification.",
"For the Calabi-Yau three-fold compactification this part takes the form $\\begin{aligned}{\\cal L}_\\text{fermion} =&+ \\lambda _- \\partial _+ \\tilde{\\lambda }_-+ \\psi _+ \\partial _\\mathcal {A} \\tilde{\\lambda }_-- \\tilde{\\lambda }_- \\partial _+ \\lambda _-- \\tilde{\\lambda }_- \\partial _\\mathcal {A}\\psi _++ \\mu _+ \\partial _- \\tilde{\\mu }_+ \\cr &- \\tilde{\\mu }_+ \\partial _-\\mu _++ \\rho _- \\bar{\\partial }\\tilde{\\mu }_+- \\tilde{\\mu }_+ \\bar{\\partial }\\rho _-- \\lambda _- \\bar{\\partial }_\\mathcal {A} \\tilde{\\psi }_+- \\psi _+\\partial _- \\tilde{\\psi }_++ \\tilde{\\psi }_+ \\bar{\\partial }_\\mathcal {A} \\lambda _- \\cr &+ \\tilde{\\psi }_+ \\partial _- \\psi _+- \\mu _+ \\partial \\tilde{\\rho }_-- \\rho _-\\partial _+ \\tilde{\\rho }_-+ \\tilde{\\rho }_-\\partial \\mu _++ \\tilde{\\rho }_- \\partial _+ \\rho _- \\cr {\\cal L}_\\text{scalar} =&- \\partial _- \\sigma \\partial _+ \\bar{\\sigma }- \\partial _- \\bar{\\sigma }\\partial _+ \\sigma + \\bar{\\partial }_\\mathcal {A} \\sigma \\partial _\\mathcal {A} \\bar{\\sigma }+ \\bar{\\partial }_\\mathcal {A} \\bar{\\sigma }\\partial _\\mathcal {A} \\sigma + \\partial _+ \\varphi \\partial _- \\varphi - \\partial \\varphi \\bar{\\partial }\\varphi \\,.\\end{aligned}$ The supersymmetry variations likewise follow by dimensional reduction, starting in 10d.",
"The result of this decomposition of the bosonic SUSY variations is $\\begin{aligned}\\sqrt{\\tau _2 } \\delta a &= 2 i \\epsilon _- \\tilde{\\psi }_{+} \\cr \\delta \\sigma &= - 2 i \\tilde{\\epsilon }_- \\tilde{\\psi }_{+} \\cr \\sqrt{\\tau _2} \\, \\delta v_- &= 2 i (\\lambda _- \\tilde{\\epsilon }_- + \\tilde{\\lambda }_- \\epsilon _-) \\cr \\delta \\varphi _{A \\dot{B}} &= - 2 i (\\epsilon _{-A} \\mu _{+\\dot{B}} + \\tilde{\\epsilon }_{-A} \\tilde{\\mu }_{+\\dot{B}}) \\,.\\end{aligned}\\qquad \\begin{aligned}\\sqrt{\\tau _2 } \\delta \\bar{a} &= 2 i \\tilde{\\epsilon }_- \\psi _{+} \\cr \\delta \\bar{\\sigma }&= 2 i \\epsilon _- \\psi _{+} \\cr \\delta v_+ &= 0\\cr \\cr \\end{aligned}$ Here we have made manifest the transformation of the fields as representations of the transverse $SO(4)_T = SU(2)_{T,1} \\times SU(2)_{T,2}$ rotation group, with $A$ and $\\dot{A}$ referring to the two $SU(2)$ factors.",
"More details on the decomposition and our conventions can be found in appendix .",
"For the fermionic variations one finds $\\begin{aligned}\\delta \\psi _{+} &= \\epsilon _- \\sqrt{\\tau _2} (- \\partial _+ \\bar{a} + \\, \\bar{\\partial }v_+ ) + \\tilde{\\epsilon }_- \\partial _+ \\bar{\\sigma }\\quad \\cr \\delta \\mu _+^{\\dot{B}} &= - \\tilde{\\epsilon }_{- A} \\partial _+ \\varphi ^{A\\dot{B}} \\cr \\delta \\rho _-^{\\dot{B}} &= \\tilde{\\epsilon }_{ - A} \\partial _{\\cal A}\\varphi ^{A \\dot{B}} \\cr \\delta \\lambda _- &= - \\epsilon _- (\\sqrt{\\tau _2} F_{01} + {\\cal F}_{\\cal A}) - \\tilde{\\epsilon }_- \\ast _C \\partial _{\\cal A} \\bar{\\sigma }\\end{aligned}\\qquad \\begin{aligned}\\delta \\tilde{\\psi }_{+} &= \\tilde{\\epsilon }_- \\sqrt{\\tau _2} (\\partial _+ a - \\partial v_+) + \\epsilon _- \\partial _+ \\sigma \\cr \\delta \\tilde{\\mu }_+^{\\dot{B}} &= + \\epsilon _{ - A} \\partial _+ \\varphi ^{A \\dot{B}}\\cr \\delta {\\tilde{\\rho }}_-^{\\dot{B}} &= \\epsilon _{- A} \\bar{\\partial }_{\\cal A} \\varphi ^{A \\dot{B}}\\cr \\delta \\tilde{\\lambda }_- &= - \\tilde{\\epsilon }_- (\\sqrt{\\tau _2} F_{01} - {\\cal F}_{\\cal A} )+ \\epsilon _- \\ast _C \\bar{\\partial }_{\\cal A} \\sigma \\,.\\end{aligned}$ Up to boundary terms the above action (REF ) is invariant off-shell under the supersymmetry transformations given by (REF ) and (REF ).",
"This result depends crucially on the properties (REF ) of the holomorphically varying axio-dilaton, as is further explained in appendix .",
"Note the factor of $\\sqrt{\\tau _2}$ in front of the external components of the gauge field; this factor arises by rescaling the 10d supersymmetry variations (REF ) such as to comply with our normalization of the 4d $N=4$ SYM action.",
"Furthermore, we have defined ${\\cal F}_{\\cal A} := \\frac{1}{2}\\sqrt{\\tau _2}(\\bar{\\partial }a - \\partial \\bar{a}) \\,.$ It turns out that this combination can be written entirely in terms of $U(1)_D$ covariant derivatives of the fields $\\sqrt{\\tau _2} a$ and $\\sqrt{\\tau _2} \\bar{a}$ with definite $U(1)_D$ charge, ${\\cal F}_{\\cal A} = \\frac{1}{2}(\\bar{\\partial }_{\\cal A}(\\sqrt{\\tau _2}a) -\\partial _{\\cal A}(\\sqrt{\\tau _2}\\bar{a})) \\,.$ This uses holomorphy of the varying axio-dilaton $\\tau $ , see equ.",
"(REF ).",
"We will come back to this important point in section REF .",
"Ignoring potential boundary terms, the equations of motion obtained from the variation of the fermionic action are + - - 2 A + = 0 + - + A + = 0 - + - 2 A - = 0 - + + 2 A - = 0 - + - 2 A - = 0 - + + 2 A - = 0 + - - 2 A + = 0 + - + 2 A + = 0 .",
"The BPS equations that follow from the supersymmetry variations are on the other hand F01 = 12(-v+ - +v-) = 0 FA = 122( a - a) = 0 v+ - +a = 0 v+ - + a = 0 += 0 += 0 A= 0 A= 0 AB = 0 AB = 0 +AB = 0 .",
"The BPS equations are similar to Hitchin equations for Yang-Mills theory on a Riemann surface, except that the fields here transform as sections of $\\mathcal {L}_D$ .",
"We shall discuss some aspects of these equations in section REF .",
"The next task here will be to deduce the cohomology groups counting the spectrum of zero-modes in the 2d effective theory along the string."
],
[
"Spectrum",
"If one performs a dimensional reduction to determine the 2d effective action each field decomposes as $\\Phi (x_\\pm ,z,\\bar{z}) = \\sum _k \\Phi ^{(k)}(x_\\pm ) \\otimes \\hat{\\Phi }^{(k)}(z, \\bar{z})\\,,$ with $\\Phi ^{(k)}(z, \\bar{z})$ an eigenmode of the internal kinetic operator.",
"From (REF ) one finds that the internal part of the 2d zero-modes are characterized by the vanishing of the second term in each equation.",
"The external fields in the 2d effective action then satisfy the massless Dirac equation given by the first terms in each equation, in agreement with their chirality.",
"Consider a field $\\hat{\\Phi }_{p,q} (z,\\bar{z})$ of charge $q_D^{\\rm twist}$ and form degree $(p,q)$ .",
"If the zero-modes are governed by the internal kinetic operator $\\bar{\\partial }_{\\cal A}$ in (REF ), i.e.",
"by $\\bar{\\partial }_{\\cal A} \\hat{\\Phi }_{p,q} (z,\\bar{z}) = (\\bar{\\partial }+ i q_D^{\\rm twist} {\\cal A}^{(0,1)}) \\hat{\\Phi }_{p,q}(z,\\bar{z}) = 0 \\,,$ then the non-trivial zero-modes correspond to $\\hat{\\Phi }^{(0)}_{p,q} (z,\\bar{z}) \\in H^{p,q}_{\\bar{\\partial }} (C, {\\cal L}_D^{-q_D^{\\rm twist}}) \\,.$ If on the other hand the equations of motion involve $\\partial _{\\cal A}$ , i.e.",
"$\\partial _{\\cal A} \\hat{\\Phi }_{p,q} (z,\\bar{z}) = (\\partial + i q_D^{\\rm twist}{\\cal A}^{(1,0)}) \\hat{\\Phi }_{p,q}(z,\\bar{z}) = 0 \\,,$ the zero-modes correspond to $\\hat{\\Phi }_{p,q}^{(0)} (z,\\bar{z}) \\in H^{p,q}_{\\partial } (C, \\bar{\\cal L}_D^{+q_D^{\\rm twist} }) = \\left(H^{q,p}_{\\bar{\\partial }} (C, {\\cal L}_D^{+q_D^{\\rm twist}})\\right)^* \\,.$ Note the different signs appearing in the powers of the bundles.",
"In the last equation we used that complex conjugation acts on the (anti-)holomorphic bundles as $(\\bar{\\cal L}_D)^\\ast = {\\cal L}_D$ .",
"For instance, in view of the $U(1)^{\\rm twist}_D$ charges, the 2d gaugino zero-modes $\\tilde{\\lambda }_-(x_\\pm )$ and $\\lambda _-(x_\\pm )$ correspond to solutions to $(\\partial - i \\tilde{\\cal A}^{(1,0)}) \\hat{\\tilde{\\lambda }}(z,\\bar{z}) = 0,\\qquad \\quad (\\bar{\\partial }+ i \\tilde{\\cal A}^{(0,1)}) \\hat{\\lambda }(z,\\bar{z})= 0 \\,,$ given by $\\hat{\\tilde{\\lambda }}^{(0)}(z,\\bar{z}) \\in H_\\partial ^{(0,0)}(C, \\bar{\\cal L}_D^{-1}) =\\left( H_{\\bar{\\partial }}^{0,0}(C,{\\cal L}_D^{-1})\\right)^*, \\qquad \\hat{\\lambda }^{(0)}(z,\\bar{z}) \\in H_{\\bar{\\partial }}^{(0,0)}(C, {\\cal L}_D^{-1}).$ This systematically leads to the counting of massless fields in the effective 2d $(0,4)$ theory summarized in table REF .",
"It is also the rationale behind the determination of the sections of which bundles the $(2+2)$ dimensional fields transform in table REF .",
"The results of our derivation agree with the spectrum stated in [31].",
"In order to evaluate the dimensions of these cohomology groups one takes into account that the duality bundle ${\\cal L}_D$ on $C$ can be viewed as a bundle on the base $B_2$ which describes the $SL(2,\\mathbb {Z})$ monodromies due to the variation of axio-dilaton $\\tau $ in F-theory.",
"At this stage we recall from the discussion around (REF ) that [47], [7] ${\\cal L}_D = K^{-1}_{B_2} |_{C} \\,,$ ${\\cal L}_D$ is of non-negative degree, and ${\\cal L}_D$ is trivial if and only if $C$ does not intersect the discriminant locus of $Y_3$ .",
"In this case, $\\tau $ does not experience any monodromies on $C$ and is therefore constant.",
"This has the following consequences: Unless ${\\cal L}_D = {\\cal O}$ , ampleness of ${\\cal L}_D$ implies that $h^0(C,{\\cal L}_D^{-1}) = 0$ .",
"Therefore the vector multiplet is projected out at the massless level and the 2d effective theory reduces to a sigma-model.",
"Only in the special case that ${\\cal L}_D = {\\cal O}$ is a vector multiplet retained and the $U(1)$ gauge symmetry unbroken.",
"This is a rather notable difference to the perturbative description in the Type IIB orientifold limit, where the D3-brane theory has a phase with an unbroken $U(1)$ symmetry despite the intersection with the 7-branes.",
"We explain this difference, along with an explanation in terms of non-perturbative effects in the vicinity of the orientifold plane, in section REF .",
"Finally, Serre duality implies that $h^1(C,{\\cal L}_D^{-1}) = h^0(C, K_C\\otimes {\\cal L}_D)$ , and for ${\\cal L}_D \\ne {\\cal O}$ Riemann-Roch implies that $h^1(C,{\\cal L}_D^{-1}) = \\chi (C, {K_C} \\otimes {\\cal L }_D) = g-1 + c_1(B_2) \\cdot C,$ where we used (REF ) and where $g$ is the genus of $C$ .",
"This explains the multiplicities in table REF , evaluated for the case of non-trivial ${\\cal L}_D$ .",
"Table: Massless 2d (0,4) multiplets of the (0,4)(0,4) theory on a D3-brane wrapping curve CC inside a Kähler base B 2 B_{2} in F-theory propagating in the bulk of the D3-brane.",
"The specific values for the Betti numbers in the last column refer to a curve CC intersecting the disciminant locus of the elliptic fibration in isolated points.",
"The representations refer to SO(4) T ×SO(1,1) L SO(4)_T \\times SO(1,1)_L.We now wish to identify the R-symmetries of the various fields.",
"A general 2d $(0,4)$ theory has an R-symmetry group $SO(4)_R = SU(2)_R \\times SU(2)_I$ .",
"If the theory flows to a superconformal theory in the infrared, only a single $SU(2)$ subgroup is preserved by the SCFT [53].",
"Each field must transform in such a way that it is consistent with the R-charges of the various $(0,4)$ multiplets as explained in section REF .",
"The supercharges of the $(0,4)$ theory must transform in the ${\\bf 2}$ of $SU(2)_R$ .",
"This suggests identifying the $SU(2)_R$ symmetry with the first $SU(2)_{T,1}$ factor in the group $SO(4)_T = SU(2)_{T,1} \\times SU(2)_{T,2}$ of rotations in the space transverse to the string.",
"The remaining $SU(2)_{T,2}$ will act as an $SU(2)$ current algebra on the worldsheet of the string [31].",
"The identification of $SU(2)_{T,1}$ as the $SU(2)_R$ symmetry is dependent on the form of the twist.",
"The second factor $SU(2)_I$ in the R-symmetry group $SO(4)_R = SU(2)_R \\times SU(2)_I$ is not generally visible from the compact geometry [31]; it is the $SU(2)$ R-symmetry of the 6d $(1,0)$ theory in which the string lives (see e.g.",
"[54]).",
"The charges of the multiplets in table REF under this $SU(2)_I$ symmetry are fixed by the structure of the $(0,4)$ supersymmetry as explained in appendix REF .",
"In table REF the states with twist charges $(\\pm 1, \\pm 1)$ comprise fermions transforming as a ${\\bf 2}$ of $SU(2)_R$ and corresponding scalars transforming as a ${\\bf 1}$ .",
"This identifies these states as a hypermultiplet, where the supersymmetry forces the fermions and scalars to transform as the ${\\bf 1}$ and ${\\bf 2}$ respectively of the $SU(2)_I$ .",
"Similarly the universal multiplet with twist charges $(0,0)$ has fermions that transform trivially and scalars which transform in the ${\\bf 2}$ of $SU(2)_R$ – this is a twisted hypermultiplet.",
"The Fermi multiplets with twist charge $(\\pm 1, 0)$ transform trivially under the R-symmetry, as expected, and finally the states with twist charges $(0, \\pm 1)$ come in the R-symmetry representations characteristic of a vector multiplet.We have seen that for a curve $C$ in generic position to the 7-branes of the F-theory background there are no zero-modes associated with such states.",
"It is instructive to collect the left- and right-moving zero-modes.",
"For definiteness we reiterate that we assume that ${\\cal L}_D$ is non-trivial, equivalent to the assumption that the coupling $\\tau $ varies non-trivially over the curve $C$ .",
"In this case the only left-handed fermion zero-modes result from the Fermi multiplets and are counted in the following way – with the charges given in the ordering $SO(1,1)_{L}\\times U(1)_C^{\\rm twist} \\times U(1)_D^{\\rm twist}$ , $\\hbox{Fermions, L:}\\qquad \\begin{array}{|c|l|c|l|}\\hline \\multicolumn{2}{|c|}{\\hbox{Representation}}& \\hbox{Cohomology Groups} & \\hbox{Multiplicity}\\cr \\hline \\rho _{-, z} & ({\\bf 1},{\\bf 2})_{-1,-1,0} & H^{1,0}(C, \\mathcal {O}) & g \\cr \\tilde{\\rho }_{-, \\bar{z}} & ({\\bf 1},{\\bf 2})_{-1,1,0} & H^{0,1}(C, \\mathcal {O}) & g \\cr \\hline \\end{array}$ For the right-handed fermion zero-modes we have $\\hbox{Fermions, R:}\\qquad \\begin{array}{|c|l|c|l|}\\hline \\multicolumn{2}{|c|}{\\hbox{Representation}}& \\hbox{Cohomology Groups} & \\hbox{Multiplicity} \\cr \\hline \\psi _{+, \\bar{z}} & ({\\bf 2},{\\bf 1})_{1,1,1} & H^{0,1}(C, \\mathcal {L}_D^{-1}) & g - 1 + c_1(B) \\cdot C \\cr \\tilde{\\psi }_{+, z}&({\\bf 2},{\\bf 1})_{1,-1,-1} & H^{1,0}(C, \\mathcal {L}_D) & g - 1 + c_1(B) \\cdot C \\cr \\mu _+ &({\\bf 1},{\\bf 2})_{1,0,0} & H^{0,0}(C, \\mathcal {O}) & 1 \\cr \\tilde{\\mu }_+ &({\\bf 1},{\\bf 2})_{1,0,0} & H^{0,0}(C, \\mathcal {O}) & 1 \\cr \\hline \\end{array}$ Finally one can consider the massless bosonic fields: $\\hbox{Bosons:}\\qquad \\begin{array}{|c|l|c|l|}\\hline \\multicolumn{2}{|c|}{\\hbox{Representation}} & \\hbox{Cohomology Groups} & \\hbox{Multiplicity}\\cr \\hline a_z& ({\\bf 1},{\\bf 1})_{0,1,1} & H^{0,1}(C, \\mathcal {L}_D^{-1}) & g - 1 + c_1(B) \\cdot C \\cr \\sigma _z & ({\\bf 1},{\\bf 1})_{0,1,1} & H^{0,1}(C, \\mathcal {L}_D^{-1}) & g - 1 + c_1(B) \\cdot C \\cr \\bar{a}_{\\bar{z}} & ({\\bf 1},{\\bf 1})_{0,-1,-1} & H^{1,0}(C, \\mathcal {L}_D) & g - 1 + c_1(B) \\cdot C \\cr \\tilde{\\sigma }_{\\bar{z}} & ({\\bf 1},{\\bf 1})_{0,-1,-1} & H^{1,0}(C, \\mathcal {L}_D) & g - 1 + c_1(B) \\cdot C \\cr \\varphi & ({\\bf 2},{\\bf 2})_{0,0,0} & H^{0,0}(C, \\mathcal {O}) & 1 \\cr \\hline \\end{array}$ Apart from this sector, extra massless fermionic states arise from strings stretched between the D3-brane and the 7-branes in the F-theory background.",
"These zero-modes are localised at the intersection of the curve $C$ with the discriminant of $Y_3$ , which, for generic position of the curve $C$ , is a set of points.",
"The existence of such 3–7 strings in the present context, found first in [24], has also been pointed out in [31], and their importance in low-dimensional compactifications for the gauge anomalies on the 7-branes was stressed independently in [16], [18].",
"The contribution from this sector consists of $ n_{37} = 8\\, c_1(B_2) \\cdot C$ (0,4) half-Fermi multiplets.",
"This might seem counterintuitive at first since the number of intersection points of the D3-brane with the discriminant locus is given by $12 \\, c_1(B_2) \\cdot C$ .",
"However, even though an $SL(2,\\mathbb {Z})$ transformation allows us to view each individual point as the intersection of the D3-brane with a (1,0) 7-brane, globally not all 7-branes can be dualised into the same $(p,q)$ -frame.",
"This is the origin of the reduction of the number of independent zero-modes to (REF ).",
"We offer three independent derivations for the specific value (REF ): First, by considering the perturbative Sen limit in section REF one realizes that the D7-brane locus is in the class $8\\,K^{-1}_{B_2}$ , the remaining $ 4\\, c_1(B_2) \\cdot C$ points being associated with the intersection with the O7-plane.",
"The latter do not host any additional 3–7 modes.",
"The extra 3–7 states are also seen, independently of a deformation to weak coupling, by comparison with the M5-brane picture in section REF , and their number is universally and uniquely predicted by anomaly cancellation as discussed in section .",
"The counting (REF ), and the arguments leading to it, are independent of the dimension of $Y_n$ and hence hold true for D3-branes wrapping curves on any base $B_{n-1}$ in F-theory.",
"Given the spectrum in table REF one can read off the right- and left-moving central charges of the effective 2d theory as $c_R = 6 g + 6 c_1(B_2) \\cdot C \\,,\\qquad c_L = 6g + 4 c_1(B_2) \\cdot C + n_{37} \\,,$ where $n_{37} = 8 c_1(B_2) \\cdot C$ is the contribution from the 3–7 string sector.",
"It can then be seen that there is a gravitational anomaly $c_L - c_R = 6 c_1(B_2) \\cdot C \\,,$ which will be cancelled by anomaly inflow from the bulk of the 6d $(1,0)$ theory as discussed in section ."
],
[
"Strings in 4d $N=1$",
"Let us now consider a D3-brane wrapping a curve $C$ inside a Kähler three-fold $B_3$ serving as the base of an elliptic Calabi-Yau 4-fold $Y_4$ on which we compactify F-theory to four dimensions.",
"This setup gives rise to a string in the extended spacetime $\\mathbb {R}^{1,3}$ and is expected to preserve $(0,2)$ supersymmetry."
],
[
"Duality Twist",
"The $SU(4)_R$ symmetry of the $N=4$ SYM theory on the D3-brane decomposes, as in (REF ), into the rotation group $SO(2)_T$ associated with the two extended dimensions transverse to the string as well as an $SU(2)_R \\times U(1)_R$ group.",
"The latter represents the structure group of the normal bundle $N_{C/B_3}$ .",
"Together with the universal decomposition (REF ) of $SO(1,3)_L$ , this results in the following decomposition of the supercharges of the $N=4$ SYM theory: $\\begin{aligned}G_{\\rm total} \\ &\\rightarrow \\ SU(2)_R \\times SO(1,1)_L \\times U(1)_C \\times U(1)_R \\times SO(2)_T \\times U(1)_ D\\cr ({\\bf 2}, {\\bf 1}, {\\bf \\bar{4}})_1 \\ &\\rightarrow \\ {\\bf 2}_{1;1,0,-1,1} \\oplus {\\bf 2}_{-1;-1,0,-1,1} \\oplus {\\bf 1}_{1;1,-1,1,1}\\cr &\\qquad \\oplus {\\bf 1}_{-1;-1,1,1,1} \\oplus {\\bf 1}_{1;1,-1,1,1} \\oplus {\\bf 1}_{-1;-1,1,1,1} \\cr ({\\bf 1}, {\\bf 2}, {\\bf 4})_{-1} \\ &\\rightarrow \\ {\\bf 2}_{1;-1,0,1,-1} \\oplus {\\bf 2}_{-1;1,0,1,-1} \\oplus {\\bf 1}_{1;-1,1,-1,-1} \\cr &\\qquad \\oplus {\\bf 1}_{-1;1,1,-1,-1} \\oplus {\\bf 1}_{1;-1,-1,-1,-1} \\oplus {\\bf 1}_{-1;1,-1,-1,-1} \\, .\\end{aligned}$ The topological and duality twist required to preserve the expected two right-moving supersymmetries take the form $\\begin{aligned}T_C^{\\text{twist}} &= \\frac{1}{2}(T_C + T_R) \\cr T_D^{\\text{twist}} &= \\frac{1}{2}(T_D + T_R) \\,.\\end{aligned}$ After this twist the supercharges decompose as in Gtotal SU(2)R SO(1,1)L U(1)twistC U(1)twistD SO(2)T (2, 1, 4)1 21;12,12,-1 2-1;-12,12,-1 11;0,0,1 1-1;-1,0,1 11;1,1,1 1-1;0,1,1 (1, 2, 4)-1 21;-12,-12,1 2-1;12,-12,1 11;0,0,-1 1-1;1,0,-1 11;-1,-1,-1 1-1;0,-1,-1 , and one ends up with two positive chirality scalar supercharges in 2d $Q_+ \\, :\\, {\\bf 1}_{1;0,0,1}\\qquad \\qquad \\widetilde{Q}_+ \\,:\\, {\\bf 1}_{1;0,0,-1} \\,,$ with $U(1)_R$ charge $\\pm 1$ .",
"To determine the bulk matter on the twisted D3-brane we analyze the decomposition of the 4d $N=4$ vector multiplet into fields transforming in the following representations SU(2)R SO(1,1)L U(1)Ctwist U(1)Dtwist SO(2)T U(1)R A : 12,0,*,0,0 1-2,0,*,0,0 10,1,*,0,0 10,-1,*,0,0 = v+ v- a a : 10,0,0,2,0 10,0,0,-2,0 20,12,12,0,1 20,-12,-12,0,-1 = g g : 21,12,12,1,0 11,1,1,-1,1 11,0,0,-1,-1 2-1,-12,12,1,0 1-1,0,1,-1,1 1-1,-1,0,-1,-1 = + + + - - - : 21,-12,-12,-1,0 11,-1,-1,1,-1 11,0,0,1,1 2-1,12,-12,-1,0 1-1,0,-1,1,-1 1-1,1,0,1,1 = + + + - - - .",
"The fields in the topologically half-twisted theory on $\\mathbb {R}^{1,1} \\times C$ transform as bundle valued differential forms on $C$ , which can be determined in a manner similar to the discussion for the string in 6d in section REF .",
"The fundamental representation ${\\bf 2}$ of the $SU(2)_R$ symmetry group indicates that the differential forms take value in the normal bundle $N_{C/B_3}$ , whose structure group this $SU(2)_R$ is.",
"The form degree in turn is determined by the topological twist charge $q_C^{\\rm twist}$ , with our conventions being that a state of topological and duality twist charges $(q_C^{\\rm twist}, q_D^{\\rm twist}) = (1,0)$ or $(-1,0)$ transforms as an element of $\\Omega ^{0,1}(C)$ or, respectively $\\Omega ^{1,0}(C)$ .",
"In addition sections of the normal bundle $N_{C/B_3}$ carry twist charges.",
"To compute these consider the adjunction formula $K_C = K_{B_3}|_C \\otimes \\wedge ^2 N_{C/B_3} = {\\cal L}^{-1}_D \\otimes \\wedge ^2 N_{C/B_3} \\,.$ Since in our conventions we associate with $K_C$ twist charge $q_C^{\\rm twist}=-1$ and with the duality twist bundle $\\mathcal {L}_D$ twist charge $q_D^{\\rm twist} = -1$ , this fixes the charges associated with sections of $N_{C/B_3} $ as $N_{C/B_3}: (q_C^{\\rm twist}, q_D^{\\rm twist}) = \\left(-\\frac{1}{2}, - \\frac{1}{2}\\right)\\,.$ Following this logic one systematically arrives at the bundle assignments displayed in the last column of table REF .",
"The results are in agreement with the way the fields enter the topologically twisted action, to which we now turn.",
"Table: The bulk field content of the (0,2)(0,2) field theory on a D3-branealong C⊂B 3 C \\subset B_3.",
"The fields are the twisted fields from thereduction of the N=4N=4 SYM spectrum as defined in ().We display only the 2d chirality, and the SU(2) R SU(2)_R-symmetryrepresentation.",
"The form type of each field is computed the action of theinternal kinetic operators on the fields determined via the equations ofmotion and the BPS equations."
],
[
"Action and BPS Equations",
"For a D3-brane wrapping a curve in a Calabi-Yau four-fold the gauge and topological parts of the action are the same as in (REF ) for Calabi-Yau three-folds, as they do not depend on the choice of decomposition of the R-symmetry.",
"The fermion and scalar Lagrangians are $\\begin{aligned}\\mathcal {L}_\\text{fermion} =&+ \\psi \\partial _-\\widetilde{\\psi }- \\psi \\partial _\\mathcal {A}\\widetilde{\\lambda }- \\beta \\bar{\\partial }\\widetilde{\\gamma }+ \\beta \\partial _+\\widetilde{\\beta }+ \\lambda \\bar{\\partial }_\\mathcal {A}\\widetilde{\\psi }- \\lambda \\partial _+\\widetilde{\\lambda }- \\gamma \\partial _-\\widetilde{\\gamma }+ \\gamma \\partial \\widetilde{\\beta }\\cr &+ \\rho \\partial _+\\widetilde{\\rho }- \\rho \\bar{\\partial }_\\mathcal {A}\\widetilde{\\mu }+ \\mu \\partial _\\mathcal {A}\\widetilde{\\rho }- \\mu \\partial _-\\widetilde{\\mu }+ \\widetilde{\\psi }\\partial _-\\psi + \\widetilde{\\psi }\\bar{\\partial }_\\mathcal {A}\\lambda - \\widetilde{\\gamma }\\partial _-\\gamma - \\widetilde{\\gamma }\\bar{\\partial }\\beta \\cr &- \\widetilde{\\lambda }\\partial _\\mathcal {A}\\psi - \\widetilde{\\lambda }\\partial _+\\lambda + \\widetilde{\\beta }\\partial \\gamma + \\widetilde{\\beta }\\partial _+\\beta - \\widetilde{\\rho }\\partial _+\\rho - \\widetilde{\\rho }\\partial _\\mathcal {A}\\mu + \\widetilde{\\mu }\\bar{\\partial }_\\mathcal {A}\\rho + \\widetilde{\\mu }\\partial _-\\mu \\,,\\end{aligned}$ and $\\begin{aligned}\\mathcal {L}_\\text{scalar} =&- \\partial _-g\\partial _+\\bar{g}- \\partial _-\\bar{g}\\partial _+g+ \\bar{\\partial }g\\partial \\bar{g}+ \\bar{\\partial }\\bar{g}\\partial g\\cr &+ \\partial _+\\varphi \\partial _-\\bar{\\varphi }+ \\partial _+\\bar{\\varphi }\\partial _-\\varphi - \\partial _\\mathcal {A}\\varphi \\bar{\\partial }_\\mathcal {A}\\bar{\\varphi }- \\partial _\\mathcal {A}\\bar{\\varphi }\\bar{\\partial }_\\mathcal {A}\\varphi \\,,\\end{aligned}$ with the total action given by the combination (REF ).",
"The supersymmetry variations, again obtained via dimensional reduction and for the fields as identified in appendix , are $\\begin{aligned}\\sqrt{\\tau _2}\\delta v_- &= -2i(\\widetilde{\\epsilon }_-\\lambda _- +\\epsilon _-\\widetilde{\\lambda }_-) \\cr \\sqrt{\\tau _2} \\delta a &= -2i\\epsilon _-\\widetilde{\\psi }_+ \\cr \\delta g &= -2 i \\epsilon _-\\gamma _+ \\cr \\delta \\varphi _A &= 2i\\widetilde{\\epsilon }_-\\widetilde{\\mu }_{+A}\\end{aligned}\\qquad \\quad \\begin{aligned}\\delta v_+ &= 0 \\cr \\sqrt{\\tau _2}\\delta \\bar{a} &= 2i\\widetilde{\\epsilon }_-\\psi _+ \\cr \\delta \\bar{g} &= -2i \\widetilde{\\epsilon }_-\\widetilde{\\gamma }_+ \\cr \\delta \\bar{\\varphi }_A &= -2i\\epsilon _-\\mu _{+ A} \\,.\\end{aligned}$ We are here making manifest the $SU(2)_R$ representations of the fields, with index $A$ referring to the 2 of $SU(2)_R$ .",
"For the fermions the variations are $\\begin{aligned}\\delta \\lambda _- &= - \\epsilon _-(\\sqrt{\\tau _2}F_{01} +\\mathcal {F}_\\mathcal {A})\\cr \\delta \\beta _- &= -\\widetilde{\\epsilon }_-\\partial g \\cr \\delta \\psi _+ &= \\epsilon _-\\sqrt{\\tau _2}(\\bar{\\partial }v_+ - \\partial _+\\bar{a})\\cr \\delta \\gamma _+ &= \\widetilde{\\epsilon }_-\\partial _+ g \\cr \\delta \\mu _{+A} &= -\\widetilde{\\epsilon }\\partial _+\\varphi _A\\cr \\delta \\rho _{-A} &= \\widetilde{\\epsilon }\\partial _\\mathcal {A}\\varphi _A\\,.\\end{aligned}\\qquad \\qquad \\begin{aligned}\\delta \\widetilde{\\lambda }&=- \\widetilde{\\epsilon }_-(\\sqrt{\\tau _2}F_{01} - \\mathcal {F}_\\mathcal {A})\\cr \\delta \\widetilde{\\beta } &=\\epsilon _-\\bar{\\partial }\\bar{g} \\cr \\delta \\widetilde{\\psi }_+& = -\\widetilde{\\epsilon }_-\\sqrt{\\tau _2}(\\partial v_+ - \\partial _+a)\\cr \\delta \\widetilde{\\gamma }_+ &=\\epsilon _-\\partial _+\\bar{g} \\cr \\delta \\widetilde{\\mu }_{+A} &=\\epsilon _-\\partial _+\\bar{\\varphi }_A \\cr \\delta \\widetilde{\\rho }_{-A} &=\\epsilon \\bar{\\partial }_\\mathcal {A}\\bar{\\varphi }_A \\,.\\end{aligned}$ The BPS equations readily follow F01 =12( -v+ - +v-) = 0 FA = 12 2 (a - a) = 0v+ - +a = 0 v+ - + a = 0 +g = 0 +g = 0 g = 0 g = 0 AA = 0 AA = 0 +A = 0 +A = 0 .",
"As in section REF the internal kinetic operator that acts on the twisted fields is read off from the BPS equations for the bosonic fields and the equations of motion for the fermionic fields.",
"The bundle sections listed in table REF which count the zero modes are consistent with this kinetic operator."
],
[
"Spectrum",
"Upon dimensional reduction on $C$ , the zero-modes of the effective 2d (0,2) theory are counted by the Dolbeault cohomology groups associated with the bundles in table REF .",
"The spectrum consists of three pairs of chiral plus conjugate-chiral multiplets, two pairs of Fermi plus conjugate Fermi-multiplets, and the vector multiplet.",
"As usual, the appearance of $\\epsilon _-$ versus $\\tilde{\\epsilon }_-$ in the supersymmetry variations fixes the notion of chiral versus conjugate-chiral as well as of Fermi versus conjugate-Fermi superfields.",
"In this regard we stick to the conventions of [53].",
"Note that in table REF the number of zero-modes for multiplets and their conjugates are equal; this follows from the discussion in appendix REF , which can be used to show that $h^i(C, N_{C/B_3}) = h^{1-i}(C, N_{C/B_3} \\otimes \\mathcal {L}_D^{-1}) \\,.$ Finally, the vector multiplet must be counted in the same way as for a D3-brane on a curve in $B_2$ , discussed in section REF .",
"Unless the curve $C$ does not intersect the discriminant locus so that ${\\cal L}_D$ is trivial the vector multiplet is projected out at the massless level.",
"The bulk spectrum for non-trivial $\\mathcal {L}_D$ is given in table REF .",
"The bulk spectrum is completed by $8 c_1(B_3) \\cdot C$ Fermi multiplets from massless 3–7 string excitations.",
"Table: Structure of the 2d (0,2)(0,2) bulk multiplets in the effective theory on ℝ 1,1 ×C\\mathbb {R}^{1,1} \\times C for CC inside the base B 3 B_3 of Y 4 Y_4.The spectrum as given in table REF allows the computation of the right- and left-moving central charges, which will now depend on the further numerical value $h^0(C, N_{C/B_3})$ .",
"This extra freedom is not present in the Calabi-Yau three-fold set-up as the normal bundle is its own determinant bundle.",
"The total numbers of right- and left-moving degrees of freedom are $n_R = 3( g + c_1(B_3) \\cdot C + h^0(C, N_{C/B_3})) \\,,\\qquad n_L = 3( g + h^0(C, N_{C/B_3})) + c_1(B_3) \\cdot C + n_{37} \\,,$ where the left-moving central charge again includes the contribution $n_{37} = 8 c_1(B_3) \\cdot C$ from the 3–7 sector.",
"The gravitational anomaly, which depends only on the line bundle $\\mathcal {L}_D = K^{-1}_{B_3}|_C$ , is $c_L - c_R = 6 c_1(B_3) \\cdot C \\,.$"
],
[
"`Strings' in 2d $(0,2)$ Theories",
"In 2d F-theory compactifications tadpole cancellation necessitates the inclusion of a D3-brane sector, describing additional `strings' filling the full spacetime.",
"F-theory compactifications on an elliptic fibration $Y_5$ have been introduced in [16], [18] and give rise to 2d $N=(0,2)$ supersymmetric theories.",
"For the above reason, the status of the D3-brane sector is rather different in 2d compared to 6d and 4d F-theory models.",
"The D3-branes are an integral component of the definition of the vacuum rather than a defect sector, and their understanding is imperative to characterize the compactification.",
"We now describe the theory on a D3-brane wrapping a curve $C$ on the base $B_4$ of an elliptically fibered Calabi-Yau five-fold $Y_5$ , and will return to the tadpole constraint in the context of the anomaly considerations in section ."
],
[
"Duality Twist",
"Since the 2d effective theory on the D3-branes along $\\mathbb {R}^{1,1}$ is now spacetime-filling, there is no group of rotations transverse to the D3-brane theory in the non-compact dimensions.",
"The $SU(4)_R$ symmetry of the $N=4$ SYM theory on the D3-branes simply decomposes into $SU(3)_R \\times U(1)_R$ , which is the structure group of the normal bundle $N_{C/B_4}$ to the curve $C$ inside the internal Kähler space $B_4$ .",
"From the decomposition of the supercharges, $\\begin{aligned}G_{\\rm total} \\ & \\rightarrow \\ SU(3)_R \\times SO(1,1)_L \\times U(1)_C \\times U(1)_R \\times U(1)_ D\\cr ({\\bf 2}, {\\bf 1}, {\\bf \\bar{4}})_{+1} \\ &\\rightarrow \\ {\\bf \\bar{3}}_{1;1,1,1} \\oplus {\\bf \\bar{3}}_{-1;-1,1,1} \\oplus {\\bf 1}_{1;1,-3,1} \\oplus {\\bf 1}_{-1;-1,-3,1} \\cr ({\\bf 1}, {\\bf 2}, {\\bf 4})_{-1} \\ &\\rightarrow \\ {\\bf 3}_{1;-1,-1,-1} \\oplus {\\bf 3}_{-1;1,-1,-1} \\oplus {\\bf 1}_{1;-1,3,-1} \\oplus {\\bf 1}_{-1;1,3,-1} \\, ,\\end{aligned}$ one infers that the topological and duality twists TCtwist = 16(3TC + TR) TDtwist = 16(3TD + TR) , result in the expected two right-moving scalar supercharges.",
"These transform under $SO(1,1)_L \\times U(1)^{\\rm twist}_C \\times U(1)^{\\rm twist}_D\\times U(1)_R$ as Q+ : 11;0,0, 3 Q+ : 11;0,0, -3 .",
"The $N=4$ fields have the following reduction: $\\begin{aligned} &\\qquad SU(3)_R \\times SO(1,1) \\times U(1)_C^{\\rm twist} \\times U(1)_D^{\\rm twist} \\times U(1)_R \\cr A \\,&:\\, \\quad {\\bf 1}_{2,0,*,0} \\oplus {\\bf 1}_{-2,0,*,0} \\oplus {\\bf 1}_{0,1,*,0} \\oplus {\\bf 1}_{0,-1,*,0} \\cr &\\, \\quad = v_+ \\oplus v_- \\oplus \\bar{a} \\oplus a \\cr \\phi \\,&:\\, \\quad {\\bf 3}_{0,\\frac{1}{3},\\frac{1}{3},2} \\oplus {\\bf \\overline{3}}_{0,-\\frac{1}{3},-\\frac{1}{3},-2} \\cr & \\, \\quad = \\varphi \\oplus \\bar{\\varphi }\\cr \\Psi \\,&:\\, \\quad {\\bf 3}_{1,\\frac{1}{3},\\frac{1}{3},-1} \\oplus {\\bf 1}_{1,1,1,3} \\oplus {\\bf 1}_{-1,0,1,3} \\oplus {\\bf 3}_{-1,-\\frac{2}{3},\\frac{1}{3},-1} \\cr &\\, \\quad = \\mu _+ \\oplus \\psi _{+} \\oplus \\lambda _- \\oplus \\rho _{-} \\cr \\widetilde{\\Psi } \\,&:\\, \\quad {\\bf \\overline{3}}_{1,-\\frac{1}{3},-\\frac{1}{3},1}\\oplus {\\bf 1}_{1,-1,-1,-3} \\oplus {\\bf 1}_{-1,0,-1,-3} \\oplus {\\bf \\overline{3}}_{-1,\\frac{2}{3},-\\frac{1}{3},1} \\cr &\\, \\quad = \\tilde{\\mu }_+ \\oplus \\tilde{\\psi }_{+} \\oplus \\tilde{\\lambda }_- \\oplus \\tilde{\\rho }_{-} \\,.\\end{aligned}$ Table: The bulk field content of the (0,2)(0,2) field theory for a D3-brane wrapping a curve C⊂B 4 C \\subset B_4.We only display the SO(1,1) L SO(1,1)_L charge and theSU(3) R SU(3)_R symmetry representation.The spectrum of the partially topologically twisted theory on $\\mathbb {R}^{1,1} \\times C$ is given in table REF .",
"The bundles appearing in the last row are determined by the representation of the fields with respect to $SU(3)_R \\times U(1)^{\\rm twist}_C \\times U(1)^{\\rm twist}_L$ in close analogy to the procedure spelled out in section REF .",
"This time, the adjunction formula $K_C = K_{B_4}|_C \\otimes \\wedge ^3 N_{C/B_4}$ and the usual assignments of twist charges imply that sections of $K_C$ , $\\mathcal {L}_D$ and of $N_{C/B_4} $ carry twist charges $\\begin{aligned}K_C: \\qquad (q_C^{\\rm twist},q_D^{\\rm twist}) &= (-1,0) \\cr \\mathcal {L}_D: \\qquad (q_C^{\\rm twist},q_D^{\\rm twist}) &= (0,-1) \\cr \\qquad N_{C/B_4}: \\qquad (q_C^{\\rm twist},q_D^{\\rm twist}) &= (-1/3,-1/3) \\,.\\end{aligned}$"
],
[
"String Action and BPS Spectrum",
"Finally, let us turn to the action and supersymmetry transformations.",
"The total action is (REF ), where the gauge and topological parts of the Lagrangian are (REF ), while the scalar and fermion parts of the Lagrangian are twist dependent and in the present case given by $\\begin{aligned}\\mathcal {L}_\\text{fermion} =&- \\lambda _-\\partial _+\\widetilde{\\lambda }_-+ \\lambda _-\\bar{\\partial }_\\mathcal {A}\\widetilde{\\psi }_+- \\rho _-\\partial _+\\widetilde{\\rho }_-+ \\rho _-\\bar{\\partial }_\\mathcal {A}\\widetilde{\\mu }_++ \\mu _+\\partial _-\\widetilde{\\mu }_+- \\mu _+\\partial _\\mathcal {A}\\widetilde{\\rho }_-\\cr &+ \\psi _+\\partial _-\\widetilde{\\psi }_+- \\psi _+\\partial _\\mathcal {A}\\widetilde{\\lambda }_-- \\widetilde{\\rho }_-\\partial _+\\rho _-- \\widetilde{\\rho }_-\\partial _\\mathcal {A}\\mu _+- \\widetilde{\\lambda }_-\\partial _+\\lambda _-- \\widetilde{\\lambda }_-\\partial _\\mathcal {A}\\psi _+\\cr &+ \\widetilde{\\mu }_+\\partial _-\\mu _++ \\widetilde{\\mu }_+\\bar{\\partial }_\\mathcal {A}\\rho _-+ \\widetilde{\\psi }_+\\partial _-\\psi _++ \\widetilde{\\psi }_+\\bar{\\partial }_\\mathcal {A}\\lambda _- \\cr \\mathcal {L}_\\text{scalar} =&- \\partial _+\\varphi \\partial _-\\bar{\\varphi }- \\partial _+\\bar{\\varphi }\\partial _-\\varphi +\\partial _\\mathcal {A}\\varphi \\bar{\\partial }_\\mathcal {A}\\bar{\\varphi }+\\partial _\\mathcal {A}\\bar{\\varphi }\\bar{\\partial }_\\mathcal {A}\\varphi \\,.\\end{aligned}$ The bosonic supersymmetry variations leaving this theory invariant take the form $\\begin{aligned}\\sqrt{\\tau _2}\\delta v_- &=-2i(\\widetilde{\\epsilon }_-\\lambda _- + \\epsilon _-\\widetilde{\\lambda }_-) \\cr \\sqrt{\\tau _2}\\delta a &= -2i\\epsilon _-\\widetilde{\\psi }_+ \\cr \\delta \\varphi _\\alpha &= 2i\\epsilon _-\\mu _{+\\alpha }\\end{aligned}\\qquad \\begin{aligned}\\delta v_+ &= 0 \\cr \\sqrt{\\tau _2}\\delta \\bar{a} &= 2i\\widetilde{\\epsilon }_-\\psi _+ \\cr \\delta \\bar{\\varphi }^{\\dot{\\alpha }} &= -2i\\widetilde{\\epsilon }_-\\widetilde{\\mu }_+^{\\dot{\\alpha }} \\,.\\end{aligned}$ Their fermionic counterparts are found to be $\\begin{aligned}\\delta \\lambda _- &= \\epsilon _-(\\sqrt{\\tau _2}F_{01} + \\mathcal {F}_\\mathcal {A}) \\cr \\delta \\psi _+ &= \\sqrt{\\tau _2}\\epsilon _-(\\bar{\\partial }v_+ - \\partial _+\\bar{a}) \\cr \\delta \\mu _{+\\alpha } &= -\\widetilde{\\epsilon }\\partial _+\\varphi _\\alpha \\cr \\delta \\rho _{-\\alpha } &=\\widetilde{\\epsilon }\\partial _{\\mathcal {A}}\\varphi _\\alpha \\end{aligned}\\qquad \\begin{aligned}\\delta \\widetilde{\\lambda }_- &= \\widetilde{\\epsilon }_-(\\sqrt{\\tau _2}F_{01} -\\mathcal {F}_\\mathcal {A}) \\cr \\delta \\widetilde{\\psi }_+ &= -\\sqrt{\\tau _2}\\widetilde{\\epsilon }(\\partial v_+ -\\partial _+ a) \\cr \\delta \\widetilde{\\mu }_+^{\\dot{\\alpha }} &=\\epsilon \\partial _+\\bar{\\varphi }^{\\dot{\\alpha }} \\cr \\delta \\widetilde{\\rho }_-^{\\dot{\\alpha }} &=\\epsilon \\bar{\\partial }_\\mathcal {A}\\bar{\\varphi }^{\\dot{\\alpha }} \\,.\\end{aligned}$ Here the indices $\\alpha $ and $\\dot{\\alpha }$ refer to the ${\\bf 3}$ and ${\\bf \\bar{3}}$ representation of $SU(3)_R$ , respectively.",
"More details are given in appendix .",
"The BPS equations are easily read off from the above variations.",
"After dimensional reduction to two dimensions, the 2d zero-modes organize into two pairs of chiral plus conjugate-chiral multiplets, a Fermi plus conjugate-Fermi multiplet, and the vector multiplet.",
"The latter is absent unless ${\\cal L}_D$ is trivial, i.e.",
"unless the curve $C$ does not intersect the discriminant locus of the elliptic fibration.",
"The number of zero-modes for each of these fields is computed by the cohomology groups associated with the bundles appearing in table REF .",
"Serre duality together with the identity $\\wedge ^2 N_{C/B_4} = N_{C/B_4}^\\vee \\otimes \\wedge ^3 N_{C/B_4}\\,,$ which follows from (REF ), guarantee that each multiplet and its conjugate indeed appear with the same multiplicity.",
"In particular, $h^{i}(C,N_{C/B_4}) = h^{1-i}(C, N_{C/B_4}^\\vee \\otimes K_C) = h^{1-i}(C,\\wedge ^2 N_{C/B_4}^\\vee \\otimes {\\cal L}_D^{-1}|_C)\\,.$ The zero-mode spectrum is given in table REF .",
"Table: Structure of 2d (0,2) bulk multiplets in effective theory on ℝ 1,1 ×C\\mathbb {R}^{1,1} \\times C inside base of CY 5 CY_5.The bulk spectrum is completed by $8 c_1(B_4) \\cdot C$ Fermi multiplets from the 3–7 sector.",
"From the spectrum the central charges can be computed as $\\begin{aligned}n_R &= 3( g + c_1(B_4) \\cdot C + h^0(C, N_{C/B_4}) - 1)\\cr n_L &= 3( g + h^0(C, N_{C/B_4}) - 1) + 9c_1(B_4) \\cdot C \\,,\\end{aligned}$ where we have included in the computation of $n_L$ the $n _{37} = 8c_1(B_4) \\cdot C$ modes from the 3–7 sector.",
"The gravitational anomaly is then $c_L - c_R = 6 \\, c_1(B_4) \\cdot C \\,.$"
],
[
"Duality Twist for 2d $N=(0, 6)$ Theories",
"So far we have focused on chiral 2d theories arising from duality twisted $N=4$ SYM on a curve with a brane realization in terms of D3-branes wrapping curves in an F-theory compactification, which naturally incorporates the varying coupling.",
"In this section we shall study one case that does not have such a brane realization, but nevertheless represents a consistent topological twist of the $N=4$ SYM theory, and upon twisted dimensional reduction results in an $N=(0,6)$ theory.",
"This may in particular be of interest in the light of recent developments on novel supersymmetric theories such as $N=3$ SYM in 4d [55].",
"For this, we consider again the R-symmetry decomposition of $N=4$ SYM which we used for the CY$_5$ twist in (REF ) $\\begin{aligned}SU(4)_R &\\ \\rightarrow \\ SU(3)_R \\times U(1)_R\\cr {\\bf 4} &\\ \\rightarrow \\ {\\bf 1}_3 \\oplus {\\bf 3}_{-1} \\cr {\\bf 6} &\\ \\rightarrow \\ {\\bf 3}_2 \\oplus {\\bf 3}_{-2} \\,,\\end{aligned}$ however unlike for the twist discussed in the CY$_5$ brane realization (REF ) we could instead consider the twist TCtwist = 12(TC - TR) TDtwist = 12(TD - TR) .",
"With the same representation theoretic decomposition in (REF ), but the twist defined as in (REF ), we end up with the following supercharges under $SU(3) \\times SO(1,1) \\times U(1)_C^{\\rm twist} \\times U(1)_D^{\\rm twist}$ $Q: \\qquad \\bar{\\bf 3}_{1; 0,0} \\oplus {\\bf 3}_{1; 0, 0} \\,,$ thus giving an $N=(0,6)$ supersymmetric theory in 2d.",
"It would be interesting to explore geometric realizations of these theories in more detail in the future."
],
[
"Hitchin Equations with $\\mathcal {L}_D$ -twist",
"The BPS equations involving the reduction of the 4d gauge field $A_\\mu $ are the same across all dimensions of Calabi-Yau compactifications that we here consider.",
"Of particular interest is the equation FA = 12 ( A(2a) - A(2a) ) = 0 .",
"This equation is expressed in terms of the $U(1)_D$ eigenstate fields $\\sqrt{\\tau _2}a$ and $\\sqrt{\\tau _2}\\bar{a}$ .",
"These fields transform as bundle-valued sections, 2 a (0,1(C, LD-1)) 2 a (0,0(C, KC LD)) , where the two bundles in questions are dual to each other.",
"The number of sections of these particular bundles, for a curve $C$ inside a Calabi-Yau $Y_n$ with $n=3, 4, 5$ intersecting the discriminant locus, is $g - 1 + \\text{deg}(\\mathcal {L}_D) = g -1 + c_1(B_{n-1})\\cdot C\\,.$ We see that the equation (REF ) is a $U(1)_D$ duality twisted version of the usual abelian Hitchin equation ${\\cal F}= 0$ , with ${\\cal F}$ the abelian field strength along $C$ .",
"Apart from the fact that $\\sqrt{\\tau _2} a$ and $\\sqrt{\\tau _2} \\bar{a}$ are sections of the bundles (REF ), the expression ${\\cal F}_{\\cal A}$ involves the $U(1)_D$ covariant derivatives defined in section (REF ).",
"The solutions will be $U(1)_D$ twisted flat connections on $C$ .",
"It would be interesting to study the extension of such a duality twisted Hitchin system to the non-abelian case and the underlying mathematics further."
],
[
"Type IIB Limit and Quantum Higgsing in F-theory",
"It is very instructive to compare our findings for the spectrum on a D3-brane wrapping a curve in F-theory to the situation in the special case of perturbative orientifolds.",
"If the F-theory setup has a smooth Type IIB orientifold limit, the base space $B_{n-1}$ of the fibration is the quotient of a Calabi-Yau $(n-1)$ -fold $X_{n-1}$ by a holomorphic involution $\\sigma $ .",
"Let us assume that the curve $C \\subset B_{n-1}$ wrapped by the D3-brane is not contained in the discriminant locus of the elliptic fibration.",
"In Type IIB two different cases are to be considered: Either the double cover $\\tilde{C} \\subset X_{n-1}$ of the curve $C$ is invariant as a whole under the involution $\\sigma $ , but not pointwise, or it splits into two irreducible curves $\\tilde{C} = C_+ \\cup C_-$ exchanged by the orientifold, $\\sigma (C_\\pm ) = C_\\mp $ .",
"In the first case the perturbative gauge group $U(1)$ is broken by the orientifold action to $O(1) = \\mathbb {Z}_2$ , and the $U(1)$ gauge potential and gaugino partners are projected out.",
"The second case is more subtle: If $[C_+] = [C_-]$ in homology, then perturbatively there exists an unbroken gauge group $U(1)_- = U(1)_{C_+} - U(1)_{C_-}$ with an associated massless gauge field and gaugino partners.",
"If $[C_+] \\ne [C_-]$ a Stückelberg mechanism involving the orientifold-odd Ramond-Ramond two-form potential $C_2$ renders the gauge boson massive and breaks the gauge symmetry to a global $U(1)$ or $\\mathbb {Z}_k$ symmetry.",
"The possibility of a massless $U(1)_- = U(1)_{C_+} - U(1)_{C_-}$ is to be contrasted with our non-perturbative F-theory result that a massless vector multiplet is found in the effective theory along the string if and only if the elliptic surface $\\widehat{C} \\subset Y_{n}$ , defined by the restriction of the elliptic fibration to $C$ , is trivially fibered, i.e.",
"$\\widehat{C} = C \\times T^2$ globally.",
"This situation occurs precisely if $C$ does not intersect the discriminant locus $\\Delta $ .",
"In the language of the Type IIB orientifold, this is equivalent to the statement that the curve $\\tilde{C} \\subset X_{n-1}$ does not intersect the divisor wrapped by the orientifold plane, whose class we denote by $[O7]$ .",
"Indeed, the 7-brane tadpole cancellation condition relates $[O7]$ to the class of the divisors wrapped by the D7-branes and their orientifold images, $8 [{\\rm O7}] = \\sum _i ([{\\rm D7}]_i + [{\\rm D7}]^{\\prime }_i)$ .",
"Therefore if the curve $\\tilde{C}$ intersects any of the 7-branes, it necessarily intersects also the O7-plane and vice versa.",
"A trivial fibration $\\widehat{C} = C \\times T^2$ then corresponds, in perturbative Type IIB orientifolds, to a D3-brane on $\\tilde{C} = C_+ \\cup C_-$ with $C_+ \\cap O7 = \\emptyset = C_- \\cap O7$ .",
"By contrast, if $\\tilde{C} = C_+ \\cup C_-$ but $C_+ \\cap C_- \\ne \\emptyset $ , then the corresponding $\\widehat{C}$ is non-trivially fibered.",
"Non-perturbatively no massless gauge bosons are found in the effective 2d action despite the fact that the perturbative limit does exhibit a $U(1)_-$ gauge group, at least if $[C_+] =[C_-]$ .",
"The key to understanding this apparent paradox is that at $C_+ \\cap C_-$ extra zero-modes localize from strings between the D3-brane on $C_+$ and its image brane on $C_-$ .",
"They carry charge $ q_- = 2$ with respect to the perturbative gauge group $U(1)_-$ (in units where the 3–7 strings carry charge $q_- = 1$ ).",
"These zero-modes assemble into chiral multiplets, and are crucial for consistency of the setup: First, since the fermions in the chiral multiplets are of positive chirality, they precisely cancel the $U(1)_-$ anomaly due to the negative chirality 3–7 strings.",
"Indeed, by assumption the $U(1)_-$ is not rendered massive by a Stückelberg mechanism and hence must be non-anomalous because massless $U(1)$ s do not participate in a Green-Schwarz mechanism in 2d.",
"Second, the chiral scalar fields correspond to the recombination moduli of $\\tilde{C} = C_+ \\cup C_-$ .",
"In the Type IIB orientifold a non-trivial VEV of the scalar component triggers a recombination of the split curve $\\tilde{C}$ into an irreducible smooth curve.",
"This recombination higgses $U(1)_- \\rightarrow \\mathbb {Z}_2$ , consistent with the fact that after the recombination $\\tilde{C}$ is orientifold invariant and carries gauge group $O(1) = \\mathbb {Z}_2$ .",
"Our results on the duality twisted D3-brane theory strongly suggest that genericallyThere is one degenerate exception described below.",
"F-theory does not distinguish between the perturbative phase at the origin of the Higgs branch with perturbative gauge group $U(1)_-$ and the phase away from the origin with gauge group $O(1)$ .",
"This will indeed be corroborated further from a geometric perspective below.",
"Our physical interpretation is that strong coupling effects dynamically higgs the perturbative gauge group in F-theory - an effect which is invisible in the perturbative description via Type IIB orientifolds.",
"Note that the string coupling $g_s$ indeed enters the non-perturbative regime near the O7-plane, which is where the recombination modes in the $C_+-C_-$ sector are localised.More precisely, the O7-plane famously splits in F-theory precisely due to strong coupling effects [56].",
"See also [57], [58] for recent discussions.",
"It is worth stressing that such a non-perturbatively triggered brane recombination process does not occur for 7-branes intersecting conventional $O7^-$ -planes.",
"A transverse intersection locus $D7_i \\cap O7$ in F-theory is always 6-dimensional, whereas the locus $D3 \\cap O7$ in the configurations considered here is only 2-dimensional.",
"The orientifold action projects out the zero-modes in the spectrum of the D7-D7' strings located at the orientifold and thus no non-perturbative recombination is possible.",
"The D7-D7' zero-modes located away from the O7-plane, if any, are not projected out by the orientifold, but since these are not located in the non- region of perturbative $g_s$ , no recombination is triggered in F-theory.",
"Indeed, for 7-branes both the recombined phase and the phase with gauge group $U(1)$ are accessible as described in [59].A stack of 7-branes in Type IIB orientifolds with gauge group $U(N)$ , on the other hand, does give rise to zero-modes in the anti-symmetric representation on top of the O7-plane, but again for non-abelian gauge groups no non-perturbative Higgsing is observed.",
"To understand this effect further, consider the Weierstrass model for an elliptic Calabi-Yau $n$ -fold $Y_n$ with base $B_{n-1} $ .",
"We can parametrise the Weierstrass model following Sen as [56] $\\begin{aligned}y^2 &= x^3 + f x + g \\\\f &= -3 h^2 + \\epsilon \\eta \\, \\qquad g = -2 h^3 + \\epsilon h \\eta - \\frac{1}{12} \\epsilon ^2 \\chi \\, \\\\\\Delta & \\simeq \\epsilon ^2 h^2 (\\eta ^2 - h \\chi ) + {\\cal O}(\\epsilon ^3) \\,,\\end{aligned}$ with $f$ and $g$ sections of $K^{-4}_{B_{n-1}}$ and $K^{-6}_{B_{n-1}}$ , respectively, and $h$ , $\\eta $ and $\\chi $ generic polynomials of appropriate degree.",
"The orientifold double-cover is described by the Calabi-Yau space $X_{n-1}$ $X_{n-1}: \\, \\xi ^2 = h $ by adding a new coordinate $\\xi $ and letting $h$ depend on the coordinates on $B_{n-1}$ .",
"The orientifold action $\\sigma : \\xi \\rightarrow - \\xi $ leaves the O7-plane located at $h=0$ invariant.",
"Consider now the family of curves $C_\\delta : h = p_1^2 + \\delta \\, p_2 \\, \\subset \\, B_{n-1}$ with $p_1$ and $p_2$ generic polynomials on $B_{n-1}$ transforming as sections of $K^{-1}_{B_{n-1}}$ and $K^{-2}_{B_{n-1}}$ and a parameter $\\delta \\in \\mathbb {R}$ .",
"The double cover of this curve on the Type IIB Calabi-Yau $X_{n-1}$ is given by $\\tilde{C}_\\delta : \\, \\xi ^2 = p_1^2 + \\delta \\, p_2 \\, \\subset \\, X_{n-1}.$ If $\\delta =0$ , $\\tilde{C}_\\delta $ splits into $\\tilde{C}_0 = C_+ \\cup C_-$ with $C_+: \\, \\xi = p_1, \\qquad C_-: \\, \\xi = - p_1.$ The two components $C_+$ and $C_-$ are exchanged by the orientifold action $\\xi \\rightarrow - \\xi $ and intersect at the orientifold locus $\\xi =0$ .",
"Hence the parameter $\\delta $ parametrizes the Higgs branch for $U(1)_-$ in the Type IIB limit: For $\\delta =0$ , the perturbative gauge potential of $U(1)_-$ is massless, while for $\\delta \\ne 0$ it is broken by a Higgs effect in which the $3-3^{\\prime }$ modes acquire a VEV.",
"The topology of the curve $C_\\delta $ in $B_{n-1}$ wrapped by the D3-brane in F-theory does not change as drastically for varying $\\delta $ .",
"Rather, the parameter $\\delta $ characterizes the intersection of $C_\\delta $ with the locus $h=0$ , given by $\\lbrace h=0\\rbrace \\cap C_\\delta : \\quad \\lbrace h=0\\rbrace \\cap \\lbrace p_1 = \\pm \\sqrt{\\delta \\, p_2} \\rbrace \\,.$ In the Type IIB limit $\\epsilon \\rightarrow 0$ , and only in this limit, $\\lbrace h=0\\rbrace $ describes the location of the O7-plane.",
"As $\\delta \\rightarrow 0$ , the pairs of intersection points $p_1 = \\pm \\sqrt{\\delta \\, p_2}$ with $h=0$ come closer together and finally coalesce as $\\delta =0$ , corresponding to vanishing Higgs VEV.",
"Therefore, in the limit $\\epsilon \\rightarrow 0$ , we can geometrically identify the VEV of the Higgs fields with the separation of pairs of intersection points of $C_\\delta $ with the O7-plane.",
"This is also intuitive because the Higgs fields correspond to $3-3^{\\prime }$ modes at the intersection of $C_+$ with $C_-$ .",
"This is precisely the intersection with the O7-plane in Type IIB, which in fact is a double point from the perspective of $C_\\delta $ at $\\delta =0$ : The two signs in (REF ) correspond to the intersection of $C_+$ and $C_-$ with the O7-plane.",
"As these points are separated for $\\delta \\ne 0$ , the $3-3^{\\prime }$ string modes acquire mass - indicating the Higgsing.",
"Figure: SL(2,ℤ)SL(2,\\mathbb {Z}) monodromies on the D3-brane in a degenerate SO(8)SO(8) model.",
"The perturbative restoration of the U(1)U(1) gauge symmetry as δ→0\\delta \\rightarrow 0 corresponds to absence of any monodromies along CC.Finally, we are in a position to address the non-perturbative effects in F-theory.",
"Such effects are parametrized by non-zero values of the parameter $\\epsilon $ in (REF ).",
"Famously, as we take the effects of order $\\epsilon ^3$ into account, the locus $h=0$ in the discriminant $\\Delta $ no longer corresponds to the true location of the orientifold plane.",
"Rather the double zeroes of $\\Delta $ at $h^2=0$ obtained by ignoring these terms split into two, with their separation being controlled by $\\epsilon $ .",
"This is Sen's celebrated splitting of the orientifold plane [56] into a mutually non-local $(p,q)$ -7-brane system.",
"If we intersect with $C_\\delta $ , we see that, independently of the effect from $\\delta $ , a splitting of the intersection points with the O7-plane system occurs due to strong coupling effects parametrized by $\\epsilon $ .",
"We claim that this separation must still be viewed as parametrizing the VEV of the Higgs fields.",
"In particular, as $\\epsilon \\ne 0$ , a Higgsing occurs in F-theory irrespective of whether or not $\\delta \\ne 0$ .",
"This is the non-perturbative Higgsing alluded to above.",
"The reason why the Higgsing is inevitable in F-theory is because the string coupling always diverges close to the orientifold plane.",
"The only exception is a situation in which the 7-brane tadpole is cancelled locally everywhere, corresponding to a configuration of $SO(8)$ gauge group.",
"Such a background is engineered by taking the vanishing orders of $(f,g,\\Delta )$ as $(2, 3, 6)$ along $h=0$ .",
"From the perspective of (REF ) this is paramount to setting $\\epsilon \\equiv 0$ in $f$ and $g$ .",
"Note that we cannot afford any gauge group different from $SO(8)$ on any other locus if we want to ensure that no strong coupling effects occur.",
"The resulting Weierstrass model is degenerate because self-intersections of the $SO(8)$ divisor in codimension-two necessarily lead to non-minimal fibers with vanishing orders ${\\rm ord}(f,g, \\delta ) \\ge (4,6,12)$ .",
"Let us nonetheless analyze the fibration over $C_\\delta $ , given by the surface $\\widehat{C}_\\delta : y^2 = x^3 - 3 (p_3^2 + \\delta \\, p_6)^2 x - 2 (p_3^2 + \\delta \\, p_6)^3 \\,.$ Now, the limit $\\delta \\rightarrow 0$ differs considerably also in the full F-theory.",
"The discriminant locus of $ \\widehat{C}_\\delta $ consists of pairs of points $p_3 = \\pm \\sqrt{\\delta \\, p_6}$ .",
"Each of these points describes a defect along the curve $C_\\delta $ due to an intersection with the $SO(8)$ locus.",
"Pairs of such defects are connected by a branch-cut associated with a duality wall for the topologically twisted theory, as studied in [7] for D3-branes wrapping surfaces.",
"As one encircles an $SO(8)$ defect and crosses a duality wall one picks up an $SL(2,\\mathbb {Z})$ monodromy given by $M_{SO(8)} = \\left(\\begin{matrix} -1 & 0 \\\\ 0 & -1 \\end{matrix}\\right) \\,.$ This monodromy acts as $\\left(\\begin{matrix} F \\\\ F_D \\end{matrix}\\right) \\rightarrow M_{SO(8)} \\left(\\begin{matrix} F \\\\ F_D \\end{matrix}\\right) \\,,$ with $F$ the $U(1)_-$ gauge field on the D3-brane and $F_D$ its magnetic dual as in (REF ).",
"This is responsible for projecting out the $U(1)_-$ gauge potential.",
"But as $\\delta \\rightarrow 0$ pairs of $SO(8)$ defects coalesce, and the monodromy disappears, see figure REF .",
"As a result the $U(1)$ gauge potential survives as a massless field.",
"To conclude, the $U(1)_-$ is massless if and only if both $\\epsilon =0$ and $\\delta =0$ .",
"These observations also suggest an interesting connection with the results of [60], [57] concerning the localisation of some of the bulk D3 modes at the intersection with the O7-plane, as required from the perspective of anomaly inflow.",
"As the brane-image brane pair recombines in Type IIB, one linear combination of the two complex recombination scalars remains as a massless modulus of the brane.",
"Typically we would expect this field to arise as a delocalized bulk field along the cycle.",
"However, it is clear that e.g.",
"for small recombination parameter $\\delta $ the avatars of the recombination modes approximately localize around the former intersection locus between the two branes, which lies on top of the O7-plane.",
"From this perspective it is natural to identify the analogue of the localised bulk modes discussed in [60], [57] as these recombination modes.",
"It would be interesting to verify or falsify this conjecture.",
"Let us also note that a similar effect occurs not only for strings from D3-branes wrapped on curves in F-theory, but also for D3-brane instantons associated with Euclidean D3-branes along surfaces.",
"As reviewed e.g.",
"in [61] the D-brane instanton literature differentiates between so-called $U(1)$ instantons and $O(1)$ instantons.",
"The $U(1)$ instantons correspond to split brane-image brane divisors and the $O(1)$ instantons to their recombined phase.",
"The results of this section explain why no such difference must be made in F-theory or M-theory: As long as an instanton intersects the discriminant locus it always behaves like an $O(1)$ instanton due to strong coupling effects.",
"Finally we turn to the microscopic counting of the rather mysterious massless Fermi multiplets from D3-7-brane intersections in the Sen limit: The discriminant takes the form $\\Delta \\simeq \\epsilon ^2 h^2 (\\eta ^2 - h \\chi ) + {\\cal O}(\\epsilon ^3)$ with $\\eta ^2 - h \\chi $ the D7-brane in class $K^{-8}_{B_{n-1}}$ and $h=0$ representing the orientifold plane.",
"The independent extra modes from the 3–7 sector are located only at the intersection of the locus $\\eta ^2 - h \\chi $ with the D3-brane, while the D3-O7 sector does not host independent, extra 3–7 modes.",
"Hence, the number of extra 3–7 Fermi multiplets is indeed given by $8 c_1(B_{n-1}) \\cdot C$ ."
],
[
"Dual Points of View: M5s and M2s",
"We now complement the analysis in the previous section by an M-theory dual point of view, given the somewhat non-standard twist that we used for the compactification of the D3-branes.",
"We now describe how using M/F-duality, this system is mapped to M5- or M2-branes wrapping suitable cycles in the elliptic fibration.",
"The background has already been summarized in sections REF and REF .",
"Apart from giving a complementary point of view, it also allows a generalization of the previous setup.",
"The duality twist that allowed us to characterize the strings from the D3-brane point of view in F-theory is a priori only defined for the abelian theory.",
"It is useful to map the system to M-theory, mapping the D3 to an M5, where the twist becomes a purely geometric topological twist [8], or an M2-brane, where a generalization to non-abelian theories is possible.",
"The M5-brane approach also allows for a completely general derivation of the total number of Fermi multiplets due to 3–7 string excitations."
],
[
"Strings and Particles in 5d",
"We begin our analysis with strings in 6d $N=(1,0)$ supersymmetric compactifications of F-theory on elliptic Calabi-Yau three-folds, which by M/F-duality map to 5d compactifications of M-theory.",
"The dual (0,4) string obtained from an M5-brane is precisely the MSW string [62], [63], [64], specialized to an elliptic Calabi-Yau.",
"For M2-branes, we find an $N=4$ SQM as the dimensional reduction of the $(0,4)$ theory to one dimension, and we match the spectra accordingly using automorphic duality."
],
[
"M5-branes and MSW-Strings",
"Following the general logic discussed in section REF , a D3-brane wrapping $C \\subset B_2$ is dual to an M5-brane with worldvolume $\\mathbb {R}^{1,1} \\times \\widehat{C}$ , where the surface $\\widehat{C}$ is the restriction of the elliptic fibration $Y_3$ to $C$ .",
"The Lorentz symmetry decomposes under the reduced holonomy of $\\mathbb {R}^{1,1} \\times \\widehat{C}$ as $\\begin{aligned}SO(1,5)_L &\\quad \\rightarrow \\quad SU(2)_l \\times SO(1,1)_L \\times U(1)_l \\cr {\\bf 4} &\\quad \\rightarrow \\quad {\\bf 2}_{1,0} \\oplus {\\bf 1}_{-1,1} \\oplus {\\bf 1}_{-1,-1} \\cr {\\bf 15} &\\quad \\rightarrow \\quad {\\bf 1}_{0,0} \\oplus {\\bf 3}_{0,0} \\oplus {\\bf 1}_{0,2} \\oplus {\\bf 1}_{0,0} \\oplus {\\bf 1}_{0,-2} \\oplus {\\bf 2}_{2,1} \\oplus {\\bf 2}_{2,-1} \\oplus {\\bf 2}_{-2,1} \\oplus {\\bf 2}_{-2,-1} \\,.\\end{aligned}$ Further we decompose the R-symmetry group $Sp(4)_R$ by the embedding $\\begin{aligned}Sp(4)_R &\\quad \\rightarrow \\quad SO(3)_T \\times U(1)_R \\cr {\\bf 4} &\\quad \\rightarrow \\quad {\\bf 2}_1 \\oplus {\\bf 2}_{-1} \\cr {\\bf 5} &\\quad \\rightarrow \\quad {\\bf 1}_2 \\oplus {\\bf 1}_{-2} \\oplus {\\bf 3}_0 \\,.\\end{aligned}$ The twist of $U(1)_l$ with $U(1)_R$ defined by $U(1)_{\\rm twist} = U(1)_l + U(1)_R$ results in the following decomposition of the fields and supercharges: SO(1,5)L Sp(4)R SU(2)l SO(3)T SO(1,1) U(1)twist B: (15, 1) (1,1)0,0 (3,1)0,0 (1,1)0,2 (1,1)0,0 (1,1)0,-2 (2,1)2,1 (2,1)2,-1 (2,1)-2,1 (2,1)-2,-1 H: (10, 1) (3,1)-2,0 (1,1)2,2 (1,1)2,0 (1,1)2,-2 (2,1)0,1 (2,1)0,-1 ij: (1, 5) (1,1)0,2 (1,1)0,-2 (1,3)0,0 Qi, i: (4, 4) (2,2)-1,1 (2,2)-1,-1 (1,2)1,2 (1,2)1,0 (1,2)1,0 (1,2)1,-2 .",
"One can see that the decomposition of the $Q^i$ gives rise to four scalar supercharges which are right-handed with respect to the 2d $SO(1,1)_L$ Lorentz group, $Q^i: ({\\bf \\overline{4}, 4}) \\supset ({\\bf 1, 2})_{1,0} \\oplus ({\\bf 1,2})_{1,0} \\,.$ We are now ready to discuss the zero-modes from each field.",
"If a state transforms in the ${\\bf 2}$ of the internal holonomy group $SU(2)_l$ , it must transform as a one-form on the compactification space, with our conventions being that a state ${\\bf 2}_{1}$ and ${\\bf 2}_{-1}$ with respect to $SU(2)_l \\times U(1)_{\\rm twist}$ transforms as an element of $\\Omega ^{0,1}(\\widehat{C})$ and, respectively $\\Omega ^{1,0}(\\widehat{C})$ .",
"This implies that a state ${\\bf 1}_{2}$ translates into an element of $\\Omega ^{0,2}(\\widehat{C})$ and ${\\bf 1}_{0}$ to an element of $\\Omega ^{0,0}(\\widehat{C})$ .",
"The zero-modes correspond to the associated cohomology groups.",
"From the 6d scalar $\\Phi $ there is a $({\\bf 1,3})_{0,0}$ field with twist charge zero.",
"Such a field contributes both three left- and three right-handed real chiral scalars since $\\Phi $ itself has no intrinsic chirality.",
"Similarly $\\Phi $ contains scalars with twist charges $\\pm 2$ , whose zero-modes are counted by $h^{2,0}(\\widehat{C})$ and $h^{0,2}(\\widehat{C})$ respectively, and also contribute to both right- and left-hand zero-modes.",
"In summary we obtain the following real bosonic zero-modes from the scalar field $\\Phi = ({\\bf 1}, {\\bf 5}): \\qquad \\begin{array}{|c|c|c|}\\hline SU(2)_l \\times SO(3)_T \\times SO(1,1)_L \\times U(1)_{\\text{twist}} & \\hbox{Multiplicity} & \\hbox{L/R}\\cr \\hline ({\\bf 1},{\\bf 3})_{0,0} & h^{0,0}(\\widehat{C}) & \\hbox{L and R}\\cr ({\\bf 1},{\\bf 1})_{0,2} & h^{0,2}(\\widehat{C})& \\hbox{L and R} \\cr ({\\bf 1},{\\bf 1})_{0,-2} & h^{2,0}(\\widehat{C})& \\hbox{L and R}\\cr \\hline \\end{array}$ The self-dual three-form $H$Recall that to compute the zero-modes of the potential $B_{\\mu \\nu }$ we can start from the decomposition of the self-dual field strength $H$ .",
"At the level of the zero-modes, the exterior derivative does not change the representation content of the internal symmetry groups.",
"Specifically the $SU(2)_l$ and $U(1)_\\text{twist}$ charges are the same for the modes associated with the potential $B_{\\mu \\nu }$ and those of the field strength.",
"contributes left-handed real scalar fields corresponding to the representation $({\\bf 3,1})_{-2,0}$ .",
"The zero-modes of this field are counted by the number $h^{1,1}(\\widehat{C}) - 1$ of anti-self-dual two-forms on $\\widehat{C}$ .",
"To understand this counting directly from the representation under $SU(2)_l \\times U(1)_{\\rm twist}$ note that elements of $H^{1,1}(\\widehat{C})$ transform under as a ${\\bf 2}_{1} \\otimes {\\bf 2}_{-1} \\rightarrow {\\bf 3}_0 \\oplus {\\bf 1}_0$ .",
"Taking into account the singlet this indeed leads to the counting of $h^{1,1}(\\widehat{C}) - 1$ massless modes transforming as a ${\\bf 3}_0$ under $SU(2)_l \\times U(1)_{\\rm twist}$ .",
"The right-handed scalar fields arise from the fields strengths with representations $({\\bf 1,1})_{2,2}$ , $({\\bf 1,1})_{2,0}$ , and $({\\bf 1,1})_{2,-2}$ , which are counted by $h^{0,2}(\\widehat{C})$ , $h^{0,0}(\\widehat{C})$ , and $h^{2,0}(\\widehat{C})$ respectively.",
"Indeed the number $2 h^{2,0}(\\widehat{C}) + 1$ counts the self-dual two-forms on $\\hat{C}$ .",
"Finally, the modes $({\\bf 2,1})_{0, \\pm 1}$ of the decomposition of $H$ in (REF ) correspond to the fields ${\\bf (2,1)}_{\\pm 2,\\pm 1}$ in $B$ .",
"The associated zero-modes would have to be represented by elements of $H^{0,1}(\\widehat{C})$ .",
"A one-form on the Kähler surface has no definite behaviour under Hodge duality and hence the appearance of such states would be in conflict with the fact that the 3-form is self-dual.",
"This implies that the would-be zero-modes must be discarded, a conclusion which has also been reached e.g.",
"in [20].",
"This is in fact as expected from the dual D3-brane perspective as the modes would give rise to additional vector potentials along the string, which are clearly absent on the D3-brane side.",
"In summary we find from the self-dual three form the following real scalar modes: $H_{\\mu \\nu \\rho }= ({\\bf \\overline{10}}, {\\bf 1}): \\qquad \\begin{array}{|c|c|c|c|}\\hline SU(2)_l \\times SO(3)_T \\times SO(1,1)_L \\times U(1)_{\\text{twist}} & \\hbox{Multiplicity}&\\hbox{L/R}\\cr \\hline ({\\bf 3},{\\bf 1})_{-2,0} & h^{1,1}(\\widehat{C})-1 & \\hbox{L}\\cr ({\\bf 1},{\\bf 1})_{2,2} & h^{0,2}(\\widehat{C}) & \\hbox{R} \\cr ({\\bf 1},{\\bf 1})_{2,0} & h^{0,0}(\\widehat{C})& \\hbox{R}\\cr ({\\bf 1},{\\bf 1})_{2,-2} & h^{2,0}(\\widehat{C})& \\hbox{R}\\cr \\hline \\end{array} $ The remaing task is to count the fermions: The states in the decomposition of the $({\\bf \\overline{4}},{\\bf 4})$ in (REF ) all come in pairs with opposite $U(1)_{\\rm twist}$ charge.",
"Each of these pairs describes a complex Weyl fermion together with its complex conjugate state.",
"The independent states are the right-handed Weyl fermions $({\\bf 1}, {\\bf 2})_{1,0}$ , $({\\bf 1}, {\\bf 2})_{1,2}$ counted by $h^{0,0} (\\widehat{C})$ and, respectively, $h^{0,2} (\\widehat{C})$ , as well as the left-moving Weyl fermions $({\\bf 2}, {\\bf 2})_{-1,1}$ counted by $h^{0,1}(\\widehat{C})$ .",
"In summary we obtain from the fermions $\\rho = (\\overline{\\bf 4}, {\\bf 4}): \\qquad \\begin{array}{|c|c|c|c|}\\hline SU(2)_l \\times SO(3)_T \\times SO(1,1)_L \\times U(1)_{\\text{twist}} & \\hbox{Multiplicity}&\\hbox{L/R}\\cr \\hline ({\\bf 1},{\\bf 2})_{1,0} + c.c.",
"& h^{0,0}(\\widehat{C}) & \\hbox{R}\\cr ({\\bf 1},{\\bf 2})_{1,2} + c.c.",
"& h^{0,2}(\\widehat{C}) & \\hbox{R}\\cr ({\\bf 2},{\\bf 2})_{-1,1} + c.c.",
"& h^{0,1}(\\widehat{C}) & \\hbox{L}\\cr \\hline \\end{array} $ These states assemble into 2d $(0,4)$ multiplets as we shall now describe.",
"It is shown in appendix REF , that the mutiplicities appearing in (REF ) satisfy the relation $h^{1,1}(\\widehat{C}) - 2 h^{0,2}(\\widehat{C}) - 2 = 8 \\, {\\rm deg}({\\cal L}_{D}) \\ge 0 \\,.$ Here we recall briefly that ${\\cal L}_D=K_{B_2}^{-1}|_C$ is the line bundle describing the elliptic fibration of the surface $\\widehat{C}$ , which is also the duality line bundle associated to the $U(1)_D$ bonus symmetry of the D3-brane theory.",
"This line bundle is always non-negative because it is the restriction of the line bundle describing the elliptic fibration of the Calabi-Yau 3-fold restricted to $\\widehat{C}$ .",
"It is trivial if and only if $\\widehat{C}$ is globally a direct product $C \\times T^2$ .",
"In particular, $2 h^{0,2}(\\widehat{C}) + 1 \\le h^{1,1}(\\widehat{C}) -1$ , and thus $2 h^{0,2}(\\widehat{C}) + 1$ of the left- and right-moving scalar modes from the self-dual 2-form can combine into $2 h^{0,2}(\\widehat{C}) + 1$ real non-chiral massless scalar fields.",
"The remaining $h^{1,1}(\\widehat{C}) - 2 h^{0,2}(\\widehat{C}) - 2$ left-moving real scalar fields can be dualised into left-handed complex Weyl fermions.",
"In this way we arrive at the 2d (0,4) multiplet structure displayed in table REF .",
"Table: The (0,4)(0,4) mutiplets and their multiplicity in a compactification ofan M5-brane wrapping an elliptic surface inside of an elliptic Calabi-Yauthree-fold.In view of (REF ) the half-Fermi multiplets in the last line are absent precisely if $\\widehat{C} = C \\times T^2$ .",
"These Fermi multiplets are the M5-brane incarnation of the zero-modes due to 3–7 strings in the dual F-theory picture.",
"As promised, their counting as $8 \\, {\\rm deg}({\\cal L}_D) = 8 \\, c_1(B_2) \\cdot C$ is completely universal.",
"In particular it remains valid for arbitrary configurations of 7-branes, corresponding to fiber degenerations of the elliptic fibration in codimension one.",
"Once we work on the resolution of the elliptic fibration, which have been studied explicitly in [65], [66], [67], [68], [69], [70], these degenerations translate into fibral curves over the 7-brane locus, which in turn restrict to fibral curves of the M5-brane surface $\\widehat{C}$ intersecting the 7-brane locus.",
"The net number of Fermi multiplets as given by $8 \\, {\\rm deg}({\\cal L}_D)$ is already taking these fibral curves into account.",
"The numerical values of the Hodge numbers appearing in (REF ), (REF ) and (REF ) are computed via a Leray spectral sequence in appendix REF .",
"In particular this allows us to compare with the D3-spectrum in table REF .",
"If we include in the latter also the 3–7 string zero-modes, perfect agreement is found."
],
[
"M2-branes and $N=4$ SQM",
"Another dual description of our string theories is in terms of an M2-brane wrapping the same curve $C \\subset B_2$ as the D3-brane, as already summarized in section REF .",
"In this case the effective theory is 1d $N=4$ SQM.",
"To deduce the effective SQM we first note that the $SO(8)_R$ -symmetry splits into the rotation group $SO(4)_T$ in the four extended directions transverse to the brane, and another $SO(4)_R$ in the four directions internal to the Calabi-Yau normal to the brane.",
"Kählerity of $Y_3$ specifies a further reduction of the structure group of the normal bundle $N_{C/Y_3}$ from $SO(4)_R$ to $U(2)_R$ .",
"The decompositions of the relevant representations of $SO(8)_R$ are then $\\begin{aligned}SO(8)_R &\\quad \\rightarrow \\quad SO(4)_T \\times SU(2)_R \\times U(1)_R \\cr {\\bf 8_v} &\\quad \\rightarrow \\quad ({\\bf 2,2,1})_0 \\oplus ({\\bf 1,1,2})_1 \\oplus ({\\bf 1,1,2})_{-1} \\cr {\\bf 8_c} &\\quad \\rightarrow \\quad ({\\bf 1,2,2})_0 \\oplus ({\\bf 2,1,1})_1 \\oplus ({\\bf 2,1,1})_{-1} \\cr {\\bf 8_s} &\\quad \\rightarrow \\quad ({\\bf 1,2,1})_1 \\oplus ({\\bf 1,2,1})_{-1} \\oplus ({\\bf 2,1,2})_0 \\,.\\end{aligned}$ Combined with the reduction of $SO(1,2)_L \\rightarrow U(1)_L$ along the M2-brane this implies that the correct twist to create scalar supercharges along $C$ is $T_{\\text{twist}} = \\frac{1}{2}(T_L + T_R) \\,,$ which is normalised appropriately.",
"Indeed with this twist the supersymmetry parameters and fields of the 3d $N=8$ theory on the M2-brane reduce as SO(1,2)L SO(8)R SO(4)T SU(2)R U(1)twist : (2, 8s) (1,2,1)1 2 (1,2,1)0(1,2,1)-1 (2,1,2)12 (2,1,2)-12 : (1, 8v) (2,2,1)0 (1,1,2)12 (1,1,2)-12 : (2, 8c) (1,2,2)12 (1,2,2)-12 (2,1,1)1 (2,1,1)-1 2 (2,1,1)0 .",
"The first line, containing the supersymmetry parameters, implies that there are $N=4$ scalar supercharges in 1d.",
"The states and the zero-mode counting are summarised in table REF .",
"The cohomology groups are determined as follows.",
"By adjunction, $K_C = \\Lambda ^2 N_{C/Y_3}$ and hence by the usual rationale zero-modes of fields forming a representation ${\\bf 2}_{-1/2}$ and ${\\bf 2}_{1/2}$ of $SU(2)_R \\times U(1)_{\\rm twist}$ transform as elements of $H^0(C,N_{C/Y_3})$ and $H^1(C,N_{C/Y_3})$ , respectively.",
"Indeed, we assign twist charge $-1$ to $K_C$ and therefore $-1/2$ to $N_{C/Y_3}$ .",
"Note that the dimensions of these groups are in fact equal, as required by the multiplet structure of $N=4$ SQM.",
"This follows from the theorem on complex vector bundles (REF ), which in this instance implies that $N_{C/Y_3} = N_{C/Y_3}^\\vee \\otimes K_C \\,.$ Together with Serre duality this can be used to show that $H^1(C, N_{C/Y_3}) = [H^0(C,K_C \\otimes N_{C/Y_3}^\\vee )]^\\vee = [H^0(C,N_{C/Y_3})]^\\vee \\,.$ Table: The field content from a single M2-brane wrapping a curve inside aCalabi-Yau three-fold.",
"The representations are in terms of the remnant symmetry groups SO(4) T ×SU(2) R ×U(1) twist SO(4)_T \\times SU(2)_R \\times U(1)_{\\rm twist}.It is necessary for the sake of comparison to associate the bundles $N_{C/Y_3}$ to the bundles $N_{C/B_2}$ that appear in the spectrum computations from the D3 branes.",
"The long exact sequence in cohomology associated to the short exact sequence of normal bundles in appendix REF yields the result that $h^0(C, N_{C/Y_3}) = h^0(C, N_{C/B_2}) = g - 1 + \\text{deg}(\\mathcal {L}_D) \\,.$ The zero-modes of the states in table REF can then be counted explicity and compared to the D3-brane spectrum in table REF .",
"The fermionic content of the spectrum readily matches the total fermion content of the D3-brane analysis summarized in table REF .",
"To understand the relation of the bosonic spectra, we must carefully compare the multiplet structure of the 2d $(0,4)$ spectrum in table REF and the structure in table REF .",
"We denote the 1d $N=4$ SQM multiplets by $(n, N, N-n)$ with $n$ the number of bosonic degrees of freedom, $N$ the number of fermions and supersymmetries, and $N-n$ the number of auxiliary fields, as explained in appendix REF .",
"These are to be compared to the circle reduction of the 2d $(0,4)$ multiplets from the D3-brane analysis.",
"The 2d twisted hypermultiplet of table REF straightforwardly reduces to the $(4,4,0)$ hypermultiplet in 1d, and similarly the Fermi multiplets in both tables map onto each other.",
"More subtle is the $(2,4,2)$ multiplet.",
"The fermions in the $(2,4,2)$ multiplet correspond to the $\\psi $ and $\\tilde{\\psi }$ modes of table REF , and the two real scalar fields are identified with the modes $\\sigma $ and $\\tilde{\\sigma }$ .",
"Both of these types of states are counted by $h^{1}(C, N_{C/Y_3}) = h^{0}(C, K_C \\otimes \\mathcal {L}_D)\\,,$ which matches the counting in table REF .",
"The crucial point concerns the Wilson line modes $a, \\bar{a}$ of table REF .",
"As discusssed in section REF , after reduction to 1d these can be dualised into the two real auxiliary fields in the $(2,4,2)$ multiplet.",
"Automorphic duality can be applied to the Wilson line scalars because these enjoy a shift symmetry.",
"In this way we find perfect match between the D3-brane and the M2-brane spectrum."
],
[
"M5-branes and $(0,2)$ -Strings",
"Let us now consider an M5-brane along $\\mathbb {R}^{1,1} \\times \\widehat{C}$ with $\\widehat{C}$ an elliptic surface inside a Calabi-Yau four-fold $Y_4$ .",
"There is only a single transverse non-compact direction, which does not admit any continuous rotation group.",
"Thus the reduction of the R-symmetry takes the form $\\begin{aligned}SO(5)_R &\\rightarrow SO(4)_R \\rightarrow SU(2)_R \\times U(1)_R \\cr {\\bf 4} &\\rightarrow {\\bf 2}_0 \\oplus {\\bf 1}_1 \\oplus {\\bf 1}_{-1} \\cr {\\bf 5} &\\rightarrow {\\bf 2}_1 \\oplus {\\bf 2}_{-1} \\oplus {\\bf 1}_0 \\,,\\end{aligned}$ with $SU(2)_R \\times U(1)_R$ being the structure group associated with the normal bundle $N_{\\widehat{C}/Y_4}$ to $\\widehat{C}$ inside $Y_4$ .",
"We recall that the Lorentz symmetry along the M5-brane decomposes into $SO(1,5)_L \\rightarrow SO(1,1)_L \\times SU(2)_l \\times U(1)_l \\,.$ The partial topological twist $ T_{\\text{twist}} = T_l + T_R$ gives rise to exactly two positive chirality supercharges transforming as scalars along $\\widehat{C}$ as required for $(0,2)$ supersymmetry along $\\mathbb {R}^{1,1}$ .",
"With this twist the field content and the supercharges decompose as SU(2)l SU(2)R SO(1,1)L U(1)twist U(1)R , Q : (2,2)-1,0,0 (2,1)-1,1,1 (2,1)-1,-1,-1 (1,2)1,-1,0 (1,1)1,0,1 (1,1)1,-2,-1 (1,2)1,1,0 (1,1)1,2,1 (1,1)1,0,-1 : (1,2)0,1,1 (1,2)0,-1,-1 (1,1)0,0,0 H : (3,1)-2,0,0 (1,1)2,2,0 (1,1)2,0,0 (1,1)2,-2,0 (2,1)0,1,0 (2,1)0,-1,0 .",
"The representations of the various fields with respect to $SU(2)_l \\times SU(2)_R \\times U(1)_{\\text{twist}} $ determine the cohomology groups which count the massless modes as follows: States transforming as a ${\\bf 2}$ of $SU(2)_l$ correspond to one-forms on $\\widehat{C}$ , while $SU(2)_l$ singlets can a priori be zero-forms or two-forms.",
"Similarly, states transforming as a ${\\bf 2}$ of $SU(2)_R$ must take values in the normal bundle $N_{\\widehat{C}/Y_4}$ .",
"The remaining ambiguities are fixed by the $U(1)_{\\rm twist}$ charge.",
"A $(0,q)$ -form contributes $+q$ to the total twist charge.",
"Importantly, the normal bundle also carries twist charge because of (REF ).",
"This can also be seen as follows: First, by adjunction $K_{\\widehat{C}} = \\wedge ^2 N_{\\widehat{C}/Y_4} \\,,$ and combined with the fact that $H^0(\\widehat{C}, K_{\\widehat{C}}) = H^{2,0}(\\widehat{C})$ , we assign the following twist charges to $K_{\\widehat{C}}$ and $N_{\\widehat{C}/Y_4}$ $q^{\\rm twist}(K_{\\widehat{C}}) = -2, \\qquad \\quad q^{\\rm twist}(N_{\\widehat{C}/Y_4}) = -1.$ Thus, for instance the zero-mode of a state transforming as a $({\\bf 1},{\\bf 2 })_{-1}$ under $SU(2)_l \\times SU(2)_R \\times U(1)_{\\text{twist}}$ is associated with an element of $H^0(\\widehat{C},N_{\\widehat{C}/Y_4})$ , while massless states of the form $({\\bf 2},{\\bf 2 })_{0}$ are associated to $H^1(\\widehat{C},N_{\\widehat{C}/Y_4})$ .",
"This puts us in a position to assemble the zero-modes.",
"From the decomposition of the scalars we find $\\Phi = ({\\bf 1}, {\\bf 5}): \\qquad \\begin{array}{|c|c|c|}\\hline SU(2)_l \\times SU(2)_R \\times SO(1,1)_L \\times U(1)^{\\text{twist}} & \\hbox{Multiplicity} & \\hbox{L/R}\\cr \\hline ({\\bf 1},{\\bf 1})_{0,0} & 1 & \\hbox{L and R}\\cr ({\\bf 1},{\\bf 2})_{0,1} & h^{2}(\\widehat{C}, N_{\\widehat{C}/Y_4}) & \\hbox{L and R} \\cr ({\\bf 1},{\\bf 2})_{0,-1} & h^{0}(\\widehat{C}, N_{\\widehat{C}/Y_4})& \\hbox{L and R}\\cr \\hline \\end{array} $ The 3-form modes lead to the same types of left- and right-moving real scalar fields as for an M5-brane on a Calabi-Yau 3-fold, discussed in detail in section REF , $H_{\\mu \\nu \\rho }= ({\\bf \\overline{10}}, {\\bf 1}): \\qquad \\begin{array}{|c|c|c|c|}\\hline SU(2)_l \\times SU(2)_R \\times SO(1,1)_L \\times U(1)_{\\text{twist}} & \\hbox{Multiplicity}&\\hbox{L/R}\\cr \\hline ({\\bf 3},{\\bf 1})_{-2,0} & h^{1,1}(\\widehat{C})-1 & \\hbox{L}\\cr ({\\bf 1},{\\bf 1})_{2,2} & h^{0,2}(\\widehat{C}) & \\hbox{R} \\cr ({\\bf 1},{\\bf 1})_{2,0} & h^{0,0}(\\widehat{C})& \\hbox{R}\\cr ({\\bf 1},{\\bf 1})_{2,-2} & h^{2,0}(\\widehat{C})& \\hbox{R}\\cr \\hline \\end{array} $ The component $({\\bf 3,1})_{-2,0,0}$ of $H$ results in $h^{1,1}(\\widehat{C}) -1$ left-moving real scalar zero-modes.",
"Of these, $h^{0,2}(\\widehat{C}) + h^{2,0}(\\widehat{C}) + 1$ combine with the right-moving scalar zero-modes into non-chiral real scalars.",
"The remaining $h^{1,1}(\\widehat{C}) - 2 h^{0,2}(\\widehat{C}) -2 = 8 \\, {\\rm deg}({\\cal L}_D)$ left-moving real scalars can be dualised into left-moving complex fermion zero-modes.",
"The states $({\\bf 2,1})_{0,\\pm 1,0}$ from $H$ do not give rise to any zero-modes for the reasons discussed in section REF .",
"From the decomposition of the fermions we obtain $\\rho = ({\\bf \\bar{4}}, {\\bf 4}): \\qquad \\begin{array}{|c|c|c|c|}\\hline SU(2)_l \\times SU(2)_R \\times SO(1,1)_L \\times U(1)_{\\text{twist}} & \\hbox{Multiplicity}&\\hbox{L/R}\\cr \\hline \\end{array}({\\bf 2,2})_{-1,0} & h^{1}(\\widehat{C}, N_{\\widehat{C}/Y_4}) & \\hbox{L}\\cr ({\\bf 2,1})_{-1,1} + c.c.",
"& h^{0,1}(\\widehat{C}) & \\hbox{L}\\cr ({\\bf 1,2})_{1,1} + c.c.",
"& h^{2}(\\widehat{C}, N_{\\widehat{C}/Y_4}) & \\hbox{R}\\cr ({\\bf 1,1})_{1,0} + c.c.",
"& h^{0}(\\widehat{C}) & \\hbox{R}\\cr ({\\bf 1,1})_{1,2} + c.c.",
"& h^{0,2}(\\widehat{C}) & \\hbox{R}\\cr \\hline $ We interpret pairs of states with opposite twist charge in the decomposition of the $({{\\bf \\overline{4}}, {\\bf 4}})$ in (REF ) as complex conjugate Weyl fermions, the independent ones being as listed in lines $2-5$ of (REF ).",
"The state $({\\bf 2,2})_{-1,0}$ is special to the extent that it forms its own complex conjugate and must therefore be counted as real.",
"The spectrum now organizes into 2d $(0,2)$ multiplets with the multiplicities as shown in table REF .",
"Note in particular that we have assembled the $h^{1}(\\widehat{C}, N_{\\widehat{C}/Y_4})$ left-moving Majorana-Weyl fermions from (REF ) into $\\frac{1}{2} h^{1}(\\widehat{C}, N_{\\widehat{C}/Y_4}) $ Fermi multiplets, each of which contains one (complex) left-moving Weyl fermion.",
"This is possible because $ h^{1}(\\widehat{C}, N_{\\widehat{C}/Y_4}) $ is in fact an even integer.",
"To see this and to compare the spectrum to the results obtained from the D3-brane on $C$ , we use the Leray spectral sequence discussed in appendix REF together with the results of appendix REF .",
"This relates the cohomology groups on $\\widehat{C}$ to cohomology groups on $C$ .",
"In particular one can see from the second line of (REF ) that $h^{1}(\\widehat{C}, N_{\\widehat{C}/Y_4}) \\in 2 \\mathbb {Z}$ .",
"Altogether we find perfect match with the D3-brane spectrum of table REF once the 3–7 Fermi zero-modes are taken into consideration.",
"Table: The (0,2)(0,2) multiplets of the 2d theory arising when an M5-branewraps an elliptic surface inside an elliptic Calabi-Yau four-fold."
],
[
"M2-branes and $N=2$ SQM",
"In the dual M2-brane setup, compactification of the M2-brane on the curve $C \\subset Y_4$ makes it necessary to reduce the Lorentz symmetry as $SO(1,2)_L \\rightarrow U(1)_L$ , which $U(1)_L$ the holonomy group on $C$ .",
"The R-symmetry $SO(8)_R$ decomposes, as appropriate for a curve in a Calabi-Yau four-fold, as $\\begin{aligned}SO(8)_R &\\quad \\rightarrow \\quad SU(3)_R \\times U(1)_R \\times SO(2)_T \\cr {\\bf 8}_{\\bf v} & \\quad \\rightarrow \\quad {\\bf 3}_{2,0} \\oplus \\bar{\\bf 3}_{-2, 0} \\oplus {\\bf 1}_{0,2} \\oplus {\\bf 1}_{0, -2} \\cr {\\bf 8}_{\\bf s} & \\quad \\rightarrow \\quad {\\bf 3}_{-1, 1} \\oplus \\bar{\\bf 3}_{1,-1} \\oplus {\\bf 1}_{3, 1} \\oplus {\\bf 1}_{-3, -1} \\cr {\\bf 8}_{\\bf c} & \\quad \\rightarrow \\quad {\\bf 3}_{-1, -1} \\oplus \\bar{\\bf 3}_{1,1} \\oplus {\\bf 1}_{3, -1} \\oplus {\\bf 1}_{-3, 1} \\,.\\end{aligned}$ Geometrically we can think of $SU(3)_R \\times U(1)_R$ as the structure group of the normal bundle $N_{C/Y_4}$ of $C$ in $Y_4$ .",
"The supersymmetry parameters transform in $({\\bf 2}, {\\bf 8}_{\\bf s})$ under $SO(1,2)_L\\times SO(8)_R$ .",
"Twisting with $T_{\\rm twist} = {1\\over 6} (3 T_L + T_R) \\,,$ where $T$ denotes the generators of the respective abelian groups, the supersymmetry variation parameters contain the modes $\\epsilon \\quad \\supset \\quad {\\bf 1}_{-1; 0} \\oplus {\\bf 1}_{+1; 0} \\,,$ under $U(1)_L \\times U(1)_{\\rm twist}$ generating the $N=2$ supersymmetry.",
"The matter fields decompose and twist into the following fields $\\begin{aligned}SO(3)_L\\times SO(8)_R &\\quad \\rightarrow \\quad SU(3)_R \\times SO(2)_T \\times U(1)_{\\rm twist} \\cr \\rho : ({\\bf 2}, {\\bf 8}_{\\bf c}) & \\quad \\rightarrow \\quad {\\bf 3}_{-1;{1\\over 3}} \\oplus {\\bf 3}_{-1; - {2\\over 3}} \\oplus \\bar{\\bf 3}_{1;{2\\over 3}} \\oplus \\bar{\\bf 3}_{1; -{1\\over 3}} \\oplus {\\bf 1}_{-1; 1} \\oplus {\\bf 1}_{-1;0} \\oplus {\\bf 1}_{1; 0} \\oplus {\\bf 1}_{1; -1} \\cr \\phi : ({\\bf 1}, {\\bf 8}_{\\bf v}) & \\quad \\rightarrow \\quad {\\bf 3}_{ 0; {1\\over 3} } \\oplus \\bar{\\bf 3}_{0; -{1\\over 3} } \\oplus {\\bf 1}_{2; 0} \\oplus {\\bf 1}_{-2, 0} \\,.\\end{aligned}$ In our conventions, the canonical bundle $K_C$ carries twist charge $-1$ .",
"In view of the adjunction formula $K_C = \\Lambda ^3 N_{C/Y_4} \\,,$ it is hence appropriate to assign the normal bundle $N_{C/Y_4}$ twist charge $-1/3$ such that a state in $\\bar{\\bf 3}_{1; - {1\\over 3}}$ gives rise to $h^0(C, N_{C/Y_4})$ zero-modes.The identification of the ${\\bf \\overline{3}}$ , as opposed to the ${\\bf 3}$ , of $SU(3)_R$ with a section of the normal bundle with structure group $SU(3)_R$ is of course a pure matter of convention.",
"The choice here is made to keep our convention for the twist charge of $K_C$ .",
"Furthermore the zero-modes associated with the state $\\bar{\\bf 3}_{1; {2\\over 3}}$ are correctly counted by $h^1(C, N_{C/Y_4})$ .",
"Serre duality guarantees that this is the same number as the number of zero-modes of ${\\bf 3}_{-1; -{2\\over 3}}$ , which a priori is counted by $h^0(C, N^\\vee _{C/Y_4} \\otimes K_C)$ .",
"The short exact sequence (REF ) and the resulting relation (REF ) in turn imply that these states receive two types of contributions, one from $h^{1}(C, N_{C/B_3})$ , and another from $h^{0}(C, K_C \\otimes \\mathcal {L}_D)$ .",
"Then the twisted component fields fall into 1d 2B supermultiplets, and are summarized in table REF .",
"We recall the notation $(n,N,N-n)$ for an SQM multiplet (in a theory with $N$ supercharges) containing $n$ scalar, $N$ fermionic and $N-n$ auxiliary field real degrees of freedom.",
"Table: Spectrum of an M2-brane on C×ℝC\\times \\mathbb {R} with C⊂Y 4 C\\subset Y_4 in terms of the 1d N=2N=2 SQM.Using the decomposition of 2d $(0,2)$ chiral/Fermi multiplets into 1d 2B chiral/Fermi multiplets outlined in appendix REF , the above spectrum matches the one determined from the F-theory computation of D3-branes with duality twist in table REF .",
"Following the logic of section REF , circle reduction of the Wilson line scalars $a, \\bar{a}$ and fermion partners $\\psi , \\tilde{\\psi }$ a priori gives rise to $(2,2,0)$ chiral multiplets, which by the automorphic duality, discussed in appendix , are dual to the 2B Fermi multiplets counted by $h^{0}(C, K_C \\otimes \\mathcal {L}_D)$ in table REF .",
"In this map the Wilson line modes are mapped to the auxiliary fields of the Fermi multiplets."
],
[
"M2-branes and SQM from M-theory on CY5",
"We now come to the M-theory dual description of F-theory compactified on a Calabi-Yau $Y_5$ .",
"The latter gives rise to a 2d (0,2) theory [16], [18].",
"Since, in the F-theory description, the strings from D3-branes on a curve $C \\subset B_4$ are already spacetime-filling, there are no transverse non-compact dimensions along which we can T-dualise such as to end up with a description of the system in terms of an M5-brane.",
"The only M-theory dual configuration to a D3-brane in F-theory on $Y_5$ is hence an M2-brane, obtained by T-dualizing along the D3-brane worldvolume to a D2-brane in IIA, and subsequent uplift to M-theory.",
"An M2-brane wrapping a curve in the base of $Y_5$ results in an SQM with $N=2$ supercharges.",
"The first step of our analysis is to note that the R-symmetry is reduced as $\\begin{aligned}SO(8)_R \\quad &\\rightarrow \\quad SU(4)_R \\times U(1)_R \\cr {\\bf 8}_{\\bf v} \\quad &\\rightarrow \\quad {\\bf 4}_{1} \\oplus \\bar{\\bf 4}_{-1 } \\cr {\\bf 8}_{\\bf c} \\quad &\\rightarrow \\quad {\\bf 4}_{-1} \\oplus \\bar{\\bf 4}_{1 } \\cr {\\bf 8}_{\\bf s} \\quad &\\rightarrow \\quad {\\bf 6}_{0} \\oplus {\\bf 1}_{2} \\oplus {\\bf 1}_{-2} \\,.\\end{aligned}$ Twisting the $U(1)_L$ from $SO(1,2)_L \\rightarrow U(1)_L$ with $U(1)_R$ by $T_{\\rm twist} = {1\\over 4} (2T_{L} + T_R)$ results in the following spectrum of the M2-brane theory: $\\begin{aligned}SO(3)_L\\times SO(8)_R &\\quad \\rightarrow \\quad SU(4)_R \\times U(1)_{\\rm twist} \\cr \\epsilon : ({\\bf 2}, {\\bf 8}_{\\bf s}) &\\quad \\rightarrow \\quad {\\bf 6}_{1\\over 2} \\oplus {\\bf 6}_{- {1\\over 2}}\\oplus {\\bf 1}_{1} \\oplus {\\bf 1}_{0}\\oplus {\\bf 1}_{-1} \\oplus {\\bf 1}_{0}\\cr \\phi : ({\\bf 2}, {\\bf 8}_{\\bf c}) &\\quad \\rightarrow \\quad {\\bf 4}_{{1\\over 4}} \\oplus {\\bf 4}_{-{3\\over 4}} \\oplus \\bar{\\bf 4}_{-{1\\over 4}} \\oplus \\bar{\\bf 4}_{{3\\over 4}} \\cr \\rho : ({\\bf 1}, {\\bf 8}_{\\bf v}) &\\quad \\rightarrow \\quad {\\bf 4}_{1\\over 4} \\oplus \\bar{\\bf 4}_{-{1\\over 4}} \\,.\\end{aligned}$ The zero-modes can be determined by recalling our usual convention that we assign twist charge $-1$ to $K_C$ .",
"Adjunction, $K_C = \\wedge ^4 N_{C/Y_5} \\,,$ hence implies that the normal bundle $N_{C/Y_5}$ with structure group $SU(4)_R \\times U(1)_R$ must have twist charge $-1/4$ .",
"For instance the state ${\\bf \\overline{4}}_{-1/4}$ leads to $h^0(C, N_{C/Y_5})$ zero-modes in the SQM.",
"As for the zero-modes from ${\\bf 4}_{1/4}$ , we identify the ${\\bf 4}$ as the triple-antisymmetric tensor product of the ${\\bf \\bar{4}}$ and conclude that the zero-modes are counted by $h^1(C, \\wedge ^3 N_{C/Y_5})$ .",
"This is in perfect agreement with their twist charge - recalling that elements of $H^1(C) = H^0(C,K_C^{-1})$ carry an additional $q_{\\rm twist} = 1$ .",
"Then Serre duality and the vector bundle formula (REF ), $N_{C/Y_5} = \\wedge ^3 N_{C/Y_5}^\\vee \\otimes K_C \\,,$ guarantee that $h^1(C, \\wedge ^3 N_{C/Y_5}) = h^0(C, N_{C/Y_5})$ .",
"This number is in fact equal to $h^0(C, N_{C/B_4})$ as explained in appendix REF around equation (REF ).",
"Analogous reasoning for the remaining fields leads to the zero-modes in table REF , where we are invoking once more equation (REF ) to express the modes counted by $h^1(C, N_{C/Y_5})$ in terms of cohomology groups on the base only.",
"Comparing the modes here with those in table REF we observe the same number of fermionic degrees of freedom.",
"The only difference at first sight is from the $h^0(C, K_C \\otimes \\mathcal {L}_D)= g - 1 + c_1(B_4) \\cdot C$ Fermi multiplets.",
"By automorphic duality these get mapped to chiral multiplets associated to the Wilson line scalars $a, \\bar{a}$ and their fermionic partners, which completes the match of both approaches.",
"Table: Spectrum of an M2-brane on C×ℝC\\times \\mathbb {R} where C⊂Y 5 C\\subset Y_5, in terms of multiplets of the 1d N=2BN=2B SQM."
],
[
"Anomalies for Strings in Various Dimensions",
"The chiral string theories analyzed so far exhibit a rich pattern of gravitational and global anomalies.",
"In this section we investigate the cancellation of the field theoretic anomalies by anomaly inflow from the ambient space to the string.",
"After a general discussion of the anomaly inflow mechanism applied to the strings under consideration in section REF , we explicitly match the field theoretic anomalies with the anomaly inflow for the strings in $d=8,6, 4$ , and 2 dimensions.",
"As we will see, the structure of anomalies has interesting implications, in particular for the rather mysterious spectrum of 3–7 strings."
],
[
"Anomaly Inflow for Strings in $\\mathbb {R}^{1,d-1}$ ",
"The contribution of a positive chirality complex Weyl fermion in representation $R$ to the field theoretic gauge and gravitational anomalies on the string is characterized by the anomaly 4-form $I_{4,R} = \\hat{A}(T) \\, {\\rm tr}_R \\, e^{i F} |_{4-{\\rm form}}= - \\frac{1}{2}{\\rm tr}_R F^2 - \\frac{1}{24} p_1(T) \\,.$ In terms of this 4-form, the chiral anomaly contribution of a complex Weyl fermion takes the form ${\\cal A}|_{R} = - 2 \\pi \\int _{\\rm string} I_{2,R}^{(1)} \\,,$ with ${I}_{4,R} = d {I}^{(0)}_{3,R} \\,, \\qquad \\delta {I}^{(0)}_{3,R} = d{I}_{2,R}^{(1)} \\,.$ For consistency these field theoretic anomalies must be cancelled by the standard mechanism of anomaly inflow from the dimensions transverse to the string.",
"For a string along $\\mathbb {R}^{1,1} \\subset \\mathbb {R}^{1,d-1}$ with $d >2$ the anomaly inflow can be formulated as an inflow from the extended transverse spacetime dimensions.The case $d=2$ requires a special treatment and will be discussed separately in section REF .",
"In the sequel we briefly summarize this standard anomaly inflow to fix our conventions and notation in a simplified version, which is valid in this form except in $d=6$ , where the string is self-dual and special care must be applied.",
"Let $c_2$ denote the 2-form sourced electrically by the string in $\\mathbb {R}^{1,d-1}$ .",
"The field strength associated with its magnetically dual $(d-4)$ -form $c_{\\rm d-4}$ is subject to the modified Bianchi identity $d F_{\\rm d-3} = \\delta ^{\\rm (d-2)}(x_{\\rm T}) \\,.$ Here $\\delta ^{\\rm (d-2)}(x_{\\rm T})$ denotes the Poincaré dual to the string at $x_{\\rm T} = 0$ .",
"The magnetic field strength $F_{\\rm d-3}$ enjoys the Chern-Simons-type couplings $S_{\\rm CS} = - 2 \\pi \\int _{\\mathbb {R}^{1,d-1}} F_{\\rm d-3} \\wedge {\\cal I}_3^{(0)} \\,.$ The standard descent relations ${\\cal I}_4 = d {\\cal I}^{(0)}_3, \\qquad \\delta {\\cal I}^{(0)}_3 = d {\\cal I}_2^{(1)},$ with $d {\\cal I}_4 = 0$ , imply that in the presence of the source term (REF ), the CS action (REF ) picks up a non-trivial gauge variation of the form $\\delta S_{\\rm CS} &=& - 2 \\pi \\int _{\\mathbb {R}^{1,d-1}} F_{\\rm d-3} \\wedge \\delta {\\cal I}^{(0)}_3 = - 2 \\pi \\int _{\\mathbb {R}^{1,d-1}} d F_{\\rm d-3} \\wedge {\\cal I}^{(1)}_2 \\\\&=& - 2 \\pi \\int _{\\rm string} {\\cal I}^{(1)}_2.$ This classical gauge variance cancels the field theoretic anomalies along the string: ${\\cal I}_4 + \\sum _{R} \\chi _R \\, I_{4,R} = 0 \\,.$ Here $\\chi _R = n_{R,+} - n_{R,-}$ , and $n_{R,\\pm }$ denotes the number of complex Weyl fermions in representation $R$ , with the upper and lower sign referring to positive and negative chirality, respectively.",
"In our case the string arises from a D3-brane wrapping a curve on an internal compactification space.",
"The 2-form $c_2$ coupling electrically to the string in $\\mathbb {R}^{1,d-1}$ is then a linear combination of the modes obtained by reducing the Ramond-Ramond 4-form along $H^{1,1}_{B_n}$ with $B_n$ the base of the F-theory elliptic fibration and $n=5-d/2$ .",
"The Chern-Simons couplings (REF ) can be determined by dimensional reduction of the CS couplings for the 7-branes in the perturbative limit.",
"The latter are expressed in terms of the A-roof and Hirzebruch L-roof genera as [71] $\\begin{aligned}S_{\\rm D7} &= 2\\pi \\int _{\\rm D7} C_4 \\wedge \\frac{1}{2 }\\mathop {\\mathrm {Tr}}\\nolimits e^{i F} \\sqrt{\\widehat{A} (T)} \\cr S_{\\rm O7} &= - 8 \\pi \\int _{\\rm O7} C_4 \\wedge \\sqrt{\\widehat{L}\\left({1\\over 4} T\\right)} \\,,\\end{aligned}$ in units where the string scale $\\ell _s=1$ and $T$ denotes the respective tangent bundles.Note that the relative factor of $-4$ in the normalization of $S_{\\rm D7}$ and $S_{\\rm O7} $ is the one relevant for computations downstairs on the F-theory base as opposed to upstairs, prior to orientifolding, on the Calabi-Yau double cover.",
"Furthermore, the A-roof and L-roof genera are expanded into the Pontrjagin classes as $\\begin{aligned}\\sqrt{\\widehat{A} (T) }&= 1 - {1\\over 48} \\, p_1(R) + \\cdots \\cr \\sqrt{\\widehat{L}\\left({1\\over 4} T\\right)} &= 1+ {1\\over 96} \\, p_1 (R) + \\cdots \\, ,\\end{aligned}$ where $p_1 (R) = - {1\\over 2} \\mathop {\\mathrm {tr}}\\nolimits R\\wedge R \\,.$ Here and in the sequel, the symbol $\\mathop {\\mathrm {tr}}\\nolimits $ refers to the trace in the fundamental representation of the structure group of the relevant bundle.",
"As for the trace over the gauge bundle in $S_{\\rm D7}$ , $\\mathop {\\mathrm {Tr}}\\nolimits F^2$ is related to the trace in the fundamental via $\\frac{1}{2} {\\rm Tr} F^2 = {\\rm tr} F^2$ for the perturbative gauge group $SU(n)$ .",
"Its normalization for other gauge groups will be discussed in more detail in section REF .",
"To compute the CS couplings for a string on $\\mathbb {R}^{1,d-1}$ from compactification of F-theory with base $B_n$ , we introduce a basis $\\omega ^{\\rm (2n-2)}_\\alpha $ of ${(2n-2)}$ -forms on $B_n$ and expand $C_4 = c^\\alpha _{\\rm d-4} \\wedge \\omega ^{\\rm (2n-2)}_\\alpha \\,,$ which is possible for $d=8,6,4$ as considered in this section.",
"We then plug this ansatz into (REF ) and use the tadpole relation $\\sum _a { D}_a = 4 \\, {O7} = 8 \\, c_1(B_n)$ for the divisor classes $D_a$ wrapped by the 7-branes and the orientifold-plane class $O7$ in the perturbative Type IIB limit.",
"Summing up the gravitational couplings of all 7-branes and the O7-plane gives $\\begin{aligned}(S_{\\rm D7} + S_{\\rm O7})|_{\\rm grav} &= 2\\pi \\int _{\\mathbb {R}^{1,d-1}} c^\\alpha _{\\rm d-4} \\wedge p_1(R) \\left(\\int _{B_n} \\omega ^{\\rm (2n-2)}_\\alpha \\wedge (- \\frac{1}{48}\\sum _a D_a - \\frac{4}{96} O7 ) \\right) \\cr &= {2\\pi } \\int _{\\mathbb {R}^{1,d-1}} c^\\alpha _{\\rm d-4} \\wedge \\left(-\\frac{1}{4}\\right) p_1(R) \\left(\\int _{B_n} \\omega ^{\\rm (2n-2)}_\\alpha \\wedge c_1(B_n) \\right)\\,.\\end{aligned}$ A similar contribution describes the CS couplings involving the field strengths $F_a$ on each individual brane on divisor $D_a$ .",
"From the perspective of the string, the gauge symmetry along the 7-branes is related to the flavor symmetry for the 3–7 modes.",
"Let us now come back to the string from a D3-brane wrapped on the curve $C$ in $B_n$ .",
"The CS coupling of the form $c_{\\rm d-4}$ which couples magnetically to it is given by $S_{\\rm CS} = 2 \\pi \\int _{\\mathbb {R}^{1,d-1}} c_{\\rm d-4} \\wedge \\left( p_1(R) \\left(- \\frac{1}{4} \\, c_1(B_n) \\cdot C\\right) - \\sum _a \\frac{1}{4}\\, {\\rm Tr} F_a^2 \\Big (D_a \\cdot C \\Big ) \\right) \\, .$ This identifies ${\\cal I}_4 = p_1(R) \\left(- \\frac{1}{4} \\, c_1(B_n) \\cdot C\\right) - \\sum _a\\frac{1}{4} \\, {\\rm Tr} F_a^2 \\, \\Big (D_a \\cdot C \\Big )\\,.$ Finally, one can decompose $p_1(R)$ with respect to the tangent bundle along the string and the normal bundle in the transverse $(d-2)$ dimensions as $p_1(R) = p_1(T) + p_1(N)\\,.$ In the special case $d=6$ extra contributions to the anomaly polynomial (REF ) arise due to the self-dual nature of the string, which will modify the normal bundle anomaly inflow.",
"Furthermore one can complete (REF ) to include also inflow of R-symmetry anomalies as will be discussed in section REF ."
],
[
"Anomalies for Strings in 8d and 4d",
"These considerations can be most directly applied to strings in 8d and 4d.",
"We begin with F-theory compactified on an elliptic K3 with base $B_1=\\mathbb {P}^1$ .",
"As discussed in appendix , a wrapped D3-brane on $C = B_1$ gives rise to a string in $\\mathbb {R}^{1,7}$ , which couples magnetically to the Type IIB 4-form $C_4$ along the extended spacetime directions.",
"Hence in (REF ) no expansion in terms of internal forms is required (i.e.",
"$n=1)$ .",
"The gravitational part of the anomaly inflow (REF ) therefore provides a contribution ${\\cal I}_{4,\\rm grav + N} = \\Big ( - \\frac{1}{4} c_1(B_1) \\cdot C \\Big ) \\,\\left(p_1(T) + p_1(N)\\right) = - \\frac{1}{2} \\left(p_1(T) + p_1(N)\\right)\\,.$ In the last equation we are using $c_1(B_1) = c_1(\\mathbb {P}^1) = 2$ and $C=B_1$ .",
"From the spectrum in equation (REF ) one can compute the contribution of each complex Weyl fermion to the anomaly following (REF ), which leads to the gravitational anomaly ${I}_{4,{\\rm grav}} = - \\frac{1}{24}p_1(T)(4 - 16) = {1 \\over 2}p_1(T) \\,.$ This is accompanied by an anomaly in the normal bundle $N$ associated with the uncompactified directions transverse to the string.",
"The positive chirality fermions in the $(0,8)$ hypermultiplet transform in the ${\\bf 4}$ spinor representation of its structure group $SO(6)$ .",
"The induced normal bundle anomaly is therefore ${I}_{4,{\\rm N}} = - {1 \\over 2} \\text{tr}_{\\bf 4} F^2 = - {1 \\over 4}\\text{tr} F^2 = {1 \\over 2} p_1(N) \\,,$ where we have related the trace over the spinor representation to the trace over the fundamental representation using [72].",
"We can see that these two terms are cancelled by the anomaly inflow (REF ).",
"The cancellation of the flavor group anomalies will be discussed in more detail in section REF .",
"A similar pattern occurs for strings along $\\mathbb {R}^{1,3}$ obtained from F-theory compactified on an elliptically fibered four-fold $Y_4$ .",
"The spectrum of the string from the pure D3-brane sector is listed in table REF and is accompanied by a further $ 8 c_1(B_3) \\cdot C $ Fermi multiplets from the 3–7 string sector.",
"The gravitational anomaly contribution from these sectors is $\\begin{aligned}I_{4,{\\rm grav}} &= - {1 \\over 24}p_1(T)(2 c_1(B_3) \\cdot C)+ \\frac{1}{24}p_1(T)(8c_1(B_3) \\cdot C) \\cr &= {1 \\over 4}p_1(T) \\Big (c_1(B_3) \\cdot C\\Big ) \\,.\\end{aligned}$ Further, the $SO(2)_T$ normal bundle anomaly takes the form $\\begin{aligned}I_{4, {\\rm N}} = 2 (c_1(B_3) \\cdot C) \\left( - {1 \\over 2} \\text{tr}_{\\text{spin}}F^2 \\right)= {1 \\over 4} \\Big (c_1(B_3) \\cdot C\\Big ) p_1(N) \\,,\\end{aligned}$ where we have used that for $SO(2)$ $\\text{tr}_{\\text{spin}} F^2 = {1 \\over 8} \\text{tr} F^2 \\,.$ Thus the gravitational and normal bundle anomaly from the spectrum of the $(0,2)$ theory on the string are both cancelled by the anomaly inflow (REF )."
],
[
"Anomalies for Self-dual Strings in 6d",
"The anomaly of strings in 6d theories is on a slightly different footing compared to the other dimensions, as the string couples to a potential with self-dual field strength.",
"The anomaly of self-dual strings in 6d were initially discussed in [73].",
"More specifically for F-theory compactifications on elliptic Calabi-Yau three-folds this has been discussed in [74] (see also [75]) for setups where the D3 is wrapped on a $\\mathbb {P}^1$ with normal bundle degree $-n$ , $n\\ge 3$ .",
"We provide a generalization of this analysis for any curve $C$ in the base $B_2$ not contained in the discriminant of the fibration.",
"Part of the anomaly polynomial derives from the expression (REF ) present in all dimensions.",
"The structure group of the normal bundle $N$ to the string in $\\mathbb {R}^{1,5}$ is $SO(4)_T = SU(2)_{T,1} \\times SU(2)_{T,2}$ , for which one can write $p_1(N) = -\\frac{1}{2} {\\rm tr}F_{SO(4)}^2 = - \\mathop {\\mathrm {tr}}\\nolimits F_{T,1}^2 - \\mathop {\\mathrm {tr}}\\nolimits F_{T,2}^2 \\,.",
"$ There exist, however, two more contributions to the anomaly polynomial special to $d=6$ : The first is an extra contribution to the normal bundle anomaly which is rooted in the self-dual nature of the string and which is derived in [74], [75] to take the form $- \\frac{1}{2} \\, (C \\cdot C) \\, \\chi _4(N) = - \\frac{1}{2} \\, (C \\cdot C) \\, \\Big ( \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F_{T,2}^2 - \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F_{T,1}^2 \\Big )\\,.$ Furthermore, there is a contribution to the field theoretic $SU(2)_I$ R-symmetry, which can be inferred on purely field theoretic grounds from the Green-Schwarz terms of the 6d $(1,0)$ supergravity [54].",
"In our conventions and notation, this yields an extra term of the form $ \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F_I^2$ in ${\\cal I}_4$ .",
"Altogether the anomaly inflow polynomial is hence $\\begin{aligned}{\\cal I}_4 &= - \\frac{1}{4} \\Big ( c_1(B_2) \\cdot C \\Big ) p_1(T) - \\frac{1}{4} \\sum _a {\\rm Tr} F_a^2 [D_a] \\cdot C + \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F_I^2 \\cr & + \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F_{T,1}^2 \\left( \\frac{1}{2} C \\cdot C + \\frac{1}{2} c_1(B_2) \\cdot C \\right) +\\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F_{T,2}^2 \\left( - \\frac{1}{2} C \\cdot C + \\frac{1}{2} c_1(B_2) \\cdot C \\right) \\,.\\end{aligned}$ To compare this to the field theoretic anomaly induced by the spectrum in table REF , we first note that all bulk fields transform in real or pseudo-real representations and should therefore be counted as Majorana-Weyl fermions.",
"This gives rise to an additional factor of $\\frac{1}{2}$ compared to (REF ).",
"The field theoretic anomaly polynomial then takes the form $\\begin{aligned}I_4 &= - \\frac{1}{24} p_1(T) \\, \\Big ( (2 \\times 2 \\times \\frac{1}{2}) \\, ( g-1 + c_1(B_2) \\cdot C + (1-g)) - ( 1 \\times 1 \\times 1)\\, 8 c_1(B_2)\\cdot C \\Big ) \\\\&+ \\frac{1}{4} \\sum _a {\\rm Tr} F_a^2 \\, \\Big (D_a \\cdot C \\Big ) - \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F^2_I \\times \\frac{1}{2} \\times 2 \\\\& - \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F^2_{T,1} \\, \\Big (g-1 + c_1(B_2) \\cdot C\\Big ) \\times \\frac{1}{2} \\times 2- \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F^2_{T,2} \\times \\Big (\\frac{1}{2} \\times 2 -g \\times \\frac{1}{2} \\times 2 \\Big ) \\\\&=\\frac{1}{4} \\Big ({c_1(B_2) \\cdot C} \\Big ) \\, p_1(T) + \\frac{1}{4} \\sum _a {\\rm Tr} F_a^2 \\Big ( D_a \\cdot C \\Big ) - \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F^2_I \\\\& - \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F^2_{T,1} \\Big (g-1 + c_1(B_2) \\cdot C\\Big ) + \\frac{1}{2} \\mathop {\\mathrm {tr}}\\nolimits F^2_{T,2} \\Big (g -1\\Big ) \\,.\\end{aligned}$ This precisely cancels the anomaly inflow (REF ) because, by adjunction, $C \\cdot (C - c_1(B_2)) = 2g -2 \\,.$ Note that in the first line we have also included the contribution of the $ 8c_1(B_2) \\cdot C$ complex $(0,4)$ half-Fermi-multiplets from the 3–7 string sector."
],
[
"Anomalies in 2d",
"Since the string in two dimensions is spacetime-filling, there is no room for anomaly inflow from non-compact dimensions transverse to the string.",
"To study the cancellation of anomalies one has to consider instead the standard anomaly inflow to the string from the bulk of the 7-branes in 10 dimensions, where the string is viewed as a defect along the eight-dimensional 7-brane worldvolume.",
"Consider a 7-brane along $\\mathbb {R}^{1,1} \\times D_a$ with $D_a$ a complex 3-cycle on the base $B_4$ .",
"Viewed as a defect $\\mathbb {R}^{1,1} \\times C$ inside this worldvolume the string couples magnetically to the Type IIB 4-form $C_4$ inside $\\mathbb {R}^{1,1} \\times D_a$ such that the Bianchi identity gets modified to $d F_5 = \\delta ^{(6)}(x_T) \\,,$ with $\\delta ^{(6)}(x_T)$ the Poincaré dual to the string inside the 7-brane.",
"The Chern-Simons couplings of $C_4$ along each brane can be summed up, as before, to $S_{CS} = 2\\pi \\int _{\\mathbb {R}^{1,1} \\times B_4} C_4 \\wedge \\Big (-\\frac{1}{4} p_1(R) \\wedge c_1(B_4) - \\frac{1}{4} \\sum _a {\\rm Tr} F_a\\wedge F_a \\wedge D_a \\Big )\\,,$ which by descent contributes to the gauge variance as $\\delta S_{CS} = 2 \\pi \\int _{\\mathbb {R}^{1,1} \\times B_4} {{\\cal I}^{(1)}_{2,R}} \\wedge C \\wedge (-\\frac{1}{4} c_1(B_4)) + \\sum _a {{\\cal I}^{(1)}_{2,a}} \\wedge C \\wedge D_a\\,.$ Here ${{\\cal I}^{(1)}_{2,R}}$ and ${{\\cal I}^{(1)}_{2,a}}$ are related by descent to $-\\frac{1}{2} {\\rm tr} R \\wedge R$ and $-\\frac{1}{4} {\\rm Tr} F_a \\wedge F_a$ respectively.",
"Of course this leads to the same anomaly form (REF ) cancelling the gauge and gravitational anomalies on the string.",
"Consistently the gravitational contribution to the anomaly polynomial from the 3–3 and 3–7 sectors, which we read off from the spectrum in table REF , is $\\begin{aligned}I_{4,T} &= - {1 \\over 24} p_1(T) (2 c_1(B_4) \\cdot C - 8 c_1(B_4) \\cdot C)\\cr &= + {1 \\over 4} p_1(T) c_1(B_4) \\cdot C \\,.\\end{aligned}$ This time there is no normal bundle anomaly.",
"In fact, much more can be said: Since the D3-brane `strings' fill all of spacetime, the curve class $C$ is fixed by a tadpole condition [16], [18] as $C = \\frac{1}{24} c_4(Y_5)|_{\\rm base}$ .",
"Tadpole cancellation implies that the sum of all anomaly inflow terms vanishes.",
"Hence the total field theoretic anomalies from all sectors of the 2d theory must sum up to zero.",
"These sectors contain the D3-brane theory as discussed here, but furthermore the moduli sector from the 2d (0,2) supergravity, the actual gravitational sector and the 7-brane sector.",
"The anomalies due the moduli sector and the gravitational sector are derived in detail in the companion paper [45], and found to be $ I_{4,{\\rm SUGRA}} = - {1\\over 24} p_1(T) \\left( - \\tau (B_4) + \\chi _1(Y_5) - 2 \\chi _1(B_4) + 24\\right) \\,.$ Here $\\tau $ is the signature of the manifold and $\\chi _i$ are the arithmetic genera.",
"The anomaly contribution from the 7-branes has been derived in [16], [18]; in the simplest case of an elliptic fibration $Y_5$ which is a smooth Weierstrass model the 7-branes only contribute moduli fields which are already part of the anomaly (REF ).",
"In [45] we furthermore determine that when one adds the D3-brane contribution (REF ), augmented with $C$ such that the tadpole is solved, to the supergravity sector then the total anomaly can be written as $-{1\\over 24} p_1(T) \\left(-24\\chi _0(B_4) + 24\\right) \\,.$ In terms of Hodge numbers, $\\chi _0(B_4)$ is defined as $\\chi _0(B_4) = h^{0,0}(B_4) - h^{1,0}(B_4) + h^{2,0}(B_4) -h^{3,0}(B_4) +h^{0,4}(B_4) \\,,$ where the Kähler structure of $B_4$ forces all but the first and last terms to be zero, and $h^{0,0} = 1$ .",
"Further it is required that $B_4$ is compatible with being the base space of a Calabi-Yau elliptic fibration, $Y_5$ , and the existence of any $(0,4)$ -forms on $B_4$ would, under the uplift to $Y_5$ , ruin the Calabi-Yau property of the total space.",
"In conclusion, it is necessary for such a $B_4$ to have $\\chi _0(B_4) = 1$ and thus the total anomaly (REF ) always vanishes."
],
[
"Consequences for Flavor Symmetry from 3–7 Strings",
"To this juncture we have been principally concerned with the gravitational sectors of the anomaly inflow.",
"In this section we shall determine constraints on the flavour sector from the anomaly inflow.",
"The flavor group along the string corresponds to the gauge group on the 7-branes.",
"Recall first expression (REF ) describing the contribution to the anomaly polynomial from a complex positive chirality Weyl fermion in a representation $R$ .",
"This is to be contrasted with the relevant anomaly inflow term $ - \\sum _a {1 \\over 4} \\text{Tr} F_a^2 (D_a \\cdot C) \\,,$ where $\\text{Tr}$ is a normalised trace fixed by requiring that the smallest topological charge of the embedded $SU(2)$ instanton is 1.",
"Such a normalised trace can be related to the trace over the fundamental representation through $\\text{tr} F^2 = s_G \\text{Tr} F^2 \\,.$ The $s_G$ are numerical factors depending only on the gauge group $G$ and we refer in particular to appendix A of [54] for details.",
"For gauge group $G= SU(k)$ , $s_G=\\frac{1}{2}$ and hence the expression (REF ) agrees with the perturbatively expected form $ - \\sum _a {1 \\over 2} \\text{tr} F_a^2 (D_a \\cdot C)$ obtained by reducing the D7-brane CS action, which is typically written in terms of the trace over the fundamental representation.",
"For non-perturbative gauge groups, however, in particular the exceptional series, $s_G \\ne \\frac{1}{2}$ .",
"In this case a direct derivation via the CS action of perturbative Type IIB theory is of course not possible, but the crucial point is that the expression (REF ) appears in the anomaly polynomial of the 6d $N=(1,0)$ supergravity theory obtained by F-theory compactifications on a Calabi-Yau three-fold [76].",
"Anomaly cancellation in 6d alone fixes the normalization of the traces as above [77], [54].",
"This raises the question how the anomaly from the spectrum may be cancelled off by the anomaly inflow.",
"Let us consider only a single simple flavour group $G$ .",
"The total flavour contribution from the spectrum is of the form $ \\left(-{1 \\over 2}\\text{tr}_R F^2 \\right)(n_{R,+} - n_{R,-}) \\,,$ where $n_{R,+/-}$ is the number of positive/negative chirality complex Weyl fermions in representation $R$ .",
"If $R$ is a real representation of $G$ then this contribution comes in addition with an overall factor of $1/2$ to account for the Majorana-Weyl nature of the fermions.",
"Since the flavour symmetry has its origin in the gauge group on the 7-branes, it is only the 3–7 string modes that are charged under it.",
"For these $n_{R,+}=0$ , $n_{R,-} = C \\cdot D_G$ with $D_G$ the divisor wrapped by the 7-brane with gauge group $G$ .",
"The anomaly inflow which should cancel off the anomaly (REF ) is $ \\left(- {1 \\over 4} \\text{Tr} F^2 \\right) (D_G \\cdot C) \\,.",
"$ If the 3–7 strings transform in the fundamental representation $R$ , then it is immediately apparent that for such a cancellation it would be necessary to have $G$ such that either $s_G = 1/2$ and a complex fundmental representation, or $s_G = 1$ and a real fundamental representation.",
"One can see from the appendix of [54] that these are the cases where $G = SU(k)$ or $USp(k)$ , and, respectively, $G = SO(k)$ or $G_2$ .",
"For any other flavour group $G$ , and for any representation other than the fundamental, the contribution from (REF ) overshoots the inflow term (REF ).",
"From this we are lead to the inescapable conclusion that the anomalies explicitly forbid any other flavour groups than the above.",
"One can see this feature already when considering the anomaly of the E-string theory, in 6d, which was written down in [74].",
"The E-string theory has a global $E_8$ flavor symmetry at the infrared fixed-point, corresponding to strong coupling.",
"One can see this symmetry by considering the E-string theory as the theory on the worldvolume of an M2-brane interpolating between an M5-brane and an end-of-the-world M9-brane.",
"The endpoint on the M9-brane gives rise to the $E_8$ symmetry.",
"Such a setup has a perturbative description in type IIA, see [28] for more details, where the E-string theory in turn has a weakly coupled description.",
"In this understanding of the theory there is not any $E_8$ symmetry any longer in the UV, it is removed as the M9-brane becomes a stack of 8 D8-branes and an O8-plane in the limit.",
"Such a brane stack has an $SO(16)$ symmetry, which is then the flavor symmetry in the UV of the E-string theory.",
"Calculating the anomaly from the spectrum here yields a successful cancellation [74], as the 3–7 string multiplets now transform in a real representation of an $SO$ group, which as we discussed above, is sufficient for the cancellation between the inflow and the spectrum.",
"From these results we conjecture that whenever a D3-brane wraps a curve intersecting a 7-brane with gauge group $G$ which is not of $SU$ , $USp$ , $SO$ , or $G_2$ type, then $G$ will not survive intact as a flavour group into the UV description of the 2d theory."
],
[
"Acknowledgements",
"We thank Benjamin Assel, Chris Couzens, Michele del Zotto, Arthur Hebecker, Neil Lambert, Ling Lin, Dario Martelli, Christoph Mayrhofer, Eran Palti, Christian Reichelt, Fabian Ruehle, Jenny Wong, and Fengjun Xu for discussions.",
"SSN acknowledges support by the ERC Consolidator Grant 682608 “Higgs bundles: Supersymmetric Gauge Theories and Geometry (HIGGSBNDL)\".",
"The work of TW and CL was partially supported by DFG under Grant TR33 'The Dark Universe' and under GK 1940 'Particle Phyiscs Beyond the Standard Model'.",
"TW and CL thank the Fields Institute Toronto for hospitality during important stages of this work."
],
[
"2d $(0,2)$ and {{formula:9475d609-f04a-4134-91a8-7509441381c9}} Supersymmetry",
"In this appendix we summarize the structure of the supermultiplets in 2d $(0,2)$ and $(0,4)$ theories.",
"$(0,2)$ theories in two dimensions have an R-symmetry group $U(1)_R$ and the following multiplets [53]: Vector multiplet: contains a gauge field, $A_\\mu $ , and an adjoint valued left-moving fermion, $\\lambda _-$ .",
"Chiral multiplet: contains a complex scalar, $\\phi $ , and a right-moving fermion, $\\psi _+$ , both valued in a representation $R$ of the gauge group.",
"If $\\mathcal {R}[-]$ denotes the R-symmetry charge of the given field then the fields in the chiral multiplet must satisfy $\\mathcal {R}[\\psi _+] =\\mathcal {R}[\\phi ] - 1$ .",
"Fermi multiplet: contains a single left-moving fermion, $\\psi _-$ , transforming in a representation $R$ of $G$ .",
"The 2d $(0,4)$ multiplets can be built out of the $(0,2)$ multiplets.",
"Such theories have an $SO(4)_R \\cong SU(2)_R \\times SU(2)_I$ R-symmetry, which, in the IR SCFT is broken to a single $SU(2)_R$ R-symmetry [78].",
"The supercharges transform in the $({\\bf 2,2})_+$ representation, where the subscript denotes the Lorentz $SO(1,1)$ chirality.",
"The possible multiplets compatible with supersymmetry are then given as follows [79], [80]: Vector multiplet: contains a $(0,2)$ vector multiplet and an additional adjoint valued $(0,2)$ Fermi multiplet.",
"It then contains a gauge field, $A_\\mu $ , and a pair of left-moving complex fermions, $\\lambda ^a_-$ , transforming in the $({\\bf 2,2})_-$ .",
"Hypermultiplet: contains a pair of $(0,2)$ chiral multiplets in conjugate representations of $G$ .",
"The pair of complex scalars transforms as $({\\bf 2,1})$ under the R-symmetry, while the pair of right-moving fermions transforms in $({\\bf 1,2})$ .",
"Twisted hypermultiplet: much like the regular hypermultiplet it consists of a pair of $(0,2)$ chiral multiplets in conjugate representations, however the pair of complex scalars transforms in the $({\\bf 1,2})$ , and the right-moving fermions as $({\\bf 2,1})$ .",
"Fermi multiplet: contains a pair of $(0,2)$ Fermi multiplets in conjugate representations.",
"The left-handed fermions transform in the $({\\bf 1,1})$ representation of the R-symmetry.",
"$1/2$ –Fermi multiplet: a single $(0,2)$ Fermi multiplet which is a singlet under the $SO(4)_R$ symmetry is also consistent with the enhanced $(0,4)$ supersymmetry.",
"We shall call this a $1/2$ –Fermi to contrast it with the “true” $(0,4)$ Fermi above.",
"For the $(0,4)$ theories we are interested in how each multiplet contributes to the central charges of the theory.",
"The right-moving central charge and the gravitational anomaly are $c_R = 3 \\text{Tr}_{\\text{Weyl ferm.}}",
"\\gamma ^3 Q_R^2 \\quad \\,,\\, \\quad c_R -c_L = \\text{Tr}_{\\text{Weyl ferm.}}",
"\\gamma ^3 \\,,$ where $Q_R$ is the R-charge of the fermion, and $\\gamma ^3$ is the chirality.",
"Since the $(0,4)$ superconformal algebra fixes that the fermions of each (twisted) hypermultiplet have R-charge 1, and of each (half-)Fermi multiplet have R-charge 0, it is easily seen that the contribution from each multiplet is $c_R = 3 n_R \\quad \\,,\\, \\quad c_L = 2n_R + n_L \\,,$ where $n_{R/L}$ is the number of right/left Weyl fermions in the multiplet.",
"The contributions, excluding the vector multiplet, are $\\begin{array}{c|c|c|c}& \\text{(Twisted) Hyper} & \\text{Fermi} & 1/2\\text{--Fermi} \\cr \\hline (c_R, c_L) & (6,4) & (0,2) & (0,1)\\end{array} \\,.$ It is not possible to give such a conclusion for the $(0,2)$ theories, as the R-charge is not determined by the superconformal algebra, one must use the expressions (REF ) for the appropriate $Q_R$ in each case."
],
[
"1d SQM",
"We briefly summarize now the 1d SQM with extended supersymmetry and its multiplet structure, which we obtain from the M2-brane reduction and which will be matched with the dimensional reduction of the 2d $(0,2)$ theories.",
"A nice review of these 1d SQM theories can be found in [35].",
"An important fact to remember in 1d is that physical scalars and auxiliary scalars can be exchanged into each other, by the so-called automorphic duality [36] (for an in-depth exposition see [37]), which corresponds here to exchanging time-derivatives of physical bosons with auxiliary fields.",
"This map can be performed as long as the theory is free or the fields take values in a target whose metric does not depend on the dualized fields.",
"Multiplets are therefore denoted by $(n,N, N-n)$ where $n$ is the number of real bosonic degrees of freedom, $N$ the number of supersymmetries and $N-n$ the number of auxiliary fields.",
"The $N=2$ SQM has two types of representations, either $2A$ , which is the dimensional reduction of $N=(1,1)$ in 2d, or, more relevant for us, $2B$ , which is the reduction of 2d $N=(0,2)$ .",
"The $N=2B$ SQM has a $U(1)_R$ symmetry, which rotates the two supercharges into each other, and descends from the R-symmetry of the 2d theory.",
"There are chiral and Fermi multiplets of $2B$ SQM: $2B$ chiral multiplet $(2,2,0)$ : this has a complex scalar and complex fermion and no auxiliary field, $\\Phi = \\varphi + \\theta \\psi + {i\\over 2 }\\theta \\bar{\\theta }\\dot{\\varphi } \\,,$ with $\\begin{aligned}\\delta \\varphi &= i\\epsilon \\psi \\,, && \\delta \\bar{\\varphi }= i \\bar{\\epsilon }\\bar{\\psi }\\\\\\delta \\psi & = \\bar{\\epsilon }\\dot{\\varphi } \\,,&& \\delta \\bar{\\psi }= \\epsilon \\dot{\\bar{\\varphi }} \\,.\\end{aligned}$ The $2B$ chiral multiplet is related, upon reduction from 2d to 1d, to a 2d $(0,2)$ chiral multiplet with one complex scalar, $\\phi $ , and fermion, $\\psi _+$ , and supersymmetry transformations (see for conventions [16]) $\\begin{aligned}\\delta \\phi &= - \\sqrt{2} \\epsilon _- \\psi _+ \\,, & & \\delta \\bar{\\phi }= \\sqrt{2} \\bar{\\epsilon }_- \\bar{\\psi }_+ \\cr \\delta \\psi _+& = i \\sqrt{2} \\partial _+ \\phi \\bar{\\epsilon }_- \\,,& & \\delta \\bar{\\psi }_+ = - i \\sqrt{2} \\partial _+ \\bar{\\phi }\\epsilon _- \\,.\\end{aligned}$ Indeed, these transformations reduce (up to rescaling of the scalars by $i/\\sqrt{2}$ ) to the $2B$ multiplet.",
"In a similar fashion one can see that the 2d $(1,1)$ multiplets reduce to $2A$ multiplets, which have real scalars and auxiliary fields.",
"$2B$ Fermi multiplet $(0,2,2)$ : the lowest component is a complex fermion $\\rho $ , and includes a complex auxiliary scalar $h$ such that $\\Lambda = \\rho + \\theta h + {i\\over 2} \\theta \\bar{\\theta }\\dot{\\rho }\\,,$ with the supersymmetry transformations under the two supercharges $\\begin{aligned}\\delta \\rho &= i\\epsilon h \\,, && \\delta \\bar{\\rho }= -i \\bar{\\epsilon }\\bar{h} \\\\\\delta h& = \\bar{\\epsilon }\\dot{\\rho } \\,,&& \\delta \\bar{h}= -\\epsilon \\dot{\\bar{\\rho }} \\,.\\end{aligned}$ As in the chiral multiplet case, the Fermi multiplet of 2d $(0,2)$ dimensionally reduces to a 1d 2B Fermi multiplet."
],
[
"Conventions and Field Identifications",
"In this appendix we shall give some more details about the reduction of the $N=4$ SYM action and supersymmetry transformations in terms of the duality twisted fields and the scalar supersymmetries.",
"Let us begin by summarizing our conventions to help us to identify the field content of the D3-brane theory on $\\mathbb {R}^{1,1} \\times C$ .",
"For all three situations, $C \\subset Y_n$ with $n=3,4,5$ , we introduce the coordinates $x^0$ , $x^1$ along $\\mathbb {R}^{1,1}$ and $x^8$ , $x^9$ along $C$ , which we shall usually write in the combinations $z =\\frac{1}{2}(x^8 - i x^9)$ and $\\bar{z} = \\frac{1}{2}(x^8 + ix^9)$ .Note that this slightly unconventional definition of holomorphic versus anti-holomorphic coordinates matches our assignment that after the twist a field of topological twist charge $q_C = +1$ corresponds to a $(0,1)$ form.",
"We furthermore define the derivatives $\\partial _{\\pm }= \\partial _0 \\pm \\partial _1$ as well as $\\partial \\equiv \\partial _z =\\partial _8 + i \\partial _9$ , $\\bar{\\partial }\\equiv \\partial _{\\bar{z}} =\\partial _8 - i \\partial _9$ .",
"In these conventions the Kähler form on $C$ becomes $J = 2 dx^8 \\wedge dx^9 = - 4 i d z \\wedge d \\bar{z}, \\qquad \\quad \\ast _C 1= J, \\qquad \\ast _C J = 1 \\,.$ The fields appearing in the decomposition of the 4d vector field $A_\\mu $ as in (REF ), (REF ) and (REF ) and can then be identified, by studying their transformation properties under $SO(1,1)$ and $U(1)_C$ : The scalar fields $a$ and $\\bar{a}$ represent the components of the $U(1)$ gauge field along the curve $C$ , $a \\equiv a_{z} = A_8 + i A_9, \\qquad \\bar{a} \\equiv \\bar{a}_{\\bar{z}} = A_8 - i A_9\\,,$ where the subscripts $z$ , $\\bar{z}$ anticipate the form bi-degree $(1,0)$ and $(0,1)$ respectively of the fields on the curve $C$ , and the external components of the gauge field organize into $v_{\\pm } = A_0 \\pm A_1 \\,.$ In particular, $i F_{89} = i (\\partial _8 A_9 - \\partial _9 A_8) =\\frac{1}{2} \\left(\\bar{\\partial }a_{ z} - \\partial \\bar{a}_{\\bar{z}} \\right) \\,.$ The treatment of the normal space to the D3 brane differs for compactifications on $Y_3$ , $Y_4$ and $Y_5$ :"
],
[
"D3 on $Y_3$",
"On the normal space to the D3-brane we introduce coordinates $x^2,\\ldots ,x^5$ for the external normal space $\\mathbb {R}^4 \\subset \\mathbb {R}^{1,5}$ as well as local coordinates $x^6,x^7$ for the normal space of $C$ inside the base $B_2$ .",
"The scalar fields $\\phi $ transforming in the ${\\bf 6}$ of $SU(4)_R$ of the $N=4$ SYM on the D3-brane can then be identified with the normal fluctuations in the transverse directions and written in components as $\\phi _m$ , $m=2, \\ldots ,7$ .",
"The normal fluctuations of the D3-brane inside the Calabi-Yau are given by $\\sigma _{z} = \\phi _6 + i \\phi _7, \\qquad \\bar{\\sigma }_{\\bar{z}} = \\phi _6 - i \\phi _7,$ while the external normal fluctuations $\\phi _2, \\ldots \\phi _5$ organize as an $SO(4)_T$ bispinor $\\varphi _{A \\dot{B}}$ defined as $\\varphi _{A \\dot{B}} = \\left( \\begin{array}{cc} - \\phi _2 + i \\phi _3 & \\phi _4 - i \\phi _5 \\cr \\phi _4 + i \\phi _5 & \\phi _2 + i \\phi _3 \\end{array} \\right).$ Finally, chiral $SO(4)_T$ spinors transforming as a $({\\bf 2,1})$ of $SO(4)_T \\simeq SU(2)_{T,1} \\times SU(2)_{T,2}$ are denoted as objects $\\psi _A$ , while anti-chiral $SO(4)_T$ spinors in the $({\\bf 1,2})$ are of the form $\\tau ^{\\dot{B}}$ .",
"We will encounter contractions among chiral spinors defined as $\\psi \\chi = \\psi ^A \\chi _A = \\chi \\psi $ (and similarly for anti-chiral spinors as $\\tau \\rho = \\tau _{\\dot{A}} \\rho ^{\\dot{A}} = \\rho \\tau $ ).Spinor indices are raised and lowered by $\\varepsilon ^{AB} = \\left(\\begin{array}{cc} 0 & 1 \\cr -1 & 0 \\end{array}\\right)=\\varepsilon ^{\\dot{A} \\dot{B}}$ and $\\varepsilon _{AB} = \\left(\\begin{array}{cc} 0 & -1 \\cr 1 & 0 \\end{array}\\right)=\\varepsilon _{\\dot{A} \\dot{B}}$ .",
"Let us denote by $x^6, x^7$ the two coordinates transverse to the D3-brane in $\\mathbb {R}^{1,3}$ and by $x^2,x^3,x^4,x^5$ the coordinates normal to $C$ inside $B_3$ .",
"In terms of the scalar field $\\phi _2, \\ldots \\phi _7$ in the ${\\bf 6}$ of $SU(4)_R$ the remaining scalar fields in the decomposition (REF ) are given by the combinations g = 6 + i 7, g = 6 - i 7 A = ( c 4 - i 5 2 + i 3 ), A = ( c -2 + i 3 4 + i 5 ).",
"The index $A$ refers to the ${\\bf 2}$ representation of $SU(2)_R$ , the non-abelian part of the structure group of the normal bundle $N_{C/B_3}$ .",
"Here the normal coordinates $x^2, \\ldots , x^7$ lie entirely inside the base $B_4$ .",
"We introduce the index $\\alpha $ and $\\dot{\\alpha }$ for the ${\\bf 3}$ and ${\\bf \\bar{3}}$ of the $SU(3)_R$ structure group of the normal bundle $N_{C/B_4}$ and define the fields $\\varphi _\\alpha = \\left(\\begin{array}{c} \\phi _2 - i \\phi _3 \\\\-\\phi _4 + i\\phi _5 \\\\\\phi _6 - i \\phi _7\\end{array} \\right)\\,, \\qquad \\bar{\\varphi }^{\\dot{\\alpha }} = \\left(\\begin{array}{c} \\phi _2 + i \\phi _3 \\\\-\\phi _4- i \\phi _5 \\\\\\phi _6 + i \\phi _7\\end{array} \\right) \\,.$ In performing the reduction we follow closely the conventions specified in appendix A of [16], to which we refer for more details."
],
[
"Action and Supersymmetry",
"We now verify that the dimensionally reduced effective action for the D3-brane on $\\mathbb {R}^{1,1} \\times C$ is indeed invariant under the twisted supersymmetry variations stated in the main text.",
"For brevity we describe this here explicitly only for the (0,4) theory obtained for a curve $C \\subset Y_3$ .",
"The action of $N=4$ SYM has four components, covering the gauge, topological, fermion, and scalar sectors.",
"We shall consider each of these terms separately, and fix the relative factors between the four terms by requiring that the action be invariant under the twisted supersymmetry.",
"The gauge and topological actions can be rewritten in terms of the twisted fields as Lg = F F = -12 (a - a)2 - 12 (- v+ - + v-)2 - - a + a - - a + a - v+ v- - v- v+ + v+ - a + v+ - a + v- + a + v- + a , Ltop = F F = 3i2 ( v+ v- - v- v+ + - a + a - - a + a + v+ - a - v+ - a + a - v+ - a - v+ - v- + a + v- + a - a + v- + a + v- ) .",
"They are agnostic towards the R-symmetry part of the $N=4$ theory as they depend only on the 4d gauge field, which is a singlet under the $SU(4)_R$ symmetry.",
"These terms and fields will then be the same regardless of the dimension of the Kähler base in which the wrapped curve is located.",
"The fermion and scalar terms will depend on the dimension of the compactification, and here is where we shall focus on the $(0,4)$ content from the Calabi-Yau three-fold compactification.",
"Now that the coupling $\\tau $ varies over the curve $C$ there are fields in the decomposition of the scalars and fermions which transform non-trivially under the $U(1)_D$ duality group.",
"Any derivatives along $C$ of these fields are, following [7], promoted to $U(1)_D$ covariant derivatives, which were described in (REF ).",
"The scalar and fermion actions from the $N=4$ Lagrangian for compactification on Calabi-Yau three-fold are then Lsc =i i = - - + - - + + A A + A A + + A B - AB - A B A B , Lf = = - + - + + A - - - + - - - A+ + + - + - + -+ + - + - + - - - A + - +- + + + A - + + - + - + - - -+ - + -+ + - + - .",
"The total action for the twisted theory on $\\mathbb {R}^{1,1} \\times C$ can be written as $\\int _{\\mathbb {R}^{1,1} \\times C} {d^4 x \\sqrt{|g|}} \\left( c_1 \\, \\tau _2 \\, {\\cal L}_{\\rm g} + c_2 \\, \\tau _1 \\, {\\cal L}_{\\rm top} + c_3\\, {\\cal L}_{\\rm sc} + c_4 \\, {\\cal L}_{\\rm f} \\right) \\,,$ where the constants will be fixed, up to an overall factor, by twisted supersymmetry invariance."
],
[
"Supersymmetry",
"The supersymmetry variations of the $N=4$ SYM theory can be reduced to variations with respect to the surviving supersymmetries and written in terms of the twisted fields.",
"The variation of the bosonic fields becomes 2 a = 2 i - + , 2 a = 2 i - + , = - 2 i - + , = 2 i - + , A B = - 2 i (-A +B + -A +B) , 2 v- = 2 i (- - + - -) , v+ = 0 .",
"For the fermionic variations one finds + = - 2 (- + a + v+ ) + - + , + = - 2 (+ a - v+) + - + , +B = - - A + A B , +B = + - A + A B , -B = - A A A B , -B = - A A A B , - = - - (2 F01 + FA) - - C A , - = - - (2 F01 - FA )+ - C A , where $F_{01} = \\frac{1}{2} \\left( \\partial _- v_+ - \\partial _+ v_-\\right)\\,, \\qquad {\\cal F}_{\\cal A} = \\frac{1}{2} \\sqrt{\\tau _2} (\\bar{\\partial }a- \\partial \\bar{a}) \\,.$ It can be verified explicity that the action (REF ) is indeed invariant under these supersymmetry variations, without use of equations of motion and as long as one ignores boundary terms, if we fix $4 i c_3 - 2 c_4 = 0 \\,, \\quad c_4 - 2 i c_1 = 0 \\,, \\quad 3 c_2 - 2c_1 = 0 \\,.$ It is vital for supersymmetry that the coupling $\\tau $ varies holomorphically along $C$ , $\\bar{\\partial }\\tau = 0 \\,,$ as this is required for a cancellation of the variations between the gauge and the topological sectors.",
"The topological terms would usually be annihilated directly by the SUSY variation, however, there are terms remaining in this variation that are proportional to the derivatives of $\\tau _1$ .",
"Holomorphicity of $\\tau $ allows one to relate $ \\partial \\tau _1 = i \\partial \\tau _2 \\quad \\,, \\quad \\overline{\\partial }\\tau _1 = -i \\overline{\\partial }\\tau _2 \\,,$ and one can then notice that these variations are cancelled by the variations proportional to the derivatives of $\\tau _2$ from the gauge action.",
"Furthermore, the boundary terms are expected to cancel against the variation of the defect action describing the 7-brane insertions along $C$ .",
"This would be in analogy to a Euclidean D3-brane wrapping a Kähler surface as was studied in [7].",
"It is observed that the righthand side of (REF ) has a definite $U(1)_D$ charge, as it is composed to a fermion whose $U(1)_D$ charged is fixed by the well-definedness of the 4d fermions with respect to the $U(1)_D$ symmetry.",
"The lefthandside is then an object of the same $U(1)_D$ charge, and as the variation is constructed to be a singlet under $U(1)_D$ it can be determined that the objects $ \\sqrt{\\tau _2} a$ and $ \\sqrt{\\tau _2}\\bar{a}$ have unambiguous $U(1)_D$ charge, despite their origin in the 4d vector field, which has no definite charge.",
"The first two and the last equations vanish identically once the BPS equations $\\partial _+ v_- - \\partial _- v_+ = 0$ and $\\bar{\\partial }a - \\partial \\bar{a} =0$ as well as $\\partial v_+ = \\partial _+ a$ and $\\partial _+ \\bar{a} = \\bar{\\partial }v_+$ are enforced.",
"The third equation involving $v_-$ does not vanish identically by means of the BPS equations.",
"This is consistent with the chiral $(0,4)$ nature of the supersymmetry and in particular the fact that $v_-$ does not appear independently in the BPS equations."
],
[
"Invariants of Elliptic Surfaces",
"In this appendix we collect some results on the topological properties of elliptic surfaces, following [81], [82], that will be useful in the computation of the zero-modes for the D3 and M5 compactifications.",
"A Weierstrass fibration, $S$ , over a projective curve $C$ can be defined by the triple $(\\mathcal {L}, f, g)$ , where $\\mathcal {L}$ is a line bundle on $C$ and $f$ ($g$ ) is a section of $\\mathcal {L}^4$ ($\\mathcal {L}^6$ ) such that $\\Delta = 4f^3 + 27g^2$ is a non-identically-zero section of $\\mathcal {L}^{12}$ .",
"The total space of the fibration is not required to be smooth but can have rational double points.",
"When the elliptic surface $S$ is chosen as the restriction of an elliptically fibered Calabi-Yau $n$ -fold $Y_n$ to a curve $C$ , with $\\mathcal {L} \\equiv {\\cal L}_D \\cong K_{B_{n-1}}^{-1}|_C$ , then we shall call the resulting surface $\\widehat{C}$ , as is frequently used in the main text.",
"However in this appendix we shall consider a general $\\mathcal {L}$ .",
"We shall first consider the Leray spectral sequence relating the vector-bundle valued cohomology groups on $S$ to those on $C$ .",
"For an elliptic surface the spectral sequence degenerates to $@R=10pt@M=4pt@H+=22pt{0 [r] & H^0(C,\\pi _\\star V) [r] &H^0(S, V) [r] &0`[rd]^<>(0.5){}`[l]`[dlll]`[d][dll] &\\\\& H^1(C,\\pi _\\star V) [r] &H^1(S, V) [r] &H^0(C, R^1 \\pi _\\star V)`[rd]^<>(0.5){}`[l]`[dlll]`[d][dll] &\\\\& 0 [r] &H^2(S, V) [r] &H^1(C, R^1 \\pi _\\star V) [r] &0,\\,}$ where $V$ is a vector bundle on $S$ .",
"In the cases of interest to us, the right derived images can be obtained with the help of the projection formula $R^q\\pi _\\star ( M \\otimes \\pi ^\\star {\\cal N} ) = R^q\\pi _\\star ( M) \\otimes {\\cal N},$ where ${\\cal N}$ is a vector bundle on $C$ .",
"In particular, for any pullback bundle we obtain $R^0\\pi _\\star ( \\pi ^\\star {\\cal N} ) &=& \\pi _\\star ( \\pi ^\\star {\\cal N} ) = {\\cal N}, \\\\R^1\\pi _\\star ( \\pi ^\\star {\\cal N} ) &=& R^1\\pi _\\star ( {\\cal O}_{S} ) \\otimes \\pi ^\\star {\\cal N} = {\\cal L}^\\vee \\otimes {\\cal N}.$ In the last equality we have used that $R^1\\pi _\\star ( {\\cal O}_{S}) ={\\cal L}^{\\vee }.$ For instance, the cohomology groups counting the massless spectrum on an M5-brane wrapping $\\widehat{C}$ , the elliptic surface over $C$ in $Y_n$ with $\\mathcal {L} = \\mathcal {L}_D \\cong K_{B_{n-1}}^{-1}|_C$ , will involve $V = N_{\\widehat{C}/Y_n} = \\pi ^*N_{C/B_{n-1}} \\, .$ As will be discussed in the next section, this particular bundle $V$ together with the fixed form of $\\mathcal {L}$ will allow the computation, via adjunction, of the dimensions of the cohomology groups explicitly as used in the main text in sections REF and REF .",
"Now let us summarise some of the results for a general elliptic surface, which follow from the above spectral sequence, and compute the numerical invariants of the surface.",
"The fibration $S$ has the form of a product of $C$ with a smooth elliptic curve if and only if $\\mathcal {L} \\cong \\mathcal {O}_C$ .",
"The global invariants, as just discussed, can be computed via the Leray spectral sequence for the projection $\\pi : S \\rightarrow C$ .",
"One finds that $q = h^{0,1}(S) = {\\left\\lbrace \\begin{array}{ll}g \\,\\,\\,\\,\\,\\qquad S \\text{ is not a product} \\cr g + 1 \\quad S \\text{ is a product}\\end{array}\\right.}",
"\\,,$ with $g$ the genus of $C$ .",
"Thus when $S$ carries a non-trivial elliptic fibration all one-forms on $S$ are pullbacks from $C$ .",
"Similarly one can compute the geometric genus $p_g = h^{0,2}(S) = {\\left\\lbrace \\begin{array}{ll}g + \\text{deg}(\\mathcal {L}) - 1 \\quad S \\text{ is not a product} \\cr g + \\text{deg}(\\mathcal {L}) \\,\\,\\,\\,\\qquad S \\text{ is a product}\\end{array}\\right.}",
"\\,.$ The Euler characteristic can be computed as the alternating sum of the above Hodge numbers and in both the trivial and non-trivial cases is $\\chi (S) = \\text{deg}(\\mathcal {L}) \\,.$ Using the Noether formula and that $K_S^2 = 0$ determines the Euler number $e(S) = 12\\text{deg}(\\mathcal {L}) \\,,$ which can be used to determine the final unknown entry in the Hodge diamond of $S$ : $h^{1,1}(S) = e(S) + 4h^{0,1} - 2h^{0,2} - 2 = {\\left\\lbrace \\begin{array}{ll}10 \\, \\text{deg}(\\mathcal {L}) + 2g \\,\\,\\,\\,\\,\\qquad S \\text{ is not a product} \\cr 10 \\, \\text{deg}(\\mathcal {L}) + 2g + 2 \\quad S \\text{ is a product}\\end{array}\\right.}",
"\\,.$ Finally we know that if $(\\mathcal {L}, f, g)$ forms Weierstrass data over a projective curve $C$ then $\\text{deg}(\\mathcal {L}) \\ge 0$ , since $\\mathcal {L}^{12}$ must have a section that is not identically zero for the discriminant $\\Delta $ to not be identically zero.",
"This can be used to make a statement purely about the cohomology on the base curve $C$ ; if $(\\mathcal {L},f, g)$ provides the Weierstrass data for a Weierstrass fibration over $C$ then $h^0(C, \\mathcal {L}^{-1}) = {\\left\\lbrace \\begin{array}{ll}0 \\quad S \\text{ is not a product} \\cr 1 \\quad S \\text{ is a product}\\end{array}\\right.}",
"\\,.$"
],
[
"Bundle Cohomology Computations",
"Consider $\\pi : Y_n \\rightarrow B_{n-1}$ an elliptically fibered Calabi-Yau $n$ -fold with curve $C \\subset B_{n-1}$ such that $\\widehat{C} = \\pi ^{-1}(C)$ is a non-trivially fibered elliptic surface.",
"As noted, in this case ${\\cal L} = {\\cal L}_D = K^{-1}_{B_{n-1}}|_C$ .",
"To determine the number of zero-modes in each of the compactifications in the main text we need to be able to relate the following Hodge numbers $\\begin{aligned} D3 \\,&:\\, & &h^0(C, K_C \\otimes \\mathcal {L}_D) \\,,\\quad h^0(C, N_{C/B_{n-1}})\\,,\\quad h^1(C, N_{C/B_{n-1}}) \\cr M2 \\,&:\\, & &h^0(C, N_{C/Y_{n}}) \\,,\\quad h^1(C, N_{C/Y_n}) \\cr M5 \\,&:\\, & &h^0(\\widehat{C}, N_{\\widehat{C}/Y_n}) \\,,\\quad h^1(\\widehat{C},N_{\\widehat{C}/Y_n}) \\,,\\quad h^2(\\widehat{C}, N_{\\widehat{C}/Y_n}) \\,.\\end{aligned}$ One can see that all of the multiplets in tbl:CY3fcmultiplets,eqn:CY4D3zm,tab:CY5D3zm,M5CY3multiplets,tbl:M2CY3fc2,M5CY4multiplets,CY4M2Spec,tab:CY5M2zm are counted by one of these Hodge numbers (or just by a Hodge number of $C$ or $\\widehat{C}$ ).",
"It may be necessary to utilize Serre duality or other bundle isomorphisms to write it in the above form.",
"If $V$ is an arbitrary vector bundle over $C$ then the Hirzebruch-Riemann-Roch theorem reads $\\chi (C, V) = \\int _C \\big (c_1(V) - \\frac{1}{2}\\text{rk}(V)c_1(K_C)\\big ) \\,.$ Since $\\mathcal {L}_D$ is an ample line bundle $h^1(C, K_C \\otimes \\mathcal {L}_D) = h^0(C, \\mathcal {L}_D^\\vee ) = 0 \\,,$ by the Kodaira vanishing theorem, and thus $h^0(C, K_C \\otimes \\mathcal {L}_D) = g - 1 + \\text{deg}(\\mathcal {L}_D) \\,.$ Furthermore the adjunction theorem in this notation reads that $\\text{det}(N_{C/B_{n-1}}) = K_C \\otimes \\mathcal {L}_D \\,.$ For $Y_3$ the determinant acts trivially and thus the dimensions of $H^i(C,N_{C/B_{2}})$ translate into the expressions in (REF ).",
"Recalling that the first Chern class is blind to the determinant, and that ${\\rm rk}(N_{C/B_{n-1}}) = n-2$ one can compute the Euler characteristic to determine that $h^0(C, N_{C/B_{n-1}}) - h^1(C, N_{C/B_{n-1}}) = \\text{deg}(\\mathcal {L}_D) + (g - 1)(4 -n) \\,.$ As such one can write all of the degrees of freedom in terms of the three quantities $\\begin{aligned}&g \\,, & &\\text{deg}(\\mathcal {L}_D) \\,, & &h^0(C, N_{C/B_{n-1}}) \\,.\\end{aligned}$ For the M2-brane quantities one can use adjunction on $Y_n$ , $K_{B_{n-1}} =N_{B_{n-1}/Y_n}$ , to write the short exact sequence for normal bundles as $ 0 \\rightarrow N_{C/B_{n-1}} \\rightarrow N_{C/Y_{n}} \\rightarrow \\mathcal {L}_D^\\vee \\rightarrow 0 \\,.$ The associated long exact sequence in cohomology yields the equivalences $ \\begin{aligned}h^0(C, N_{C/Y_n}) &= h^0(C, N_{C/B_{n-1}}) \\cr h^1(C, N_{C/Y_n}) &= h^1(C, N_{C/B_{n-1}}) + h^0(C, K_C \\otimes \\mathcal {L}_D) \\,.\\end{aligned}$ The last expression can be simplified using (REF ) to $h^1(C, N_{C/Y_n}) = h^0(C, N_{C/B_{n-1}}) + (g - 1)(n - 3) \\,.$ For the M5-brane the bundle of interest is $N_{\\widehat{C}/Y_{n}} = \\pi ^*N_{C/B_{n-1}} \\,,$ whenceforth one can use the Leray spectral sequence for elliptic surfaces (REF ) to determine that $\\begin{aligned}h^0(\\widehat{C}, N_{\\widehat{C}/Y_n}) &= h^0(C, N_{C/B_{n-1}}) \\cr h^1(\\widehat{C}, N_{\\widehat{C}/Y_n}) &= h^0(C, N_{C/B_{n-1}} \\otimes \\mathcal {L}_D^\\vee ) + h^1(C, N_{C/B_{n-1}}) \\cr h^2(\\widehat{C}, N_{\\widehat{C}/Y_n}) &= h^1(C, N_{C/B_{n-1}} \\otimes \\mathcal {L}_D^\\vee ) \\,.\\end{aligned}$ At this point we note that the only situation where these particular cohomology groups are of interest is when $Y_4$ is a four-fold; as such we may restrict now to this case without consequence.",
"For a general complex vector bundle $\\mathcal {G}$ of rank $r$ it follows [83] that $\\wedge ^n \\mathcal {G} = \\wedge ^{r - n} \\mathcal {G}^\\vee \\otimes \\wedge ^r\\mathcal {G} \\,.$ By combining this relation with Serre duality one can compute that for Calabi-Yau four-folds there exists the relationship $h^i(C, N_{C/B_3} \\otimes \\mathcal {L}_D^\\vee ) = h^{1-i}(C, N_{C/B_3}) \\,.$ One can thus rewrite the relations (REF ) as $\\begin{aligned}h^0(\\widehat{C}, N_{\\widehat{C}/Y_4}) &= h^0(C, N_{C/B_3}) \\cr h^1(\\widehat{C}, N_{\\widehat{C}/Y_4}) &= 2(h^0(C, N_{C/B_3}) -\\text{deg}(\\mathcal {L}_D)) \\cr h^2(\\widehat{C}, N_{\\widehat{C}/Y_4}) &= h^0(C, N_{C/B_3}) \\,.\\end{aligned}$"
],
[
"Strings in 8d...",
"This section is devoted to strings in 8d compactifications of F-theory on an elliptically fibered Calabi-Yau twofold, i.e.",
"an elliptic K3 surface with base isomorphic to a smooth rational curve.",
"An M5-brane wrapping a K3 surface is well-known to be the heterotic string [84] and the low-energy 2d theory on the worldvolume of the M5-brane has $(0,8)$ supersymmetry.",
"In contradistinction to the situation studied in the previous section, it is of course not necessary to twist here as the elliptic surface wrapped by the M5-brane is itself Calabi-Yau.",
"This $(0,8)$ theory has an F-theory description in terms of a D3-brane wrapping $C$ , the smooth rational curve forming the base of the elliptic fibration.",
"As we will see, the treatment of this theory deviates in an interesting way from its counterparts investigated in section .",
"Finally we will consider the $N=8$ SQM that arises from an M2-brane on the base $C$ of $K3$ ."
],
[
"...from D3-branes",
"If one tries to compactify the D3-brane theory, $N=4$ SYM theory, on a 2-sphere one might be be tempted to conclude that there is no obvious mechanism with which to twist the theory, as the entire $SO(6)_R$ symmetry group is used up to describe rotations in the transverse non-compact space to the brane.",
"Naively one would be lead to believe that in this setup all supersymmetry on the worldvolume of the D3-brane is broken.",
"The resolution to this puzzle is that in F-theory on $K3$ the D3-brane intersects 24 7-branes at various points in its worldvolume, and the resulting variation of the axio-dilaton provides an additional duality twist which ensures that 8 supercharges are preserved.",
"Indeed, the topological twist giving rise to chiral supersymmetry along the string twists the $U(1)_C$ holonomy group along curve $C$ by the $N=4$ bonus symmetry $U(1)_D$ such that $T_{\\rm twist} = {1\\over 2} \\left(T_{C} - T_{D} \\right) \\,.$ Since the D3-brane wraps the entire base $C = B_1 = \\mathbb {P}^1$ , there is no $U(1)_R$ symmetry available which would be associated with the transverse directions of $C$ within the base.",
"The twist (REF ) can hence be viewed as the difference of the would-be individual twists of $U(1)_C$ and $U(1)_D$ by $U(1)_R$ , which is independent of $U(1)_R$ .",
"This is the only option in absence of a well-defined $U(1)_R$ .",
"The appearance of a single twist is clear also from the M5-brane point of view, where the M5 wraps the full K3 and thus, as pointed out already, does not require any topological twist at all.",
"The twist (REF ) is the remnant of the non-triviality of the elliptic fibration in the M5-brane picture.",
"We obtain the following decomposition of the supercharges under the symmetry groups $\\begin{aligned}SO(1,3)_L \\times SU(4)_R \\times U(1)_D \\quad &\\rightarrow \\quad SO(1,1) \\times U(1)_C \\times U(1)_D \\cr ({\\bf 2}, {\\bf 1}, \\bar{\\bf 4})_1 \\quad & \\rightarrow \\quad \\bar{\\bf 4}_{1, 1, 1} \\oplus \\bar{\\bf 4}_{-1, -1, 1} \\cr ({\\bf 1}, {\\bf 2}, {\\bf 4})_{-1} \\quad & \\rightarrow \\quad {\\bf 4}_{1, -1, -1} \\oplus {\\bf 4}_{-1, 1, -1} \\,.\\end{aligned}$ The twist (REF ) results in two positive chirality scalar supercharges which transform as $SU(4)_R \\times SO(1,1)_L \\times U(1)_{\\rm twist}: \\qquad {\\bf 4}_{1, 0} \\oplus \\bar{\\bf 4}_{1, 0} \\,.$ The resulting theory is an $N=(0,8)$ supersymmetric theory with the following massless spectrum: The scalars remain in a ${\\bf 6}$ of the R-symmetry $SU(4)_R$ and are trivially charged under $U(1)_{\\rm twist}$ , whereby their multiplicity is counted by $h^{0}(C)=1$ .",
"The fermions decompose as $\\begin{aligned}SO(1,3)_L \\times SU(4)_R \\times U(1)_D \\quad &\\rightarrow \\quad SO(1,1) \\times U(1)_{\\rm twist} \\cr \\Psi : \\quad ({\\bf 2}, {\\bf 1}, {\\bf 4})_{1} \\quad & \\rightarrow \\quad {\\bf 4}_{1, 0} \\oplus {\\bf 4}_{-1, -1} \\cr \\widetilde{\\Psi }:\\quad ({\\bf 1}, {\\bf 2}, \\bar{\\bf 4})_{-1} \\quad & \\rightarrow \\quad \\bar{\\bf 4}_{1, 0} \\oplus \\bar{\\bf 4}_{-1, 1} \\,.",
"\\cr \\end{aligned}$ These are counted by $h^0(C) = 1$ for $q_{\\rm twist} =0$ , and by $h^{0,1}(C)=0$ and $h^{1,0}(C) = 0$ for $q_{\\rm twist} = \\pm 1$ .",
"Finally, the gauge field decomposes as in (REF ).",
"This gives rise to scalar fields $a_z$ and $\\bar{a}_{\\bar{z}}$ of twist charge $q_{\\rm twist} = 0$ , i.e.",
"transforming as ${\\bf 1}_{0}$ , of multiplicity 1.",
"The twist charge can either be inferred directly from the supersymmetry variation, or, more heuristically perhaps, by noting that $U(1)_{\\rm twist}$ corresponds to the difference of the individual $U(1)^{\\rm twist}_{C}$ and $U(1)^{\\rm twist}_{D}$ which would arise in presence of a $U(1)_R$ symmetry on higher dimensional base spaces; since the Wilson line scalars $a_z$ and $\\bar{a}_{\\bar{z}}$ have equal charge under these two twisted $U(1)$ symmetries in the higher dimensional examples, this explains $q_{\\rm twist}=0$ .",
"The remaining gauge field components $v_{\\pm }$ have zero multiplicity, again by supersymmetry as in the lower-dimensional examples.",
"Finally, the 3–7 sector gives rise to the remaining $8 c_1(B_1) \\cdot C=16$ complex Weyl fermions of negative chirality, where we are using that $C=B_1 = \\mathbb {P}^1$ .",
"The complete spectrum is thus: $\\begin{array}{c|c|c}\\text{Bosons} & \\text{Fermions} & \\text{Zero-modes} \\cr \\hline {\\bf 6}_0 , \\ 2 \\times {\\bf 1}_{0} & {\\bf 4}_{1} \\oplus \\overline{\\bf 4}_{1} & h^{0}({C}) = 1 \\cr & {\\bf 4}_{1-} \\oplus \\overline{\\bf 4}_{-1} & h^{1}({C}) = 0 \\cr & {\\bf 1}_{-1} & 8 c_1(B_1) \\cdot C = 16\\end{array}$ The first row provides the field content of a 2d $(0,8)$ hypermultiplet and the 16 complex Weyl fermions from the 3–7 sector represent purely fermionic $N=(0,8)$ multiplets."
],
[
"...from M5-branes",
"When considering an M5-brane wrapping a K3 surface there are five transverse non-compact directions, which are rotated by the $SO(5)$ R-symmetry of the $(0,2)$ theory on the worldvolume of the M5.",
"It is only the 6d Lorentz symmetry along the M5-brane which decomposes since the 6d spacetime has the form $\\mathbb {R}^{1,1} \\times K3$ , $\\begin{aligned}SO(1,5)_L &\\ \\rightarrow \\ SO(1,1)_L \\times SU(2)_l \\cr {\\bf 4} &\\ \\rightarrow \\ {\\bf 2}_1 \\oplus 2 \\times {\\bf 1}_{-1} \\cr {\\bf 10} &\\ \\rightarrow \\ {\\bf 3}_2 \\oplus 3 \\times {\\bf 1}_{-2} \\oplus 2 \\times {\\bf 2}_0 \\,.\\end{aligned}$ The full field content of the abelian tensor multiplet can then be decomposed under this holonomy reduction, and one finds: $\\begin{aligned}SO(1,5) \\times Sp(4) &\\ \\rightarrow \\ SO(1,1)_L \\times SU(2)_l \\times Sp(4)\\cr H: ({\\bf \\overline{10}, 1}) &\\ \\rightarrow \\ ({\\bf 3, 1})_{-2} \\oplus 3 \\times ({\\bf 1,1})_2 \\oplus 2 \\times ({\\bf 2,1})_0 \\cr \\Phi : ({\\bf 1, 5}) &\\ \\rightarrow \\ ({\\bf 1, 5})_0 \\cr Q, \\rho : ({\\bf \\overline{4}, 4}) &\\ \\rightarrow \\ ({\\bf 2,4})_{-1} \\oplus 2 \\times ({\\bf 1,4})_1 \\,.\\end{aligned}$ The theory has eight right-moving supercharges which are scalars under the internal $SU(2)_l$ holonomy.",
"The counting of zero-modes is determined entirely by the internal $SU(2)_l$ representation under which the twisted fields transform: $\\begin{array}{c|c|c|c}SU(2)_l & \\text{Bosons} & \\text{Fermions} & \\text{Zero-modes} \\cr \\hline {\\bf 1} & 3 \\times ({\\bf 1,1})_2 \\,\\,,\\, ({\\bf 1, 5})_0 & 2 \\times ({\\bf 1,4})_1 & h^{0,0}(\\widehat{C}) = 1 \\cr {\\bf 2} & 2 \\times ({\\bf 2,1})_0 & ({\\bf 2,4})_{-1} & h^{1,0}(\\widehat{C})= 0 \\cr {\\bf 3} & ({\\bf 3,1})_{-2} & & h^{1,1}(\\widehat{C}) - 1 = 19\\end{array}$ Note that the three $({\\bf 1,1})_2$ scalar bosons arising from the decomposition of the self-dual two-form are purely right-moving because of the chirality of $B$ .",
"To complete the top row of (REF ) into a full $(0,8)$ hypermultiplet, where all of the eight real scalar degrees of freedom are both left- and right-moving, it is necessary to combine them with three of the left-moving bosonic degrees of freedom from the $({\\bf 3,1})_{-2}$ zero-modes.",
"In addition to the single $(0,8)$ hypermultiplet this leaves us, after fermionization, with 16 left-moving complex Weyl fermions in the 2d theory, which is consistent with the heterotic string [84].",
"From the field content described in table (REF ) one can compute the left- and right-moving central charges.",
"A complete $(0,8)$ hypermultiplet contains two $(0,4)$ hypermultiplets, whose central charge we know from appendix .",
"Each hypermultiplet thus contributes $(c_R, c_L) =(12, 8)$ .",
"The $19-3$ remaining left-moving Weyl fermions each contribute $+1$ to $c_L$ .",
"As such the central charges are $c_L = 24 \\,, \\qquad c_R = 12 \\,.$ The resulting gravitational anomaly $c_L - c_R = 12$ along the string will be discussed from the point of view of anomaly inflow in the next section."
],
[
"...from M2-branes",
"For an M2-brane wrapping the base inside the K3 surfaceM2-branes on general K3-surfaces have been discussed in [34], [35].",
"there are six non-compact transverse directions, and two real directions related to how the curve can move inside the K3.",
"As such the R-symmetry of the M2-brane theory is broken as $\\begin{aligned}SO(8)_R &\\ \\rightarrow \\ SO(6)_T \\times U(1)_R \\cr {\\bf 8}_v &\\ \\rightarrow \\ {\\bf 6}_0 \\oplus {\\bf 1}_{\\pm 2} \\cr {\\bf 8}_c &\\ \\rightarrow \\ {\\bf 4}_1 \\oplus {\\bf \\overline{4}}_{-1} \\cr {\\bf 8}_s &\\ \\rightarrow \\ {\\bf 4}_{-1} \\oplus {\\bf \\overline{4}}_1 \\,.\\end{aligned}$ Combining this decomposition with the field content of the $3d$ theory one finds $\\begin{aligned}SO(1,2) \\times SO(8) &\\ \\rightarrow \\ SO(6)_T \\times U(1)_L \\times U(1)_R \\cr \\epsilon : ({\\bf 2, 8}_s) &\\ \\rightarrow \\ {\\bf 4}_{1,-1} \\oplus {\\bf 4}_{-1,-1} \\oplus {\\bf \\overline{4}}_{1,1} \\oplus {\\bf \\overline{4}}_{-1,1} \\cr \\rho : ({\\bf 2, 8}_c) &\\ \\rightarrow \\ {\\bf 4}_{1,1} \\oplus {\\bf 4}_{-1,1} \\oplus {\\bf \\overline{4}}_{1,-1} \\oplus {\\bf \\overline{4}}_{-1,-1} \\cr \\Phi : ({\\bf 1, 8}_v) &\\ \\rightarrow \\ {\\bf 6}_{0,0} \\oplus {\\bf 1}_{0,\\pm 2} \\,.\\end{aligned}$ With the obvious additive twist $T_{\\rm twist} = \\frac{1}{2} (T_L + T_R)$ between the two $U(1)$ s one can see that the supersymmetry parameters in the $({\\bf 2, 8}_s)$ contain eight scalar modes with respect to $U(1)_{\\rm twist}$ , as required for an $N=8$ SQM.",
"The cohomology groups counting the zero-modes follow from the $U(1)_\\text{twist}$ charges of the fields: $\\begin{array}{c|c|c|c}U(1)_{\\text{twist}} & \\text{Bosons} & \\text{Fermions} &\\text{Zero-modes} \\cr \\hline 0 & {\\bf 6}_0 & {\\bf 4}_0 \\,\\,,\\, {\\bf \\overline{4}}_0 & h^{0,0}(C) = 1\\cr +1 & {\\bf 1}_{1} & {\\bf 4}_1 & h^{1,0}(C) = 0 \\cr -1 & {\\bf 1}_{-1} & {\\bf \\overline{4}}_{-1} & h^{0,1}(C) = 0\\end{array}$ The scalars and fermions in the first line assemble into a $(6,8,2)$ multiplet of $N=8$ SQM.",
"This multiplet is matched with the 2d $N=(0,8)$ hypermultiplet along the string obtained from the dual D3-brane configuration by automorphic duality: After circle reducing the 2d theory the two Wilson line degrees of freedom, $a_z$ and $\\bar{a}_{\\bar{z}}$ , from decomposition of the $N=4$ SYM gauge field can be dualised into the auxiliary fields of the $(6,8,2)$ multiplet."
]
] | 1612.05640 |
[
[
"Gas pressure in bubble attached to tube circular outlet"
],
[
"Abstract In the present Supplementary notes to our work \"Arresting bubble coarsening: A two-bubble experiment to investigate grain growth in presence of surface elasticity\" (accepted in EPL), we derive the expression of the gas pressure inside a bubble located above and attached to the circular outlet of a vertical tube."
],
[
"Contents of the present notes",
"The calculation presented in the present notes is performed in the limit of low gravity and is obtained in the following form: $p_{\\rm gas}&=& p_{\\rm liq}^{\\rm out}+ P_0(\\beta ) \\frac{\\gamma }{R}+ P_1(\\beta ) \\rho g R+ ...$ where $\\beta $ is the ratio of the outlet to the bubble size: $\\beta &=& \\frac{r_{\\rm out}}{R}$ More generally, let us define $P(\\alpha ,\\beta )$ through: $p_{\\rm gas}&=& p_{\\rm liq}^{\\rm out}+ \\frac{\\gamma }{R} P(\\alpha ,\\beta )$ where $\\alpha $ is the non-dimensionalized gravity: $\\alpha &=& \\frac{\\rho g R^2}{\\gamma } = \\frac{R^2}{l_{\\rm cap}^2}$ Note that for a given value of the outlet radius $r$ (or $\\beta $ ), there exists a maximum gravity (or $\\alpha $ ) that must not be exceeded for the bubble to remain attached to the tube outlet: $0 \\le \\alpha < \\alpha _{\\rm max}(\\beta )$ In the limit of low gravity ($\\alpha \\rightarrow 0$ ), Eq.",
"(REF ) can be expanded as: $p_{\\rm gas}&=& p_{\\rm liq}^{\\rm out}+ \\frac{\\gamma }{R} \\left(P_0(\\beta ) + \\alpha P_1(\\beta ) + {\\cal O}(\\alpha ^2)\\right)\\qquad $ where: $P_0(\\beta ) &=& \\lim _{\\alpha \\rightarrow 0}{P(\\alpha ,\\beta )}\\\\P_1(\\beta ) &=& \\lim _{\\alpha \\rightarrow 0}{\\frac{\\partial P}{\\partial \\alpha }(\\alpha ,\\beta )}$ Functions $P_0$ and $P_1$ are the output of the calculation performed in the present Supplementary notes.",
"They are plotted on Fig.",
"REF b.",
"The small outlet limit ($\\beta \\rightarrow 0$ ) is obtained in Appendix REF and is consistent with Fig.",
"REF b: $P_0(0) &=& 2\\\\P_1(0) &=& -\\frac{4}{3}$ These coefficients are those shown in Eq.",
"(REF ).",
"Note: this outlet limit ($\\beta \\rightarrow 0$ ) is to be taken after the small gravity limit ($\\alpha \\rightarrow 0$ ) which is the basis of the expansion of Eq.",
"(REF ) and of the whole calculation of Appendix .",
"If the outlet radius goes to zero ($\\beta \\rightarrow 0$ ) before gravity goes to zero, the bubble detaches!",
"These notes are organized as follows.",
"In Section , we express the gas pressure in terms of the bubble geometry.",
"In Section , we calculate the bubble shape equation and solve it first analytically, then numerically, and thus obtain the expressions and data used in Fig.",
"REF and in Eq.",
"(REF )."
],
[
"Expression of the gaz pressure",
"As compared to the pressure $p_{\\rm liq}^{\\rm out}$ in the liquid at the same altitude as the tube outlet, the gas pressure $p_{\\rm gas}$ can be expressed for instance in terms of the apex radius of curvature and altitude: $p_{\\rm gas}= p_{\\rm liq}^{\\rm out}+ \\frac{2\\gamma }{R_{\\rm apex}} + \\rho g (z_{\\rm out}- z_{\\rm apex})$ where the last term provides the pressure difference between the liquid pressure at the tube outlet and apex altitudes while the middle term expresses the pressure jump across the interface at the apex, due to the interfacial tension $\\gamma $ and the total curvature $2/R_{\\rm apex}$ .",
"When the bubble is attached to the tube outlet ($\\alpha < \\alpha _{\\rm max}(\\beta )$ ), let us define functions $A$ and $B$ through the apex radius of curvature and altitude that appear in Eq.",
"(REF ): $R_{\\rm apex}&=& \\frac{R}{A(\\alpha ,\\beta )}\\\\z_{\\rm apex}-z_{\\rm out}&=& 2R \\, B(\\alpha ,\\beta )$ Substituting Eqs.",
"(REF ,) into Eq.",
"(REF ) and combining with Eq.",
"(REF ): $P(\\alpha ,\\beta ) = 2 A(\\alpha ,\\beta ) -2 \\alpha B(\\alpha ,\\beta )$ Let us expand functions $A$ and $B$ in the limit of low gravity: $A(\\alpha ,\\beta ) &=& A_0(\\beta ) + A_1(\\beta )\\alpha + {\\cal O}(\\alpha ^2)\\\\B(\\alpha ,\\beta ) &=& B_0(\\beta ) + B_1(\\beta )\\alpha + {\\cal O}(\\alpha ^2)$ In other words: $P_0(\\beta ) &=& 2 A_0(\\beta )\\\\P_1(\\beta ) &=&2A_1(\\beta )-2B_0(\\beta )$"
],
[
"Calculating the bubble shape",
"Functions $A_0(\\beta )$ and $B_0(\\beta )$ correspond to vanishing gravity ($\\alpha =0$ ) and can thus be determined by simply considering a spherical bubble (Section REF ).",
"However, determining $A_1(\\beta )$ (or equivalently $P_1(\\beta )$ ) requires to calculate the non-trivial bubble shape in the presence of gravity (Sections REF -REF )."
],
[
"Spherical bubble",
"The spherical bubble case (zero gravity, $\\alpha =0$ ) is treated geometrically in Appendix and provides functions $A_0(\\beta )$ and $B_0(\\beta )$ : $A_0(\\beta ) &=& \\frac{ (U+4)^{2/3} - (U-4)^{2/3} }{U}\\\\&&= 1 +{\\cal O}(\\beta ^4)\\\\&& {\\rm where}\\,\\, U = \\sqrt{16 + \\beta ^6}\\nonumber \\\\B_0(\\beta ) &=& \\frac{1+\\sqrt{1 -\\beta ^2 A_0^2}}{2A_0}\\\\&&= 1 -\\frac{1}{4}\\beta ^2 +{\\cal O}(\\beta ^4)$ The quantity $2A_0(\\beta )$ is equal to the zero-gravity component $P_0$ of the pressure, as expressed by Eq.",
"(REF ).",
"It is plotted as circles on Fig.",
"REF b.",
"Note that when the tube outlet radius goes to zero, the above expressions go to unity: $A_0(0)=B_0(0)=1$ ."
],
[
"Equation for the bubble shape",
"In order to determine $A_1(\\beta )$ (or $P_1(\\beta )$ ), let us generalize Eq.",
"(REF ) as: $p_{\\rm gas}= p_{\\rm liq}^{\\rm out}+ \\gamma \\,C(s) + \\rho g (z_{\\rm out}- z(s))$ where $C$ is the total curvature and $z$ the altitude at point $s$ , where $s$ is for instance the curvilinear distance from the top of the bubble.",
"Let $r(s)$ be the distance from the bubble axis and $\\psi (s)$ the angle between the tangent to the bubble contour at point $s$ and the horizontal (with the convention that $\\psi (s)$ is positive).",
"The total curvature of such a axisymmetric shape can be shown to be: $C(s) = \\frac{{\\rm d}\\psi }{{\\rm d}s} + \\frac{\\sin \\psi (s)}{r(s)}$ The evolution of $r$ and $z$ along the contour are trivially related to $\\psi $ : $\\frac{{\\rm d}r}{{\\rm d}s} &=& \\cos \\psi \\\\\\frac{{\\rm d}z}{{\\rm d}s} &=& -\\sin \\psi $ The evolution of $\\psi $ results from Eqs.",
"(REF ,REF ): $\\frac{{\\rm d}\\psi }{{\\rm d}s}&=& \\frac{2Q}{R} - \\frac{\\sin \\psi }{r}+ \\frac{\\alpha }{R^2}(z-z_{\\rm apex})$ where the constant $Q$ is defined by: $\\frac{2Q}{R} &=&\\frac{p_{\\rm gas}- p_{\\rm liq}^{\\rm out}}{\\gamma }+ \\frac{\\alpha }{R^2}(z_{\\rm apex}-z_{\\rm out})$ The evolution of the volume $V(s)$ of the bubble above altitude $z(s)$ is simply: $\\frac{{\\rm d}V}{{\\rm d}s}= \\pi r^2 \\left|\\frac{{\\rm d}z}{{\\rm d}s}\\right|= \\pi r^2 \\sin \\psi $ The boundary conditions at the apex ($s=0$ ) and at the outlet ($s=s_{\\rm out}$ ) are: $r(0) &=& 0\\\\z(0) &=& z_{\\rm apex}= 0\\\\\\psi (0) &=& 0\\\\V(0) &=& 0\\\\r(s_{\\rm out}) &=& r_{\\rm out}= \\beta R\\\\z(s_{\\rm out}) &=& z_{\\rm out}\\\\V(s_{\\rm out}) &=& \\frac{4\\pi }{3} R^3$ where $z_{\\rm apex}=0$ by convention and where $r_{\\rm out}$ is related to $\\beta $ through Eq.",
"(REF )."
],
[
"Non-dimensional bubble shape",
"Note that in Eq.",
"(REF ), $Q$ is unknown since it contains $p_{\\rm gas}-p_{\\rm liq}^{\\rm out}$ and $z_{\\rm out}-z_{\\rm apex}$ , see Eq.",
"(REF ).",
"The curvilinear position $s_{\\rm out}$ of the outlet is also unknown, and only for the correct value of $Q$ will boundary conditions (REF ) and () be satisfied for the same value of $s_{\\rm out}$ .",
"Thus, for every value of $\\alpha $ and $\\beta $ , the system of Eqs.",
"(REF –) needs to be integrated a number of times with different values of $Q$ to obtain the correct $Q$ and hence a correct bubble shape and gas pressure.",
"In order to avoid these complications, let us renormalize all distances with $R/Q$ (even though $Q$ is yet unknown): $\\hat{r}&=& r\\,Q/R\\\\\\hat{z}&=& (z-z_{\\rm apex})\\,Q/R\\\\\\hat{s}&=& s\\,Q/R\\\\\\hat{V}&=& V\\,Q^3/R^3\\\\\\hat{\\alpha }&=& \\alpha /Q^2$ In terms of these new variables, the system of differential equations reads: $\\frac{{\\rm d}\\hat{r}}{{\\rm d}\\hat{s}} &=& \\cos \\psi \\\\\\frac{{\\rm d}\\hat{z}}{{\\rm d}\\hat{s}} &=& -\\sin \\psi \\\\\\frac{{\\rm d}\\psi }{{\\rm d}\\hat{s}}&=& 2 - \\frac{\\sin \\psi }{\\hat{r}}+ \\hat{\\alpha }\\,\\hat{z}\\\\\\frac{{\\rm d}\\hat{V}}{{\\rm d}\\hat{s}}&=& \\pi \\hat{r}^2 \\sin \\psi $ For any given $\\hat{\\alpha }$ , the above system can be solved starting from the initial conditions: $\\hat{r}(0) &=& 0\\\\\\hat{z}(0) &=& 0\\\\\\psi (0) &=& 0\\\\\\hat{V}(0) &=& 0$ The solution is obtained in the form of functions $\\hat{r}(\\hat{\\alpha },\\hat{s})$ , $\\hat{z}(\\hat{\\alpha },\\hat{s})$ , $\\psi (\\hat{\\alpha },\\hat{s})$ , $\\hat{V}(\\hat{\\alpha },\\hat{s})$ ."
],
[
"Outlet position and gas pressure",
"Let us now define: $\\hat{\\beta }(\\hat{\\alpha },\\hat{s})\\equiv \\frac{\\hat{r}(\\hat{\\alpha },\\hat{s})}{\\left(\\frac{3}{4\\pi }\\hat{V}(\\hat{\\alpha },\\hat{s})\\right)^{1/3}}$ Using this new function $\\hat{\\beta }$ and definitions (REF ) and (), the position $\\hat{s}_{\\rm out}(\\hat{\\alpha },\\beta )$ of the outlet is obtained very simply as the value of $\\hat{s}$ where: $\\hat{\\beta }(\\hat{\\alpha },\\hat{s})\\equiv \\frac{r_{\\rm out}}{\\left(\\frac{3}{4\\pi }V\\right)^{1/3}}= \\beta $ Once $\\hat{s}_{\\rm out}(\\hat{\\alpha },\\beta )$ is thus determined, we define: $\\hat{r}_{\\rm out}(\\hat{\\alpha },\\beta ) &=& \\hat{r}(\\hat{\\alpha },\\hat{s}_{\\rm out}(\\hat{\\alpha },\\beta ))\\\\\\hat{z}_{\\rm out}(\\hat{\\alpha },\\beta ) &=& \\hat{z}(\\hat{\\alpha },\\hat{s}_{\\rm out}(\\hat{\\alpha },\\beta ))$ And we obtain: $Q(\\hat{\\alpha },\\beta ) &=& \\hat{r}_{\\rm out}(\\hat{\\alpha },\\beta ) \\, R / r_{\\rm out}\\nonumber \\\\&=& \\hat{r}_{\\rm out}(\\hat{\\alpha },\\beta ) / \\beta $ Using Eqs.",
"(REF ,REF ,,), the pressure $P$ and the gravity parameter $\\alpha $ can be expressed from the results of Eqs.",
"(REF ,,REF ) in terms of parameter $\\hat{\\alpha }$ : $P(\\hat{\\alpha },\\beta ) &=& \\frac{R}{\\gamma } \\, (p_{\\rm gas}- p_{\\rm liq}^{\\rm out})\\nonumber \\\\&=& (2 + \\hat{\\alpha }\\, \\hat{z}_{\\rm out}(\\hat{\\alpha },\\beta )) \\, \\hat{r}_{\\rm out}(\\hat{\\alpha },\\beta ) / \\beta \\\\\\alpha (\\hat{\\alpha },\\beta )&=& \\hat{\\alpha }\\, Q^2(\\hat{\\alpha },\\beta )= \\hat{\\alpha }\\, \\hat{r}_{\\rm out}^2(\\hat{\\alpha },\\beta ) / \\beta ^2\\qquad $ Expressions (REF ,) are valid for all $\\alpha $ values within some range defined by Eq.",
"(REF ) where the bubble remains attached to the tube outlet.",
"Let us now first show the results of an analytic derivation of $P_0(\\beta )$ and $P_1(\\beta )$ (Section REF ) and of its numeric counterpart (Section REF )."
],
[
"Analytic (near-spherical) shape",
"Let us now decompose the functions that appear in the system of Eqs.",
"(REF -REF ) into the trivial solution when $\\hat{\\alpha }=0$ and a term that depends on $\\hat{\\alpha }$ : $\\hat{r}(\\hat{\\alpha },\\hat{s}) &=& \\sin \\hat{s}+ \\hat{\\alpha }\\, \\hat{r}_1(\\hat{\\alpha },\\hat{s})\\\\\\hat{z}(\\hat{\\alpha },\\hat{s}) &=& \\cos \\hat{s}-1 + \\hat{\\alpha }\\, \\hat{z}_1(\\hat{\\alpha },\\hat{s})\\\\\\psi (\\hat{\\alpha },\\hat{s}) &=& \\hat{s}+ \\hat{\\alpha }\\, \\psi _1(\\hat{\\alpha },\\hat{s})\\\\\\hat{V}(\\hat{\\alpha },\\hat{s}) &=& \\frac{\\pi }{3}\\left(2-3\\cos \\hat{s}+\\cos ^3\\hat{s}\\right)\\nonumber \\\\&&+ \\hat{\\alpha }\\, \\hat{V}_1(\\hat{\\alpha },\\hat{s})$ where the initial conditions imply: $\\hat{r}_1(0) = \\hat{z}_1(0) = \\psi _1(0) = \\hat{V}_1(0) = 0$ It is shown in Appendix that: $\\hat{r}_1 &=&\\frac{1}{3} \\sin \\hat{s}\\cos \\hat{s}+\\frac{1}{6} \\sin \\hat{s}-\\frac{1}{2} \\hat{s}\\cos \\hat{s}\\qquad \\\\\\hat{z}_1 &=& -\\frac{1}{3}\\sin ^2\\hat{s}+\\frac{1}{2}\\hat{s}\\sin \\hat{s}\\nonumber \\\\&&+\\frac{2}{3} \\log \\cos \\frac{\\hat{s}}{2}-\\frac{1}{3}\\sin ^2\\frac{\\hat{s}}{2}$ The dashed contours on Fig.",
"REF a correspond to the dimensional version of Eqs.",
"(REF ,) obtained through the non-dimensionalizing factor $Q$ provided by Eq.",
"(REF ) and plotted parametrically as a function of $\\beta $ using Eq.",
"(REF ) to express it in terms of the same parameter $\\hat{\\delta }_{\\rm out}$ .",
"Similarly, concerning the pressure $P$ defined by Eqs.",
"(REF ,REF ), as shown in Appendix REF , explicit expressions for both the zero gravity limit $P_0$ and the first derivative $P_1$ are provided respectively by Eqs.",
"(REF ) and (REF ).",
"Using Eq.",
"(REF ) again, $P_0$ and $P_1$ can be plotted, respectively as the solid and the dashed curves on Fig.",
"REF b.",
"The limits $P_0$ and $P_1$ can be obtained easily, as shown in Appendix REF : $P_0(0) &=& 2\\\\P_1(0) &=& -\\frac{4}{3}$ These two values can be read out on Fig.",
"REF b as the value reached by both curves when they meet the vertical axis $\\beta =0$ .",
"They are used in the approximate expression announced as Eq.",
"(REF ).",
"A more elaborate expansion of the same expressions is presented in Appendix REF and yields Eq.",
"(REF ) which can be expressed as: $P_0(\\beta ) &=& 2 + {\\cal O}(\\beta ^4)\\\\P_1(\\beta ) &=& -\\frac{4}{3} + {\\cal O}(\\beta ^2)$"
],
[
"Numeric bubble shape",
"As a complement to the analytic approach of Section REF , one can integrate numerically Eqs.",
"(REF ) to ().",
"Because of the structure of Eq.",
"() which contains the ratio of $\\sin \\psi $ and $\\hat{r}$ , both going to zero at $\\hat{s}=0$ , we start with the following initial conditions: $\\hat{r}&=& \\hat{s}_1\\\\\\hat{z}&=& -\\frac{1}{2} \\, \\hat{s}_1^2\\\\\\psi &=& \\hat{s}_1\\\\\\hat{V}&=& \\frac{\\pi }{4} \\, \\hat{s}_1^4$ We integrate using the explicit Runge–Kutta method of order (4,5), more precisely GNU Octave's [1] ode45 function, with a maximum integration step taken as equal to $\\hat{s}_1$ .",
"We stop integration at the outlet position defined by $\\beta $ as stated in Section REF , then read $Q$ , $P$ and $\\alpha $ as prescribed by Eqs.",
"(REF ), (REF ) and () respectively.",
"Each integration is performed for a given triplet $(\\hat{\\alpha },\\beta ,\\hat{s}_1)$ .",
"For every pair of values $(\\hat{\\alpha },\\beta )$ , three integrations have been performed, with $\\hat{s}_1$ equal to $10^{-3}$ , $10^{-4}$ and $3.10^{-5}$ .",
"The values $P(\\hat{\\alpha },\\beta )$ and $\\alpha (\\hat{\\alpha },\\beta )$ have then been extrapolated to the limit $\\hat{s}_1\\rightarrow 0$ .",
"For each value of $\\beta $ , three such processes have been performed with $\\hat{\\alpha }$ equal to $5.10^{-4}$ , $2.10^{-4}$ and $10^{-4}$ .",
"The resulting values of $P(\\hat{\\alpha },\\beta )$ and $\\alpha (\\hat{\\alpha },\\beta )$ have been used to extrapolate $P(\\alpha ,\\beta )$ and $\\partial P(\\alpha ,\\beta )/\\partial \\alpha $ to the limit $\\alpha \\rightarrow 0$ , so as to obtain $P_0(\\beta )$ and $P_1(\\beta )$ .",
"This whole process has been carried out for $\\beta $ equal to $0.2$ , $0.1$ and $0.05$ and the corresponding values of $P_0$ and $P_1$ are plotted on Fig.",
"REF b as large circles and diamonds respectively (purple color).",
"Finally, values for $P_0(0)$ and $P_1(0)$ are shown in black color.",
"They were extrapolated from the corresponding values for the three non-zero values of $\\beta $ .",
"The values thus obtained confirm the values $P_0(0)=2$ and $P_1(0)=-4/3$ adopted for the approximate expression announced in Eq.",
"(REF )."
],
[
"Truncated sphere",
"In this Appendix, we consider the situation with zero gravity ($\\alpha =0$ ), hence with a purely spherical drop, and calculate the drop radius of curvature and apex altitude as a function of the outlet radius $r_{\\rm out}$ .",
"The result is expressed in the form of $A=A_0(\\beta )$ and $B=B_0(\\beta )$ defined by Eqs.",
"(REF ,,REF ,) with $\\alpha =0$ , where $\\beta $ is defined by Eq.",
"(REF ).",
"The bubble, whose radius is $R$ when purely spherical, becomes a truncated sphere when attached to an outlet of radius $r_{\\rm out}$ .",
"Let $R_{\\rm apex}$ be the radius of the truncated sphere.",
"The height of the truncated part is: $H &=& 2R_{\\rm apex}-(z_{\\rm apex}-z_{\\rm out})$ where $z_{\\rm apex}$ (resp.",
"$z_{\\rm out}$ ) is the altitude of the bubble apex (resp.",
"tube outlet), see Fig.",
"REF .",
"Pythagore: $R_{\\rm apex}^2 &=& r_{\\rm out}^2 + (R_{\\rm apex}-H)^2\\\\H &=& R_{\\rm apex}- \\sqrt{R_{\\rm apex}^2-r_{\\rm out}^2}$ Using Eqs.",
"(REF ) and (REF ) to reformulate Eq.",
"(REF ): $\\frac{H}{R_{\\rm apex}} &=& 1 - \\sqrt{1 - A_0^2\\beta ^2}$ The volume of the truncated part is that of a spherical cap of height $H$ and radius of curvature $R_{\\rm apex}$ : $\\frac{\\pi }{3}\\,H^2\\,(3R_{\\rm apex}- H)$ The condition that the initial drop of radius $R$ has the same volume as the truncated sphere of radius $R_{\\rm apex}$ can be expressed as: $\\frac{4\\pi }{3} R^3= \\frac{4\\pi }{3} R_{\\rm apex}^3- \\frac{\\pi }{3}\\,H^2\\,(3R_{\\rm apex}- H)$ Using Eqs.",
"(REF ,REF ) and noting $Z=\\sqrt{1-A_0^2\\beta ^2}$ , Eq.",
"(REF ) can be transformed as follows: $4(1-A_0^3)&=&(1-Z)^2(2+Z)\\\\4(1-A_0^3)&=&2-(2+A_0^2\\beta ^2)Z\\\\4A_0^3-2&=&(2+A_0^2\\beta ^2)\\sqrt{1-A_0^2\\beta ^2}\\\\(4A_0^3-2)^2&=&4-3A_0^4\\beta ^4-A_0^6\\beta ^6$ and finally, after dividing by $A_0^3$ : $(16+\\beta ^6)\\,A_0^3 +3\\beta ^4\\,A_0 -16 = 0$ Defining: $U = \\sqrt{16 + \\beta ^6},$ the solution to the third order polynomial equation (REF ) is: $A_0(\\beta ) &=& \\frac{ (U+4)^{2/3} - (U-4)^{2/3} }{U}\\\\&=& 1 - \\frac{1}{16}\\beta ^4 -\\frac{1}{48}\\beta ^6 +o(\\beta ^6)$ Using Eqs.",
"(REF ,,REF ,REF ,REF ): $B_0(\\beta ) &=& \\frac{1+\\sqrt{1 -\\beta ^2 A_0^2}}{2A_0}\\\\&=& 1 - \\frac{1}{4}\\beta ^2 +\\frac{1}{192}\\beta ^6 +o(\\beta ^6)$"
],
[
"Analytic (near-spherical) shape",
"In the present Appendix, we derive the results presented in Section REF ."
],
[
"First order functions",
"Using Eq.",
"(), $\\sin \\psi $ and $\\cos \\psi $ can be expressed to first order in $\\hat{\\alpha }$ : $\\sin \\psi = \\sin \\hat{s}+ \\hat{\\alpha }\\, \\psi _1\\cos \\hat{s}+ {\\cal O}(\\hat{\\alpha }^2)\\\\\\cos \\psi = \\cos \\hat{s}- \\hat{\\alpha }\\, \\psi _1\\sin \\hat{s}+ {\\cal O}(\\hat{\\alpha }^2)$ Inserting Eqs.",
"(REF –) and Eqs.",
"(REF ,) into Eqs.",
"(REF –): $\\frac{{\\rm d}\\hat{r}_1}{{\\rm d}\\hat{s}} &=& -\\psi _1 \\sin \\hat{s}+ {\\cal O}(\\hat{\\alpha })\\\\\\frac{{\\rm d}\\hat{z}_1}{{\\rm d}\\hat{s}} &=& -\\psi _1 \\cos \\hat{s}+ {\\cal O}(\\hat{\\alpha })\\\\\\sin \\hat{s}\\,\\frac{{\\rm d}\\psi _1}{{\\rm d}\\hat{s}}&=& \\hat{r}_1 -\\psi _1\\cos \\hat{s}\\nonumber \\\\&&-\\sin \\hat{s}+\\sin \\hat{s}\\cos \\hat{s}+ {\\cal O}(\\hat{\\alpha }) \\qquad \\\\\\frac{{\\rm d}\\hat{V}_1}{{\\rm d}\\hat{s}}&=& \\pi (\\psi _1 \\cos \\hat{s}+2 \\hat{r}_1) \\sin ^2\\hat{s}+ {\\cal O}(\\hat{\\alpha }) \\qquad $ Let us differentiate Eq.",
"() and combine it with Eq.",
"(REF ): $&&\\sin \\hat{s}\\,\\frac{{\\rm d^2}\\psi _1}{{\\rm d}\\hat{s}^2}+2\\cos \\hat{s}\\,\\frac{{\\rm d}\\psi _1}{{\\rm d}\\hat{s}}\\nonumber \\\\&&\\qquad \\qquad =-\\cos \\hat{s}+2\\cos ^2\\hat{s}-1\\qquad \\qquad $ Multiplyling by $\\sin \\hat{s}$ : $&&\\frac{{\\rm d}}{{\\rm d}\\hat{s}}\\left(\\sin ^2\\hat{s}\\,\\frac{{\\rm d}\\psi _1}{{\\rm d}\\hat{s}}\\right)\\nonumber \\\\&&\\qquad =-\\cos \\hat{s}\\sin \\hat{s}+2\\cos ^2\\hat{s}\\sin \\hat{s}-\\sin \\hat{s}\\qquad $ Integrating with respect to $\\hat{s}$ : $\\sin ^2\\hat{s}\\,\\frac{{\\rm d}\\psi _1}{{\\rm d}\\hat{s}}&=&\\frac{1}{2}\\cos ^2\\hat{s}-\\frac{2}{3}\\cos ^3\\hat{s}+\\cos \\hat{s}-\\frac{5}{6}\\qquad \\nonumber \\\\&=&\\left[\\frac{2}{3}\\cos \\hat{s}-\\frac{1}{2}\\right]\\sin ^2\\hat{s}-\\frac{2}{3}\\sin ^2\\frac{\\hat{s}}{2}$ where the integration constant was chosen to obtain zero when $\\hat{s}=0$ .",
"Dividing by $\\sin ^2\\hat{s}$ : $\\frac{{\\rm d}\\psi _1}{{\\rm d}\\hat{s}}&=&\\frac{2}{3}\\cos \\hat{s}-\\frac{1}{2}-\\frac{1}{6\\cos ^2\\frac{\\hat{s}}{2}}$ By integration: $\\psi _1 &=& \\frac{2}{3}\\sin \\hat{s}- \\frac{\\hat{s}}{2}- \\frac{1}{3} \\tan \\frac{\\hat{s}}{2}$ Multiplying Eq.",
"(REF ) by $\\sin \\hat{s}$ and integrating as suggested by Eq.",
"(REF ) with the condition $\\hat{r}_1(0)=0$ , we obtain: $\\hat{r}_1 &=&\\frac{1}{3} \\sin \\hat{s}\\cos \\hat{s}+\\frac{1}{6} \\sin \\hat{s}-\\frac{1}{2} \\hat{s}\\cos \\hat{s}\\qquad $ Similarly, multiplying Eq.",
"(REF ) by $-\\cos \\hat{s}$ or $1-2\\cos ^2\\frac{\\hat{s}}{2}$ , as suggested by Eq.",
"(), we obtain: $-\\psi _1 \\cos \\hat{s}&=& -\\frac{2}{3}\\sin \\hat{s}\\cos \\hat{s}+ \\frac{1}{2}\\hat{s}\\cos \\hat{s}\\nonumber \\\\&&- \\frac{1}{3} \\tan \\frac{\\hat{s}}{2}+ \\frac{2}{3} \\sin \\frac{\\hat{s}}{2}\\cos \\frac{\\hat{s}}{2}\\\\&=&-\\frac{1}{3}(\\sin ^2\\hat{s})^\\prime +\\frac{1}{2}(\\hat{s}\\sin \\hat{s}+\\cos \\hat{s})^\\prime \\qquad \\nonumber \\\\&&+\\frac{2}{3}(\\log \\cos \\frac{\\hat{s}}{2})^\\prime +\\frac{2}{3}(\\sin ^2\\frac{\\hat{s}}{2})^\\prime $ Injecting Eq.",
"(REF ) into Eq.",
"() and integrating with respect to $\\hat{s}$ while imposing that $\\hat{z}_1=0$ when $\\hat{s}=0$ , we obtain: $\\hat{z}_1 &=& -\\frac{1}{3}\\sin ^2\\hat{s}+\\frac{1}{2}\\hat{s}\\sin \\hat{s}+\\frac{1}{2}(\\cos \\hat{s}-1)\\nonumber \\\\&&+\\frac{2}{3} \\log \\cos \\frac{\\hat{s}}{2}+\\frac{2}{3}\\sin ^2\\frac{\\hat{s}}{2}\\\\\\hat{z}_1 &=& -\\frac{1}{3}\\sin ^2\\hat{s}+\\frac{1}{2}\\hat{s}\\sin \\hat{s}\\nonumber \\\\&&+\\frac{2}{3} \\log \\cos \\frac{\\hat{s}}{2}-\\frac{1}{3}\\sin ^2\\frac{\\hat{s}}{2}$ Inserting Eqs.",
"(REF ,REF ) into Eq.",
"(), we obtain: $\\frac{1}{\\pi }\\frac{{\\rm d}\\hat{V}_1}{{\\rm d}\\hat{s}}&=& \\left[\\frac{4}{3}\\sin \\hat{s}\\cos \\hat{s}-\\frac{3}{2}\\hat{s}\\cos \\hat{s}\\right.\\nonumber \\\\&&\\left.+\\frac{1}{3}\\tan \\frac{\\hat{s}}{2}\\right]\\sin ^2\\hat{s}+ {\\cal O}(\\hat{\\alpha }) \\qquad $ Integrating with respect to $\\hat{s}$ while imposing that $\\hat{V}_1=0$ when $\\hat{s}=0$ , we obtain: $\\hat{V}_1 &=&\\frac{4\\pi }{3}\\sin ^2\\frac{\\hat{s}}{2}+\\frac{\\pi }{3}\\sin ^2\\hat{s}\\sin ^2\\frac{\\hat{s}}{2}\\nonumber \\\\&&-\\frac{\\pi }{2}\\hat{s}\\sin ^3\\hat{s}-\\frac{\\pi }{12}\\sin ^2(2\\hat{s})\\qquad $"
],
[
"Outlet position",
"Since the outlet position is close to the lower pole of the sphere, let us define: $\\hat{s}= \\pi - \\hat{\\delta }$ Using Eqs.",
"(REF ,REF ,REF ), $\\hat{r}$ can be expressed as: $\\hat{r}&=& \\hat{r}_0 + \\hat{\\alpha }\\, \\hat{r}_1+{\\cal O}(\\hat{\\alpha }^2)\\qquad \\\\\\hat{r}_0 &=& \\sin \\hat{\\delta }\\\\\\hat{r}_1 = \\frac{\\partial \\hat{r}}{\\partial \\hat{\\alpha }}&=& -\\frac{1}{3} \\sin \\hat{\\delta }\\cos \\hat{\\delta }+\\frac{1}{6} \\sin \\hat{\\delta }\\nonumber \\\\&&+\\frac{\\pi }{2} \\cos \\hat{\\delta }-\\frac{1}{2} \\hat{\\delta }\\cos \\hat{\\delta }\\\\\\hat{r}_0^\\prime = \\frac{\\partial \\hat{r}}{\\partial \\hat{\\delta }}&=& \\cos \\hat{\\delta }$ Using Eqs.",
"(,REF ), the leading order of $\\hat{z}$ is: $\\hat{z}&=& \\hat{z}_0 +{\\cal O}(\\hat{\\alpha })= -1 -\\cos \\hat{\\delta }+{\\cal O}(\\hat{\\alpha })\\qquad $ Using Eqs.",
"(,REF ,REF ), the volume $\\hat{\\Omega }=\\frac{3}{4\\pi }\\hat{V}$ can be expressed as: $\\hat{\\Omega }&=& \\hat{\\Omega }_0 + \\hat{\\alpha }\\, \\hat{\\Omega }_1+{\\cal O}(\\hat{\\alpha }^2)\\qquad \\\\\\hat{\\Omega }_0 &=& \\frac{1}{2} +\\frac{3}{4}\\cos \\hat{\\delta }-\\frac{1}{4}\\cos ^3\\hat{\\delta }\\\\\\hat{\\Omega }_1 &=& \\cos ^2\\frac{\\hat{\\delta }}{2}+\\frac{1}{4} \\sin ^2\\hat{\\delta }\\cos ^2\\frac{\\hat{\\delta }}{2}-\\frac{3\\pi }{8}\\sin ^3\\hat{\\delta }\\qquad \\nonumber \\\\&&+\\frac{3}{8}\\hat{\\delta }\\sin ^3\\hat{\\delta }-\\frac{1}{16}\\sin ^2(2\\hat{\\delta })\\\\\\hat{\\Omega }_0^\\prime &=& \\frac{\\partial \\hat{\\Omega }_0}{\\partial \\hat{\\delta }}= -\\frac{3}{4}\\sin \\hat{\\delta }+\\frac{3}{4}\\sin \\hat{\\delta }\\cos ^2\\hat{\\delta }$ The position $\\hat{\\delta }_{\\rm out}$ of the tube outlet is defined by Eq.",
"(REF ) and can be expressed using Eq.",
"(REF ): $\\beta &=& \\frac{\\hat{r}}{\\hat{\\Omega }^{1/3}}(\\hat{\\alpha },\\hat{\\delta }_{\\rm out})$ Eq.",
"(REF ) can be used with Eq.",
"(REF ) to express the non-dimensionalization factor: $Q &=& \\frac{\\hat{r}_{\\rm out}}{\\beta }= \\hat{\\Omega }^{1/3}(\\hat{\\alpha },\\hat{\\delta }_{\\rm out})$ where $\\hat{\\Omega }$ is provided by Eqs.",
"(REF ,,)."
],
[
"Pressure and derivative",
"Using Eq.",
"(REF ), Eqs.",
"(REF ,) can be transformed into: $\\alpha &=& \\hat{\\alpha }\\, \\hat{\\Omega }^{2/3}(\\hat{\\alpha },\\hat{\\delta }_{\\rm out})\\\\P &=& (2+\\hat{\\alpha }\\hat{z}_0)\\,\\hat{\\Omega }^{1/3}(\\hat{\\alpha },\\hat{\\delta }_{\\rm out})$ In the zero gravity limit ($\\hat{\\alpha }\\rightarrow 0$ ), Eq.",
"(REF ) simplifies into: $P_0 &=& 2\\hat{\\Omega }_0^{1/3}(\\hat{\\delta }_{\\rm out})$ where $\\hat{\\Omega }_0$ is given by Eq. ().",
"Similarly, $\\beta $ is then given by: $\\beta _0 &=& \\frac{\\hat{r}_0(\\hat{\\delta }_{\\rm out})}{\\hat{\\Omega }_0^{1/3}(\\hat{\\delta }_{\\rm out})}$ where $\\hat{r}_0$ is given by Eq. ().",
"Then, using Eqs.",
"(REF ) and (REF ), the pressure $P_0$ can be plotted as a function of $\\beta _0$ using $\\hat{\\delta }_{\\rm out}$ as a parameter, which yields the solid curve $P_0(\\beta )$ on Fig.",
"REF b.",
"In order to obtain $2A_1(\\beta )-2B_0(\\beta )= \\frac{\\partial P}{\\partial \\alpha }(\\alpha =0,\\beta )$ , let us write the differentials of $P(\\hat{\\alpha },\\hat{\\delta })$ , $\\beta (\\hat{\\alpha },\\hat{\\delta })$ and $\\alpha (\\hat{\\alpha },\\hat{\\delta })$ : ${\\rm d}P&=& \\frac{\\partial P}{\\partial \\hat{\\alpha }} {\\rm d}\\hat{\\alpha }+\\frac{\\partial P}{\\partial \\hat{\\delta }} {\\rm d}\\hat{\\delta }\\\\{\\rm d}\\beta &=& \\frac{\\partial \\beta }{\\partial \\hat{\\alpha }} {\\rm d}\\hat{\\alpha }+\\frac{\\partial \\beta }{\\partial \\hat{\\delta }} {\\rm d}\\hat{\\delta }\\\\{\\rm d}\\alpha &=& \\frac{\\partial \\alpha }{\\partial \\hat{\\alpha }} {\\rm d}\\hat{\\alpha }+\\frac{\\partial \\alpha }{\\partial \\hat{\\delta }} {\\rm d}\\hat{\\delta }\\qquad $ Solving the system of Eqs.",
"(,) for ${\\rm d}\\hat{\\alpha }$ and ${\\rm d}\\hat{\\delta }$ and injecting them into Eq.",
"(REF ), one obtains: ${\\rm d}P&=& \\frac{\\frac{\\partial \\beta }{\\partial \\hat{\\alpha }}\\frac{\\partial P}{\\partial \\hat{\\delta }}-\\frac{\\partial \\beta }{\\partial \\hat{\\delta }}\\frac{\\partial P}{\\partial \\hat{\\alpha }}}{\\frac{\\partial \\beta }{\\partial \\hat{\\alpha }}\\frac{\\partial \\alpha }{\\partial \\hat{\\delta }}-\\frac{\\partial \\beta }{\\partial \\hat{\\delta }}\\frac{\\partial \\alpha }{\\partial \\hat{\\alpha }}}\\,{\\rm d}\\alpha \\qquad \\nonumber \\\\&+& \\frac{\\frac{\\partial P}{\\partial \\hat{\\alpha }}\\frac{\\partial \\alpha }{\\partial \\hat{\\delta }}-\\frac{\\partial P}{\\partial \\hat{\\delta }}\\frac{\\partial \\alpha }{\\partial \\hat{\\alpha }}}{\\frac{\\partial \\beta }{\\partial \\hat{\\alpha }}\\frac{\\partial \\alpha }{\\partial \\hat{\\delta }}-\\frac{\\partial \\beta }{\\partial \\hat{\\delta }}\\frac{\\partial \\alpha }{\\partial \\hat{\\alpha }}}\\,{\\rm d}\\beta $ In particular: $P_1(\\beta ) =\\left.\\frac{\\partial P}{\\partial \\alpha }\\right|_{\\alpha =0}=\\left.\\frac{\\frac{\\partial \\beta }{\\partial \\hat{\\alpha }}\\frac{\\partial P}{\\partial \\hat{\\delta }}-\\frac{\\partial \\beta }{\\partial \\hat{\\delta }}\\frac{\\partial P}{\\partial \\hat{\\alpha }}}{\\frac{\\partial \\beta }{\\partial \\hat{\\alpha }}\\frac{\\partial \\alpha }{\\partial \\hat{\\delta }}-\\frac{\\partial \\beta }{\\partial \\hat{\\delta }}\\frac{\\partial \\alpha }{\\partial \\hat{\\alpha }}}\\right|_{\\hat{\\alpha }=0}$ In order to express Eq.",
"(REF ) more explicitely, one needs to evaluate partial derivatives of $P$ , $\\beta $ and $\\alpha $ with respect to $\\hat{\\alpha }$ and $\\hat{\\delta }$ .",
"Here, primes denote derivatives with respect to $\\hat{\\delta }$ : $\\frac{\\partial P}{\\partial \\hat{\\alpha }}(0,\\hat{\\delta }_{\\rm out})&=& \\hat{z}_0\\hat{\\Omega }_0^{1/3}+\\frac{2}{3}\\frac{\\hat{\\Omega }_1}{\\hat{\\Omega }_0^{2/3}}\\\\\\frac{\\partial P}{\\partial \\hat{\\delta }}(0,\\hat{\\delta }_{\\rm out})&=& \\frac{\\partial P_0}{\\partial \\hat{\\delta }}= \\frac{2}{3}\\frac{\\hat{\\Omega }_0^\\prime }{\\hat{\\Omega }_0^{2/3}}$ $\\frac{\\partial \\beta }{\\partial \\hat{\\alpha }}(0,\\hat{\\delta }_{\\rm out})&=& \\frac{\\hat{r}_1}{\\hat{\\Omega }_0^{1/3}}-\\frac{\\hat{r}_0\\hat{\\Omega }_1}{3\\hat{\\Omega }_0^{4/3}}\\nonumber \\\\&=& \\left( \\frac{\\hat{r}_1}{\\hat{r}_0}-\\frac{\\hat{\\Omega }_1}{3\\hat{\\Omega }_0} \\right) \\beta _0\\\\\\frac{\\partial \\beta }{\\partial \\hat{\\delta }}(0,\\hat{\\delta }_{\\rm out})&=& \\frac{\\hat{r}^\\prime }{\\hat{\\Omega }^{1/3}}-\\frac{\\hat{r}\\hat{\\Omega }^\\prime }{3\\hat{\\Omega }^{4/3}}\\nonumber \\\\&=& \\frac{\\hat{r}_0^\\prime }{\\hat{\\Omega }_0^{1/3}}-\\frac{\\hat{r}_0\\hat{\\Omega }_0^\\prime }{3\\hat{\\Omega }_0^{4/3}}$ $\\frac{\\partial \\alpha }{\\partial \\hat{\\alpha }}(0,\\hat{\\delta }_{\\rm out})&=& \\hat{\\Omega }^{2/3}(0,\\hat{\\delta }_{\\rm out})+ \\frac{2}{3}\\hat{\\alpha }\\frac{\\hat{\\Omega }_1}{\\hat{\\Omega }^{1/3}}(0,\\hat{\\delta }_{\\rm out})\\nonumber \\\\&=& \\hat{\\Omega }_0^{2/3}\\\\\\frac{\\partial \\hat{\\alpha }}{\\partial \\hat{\\delta }}(0,\\hat{\\delta }_{\\rm out})&=& \\left(\\hat{\\alpha }\\frac{\\partial \\hat{\\Omega }^{2/3}}{\\partial \\hat{\\delta }}\\right)(0,\\hat{\\delta }_{\\rm out})=0$ Using the above expressions, Eq.",
"(REF ) becomes: $P_1(\\beta )&=&(6\\hat{\\Omega }_0\\hat{r}_1\\hat{\\Omega }_0^\\prime -4\\hat{r}_0\\hat{\\Omega }_0^\\prime \\hat{\\Omega }_1 \\nonumber \\\\&&+3\\hat{r}_0\\hat{z}_0\\hat{\\Omega }_0\\hat{\\Omega }_0^\\prime -9\\hat{\\Omega }_0^2\\hat{z}_0\\hat{r}_0^\\prime -6\\hat{\\Omega }_0\\hat{r}_0^\\prime \\hat{\\Omega }_1)\\nonumber \\\\&&\\times \\frac{1}{3\\hat{\\Omega }_0^{4/3}\\,(\\hat{r}_0\\hat{\\Omega }_0^\\prime -3\\hat{\\Omega }_0\\hat{r}_0^\\prime )}$ Once expressions for $\\hat{r}_0^\\prime (\\hat{\\delta })$ and $\\hat{\\Omega }_0^\\prime (\\hat{\\delta })$ , given by Eqs.",
"() and (), as well as those for $\\hat{\\Omega }_0$ , $\\hat{r}_0$ , $\\hat{z}_0$ , $\\hat{\\Omega }_1$ and $\\hat{r}_1$ , have been substituted into Eq.",
"(REF ), it provides an explicit expression of $P_1$ in terms of $\\hat{\\delta }_{\\rm out}$ .",
"In the same way as $P_0$ , again using $\\beta (\\hat{\\delta }_{\\rm out})$ given by Eq.",
"(REF ), $P_1$ can then be plotted parametrically as a function of $\\beta $ , as shown on Fig.",
"REF b (dashed curve)."
],
[
"Small outlet radius limit",
"Let us now take the limit of a small needle outlet ($\\beta \\rightarrow 0$ ).",
"The following functions, provided in Section REF , can be evaluated at $\\beta =0$ : $\\hat{r}_0(0)=0 & \\hat{r}_1(0)=\\frac{\\pi }{2} & \\hat{r}_0^\\prime (0)=1\\\\\\hat{z}_0(0)=-2&&\\\\\\hat{\\Omega }_0(0)=1 & \\hat{\\Omega }_1(0)=1 & \\hat{\\Omega }_0^\\prime (0)=0$ Using these values, Eqs.",
"(REF ) and (REF ) yield the values of $P_0$ and $P_1$ in the limit of a very small tube outlet: $P_0(0) &=& 2\\hat{\\Omega }_0^{1/3}(0) = 2\\\\P_1(0) &=& -\\frac{4}{3}$ These values are used as coefficients in Eq.",
"(REF ).",
"A proper expansion at small $\\beta $ is provided below, in Appendix REF ."
],
[
"Small outlet radius expansion",
"Let us now use the decompositions expressed by Eqs.",
"(REF –) and inject them into Eqs.",
"(REF ,REF ,) in order to obtain an expansion for $P(\\alpha ,\\beta )$ to be compared with Eqs.",
"(REF –).",
"Using Eqs.",
"(REF ,REF ): $\\hat{r}_1 &\\simeq & \\frac{\\pi }{2} -\\frac{2}{3}\\hat{\\delta }-\\frac{\\pi }{4}\\hat{\\delta }^2 +{\\cal O}(\\hat{\\delta }^3)\\\\\\hat{r}&\\simeq & \\hat{\\delta }-\\frac{1}{6}\\hat{\\delta }^3+\\hat{\\alpha }\\,\\left(\\frac{\\pi }{2}-\\frac{2}{3}\\hat{\\delta }-\\frac{\\pi }{4}\\hat{\\delta }^2\\right)\\nonumber \\\\&&+{\\cal O}(\\hat{\\delta }^5,\\hat{\\alpha }\\hat{\\delta }^3,\\hat{\\alpha }^2)\\qquad $ Using Eqs.",
"(REF ,REF ): $\\frac{3}{4\\pi } \\hat{V}_1 &\\simeq & \\left(1-\\frac{1}{4}\\hat{\\delta }^2+{\\cal O}(\\hat{\\delta }^3)\\right)\\\\\\frac{3}{4\\pi } \\hat{V}&\\simeq & \\left(1+{\\cal O}(\\hat{\\delta }^4)\\right)+\\hat{\\alpha }\\,\\left(1+{\\cal O}(\\hat{\\delta }^2)\\right)\\qquad \\\\\\left(\\frac{3}{4\\pi } \\hat{V}\\right)^{1/3}&\\simeq & 1+\\frac{1}{3}\\hat{\\alpha }+{\\cal O}(\\hat{\\delta }^4,\\hat{\\alpha }\\hat{\\delta }^2,\\hat{\\alpha }^2)\\qquad $ The position of the outlet is defined by Eq.",
"(REF ), which can be expressed using Eqs.",
"(REF ,REF ): $0&=&\\hat{r}(\\hat{\\alpha },\\hat{\\delta }_{\\rm out})- \\beta \\, \\left(\\frac{3}{4\\pi } \\hat{V}(\\hat{\\alpha },\\hat{\\delta }_{\\rm out})\\right)^{1/3}\\\\0&=& \\hat{\\delta }_{\\rm out}-\\frac{1}{6}\\hat{\\delta }_{\\rm out}^3+\\hat{\\alpha }\\,\\left(\\frac{\\pi }{2}-\\frac{2}{3}\\hat{\\delta }_{\\rm out}-\\frac{\\pi }{4}\\hat{\\delta }_{\\rm out}^2\\right)\\nonumber \\\\&&+{\\cal O}(\\hat{\\delta }_{\\rm out}^5,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^3,\\hat{\\alpha }^2)\\nonumber \\\\&&-\\beta -\\frac{1}{3}\\hat{\\alpha }\\beta +{\\cal O}(\\hat{\\delta }_{\\rm out}^4\\beta ,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2\\beta ,\\hat{\\alpha }^2)\\\\0&=& \\hat{\\delta }_{\\rm out}-\\frac{1}{6}\\hat{\\delta }_{\\rm out}^3 -\\beta \\nonumber \\\\&&+\\hat{\\alpha }\\left( \\frac{\\pi }{2} -\\frac{2}{3}\\hat{\\delta }_{\\rm out}-\\frac{\\pi }{4}\\hat{\\delta }_{\\rm out}^2 -\\frac{1}{3}\\beta \\right)\\quad \\nonumber \\\\&&+{\\cal O}(\\hat{\\delta }_{\\rm out}^5,\\beta \\hat{\\delta }_{\\rm out}^4,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^3,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2\\beta ,\\hat{\\alpha }^2)\\nonumber \\\\0&=& \\hat{\\delta }_{\\rm out}\\left(1 -\\frac{1}{6}\\hat{\\delta }_{\\rm out}^2 -\\frac{2}{3}\\hat{\\alpha }-\\frac{\\pi }{4}\\hat{\\alpha }\\hat{\\delta }_{\\rm out}\\right)\\nonumber \\\\&&-\\left( \\beta -\\hat{\\alpha }\\frac{\\pi }{2} +\\frac{1}{3}\\hat{\\alpha }\\beta \\right)\\nonumber \\\\&&+{\\cal O}(\\hat{\\delta }_{\\rm out}^5,\\beta \\hat{\\delta }_{\\rm out}^4,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^3,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2\\beta ,\\hat{\\alpha }^2)\\qquad $ Multiplying by $&&1+\\frac{1}{6}\\hat{\\delta }_{\\rm out}^2 +\\frac{2}{3}\\hat{\\alpha }+\\frac{\\pi }{4}\\hat{\\alpha }\\hat{\\delta }_{\\rm out}\\nonumber \\\\&&\\qquad +{\\cal O}(\\hat{\\delta }_{\\rm out}^4,\\beta \\hat{\\delta }_{\\rm out}^3,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2,\\hat{\\alpha }\\hat{\\delta }_{\\rm out}\\beta ,\\hat{\\alpha }^2)\\qquad $ and neglecting terms of order $\\hat{\\alpha }^2$ , one obtains: $0 &=& \\hat{\\delta }_{\\rm out}\\left( 1 -\\frac{2}{9}\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2 -\\frac{\\pi }{12}\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^3 \\right)\\nonumber \\\\&&-\\left( \\beta -\\frac{\\pi }{2}\\hat{\\alpha }+\\frac{1}{3}\\hat{\\alpha }\\beta +\\frac{1}{6}\\beta \\hat{\\delta }_{\\rm out}^2 -\\frac{\\pi }{12}\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2 \\right.\\nonumber \\\\&&\\left.",
"+\\frac{1}{18}\\hat{\\alpha }\\beta \\hat{\\delta }_{\\rm out}^2 +\\frac{2}{3}\\hat{\\alpha }\\beta +\\frac{\\pi }{4}\\hat{\\alpha }\\beta \\hat{\\delta }_{\\rm out}\\right)\\nonumber \\\\&&+{\\cal O}(\\hat{\\delta }_{\\rm out}^5,\\beta \\hat{\\delta }_{\\rm out}^4,\\beta ^2\\hat{\\delta }_{\\rm out}^3,\\nonumber \\\\&&\\qquad \\hat{\\alpha }\\hat{\\delta }_{\\rm out}^3,\\hat{\\alpha }\\beta \\hat{\\delta }_{\\rm out}^2,\\hat{\\alpha }\\beta ^2\\hat{\\delta }_{\\rm out},\\hat{\\alpha }^2)$ Hence: $\\hat{\\delta }_{\\rm out}&=& \\beta +\\frac{1}{6}\\beta \\hat{\\delta }_{\\rm out}^2-\\frac{\\pi }{2}\\hat{\\alpha }+\\hat{\\alpha }\\beta \\nonumber \\\\&&+\\frac{\\pi }{4}\\hat{\\alpha }\\beta \\hat{\\delta }_{\\rm out}-\\frac{\\pi }{12}\\hat{\\alpha }\\hat{\\delta }_{\\rm out}^2\\nonumber \\\\&&+{\\cal O}(\\hat{\\delta }_{\\rm out}^5,\\beta \\hat{\\delta }_{\\rm out}^4,\\beta ^2\\hat{\\delta }_{\\rm out}^3,\\nonumber \\\\&&\\qquad \\hat{\\alpha }\\hat{\\delta }_{\\rm out}^3,\\hat{\\alpha }\\beta \\hat{\\delta }_{\\rm out}^2,\\hat{\\alpha }\\beta ^2\\hat{\\delta }_{\\rm out},\\hat{\\alpha }^2)$ This shows that the dominant terms are $\\hat{\\delta }_{\\rm out}\\simeq \\beta -\\frac{\\pi }{2}\\hat{\\alpha }$ .",
"Hence, the terms containing $\\hat{\\delta }_{\\rm out}$ and the neglected terms can be expressed in terms of $\\beta $ : $\\hat{\\delta }_{\\rm out}&=& \\beta +\\frac{1}{6}\\beta \\left(\\beta -\\frac{\\pi }{2}\\hat{\\alpha }\\right)^2-\\frac{\\pi }{2}\\hat{\\alpha }+\\hat{\\alpha }\\beta \\nonumber \\\\&&+\\frac{\\pi }{4}\\hat{\\alpha }\\beta ^2-\\frac{\\pi }{12}\\hat{\\alpha }\\beta ^2\\nonumber \\\\&&+{\\cal O}(\\beta ^5,\\hat{\\alpha }\\beta ^3,\\hat{\\alpha }^2)\\\\\\hat{\\delta }_{\\rm out}&=& \\beta +\\frac{1}{6}\\beta ^3-\\frac{\\pi }{2}\\hat{\\alpha }+\\hat{\\alpha }\\beta \\nonumber \\\\&&+{\\cal O}(\\beta ^5,\\hat{\\alpha }\\beta ^3,\\hat{\\alpha }^2)$ Injecting Eq.",
"(REF ) into Eqs.",
"(,REF ,REF ), we obtain: $\\hat{\\alpha }\\hat{z}_{\\rm out}&=& -\\hat{\\alpha }-\\hat{\\alpha }\\cos \\beta +{\\cal O}(\\hat{\\alpha }^2)\\nonumber \\\\&=& -2\\hat{\\alpha }+\\frac{1}{2}\\hat{\\alpha }\\beta ^2+{\\cal O}(\\hat{\\alpha }\\beta ^4,\\hat{\\alpha }^2)\\\\\\hat{r}_{\\rm out}&=&\\left(\\beta +\\frac{1}{6}\\beta ^3 -\\frac{\\pi }{2}\\hat{\\alpha }+\\hat{\\alpha }\\beta \\right)\\nonumber \\\\&&-\\frac{1}{6}\\left( \\beta -\\frac{\\pi }{2}\\hat{\\alpha }\\right)^3+\\hat{\\alpha }\\,\\left(\\frac{\\pi }{2}-\\frac{2}{3}\\beta -\\frac{\\pi }{4}\\beta ^2\\right)\\nonumber \\\\&&+{\\cal O}(\\beta ^5,\\hat{\\alpha }\\beta ^3,\\hat{\\alpha }^2)\\nonumber \\\\&=& \\beta +0\\beta ^3 +\\frac{1}{3}\\hat{\\alpha }\\beta +0\\hat{\\alpha }\\beta ^2\\nonumber \\\\&&+{\\cal O}(\\beta ^5,\\hat{\\alpha }\\beta ^3,\\hat{\\alpha }^2)\\\\\\frac{\\hat{r}_{\\rm out}}{\\beta } &=& 1 +\\frac{1}{3}\\hat{\\alpha }+{\\cal O}(\\beta ^4,\\hat{\\alpha }\\beta ^2,\\hat{\\alpha }^2)\\qquad $ Injecting Eqs.",
"(REF ,) into Eqs.",
"(REF ,): $P(\\hat{\\alpha },\\beta )&=& 2 -\\frac{4}{3}\\hat{\\alpha }+{\\cal O}(\\beta ^4,\\hat{\\alpha }\\beta ^2,\\hat{\\alpha }^2)\\\\\\alpha (\\hat{\\alpha },\\beta )&=& \\hat{\\alpha }+{\\cal O}(\\hat{\\alpha }\\beta ^4,\\hat{\\alpha }^2)$ Substituting Eq.",
"() into Eq.",
"(REF ): $P(\\alpha ,\\beta )&=& 2 -\\frac{4}{3}\\alpha +{\\cal O}(\\beta ^4,\\alpha \\beta ^2,\\alpha ^2)$ In other words, $P_0(\\beta )=P(0,\\beta )$ and $P_1(\\beta )=(\\partial P/\\partial \\alpha )|_{(0,\\beta )}$ are given by Eqs.",
"(REF ) and (), as announced."
]
] | 1612.05754 |
[
[
"Contrasting Prediction Methods for Early Warning Systems at\n Undergraduate Level"
],
[
"Abstract In this study, we investigate prediction methods for an early warning system for a large STEM undergraduate course.",
"Recent studies have provided evidence in favour of adopting early warning systems as a means of identifying at-risk students.",
"Many of these early warning systems rely on data from students' engagement with Learning Management Systems (LMSs).",
"Our study examines eight prediction methods, and investigates the optimal time in a course to apply an early warning system.",
"We present findings from a statistics university course which has a large proportion of resources on the LMS Blackboard and weekly continuous assessment.",
"We identify weeks 5-6 of our course (half way through the semester) as an optimal time to implement an early warning system, as it allows time for the students to make changes to their study patterns whilst retaining reasonable prediction accuracy.",
"Using detailed (fine-grained) variables, clustering and our final prediction method of BART (Bayesian Additive Regressive Trees) we are able to predict students' final grade by week 6 based on mean absolute error (MAE) to 6.5 percentage points.",
"We provide our R code for implementation of the prediction methods used in a GitHub repository."
],
[
"Introduction",
"Early warning systems to identify at-risk students (of dropping out or failing) are in practical use in large classes and online courses [6], [29], [23].",
"We provide findings from a large first year statistics course in which most of the learning materials are available online and therefore student engagement with them can be measured via the LMS Blackboard.",
"We acknowledge the impact course design, in particular weekly continuous assessment, has on developing early warning systems.",
"We contrast results from eight prediction methods (Random Forest; BART; XGBoost; Principal Components Regression; Support Vector Machine; Neural Network; Multivariate Adaptive Regression Splines; and K-Nearest Neighbours) and the impact of cluster membership (based on student engagement) on reducing prediction error.",
"We reasonably predict a student's final grade as early as week five of a twelve week semester.",
"This study was completed using R software and we have provided our R code on GitHub at https://github.com/ehoward1/Early-Warning-System, and in the appendix.",
"This study forms part of a larger goal to use the predictions we create to allow for more precisely targeted interventions for poorly performing students.",
"Determining the timing at which these interventions should occur is one of the key goals of this study.",
"We would like to intervene as early as possible, but with little information from the LMS and necessarily limited continuous assessment at the start of the semester the predictions are inaccurate.",
"This accuracy increases as we move through the semester but at the price of intervening later and so lessening the impact of any interventions.",
"We monitor the performance of the predictive models on a week by week cumulative basis.",
"For each week, we aim to predict the final percentage mark of the student based on all current information.",
"We do not dichotomise students’ performance to pass/fail unlike many other studies [21], [1], [20] which would lessen the accuracy.",
"At week 6 (of a 12 week semester) we obtain a mean absolute error (MAE) of approximately 6.5 percentage points.",
"The structure of our paper is as follows: in Section 2 we discuss the rationale and prediction methods behind current early warning systems.",
"In Section 3 we outline our approach to developing an accurate prediction method for an early warning system.",
"We extend current research on the development of early warning systems through: using `new' prediction methods including BART; identifying an `optimal time'; and including cluster membership.",
"In Section 4, we discuss the data analytics decisions made and present the results for our course Practical Statistics.",
"Finally we progress to the discussion and conclusion of these results in Section 5."
],
[
"Prediction Modelling for Early Warning Systems",
"In this section, we examine the stages in creating a prediction model for an early warning system (detailed data collection; variable selection; prediction modelling; and clustering).",
"Advancements in learning interfaces allow for fine-grained collection of data.",
"[1] highlight that “fine-grained (microscopic) analytics data should yield better results than coarse-grained (macroscopic)” (p. 589).",
"An example of a coarse-grained variable is total count of resources accessed online.",
"In comparison fine-grained data analytics refers to extracting each log entry of a student, and all the information it contains for example: the number of slides visited; number of successful compilations; and time spent on platform [1].",
"Their argument is that through using more detailed variables, more powerful prediction models can be created.",
"However this trades off against simplicity; simple models with a small number of variables are easier to interpret and understand.",
"Variables based on students' demographic/historic data, continuous assessment results and LMS usage have been collected for early warning systems [31], [23].",
"LMS data can include length of time on a LMS system, number of visits to a module page, contributions to a module discussion thread et cetera.",
"Depending on the prediction models selected, the dataset is reduced to a small number of `important' variables.",
"There are numerous types of prediction models used for learning analytics.",
"[10] note that researchers have produced prediction models by using classification algorithms such as EM, C4.5, Naive Bayes Classifier, and Support Vector Machines.",
"Logistic regression and multiple regression modelling are often used as prediction models [20], [27], with logistic regression being considered the most popular prediction method for educational settings [21].",
"Hierarchical mixed models [15], [31], K-nearest neighbor [21], neural network models [3], and decision tree methods [1] are also methods employed.",
"A common use of prediction models in learning analytics is to identify whether a student will pass or fail the course based on the binary response variable `pass/fail'.",
"The use of a binary response variable dichotomises students' performance percentage marks.",
"Studies using binary response variable include [1], [20], [3].",
"A key point to note is that predictive models are usually applied to a single course rather than used for several courses.",
"Pantucek2013 propose that this may be because each course is structured differently, and therefore dictates what learners are doing.",
"[10] investigate generalised predictive models that can be applied to multiple courses, however they note that the inherent differences in disciplines cause specific variables to be strong for some courses, and weak for other courses.",
"Hence, the nature of the course should be considered before selecting variables for an early warning system.",
"[10] believe “the understanding of practical needs in specific instructional and learning contexts is the primary driver for the development and deployment of learning analytics methods” (p. 83).",
"Clustering also plays a significant role in learning analytics through its ability to identify students' engagement levels or learning strategies statistically.",
"When investigating a blended course [19] identify three patterns of tool-use using k-means clustering: the no-users; the intensive users; and the incoherent users.",
"In comparison, [13] used model-based clustering to identify four clusters of behavioural engagement in a large mathematics course where students have the choice to use lectures or/and online videos.",
"White2016 use Latent Class Analysis to identify four clusters of engagement in a large blended business course.",
"In their discussion they identify what resources each cluster engaged with, and when during the semester these resources were engaged with."
],
[
"Early Warning Systems in Practice",
"One of the best known examples of an early warning system is in Purdue University [25], [8], [23] who introduced `Course Signals' (CS) or a `traffic light system' whereby students can see whether they are likely to succeed in their course based on a traffic light colour on their learner interface.",
"For example a green colour indicates a high likelihood of succeeding.",
"This prediction of success is based on prediction models using all available student background information and LMS interactions.",
"If a student is identified as at-risk, the lecturer has the option of providing corrective measures including: posting of a traffic signal indicator on the student's CMS home page; sending e-mail messages or reminders; sending text messages; referring the student to an academic advisor or academic resource centre; or organising a face-to-face meeting.",
"[22] found that the results of their interventions (based on a control group versus an experimental group) were: students seeking help earlier; lower D's and F's recorded; more B's and C's; and students felt more than a `number', that is less isolated.",
"Other benefits of Course Signals discussed by [25] are students using the subject help desks more, and greater attendance at additional tutorials.",
"One prime reason for the implementation of an early warning system is to detect students at-risk of dropping out of courses.",
"[22] state that most early warning systems rely on midterm grades reported by lecturers.",
"By the time midterms have been corrected it is often far into the semester, and students may have already dropped out.",
"It is crucial that early warning systems operate in the early stages of the semester.",
"However, a balance has to be achieved with the accuracy of the model.",
"As the methods, models, variables and response variable used in identifying at-risk students vary from study to study, it remains difficult to contrast the studies and identify which study has obtained the most accurate results.",
"Results are impacted by the truncating of students' performances to the binary pass/fail variable.",
"Dichotomizing is usually performed for simplicity however this can lead to: lower accuracy through loss of valuable information; a decrease in the predictive power; and in general there is a risk of getting results that may not make sense [7], [24].",
"Many studies have reported results of identifying at-risk students at the end of the course/semester however for early warning systems this is impractical.",
"Ideally we wish to support all students from the beginning of the semester.",
"For a prediction model, the beginning of the semester is too early to identify at-risk students.",
"For early warning systems, a balance needs to be obtained between the increasing accuracy of the system and the diminishing impact of intervening as we move through the semester.",
"In this paper we refer to the balance between the two as the `optimal time'."
],
[
"Research Questions",
"Our study aims to explore developing a prediction model for an early warning system taking into account the benefits of cluster analysis.",
"Furthermore our study aims to identify an `optimal time' in the semester when an early warning system could be implemented.",
"Hence our research questions, in context of Practical Statistics, are: Which prediction methods work best for predicting students' final grades?",
"How do we identify a stage in the semester that can adequately balance the required timing of intervention with the quality of the prediction?",
"What effect do cluster memberships based on student engagement have on prediction error?"
],
[
"Method",
"In this section we discuss the course background of Practical Statistics, as well as the data collection process and analysis used in this study."
],
[
"Course Background Information",
"This study took place in University Dublin (UCD).",
"Many of the large first year courses in UCD start in week 1 with material which links to the country's main State Examination and builds from there.",
"Owing to the large class sizes with mixed ability and the progression of material beyond prior knowledge, it may be several weeks before we can identify students who are struggling with the course.",
"Practical Statistics, a large online undergraduate course aimed at first years, was selected as an example of a STEM course with weekly continuous assessment.",
"It is designed as an introductory course in statistics for a class of mixed ability students.",
"The lecturer allocates 40% of the final mark to continuous assessment and distributes the continuous assessment throughout the course semester to encourage students to continuously engage with the course.",
"Practical Statistics' lectures are completely online but the students have 24 hours of software labs.",
"The continuous assessment is achieved through: lecture questions based on the course material (weeks 1-12; 0.5% per week; included in model from week 3); watching all of the online videos (2%); Minitab lab questions (weeks 3-5; 1% per week; included in model at week 5); R lab questions (weeks 7-11 excluding week 8; 1% per week; included in model at week 11); Minitab lab examination (week 6; 10%; included in model at week 6); and R lab examination (week 12; 15%; included in model at week 12).",
"Answers to lecture questions and lab sheets are submitted to the LMS and automatically marked by the system, with the marks being returned instantaneously.",
"Students have until midnight of the following Sunday to submit answers."
],
[
"Participants",
"In the first semester of 2015/16 there were a total of 144 students registered for Practical Statistics.",
"Students' data was removed from the study if: students opted out of the research study; students did not take the end of semester examination; or students had personal circumstances which affected how they were officially graded for the course.",
"Students with extenuating circumstances were excluded as these circumstances could impact students' continuous assessment and LMS use.",
"This could impact predictions.",
"In accordance with our ethical permissions from UCD, we removed these students rather than investigating individual student's circumstances.",
"Subsequently our analysis sample included 136 participants from Practical Statistics."
],
[
"Data Collection and Measurements",
"Data were recorded for students in regards to three categories: students' background information; continuous assessment; and LMS usage on a fine-grained scale.",
"Background information of students (gender, course type (elective, option or core), registration of students (repeating course etc.",
"), students' year of study, students' programme and Irish/non-Irish) were included as variables to account for differences in educational background and prior experience of students.",
"Online resources (for example videos, lectures slides, pdfs) were grouped into folders based on the material content.",
"In total, there were 15 folders (week 1 course material, ..., week 12 course material, lecture questions solutions, course information, and past examination solutions).",
"For each folder, we included the activity level for the folder for a given week as a variable, for example, in week one student `8979' had an activity level of 12 for the `week 1 course material' (see Table REF ).",
"The dataset was designed to be flexible whereby statistical analysis could be performed to incorporate data up to any stage/point in a semester.",
"We performed statistical analysis for the end of each week in the semester (12 teaching weeks) as well as initially (when only background information was available), the end of revision week, and for the end of semester when the written examination was completed.",
"In total this forms fifteen stages.",
"Table: Example dataset to be used to predict students' final module mark"
],
[
"Prediction Methods",
"K-fold cross-validation has been used in multiple prediction model studies (Wolff et al., 2013; Azcona & Casey, 2015).",
"Our prediction models (Random Forest; BART; XGBoost; Principal Components Regression; Support Vector Machine; Neural Network; Multivariate Adaptive Regression Splines; and K-Nearest Neighbours) were run using 10-fold cross-validation for the same folds.",
"The final percentage grade was used as the response variable.",
"Random Forest (RF) is an ensemble learning method.",
"[2] state “random forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest” (p. 1).",
"For RF regression prediction, the mean prediction of the individual trees is returned.",
"[16] explain that BART is a Bayesian approach to nonparametric function estimation using sums of regression trees which allows for flexibility between non-linear interactions.",
"BART differs from other tree ensemble methods (for example RF) owing to the underlying probability model and use of priors.",
"A benefit of this is that we can create confidence intervals for our predicted values.",
"XGBoost is a popular scalable machine learning system for tree boosting [4].",
"It can handle large datasets as well as sparse matrices.",
"As XGBoost cannot be applied to categorical data, any categorical variables were recoded as binary variables.",
"For XGBoost modelling $\\sqrt{n}$ iterations were run where $n$ is the number of variables.",
"XGBoost was applied to 15 stages in the semester (Initially, week 1, ...).",
"Initially XGBoost was used on 18 variables (where categorical variables were transformed to multiple binary variables).",
"The number of variables and iterations increased on a week by week basis as additional Blackboard data became available.",
"Principal Components Regression (PCR) is a technique that reduces a high dimensional dataset to a lower dimension dataset and then performing regression.",
"It does this by finding linear transformations of the data whereby the maximal amount of variance is retained [14].",
"“Kernel-based learning methods (including Support Vector Machines (SVM)) use an implicit mapping of the input data into a high dimensional feature space defined by a kernel function\" [17].",
"The training of the model is then performed in the feature space.",
"Feedforward Neural Network (NN) is a system of nodes which is an imitation of the human brain.",
"A feedforward neural network consists of nodes in layers providing information forward through the layers using the equation $y_{i}=wx_{i}+b$ .",
"Training neural networks is considered to be difficult [18].",
"Multivariate Adaptive Regression Splines is a non-parametric stepwise regression procedure [9] When including variables, the range of the variable is partitioned into subsets and a constant is applied to each subset for regression.",
"In the backward pass, the model is pruned to limit overfitting.",
"K-Nearest Neighbours (KNN) is a nonparametric method whereby the `k' nearest neighbours or `k' most similar cases impact the prediction/classification of the case of interest [11].",
"In the case of regression, the `k' nearest neighbours response values are averaged with importance weightings being considered.",
"We used MAE between the predicted grade and actual grade as a comparison basis to observe the improvement in the accuracy of the prediction model on a week-to-week basis.",
"This allowed us to identify an `optimal time' for an early warning system to be employed.",
"To improve the accuracy of the initial models, our prediction models were applied to different feature sets which included a combination of: continuous assessment data; background information; as well as varying the levels of LMS data.",
"To further reduce the prediction error we considered: Sunday count variablesThis is discussed further in Section 4.3.; cumulative count variables; and resultant cluster analysis.",
"The feature sets discussed in Section 4 are: Initial Model - Variables include background information, continuous assessment, and LMS activity level per folder No LMS Variables - Variables include background information and continuous assessment Cumulative Variables - Variables include background information, continuous assessment, and cumulative activity level for each individual folder (for Sundays and for weekdays) Cluster Variables - Variables include background information, continuous assessment, cumulative counter of views for each individual folder (for Sundays and for weekdays), and cluster membership variables"
],
[
"Clustering Methods",
"The dataset used for clustering contained fine-grained LMS data (the activity level for each individual folder per week and per Sunday).",
"We use the model-based clustering package mclust [26] to create an additional clustering of our variables.",
"We use this package because of its repeated superior performance compared to other clustering algorithms [26], and its ability to model a wide variety of cluster sizes and shapes.",
"Owing to its model-based nature, an advantage of using mclust is its ability to calculate probability memberships for each individual to each cluster.",
"Clustering was performed for each stage in the semester.",
"The estimated Bayesian Information Criterion (BIC) was compared for the different combinations, and the combination which maximised the BIC was selected.",
"The resultant cluster membership was considered as a variable for prediction modelling."
],
[
"Results",
"We now describe the development of prediction methods for an early warning system.",
"Continuous assessment played an important role in our modelling.",
"When developing an early warning system, we need to account for any delays in the correction of continuous assessment or collection of data for example if a midterm in week 5 takes two weeks to correct, we should include it in week 7.",
"Practical Statistics benefits from the instantaneous nature of online LMS assignments.",
"Through the development of our early warning system, we are able to identify an optimal time (week 5-6) in the Practical Statistics' semester to apply an early warning system."
],
[
"Student Engagement and Continuous Assessment",
"[12] and [5] have suggested that continuous assessment encourages student engagement.",
"As previously mentioned, Practical Statistics was designed to ensure consistent student engagement through having continuous assessment on a weekly basis throughout the course.",
"Figure REF shows that online materials were accessed throughout the semester, however the level of activity, not surprisingly, varied across the semester.",
"Figure: Activity level of online resources per day over the semesterThe deadline for weekly online lecture questions for credit was on Sunday nights, and this corresponds with the weekly peak in online resource activity.",
"These peaks might suggest two types of students: students who study immediately prior to assessments; and students who study in advance of assessment.",
"Similarly, as expected, the time when the greatest number of resources was accessed corresponds to the day of the R lab examination (the Monday of week 12).",
"A similar peak occurs on the Monday of the Minitab lab examination in week 6.",
"This connection between online views and continuous assessment suggests that a key driver of students' interaction with online resources is continuous assessment."
],
[
"Clustering Analysis",
"mclust was applied to several variations of the dataset.",
"Considering the high number of view counts on Sunday, this included investigating the potential of Sunday online activity as separate to weekdayIn this study weekday view counts includes Saturday view counts.",
"online activity.",
"After investigating resultant clusters, mclust was only applied to fine-grained LMS data (the activity level for each individual folder per week and per Sunday).",
"Continuous assessment variables and background information of students, were not included as cluster variables.",
"The resultant clusters identified differences in students' frequency levels of using online resources.",
"In comparison to [19] who divides online resources into tool types, this method is cruder as the clustering is unlikely to pick up subtle differences in students' learning strategies.",
"For example, for week 5 (identified optimal time) the variables used were fine-grained LMS data (the activity level for each folder per week and per Sunday) for weeks 1-5. mclust identified 3 clusters ($n_1=61 $ $(44.9\\%)$ , $n_2=69$ $(50.7\\%)$ and $n_3=6$ $(4.4\\%)$ ).",
"The distinct clusters are best represented in 2D format by boxplots (see Figure REF ) showing the standardised means and spread of the selected variables for each cluster.",
"Three variables (Total Weekday Views (up to week 5), Total Sunday Views (up to week 5), and Final Grade) were selected to show the distinct clusters (see Figure REF ).",
"Figure: Identifying engagement patterns of Practical Statistics through boxplots of selected standardised variables for week 5. for example cluster 3 contains six students who have below average resource usage.Cluster 3 students are students who display below average engagement with resources and have the widest final grade range.",
"[19] categorize these as no-users or low frequency users.",
"Cluster 2 represents the students who have below average resource use on Sunday, and average resource use during the week.",
"In comparison, Cluster 1 represents students who engage above average with resources overall.",
"Despite this high engagement, they have the median final grade.",
"[19] would describe these as the intensive users.",
"Subsequently, as the cluster analysis displayed distinct clusters with various engagement patterns, students' cluster group membership was used as variables in the prediction analysis."
],
[
"Prediction Modelling",
"Initial prediction modelling was performed on the dataset for each week (all variables available up to that date were included - see Initial Model Section 3.3.1) to determine an optimal time for corrective measures.",
"The initial stage (before teaching semester began) and final stage of the semester acted as a baseline for comparison for the power of the prediction model (see Figure REF ).",
"Out of the methods investigated, Neural Networks is clearly the inferior method.",
"Figure: Average MAE per student on a week-by-week basis from multiple out of sample prediction methodsIn Figure REF Principal Components Regression achieves the lowest MAE value (approximately 6 points at week 12).",
"PCR reduces the number of variables before performing regression.",
"An interesting feature of Figure REF is the substantial decrease in error from week 2 to week 3.",
"This decrease in error coincides with the inclusion of continuous assessment in the prediction model (the deadline for week 1 lecture questions was in week 3).",
"This emphasises the role continuous assessment plays as a predictor in online STEM courses.",
"To confirm the importance of continuous assessment we investigated the variable importance of the models.",
"Every model selected continuous assessment variables as the main variables in the model.",
"Figure REF also shows that between weeks 7 and 11 there is relatively little change in the predictive power of the models.",
"Subsequently, the stages up to week 6 were identified as important for further data analysis.",
"For early warning systems, a balance is required between the accuracy of the prediction models and the stage in the semester.",
"The stage in the semester needs to reflect where corrective measures could most effectively be given to students.",
"For Practical Statistics, week 5 is potentially the optimal time for implementing an early warning system.",
"We have included our R code for this in the Appendix with more detailed R code and fictitious datasets available on GitHub at https://github.com/ehoward1/Early-Warning-System.",
"We considered alternative feature sets including removal of the LMS data (which provided slightly less accurate predictions), including cumulative activity level for each folder (Cumulative Variables dataset), and including cluster membership variables.",
"Progressing, we will look at the Cluster Variable dataset in further detail.",
"The Cluster Variables dataset for each student consists of: background information; continuous assessment; and cumulative counter for the activity level of each individual folder (for Sundays and for weekdays) as well as cluster membership variables.",
"While including the cluster variable (in most cases) does not alter the MAE significantly, clustering can provide us with information about student engagement in general which may be of value (see Section 4.2 Clustering Analysis).",
"Figure REF gives the average MAE per student for the Cluster Variables dataset up to the optimal time of week 6.",
"The second substantial decrease in MAE between weeks 4 and 5 corresponds to the second inclusion of continuous assessment (Minitab lab results).",
"Using our BART predictive model we can identify the final mark the student will obtain to approximately a MAE of 6.5 at week 6.",
"We will proceed by discussing in further detail the BART prediction model at week 5 using the Cluster Variable dataset.",
"This dataset consists of 29 explanatory variables.",
"Figure: Improving the Prediction Model by changing the feature dataset to include cumulative variables for LMS interactions and cluster membership variablesWe can visually determine the performance of our predictive model by plotting the predicted final grade against the true final grade for each student.",
"Figure REF shows the predicted grade plotted against the actual grade of each student, both initially and at the end of week 5.",
"An identity line, showing when the predicted grade equals the actual grade (i.e.",
"a perfect prediction), has been included in Figure REF .",
"The initial plot acts as a baseline, displaying how the initial prediction of final grade has very low correlation with the actual grade of students i.e.",
"a poor predictive performance.",
"The initial model relies on a limited number of background/demographic variables.",
"As several students have the same background information, this has resulted in multiple students receiving the same predicted grade.",
"This has resulted in `bands' of predicted grades.",
"In comparison, the second plot's data is quite linear ($R^2$ = 0.74) and tighter to the identity line, with some outliers.",
"It suggests that by week 5 we can make reasonable grade predictions as the grade predictions are strongly correlated to the actual grade.",
"This supports the belief that week 5 is an optimal time to implement an early warning system, and that the selected BART model (Method - Cluster Variables) performs competently.",
"Figure: Scatter Plots showing Predicted Grade versus Actual Grade Initially and at Week 5 via an Out of Sample 10-Fold Cross-Validation"
],
[
"Continuous Assessment",
"The variables used in this study were divided into three categories: students' background information; students' engagement with LMS; and continuous assessment results.",
"Continuous assessment proved unsurprisingly the most important category.",
"Continuous assessment variables were repeatedly chosen as the most important variables by all of the prediction models.",
"Continuous assessment encourages students to engage with a course (Holmes, 2015) and partially accounts for the different levels of LMS interaction throughout the semester.",
"This is observable from the spikes in LMS resource use prior to continuous assessment tests and deadlines (Figure REF ).",
"We suggest the inclusion of consistent continuous assessment in online courses encourages students' engagement over the entire semester (as stated by [5]), and limits the number of students studying only in the weeks prior to the final examination.",
"The addition of continuous assessment also contributes to minimising prediction error when building early warning systems however this should not be the main reason for its inclusion.",
"We hypothesize that a low percentage for continuous assessment would also achieve the same effect provided that the continuous assessment is throughout the semester.",
"This study investigates how to approach developing an accurate prediction model for an early warning systems.",
"The dataset which only had continuous assessment and background information variables, performed comparatively well to the other feature sets, and enjoyed the benefit of being the simplest model.",
"However, by using this model we fail to identify areas of the curriculum where students struggled.",
"A key element in learning analytics is using the resultant analysis for the benefit of the student and teacher.",
"By including the extra LMS variables, we are able to investigate for individual students aspects of the curriculum that they failed to engage with or had overly high engagement with (potentially a sign of a harder concept or an area with which the student struggled).",
"This advantage for the inclusion of LMS variables is considerable, and should be weighed against the simplicity of the `No LMS Variables' data set."
],
[
"Advancements in Developing Early Warning Systems",
"This study summarises the methods employed in developing prediction models for early warning systems, and builds upon the current work.",
"Unlike other studies, we do not dichotomise students' final marks to pass/fail.",
"We discuss how one may reduce the prediction error through: use of fine-grained variables; manipulation of variables; and the inclusion of cluster membership in prediction modelling.",
"The detail provided by fine-grained variables gives more information on students' engagement patterns.",
"Subsequently, we hypothesize that analysis of fine-grained variables will allow for more personalised corrective measures.",
"We have used predictive methods (BART and XGBoost) which are uncommon in the data analytics literature as well as common predictive methods (Neural Networks, K-Nearest Neighbours and Random Forest).",
"We found that decision tree methods perform particularly well (BART and Random Forest).",
"Decision tree methods are suitable when using a large number of variables.",
"Hence BART, a decision tree method, is appropriate when using fine-grained variables.",
"BART may be preferable over other decision tree methods, for example Random Forests, owing to its Bayesian nature which allows for the inclusion of error variance which is independent of tree structure and leaf parameters [16].",
"In our study BART outperformed the other prediction models tested at the optimal time of weeks 5-6.",
"Clustering is not a necessary step in developing prediction models.",
"However, we have shown that clustering can be used to identify distinct student patterns of engagement which can be used to further reduce the prediction error.",
"Also, clustering may help to identify how students approach learning and subsequently be used to provide corrective measures.",
"The method outlined in this study is appropriate for both online courses and large classes with a significant amount of online material.",
"Through combining these methods, we obtain an average prediction error (based on out of sample 10-fold cross validation and MAE) of 6.5 percentage points by week 6.",
"A key part of this study was identifying an `optimal time' to implement an early warning system.",
"Implementing an early warning system too early would result in inaccurate identification of (at-risk) students.",
"In contrast, implementing it too late would diminish the effect of supporting and helping students.",
"Data analysis of prediction models identify week 5/6 as the critical time in the semester for Practical Statistics whereby prediction models have reasonably accurate forecasts balanced with sufficient time to intervene and support at-risk students.",
"Identifying at-risk students is only one stage in an early warning system, another stage is understanding what effective supports should be provided to students.",
"Consequently, our current research involves identifying at-risk students during the `optimal time' in Practical Statistics and examining which feedback/intervention measures are effective for large STEM courses."
],
[
"Limitations",
"The method outlined in this study discusses how to develop an accurate predication model for an early warning system for a course, and how to recognise an optimal time to provide students with corrective measures during a course.",
"Practical Statistics is an example of a STEM course which has continuous assessment distributed weekly throughout the semester.",
"The method discussed in this study may not be an optimal method for other online courses, particularly if the course is from a significantly different academic field.",
"Each course is unique and will have its own unique feature set.",
"STEM based courses, particularly early undergraduate courses, tend to have continuous assessment which ties to the final examination.",
"We believe BART is applicable for these STEM courses.",
"For the purpose of reproducibility, the R code for comparison of the prediction models has been included in Appendix A.",
"Further code for for this study is available on GitHub at https://github.com/ehoward1/Early-Warning-System with fictitious datasets (owing to ethical constraints)."
],
[
"Acknowledgements",
"We would like to thank UCD IT services for providing us with Blackboard data.",
"This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.",
"This study was conducted in accordance with UCD ethics guidelines, and approved by the UCD Ethics Committee under application number LS-15-53-Meehan.",
"Function to run and compare 10-cross fold validation for all prediction methods used in the paper.",
"Fictitious datasets for this function and further R code for this paper are available on GitHub at https://github.com/ehoward1/Early-Warning-System.",
"require(xgboost) require(randomForest) require(bartMachine) require(pls) require(caret) require(magrittr) require(earth) require(nnet) require(car) require(kknn) require(kernlab) prediction_function <- function(dataset, dataset_boost){ # dataset boost is for xgboost \tset.seed(123) \tfolds = createFolds(1:nrow(dataset), k = 10, list = FALSE) \tdataset_boost = apply(dataset_boost, 2, as.numeric) # XGBoost runs for numeric data, not integers \t# Vectors to store error for each prediction methods \tpred_bm = vector(\"numeric\") \tpred_rf = vector(\"numeric\") \tpred_pcr = vector(\"numeric\") \tpred_xg = vector(\"numeric\") \tpred_kknn = vector(\"numeric\") \tpred_svm = vector(\"numeric\") \tpred_nnet = vector(\"numeric\") \tpred_earth = vector(\"numeric\") \tgrades = vector(\"numeric\") \t# Loop through the folds for each prediction method \tfor(i in 1:10){ \t\t# Setting up the data \t\ttrain = dataset[folds!=i,] train_b = dataset_boost[folds!=i,] test = dataset[folds==i,] test_b = dataset_boost[folds==i,] \t\t# BART \t\tbm = bartMachine(train[,-1], train[,1], seed = 123, alpha = 0.95, num_burn_in = 400, \t\tnum_tree = 100, num_rand_samps_in_library = 20000, k = 2, q = 0.9, nu = 3) \t\tpred_bm = c(pred_bm, predict(bm, test[,-1])) \t\t# Random Forest (RF) \t\trf = randomForest(train[,-1], train[,1], ntree = 100) \t\tpred_rf = c(pred_rf, predict(rf, test[,-1])) \t\t# Principle Components Regression (PCR) \t\tpcr = pcr(FINAL~., data = train) \t\tvar_exp = compnames(pcr, explvar = TRUE) \t\tvar_e = unlist(strsplit(var_exp, \"[ (]\")) var_total = 0 # Calculating number of variables to include based on variation explained for(j in 1:length(var_e)) { var_total = var_total + var_e[j] if(var_e[j] < 1 || var_total > 90) { n_comp = j break } } pred_pcr = c(pred_pcr, predict(pcr, test[,-1], ncomp = n_comp)) # Xgboost iter = train_b xg = xgboost(data = as.matrix(train_b[,-1]), label = train_b[,1], eta = 0.5, nround = iter, max.depth = 4, objective = \"reg:linear\") pred_xg = c(pred_xg, predict(xg, as.matrix(test_b[,-1]))) # K-Nearest Neighbours (KNN) kknn = train.kknn(FINAL ~., kmax = 15, distance = 1, data = train) pred_kknn = c(pred_kknn, predict(kknn, test[,-1])) # Neural Network (NN) my.grid = expand.grid(.decay = c(0.05, 0.5, 0.75), .size = c(4, 9)) nnet = train(FINAL~., data = train, linout = 1, method = \"nnet\", maxit = 500, tuneGrid = my.grid, trace = FALSE) pred_nnet = c(pred_nnet, predict(nnet, test[,-1])) # Support Vector Machine (SVM) svm = ksvm(FINAL ~., data = train, C = 5) pred_svm = c(pred_svm, predict(svm, test[,-1])) # Multivariate Adaptive Regression Splines (Splines) earth = train(FINAL~., data = train, method = \"earth\", tuneGrid = data.frame(degree = c(1,2), nprune = 5)) pred_earth = c(pred_earth, predict(earth, test[,-1])) grades = c(grades, test$FINAL) } # Calculating the error for each method error_rf = sum(abs(pred_rf - grades))/nrow(dataset) error_pcr = sum(abs(pred_pcr - grades))/nrow(dataset) error_xg = sum(abs(pred_xg - grades))/nrow(dataset_boost) error_bm = sum(abs(pred_bm - grades))/nrow(dataset) error_earth = sum(abs(pred_earth - grades))/nrow(dataset) error_kknn = sum(abs(pred_kknn - grades))/nrow(dataset) error_nnet = sum(abs(pred_nnet - grades))/nrow(dataset) error_svm = sum(abs(pred_svm - grades))/nrow(dataset) # Returning Values my_list = list(\"MAE_bm\" = error_bm, \"MAE_rf\" = error_rf, \"MAE_pcr\" = error_pcr, \"MAE_xg\" = error_xg, \"MAE_kknn\" = error_kknn, \"MAE_nnet\" = error_nnet, \"MAE_svm\" = error_svm, \"MAE_earth\" = error_earth) return(my_list) }"
]
] | 1612.05735 |
[
[
"Parseval space-frequency localized frames on sub-Riemannian compact\n homogeneous manifolds"
],
[
"Abstract The objective of this article is to describe a construction of Parseval bandlimited and localized frames on sub-Riemannian compact homogeneous manifolds."
],
[
"Introduction",
"The objective of this chapter is to describe a construction of Parseval bandlimited and localized frames in $L_{2}$ -spaces on a class of sub-Riemannian compact homogeneous manifolds.",
"The chapter begins with a brief review in section 2 of some results obtained in [5] where a construction of Parseval bandlimited and localized frames was performed in $L_{2}({\\bf M}),\\:\\:\\:{\\bf M}$ being a compact homogeneous manifold equipped with a natural Riemannian metric.",
"In section we are using a sub-Riemannian structure on the two-dimensional standard unit sphere $\\mathbf {S}^{2}$ to explain the main differences between Riemannian and sub-Riemannian settings.",
"Each of these structures is associated with a distinguished second-order differential operator which arises from a metric.",
"These operators are self-adjoint with respect to the usual normalized invariant (with respect to rotations) measure on $\\mathbf {S}^{2}$ .",
"The major difference between these operators is that in the case of Riemannian metric the operator is elliptic (the Laplace-Beltrami operator $\\mathbf {L}$ ) and in the sub-Riemannian case it is not (the sub-Laplacian $\\mathcal {L}$ ).",
"As a result, the corresponding Sobolev spaces which are introduced as domains of powers of these operators are quite different.",
"In the elliptic case one obtains the regular Sobolev spaces and in sub-elliptic one obtains function spaces (sub-elliptic Sobolev spaces) in which functions have variable smoothness (compared to regular (elliptic) Sobolev smoothness).",
"In section we describe a class of sub-Riemannian structures on compact homogeneous manifolds and consider a construction of Parseval bandlimited and localized frames associated with such structures.",
"Leaving a detailed description of sub-Riemannian structures for later sections we will formulate our main result now.",
"We consider compact homogeneous manifolds ${\\bf M}$ equipped with the so-called sub-Riemannian metric $\\mu (x,y),\\:x,y\\in {\\bf M}$ (see Definition REF ).",
"To formulate our main result we need a definition of a sub-Riemannian lattice on a manifold ${\\bf M}$ .",
"The precise definitions of all the notions used below will be given in the text.",
"Lemma 1.1 Let $ {\\bf M}$ be a compact sub-Riemannian manifold and $\\mu (x, y), \\:x,y\\in {\\bf M}$ be a sub-Riemannian metric.",
"Let $B^{\\mu }(x, r)$ be a ball in this metric with center $x\\in {\\bf M}$ and radius $r$ .",
"There exists a natural number $N^{\\mu }_{{\\bf M}}$ such that for any sufficiently small $r>0$ there exists a set of points $\\mathcal {M}_{r}^{\\mu }=\\lbrace x_{i}\\rbrace $ with the following properties the balls $B^{\\mu }(x_{i}, r/4)$ are disjoint, the balls $B^{\\mu }(x_{i}, r/2)$ form a cover of ${\\bf M}$ , every point of ${\\bf M}$ is covered by not more than $N^{\\mu }_{{\\bf M}}$ balls $B^{\\mu }(x_{i},r)$ .",
"Definition 1 A set $\\mathcal {M}_{r}^{\\mu }=\\lbrace x_{i}\\rbrace $ constructed in the previous lemma will be called a metric $r$ -lattice.",
"The meaning of this definition is that points $\\lbrace x_{i}\\rbrace $ are distributed over ${\\bf M}$ \"almost uniformly\" in the sense of the metric $\\mu $ .",
"We will consider compact homogeneous manifolds ${\\bf M}={\\bf G}/{\\bf H}$ where ${\\bf G} $ is a compact Lie group and ${\\bf H}\\subset {\\bf G}$ is a closed subgroup.",
"Let $dx$ be an invariant (with respect to natural action of ${\\bf G}$ on ${\\bf M}$ ) measure on ${\\bf M}$ and $L_{2}({\\bf M})=L_{2}({\\bf M}, dx)$ the corresponding Hilbert space of complex-valued functions on ${\\bf M}$ with the inner product $\\left<f,g\\right>=\\int _{{\\bf M}}f\\overline{g}dx.$ The notation $\\left|B^{\\mu }(x, r)\\right|$ will be used for the volume of the ball with respect to the measure $dx$ .",
"An interesting feature of sub-Riemann structures is that balls of the same radius may have essentially different volumes (in contrast to the case of the Riemann metric and Riemann measure).",
"In the next Theorem we will mention a sub-elliptic operator (sub-Laplacian) $\\mathcal {L}$ (see the precise definition in (REF )) which is hypoelliptic [7], self-adjoint and non-negative in $L_{2}(\\bf {M})$ .",
"This operator is a natural analog of a Laplace-Beltrami operator in the case of a Riemannian manifold.",
"Theorem 1.2 We assume that ${\\bf M}$ is a compact homogeneous manifold equipped with a sub-Riemann metric $\\mu $ (see section ).",
"Set $r_{j}=2^{-j-1},\\:\\:j=0, 1, 2, ...,$ and let $\\mathcal {M}_{r_{j}}^{\\mu }=\\lbrace x^{j}_{k}\\rbrace _{k=1}^{m_{j}}, \\:\\:\\:x^{j}_{k} \\in {\\bf M},\\:\\:\\:j=0,1,2,..$ be a sequence of metric lattices.",
"With every point $x_{k}^{j}$ one can associate a function $\\Theta _{k}^{j}$ such that: every $ \\Theta ^{j}_{k}$ is bandlimited in the sense that $ \\Theta ^{j}_{k}$ belongs to the space $\\mathbf {E}_{[2^{2j-2}, 2^{2j+2}]}(\\mathcal {L})$ which is the span of all eigenfunctions of $\\mathcal {L}$ whose corresponding eigenvalues belong to the interval $[2^{2j-2}, 2^{2j+2})$ , every $ \\Theta ^{j}_{k}$ is essentially supported around $x_{k}^{j}$ in the sense that for any $N>0$ there exists a constant $C(N)>0$ such that for all $j,k$ one has $\\left| \\Theta ^{j}_{k}(y)\\right|\\le C( N)\\left|B^{\\mu }\\left(x^{j}_{k}, 2^{-j}\\right)\\right|^{-1/2}\\left(1+2^{j}\\mu (x^{j}_{k},y)\\right)^{-N},$ $\\:\\:\\lbrace \\Theta ^{j}_{k}\\rbrace \\:\\:$ is a Parseval frame i.e.",
"for all $f\\in L_{2}({\\bf M})$ $\\sum _{j\\ge 0}\\:\\: \\sum _{1\\le k\\le m_{j}} \\left|\\left<f, \\Theta ^{j}_{k}\\right>\\right|^{2}= \\Vert f\\Vert ^{2}_{L_{2}({\\bf M})},$ and as a consequence of the Parseval property one has the following reconstruction formula $f= \\sum _{j\\ge 0}\\:\\: \\sum _{1\\le k\\le m_{j}} \\left<f, \\Theta ^{j}_{k}\\right>\\Theta ^{j}_{k}.$ In Theorem REF this frame is used to obtain characterization of sub-elliptic Besov spaces in terms of the frame coefficients."
],
[
"Hilbert frames",
"Frames in Hilbert spaces were introduced in [2].",
"Definition 2 A set of vectors $\\lbrace \\psi _{v}\\rbrace $ in a Hilbert space $\\mathcal {H}$ is called a frame if there exist constants $A, B>0$ such that for all $f\\in \\mathcal {H}$ $A\\Vert f\\Vert ^{2}_{2}\\le \\sum _{v}\\left|\\left<f,\\psi _{v}\\right>\\right|^{2} \\le B\\Vert f\\Vert _{2}^{2}.$ The largest $A$ and smallest $B$ are called lower and upper frame bounds.",
"The set of scalars $\\lbrace \\left<f,\\psi _{v}\\right>\\rbrace $ represents a set of measurements of a signal $f$ .",
"To synthesize the signal $f$ from this set of measurements one has to find another (dual) frame $\\lbrace \\Psi _{v}\\rbrace $ and then a reconstruction formula is $f=\\sum _{v}\\left<f,\\psi _{v}\\right>\\Psi _{v}.$ Dual frames are not unique in general.",
"Moreover it is difficult to find a dual frame.",
"However, for frames with $A=B=1$ the decomposition and synthesis of functions can be done with the same frame.",
"In other words $f=\\sum _{v}\\left<f,\\psi _{v}\\right>\\psi _{v}.$ Such frames are known as Parseval frames.",
"For example, three vectors in $\\mathbf {R}^{2}$ with angles $2\\pi /3$ between them whose lengths are all $\\sqrt{2/3}$ form a Parseval frame."
],
[
" Compact homogeneous manifolds",
"The basic information about compact homogeneous manifolds can be found in [8], [9].",
"A homogeneous compact manifold $\\mathbf {{\\bf M}}$ is a $C^{\\infty }$ -compact manifold on which a compact Lie group $\\mathbf {G}$ acts transitively.",
"In this case $\\mathbf {{\\bf M}}$ is necessarily of the form $\\mathbf {G}/\\mathbf {H}$ , where $\\mathbf {H}$ is a closed subgroup of $\\mathbf {G}$ .",
"The notation $L_{2}(\\mathbf {{\\bf M}}),$ is used for the usual Hilbert spaces, where $dx$ is the normalized invariant measure on ${\\bf M}$ .",
"The best known example of such manifold is a unit sphere $\\mathbf {S}^{n}$ in $\\mathbf {R}^{n+1}$ : $\\mathbf {S}^{n}=\\mathbf {SO}(n+1)/\\mathbf {SO}(n)=\\mathbf {G}/\\mathbf {H}.$ If $\\mathbf {g}$ is the Lie algebra of a compact Lie group $\\mathbf {G}$ then there exists a such choice of basis $X_{1},...,X_{d}$ in $\\mathbf {g}$ , for which the operator $-\\mathbf {L}=X_{1}^{2}+X_{2}^{2}+\\ ... +X_{d}^{2}, \\ d=dim\\ \\mathbf {G}$ is a bi-invariant operator on $\\mathbf {G}$ .",
"Here $X_{j}^{2}$ is $X_{j}\\circ X_{j}$ where we identify each $X_{j}$ with a left-invariant vector field on ${\\bf G}$ .",
"We will use the same notation for its image under differential of the quasi-regular representation of $\\mathbf {G}$ in $L_{2}({\\bf M})$ .",
"This operator $\\mathbf {L}$ , which is known as the Casimir operator is elliptic.",
"There are situations in which the operator $\\mathbf {L}$ is, or is proportional to, the Laplace-Beltrami operator of an invariant metric on ${\\bf M}$ .",
"This happens for example, if ${\\bf M}$ is a $n$ -dimensional torus, a compact semi-simple Lie group, or a compact symmetric space of rank one.",
"Since ${\\bf M}$ is compact and the operator $\\mathbf {L}$ is elliptic it has a discrete spectrum $0=\\lambda _{0}<\\lambda _{1}\\le \\lambda _{2}\\le ......$ which goes to infinity without any accumulation points and there exists a complete family $\\lbrace u_{j}\\rbrace $ of orthonormal eigenfunctions which form a basis in $L_{2}({\\bf M})$ .",
"The elliptic differential self-adjoint (in $L_{2}(\\mathbf {M})$ ) operator $\\mathbf {L}$ and its powers $\\mathbf {L}^{s/2}, \\ k\\in {\\bf R_{+}},$ can be extended from $C^{\\infty }(\\mathbf {M})$ to distributions.",
"The family of Sobolev spaces $W_{p}^{s}({\\bf M}),\\ 1\\le p< \\infty ,\\ s\\in {\\bf R},$ can be introduced as subspaces of $L_{p}({\\bf M})$ with the norm $\\Vert f\\Vert _{p}+\\Vert \\mathbf {L}^{s/2}f\\Vert _{p}.$ One can show that when $s=k$ is a natural number this norm is equivalent to the norm $|||f|||_{k,p}=\\Vert f\\Vert _{p}+\\sum _{1\\le i_{1},..., i_{k}\\le d}\\Vert X_{i_{1}}...X_{i_{k}}f\\Vert _{p},\\ 1\\le p<\\infty .$ We assume now that ${\\bf M}$ is equipped with a ${\\bf G}$ -invariant Riemann metric $\\rho $ .",
"The Sobolev spaces can also be introduced in terms of local charts [26].",
"We fix a finite cover $\\left\\lbrace B^{\\rho }(y_{\\nu }, r_{0})\\right\\rbrace $ of ${\\bf M}$ ${\\bf M}=\\bigcup _{\\nu } B^{\\rho }(y_{\\nu }, r_{0}),$ where $B^{\\rho }(y_{\\nu }, r_{0})$ is a ball centered at $y_{\\nu }\\in {\\bf M}$ of radius $r_{0}$ contained in a coordinate chart.",
"Let consider $\\Psi =\\lbrace \\psi _{\\nu }\\rbrace $ be a partition of unity $\\Psi =\\lbrace \\psi _{\\nu }\\rbrace $ subordinate to this cover.",
"The Sobolev spaces $W^{k}_{p}({\\bf M}), k\\in \\mathbf {N}, 1\\le p<\\infty ,$ are introduced as the completion of $C^{\\infty }({\\bf M})$ with respect to the norm $\\Vert f\\Vert _{W^{k}_{p}({\\bf M})}=\\left(\\sum _{\\nu }\\Vert \\psi _{\\nu }f\\Vert ^{p}_{W^{k}_{p}(B^{\\rho }(y_{\\nu }, r_{0}))}\\right) ^{1/p}.$ Remark 1 Spaces $W_{p}^{k}({\\bf M})$ are independent of the choice of elliptic self-ajoint second order differential operator.",
"For every choice of such operators corresponding norms (REF ) will be equivalent.",
"Also, any two norms of the form (REF ) are equivalent [26].",
"The Besov spaces can be introduced via the formula $\\mathcal {B}_{p, q}^{\\alpha }({\\bf M}):= \\left(L_{p}({\\bf M}),W^{r}_{p}({\\bf M})\\right)^{K}_{\\alpha /r,q},$ where $0<\\alpha <r\\in {\\bf N},\\ 1\\le p< \\infty ,\\:\\:\\:1\\le q\\le \\infty .$ Here $K$ is the Peetre's interpolation functor.",
"An explicit norm in these spaces was given in [12]-[17].",
"For the same operators as above $\\lbrace X_{1},...,X_{d}\\rbrace ,\\ d=dim \\ {\\bf G}$ , let $T_{1},..., T_{d}$ be the corresponding one-parameter groups of translation along integral curves of the corresponding vector fields i.e.",
"$T_{j}(\\tau )f(x)=f(\\exp \\tau X_{j}\\cdot x),x\\in {\\bf M}, \\tau \\in \\mathbf {R}, f\\in L_{2}({\\bf M});$ here $\\exp \\tau X_{j}\\cdot x$ is the integral curve of the vector field $X_{j}$ which passes through the point $x\\in {\\bf M}$ .",
"The modulus of continuity is introduced as $\\Omega _{p}^{r}( s, f)= \\sum _{1\\le j_{1},...,j_{r}\\le d}\\sup _{0\\le \\tau _{j_{1}}\\le s}...\\sup _{0\\le \\tau _{j_{r}}\\le s}\\Vert \\left(T_{j_{1}}(\\tau _{j_{1}})-I\\right)...\\left(T_{j_{r}}(\\tau _{j_{r}})-I\\right)f\\Vert _{L_{p}({\\bf M})},$ where $f\\in L_{p}({\\bf M}),\\ r\\in \\mathbf {N}, $ and $I$ is the identity operator in $L_{p}({\\bf M}).$ We consider the space of all functions in $L_{p}({\\bf M})$ for which the following norm is finite: $\\Vert f\\Vert _{L_{p}({\\bf M})}+\\left(\\int _{0}^{\\infty }(s^{-\\alpha }\\Omega _{p}^{r}(s,f))^{q} \\frac{ds}{s}\\right)^{1/q} , 1\\le p<\\infty , 1\\le q\\le \\infty ,$ with the usual modifications for $q=\\infty $ .",
"Theorem 2.1 The norm of the Besov space $B_p^{\\alpha q}({\\bf M})=(L_{p}({\\bf M}),W^{r}_{p}({\\bf M}))^{K}_{\\alpha /r,q},\\ 0<\\alpha <r\\in \\mathbf {N},\\ 1\\le p<\\infty , 1\\le q\\le \\infty ,$ is equivalent to the norm (REF ).",
"Moreover, the norm (REF ) is equivalent to the norm $\\Vert f\\Vert _{W_{p}^{[\\alpha ]}({\\bf M})}+\\sum _{1\\le j_{1},...,j_{[\\alpha ] }\\le d}\\left(\\int _{0}^{\\infty }\\left(s^{[\\alpha ]-\\alpha }\\Omega _{p}^{1}(s,X_{j_{1}}...X_{j_{[\\alpha ]}}f)\\right)^{q}\\frac{ds}{s}\\right)^{1/q}$ if $\\alpha $ is not integer ($[\\alpha ]$ is its integer part).",
"If $\\alpha =k\\in \\mathbf {N}$ is an integer then the norm (REF ) is equivalent to the norm (Zygmund condition) $\\Vert f\\Vert _{W_{p}^{k-1}({\\bf M})}+ \\sum _{1\\le j_{1}, ... ,j_{k-1}\\le d }\\left(\\int _{0}^{\\infty }\\left(s^{-1}\\Omega _{p}^{2}(s,X_{j_{1}}...X_{j_{k-1}}f)\\right)^{q}\\frac{ds}{s}\\right)^{1/q}.$ Definition 3 The space of $\\omega $ -bandlimited functions $\\mathbf {E}_{\\omega }(\\mathbf {L})$ is defined as the span of all eigenfunctions of $\\mathbf {L}$ whose eigenvalues are not greater than $\\omega .$ To describe our construction of frames we need the notion of a lattice on a manifold ${\\bf M}$ equipped with a Riemann metric $\\rho $ .",
"This notion is similar to the corresponding notion introduced in Lemma REF .",
"Lemma 2.2 If ${\\bf M}$ is a compact Riemannian manifold then there exists a natural $N^{\\rho }_{{\\bf M}}$ such that for any sufficiently small $r$ there exists a set of points $\\mathcal {M}^{\\rho }_{r}=\\lbrace x_{i}\\rbrace $ with the following properties the balls $B^{\\rho }(x_{i}, r/4)$ are disjoint, the balls $B^{\\rho }(x_{i}, r/2)$ form a cover of ${\\bf M}$ , the height of the cover by the balls $B^{\\rho }(x_{i},r)$ is not greater than $N^{\\rho }_{{\\bf M}}.$ The meaning of this definition is that points $\\lbrace x_{k}\\rbrace $ distributed over ${\\bf M}$ almost uniformly.",
"In [5] the following theorem was proved for compact homogeneous manifolds considered with invariant Riemann metric.",
"Theorem 2.3 Set $r_{j}=2^{-j-1},\\:\\:j=0, 1, 2, ...,$ and let $\\mathcal {M}_{r_{j}}^{\\rho }=\\lbrace x^{j}_{k}\\rbrace _{k=1}^{m_{j}}, \\:\\:\\:x^{j}_{k} \\in {\\bf M},\\:\\:\\:j=0,1,2,..$ be a sequence of metric lattices.",
"With every point $x_{k}^{j}$ we associate a function $\\Psi _{k}^{j}$ such that: every $\\Psi _{k}^{j}$ is bandlimited in the sense that $ \\Psi ^{j}_{k}$ belongs to the space $\\mathbf {E}_{[2^{2j-2}, 2^{2j+2}]}(\\mathbf {L})$ which is the span of all eigenfunction of $\\mathbf {L}$ whose corresponding eigenvalues belong to the interval $[2^{2j-2}, 2^{2j+2})$ , every $\\Psi _{k}^{j}$ is essentially supported around $x_{k}^{j}$ in the sense that the following estimate holds for every $N >n$ : $\\left| \\Psi ^{j}_{k}(y)\\right|\\le C( N)2^{jn}\\left(1+ 2^{j}\\rho \\ (x_{k}^{j}, y)\\right)^{-N},\\:\\:\\: dim \\ {\\bf M} = n,$ $\\:\\:\\lbrace \\Psi ^{j}_{k}\\rbrace \\:\\:$ is a Parseval frame i.e.",
"for all $f\\in L_{2}({\\bf M})$ $\\sum _{j\\ge 0}\\:\\: \\sum _{1\\le k\\le m_{j}} \\left|\\left<f, \\Psi ^{j}_{k}\\right>\\right|^{2}= \\Vert f\\Vert ^{2}_{L_{2}({\\bf M})},$ and $f= \\sum _{j\\ge 0}\\:\\: \\sum _{1\\le k\\le m_{j}} \\left<f, \\Psi ^{j}_{k}\\right>\\Psi ^{j}_{k}.$ As an important application of Theorem REF one can describe Besov spaces in therms of the frame coefficients [5].",
"Theorem 2.4 The norm of the Besov space $\\Vert f\\Vert _{\\mathcal {B}^{\\alpha }_{p,q}({\\bf M})} ,\\:\\:\\:1\\le p< \\infty , 0<q\\le \\infty $ is equivalent to the norm $\\Vert \\tau (f)\\Vert _{{\\bf b}_{p,q}^{\\alpha }}=\\left(\\sum _{j = 0}^{\\infty }2^{jq(\\alpha -n/p+n/2)}\\left(\\sum _k |\\langle f, \\Psi ^{j}_{k} \\rangle |^p\\right)^{q/p}\\right)^{1/q}.$"
],
[
"Example of $\\mathbf {S}^{2}$ with Riemannian metric",
"We consider ${\\bf M}=\\mathbf {S}^{2}$ .",
"In this case the Casimir operator coincides with the Laplace-Beltrami operator $\\mathbf {L}$ on $\\mathbf {S}^{2}$ and it can be written as a sum of the vector fields on $\\mathbf {S}^{2}$ : $\\mathbf {L}=\\sum _{i,j=1; i<j}^{3} X_{i,j}^{2}=\\sum _{i,j=1; i<j}^{3}(x_{i}\\partial _{x_{j}}-x_{j}\\partial _{x_{i}})^{2}=\\mathbf {L}.$ Let $\\mathcal {P}_{l}$ denote the space of spherical harmonics of degree $l$ , which are restrictions to ${\\bf S}^{2}$ of harmonic homogeneous polynomials of degree $l$ in ${\\bf R}^{3}$ .",
"Each $\\mathcal {P}_{l}$ is the eigenspace of $\\mathbf {L}$ that corresponds to the eigenvalue $-l(l+1)$ .",
"Let $\\mathcal {Y}_{n,l},\\:\\:n=1,...,2l+1$ be an orthonormal basis in $\\mathcal {P}_{l}$ .",
"One has $\\mathbf {L}\\mathcal {Y}_{m,l}=-l(l+1)\\mathcal {Y}_{m,l}.$ Sobolev spaces $W_{p}^{k}(\\mathbf {L}), 1\\le p< \\infty $ , can be introduced as usual by using a system of local coordinates or by using vector fields $X_{i,j}$ : $\\Vert f\\Vert _{W_{p}^{k}({\\bf M})}=\\Vert f\\Vert _{p}+\\sum \\sum \\Vert X_{i,j}....X_{i,j}f\\Vert _{p}$ Corresponding Besov spaces $\\mathcal {B}_{p,q}^{\\alpha }(\\mathbf {L})$ can be described either using local coordinates or in terms of the modules of continuity constructed in terms of one-parameter groups of rotations $e^{\\tau X_{i,j}}$ [12]-[16].",
"In particular, when $p=2$ the Parseval identity for orthonormal bases and the theory of interpolation spaces imply descriptions of the norms of $W_{2}^{k}(\\mathbf {L})$ and $\\mathcal {B}_{2,2}^{\\alpha }(\\mathbf {L})$ in terms of Fourier coefficients: $\\left(\\sum _{l=0}^{\\infty }\\sum _{n=1}^{2l+1}(l+1)^{2\\alpha }|c_{n,l}(f)|^{2}\\right)^{1/2},$ where $c_{n,l}(f)=\\int _{{\\bf S}^{d}}f\\mathcal {Y}_{n,l},\\:\\:\\:f\\in L_{2}({\\bf S}^{d}).$"
],
[
"Sphere $S^{2}$ with a sub-Riemannian metric. A sub-Laplacian and sub-elliptic spaces on {{formula:48dee0b9-8544-41d9-be17-6b9af45823d1}}",
"To illustrate nature of sub-elliptic spaces we will consider the case of two-dimensional sphere ${\\bf S}^{2}$ .",
"We consider on ${\\bf S}^{2}$ two vector fields $Y_{1}=X_{2,3}$ and $Y_{2}=X_{1,3}$ and the corresponding sub-Laplace operator $\\mathcal {L}=Y_{1}^{2}+Y_{2}^{2}.$ Note that since the operators $Y_{1},\\: Y_{2}$ do not span the tangent space to ${\\bf S}^{2}$ along a great circle with $x_{3}=0$ the operator $\\mathcal {L}$ is not elliptic on $\\mathbf {S}^{2}$ .",
"However, this operator is hypoelliptic [7] since $Y_{1},\\:Y_{2}, $ and their commutator $Y_{3}=Y_{1}Y_{2}-Y_{2}Y_{1}=X_{1, 2}$ span the tangent space at every point of ${\\bf S}^{2}$ .",
"Let's compute its corresponding eigenvalues.",
"In the standard spherical coordinates $(\\varphi , \\vartheta )$ spherical harmonics $\\mathcal {Y}_{m,l}(\\varphi , \\vartheta ),\\:\\:l=0,1,...,\\:\\:|m|\\le l$ are proportional to $e^{im\\varphi }P_{l}^{m}(\\cos \\:\\vartheta )$ , where $P_{l}^{m}$ are associated Legendre polynomials.",
"This representation shows that for $Y_{3}=X_{1,2}$ one has $Y_{3}^{2}\\mathcal {Y}_{m,l}=-m^{2}\\mathcal {Y}_{m,l}.$ Since $\\mathcal {Y}_{m,l}$ is an eigenfunction of $\\mathbf {L}$ with the eigenvalue $-l(l+1)$ we obtain $\\mathbf {L}\\mathcal {Y}_{m,l}=-l(l+1)\\mathcal {Y}_{m,l}$ and $\\mathcal {L}\\mathcal {Y}_{m,l}=\\mathbf {L}\\mathcal {Y}_{m,l}-Y_{3}^{2}\\mathcal {Y}_{m,l}=-\\left(l(l+1)-m^{2}\\right)\\mathcal {Y}_{m,l}.$ It shows that spherical functions are eigenfunctions of both $\\mathcal {L}$ and $\\mathbf {L}$ .",
"The graph norm of a fractional power of $\\mathcal {L}$ is equivalent to the norm $\\left(\\sum _{l=0}^{\\infty }\\sum _{|m|\\le l}\\left((l+1)^{2}-m^{2}\\right)^{\\alpha }|c_{m,l}(f)|^{2}\\right)^{1/2},c_{m,l}(f)=\\int _{{\\bf S}^{d}}f\\mathcal {Y}_{m,l},\\:\\:\\:f\\in L_{2}(\\mathbf {L}).$ Note that these spaces $W_{2}^{\\alpha }(\\mathcal {L})$ are exactly the Besov spaces $\\mathcal {B}^{\\alpha }_{2, 2}(\\mathcal {L})$ .",
"We introduce subelliptic (anisotropic) Sobolev space $W_{2}^{\\alpha }(\\mathcal {L}),\\:\\:\\alpha \\ge 0,$ as the domain of $\\mathcal {L}^{\\alpha }$ with the graph norm and define Besov spaces $\\mathcal {B}_{2,q}^{\\alpha }(\\mathcal {L})$ as $\\mathcal {B}^{\\alpha }_{2,q}(\\mathcal {L})=( L_{2}({\\bf S}^{2}),W^{r}_{2}(\\mathcal {L}))^{K}_{\\theta , q},\\:\\:\\: 0<\\theta =\\alpha /r<1,\\:\\:\\:1\\le q\\le \\infty .$ where $K$ is the Peetre's interpolation functor.",
"Note that vector fields $Y_{1}, Y_{2}$ span the tangent space to $\\mathbf {S}^{2}$ at every point away from a great circle $x_{3}=0$ .",
"For this reason around such points a function belongs to the domain of $\\mathcal {L}$ if and only if it belongs to the regular Sobolev space $W_{2}(\\mathbf {L})$ .",
"At the same time the fields $Y_{1},\\: Y_{2}$ do not span the tangent space to ${\\bf S}^{2}$ along a great circle with $x_{3}=0$ .",
"However, the fields $Y_{1}, Y_{2}$ and their commutator $ Y_{3}=Y_{1}Y_{2}-Y_{2}Y_{1}=X_{1, 2}$ do span the tangent space along $x_{3}=0$ .",
"This fact implies that along the circle $x_{3}=0$ , functions in the spaces $W_{2}^{r}(\\mathcal {L})$ and $\\mathcal {B}^{\\alpha }_{2,q}(\\mathcal {L})$ are loosing $1/2$ in smoothness compared to their smoothness at other points on ${\\bf S}^{2}$ .",
"In other words, the following embeddings hold true $W_{2}^{\\alpha }(\\mathbf {L})\\subset W_{2}^{\\alpha }(\\mathcal {L})\\subset W_{2}^{\\alpha /2}(\\mathbf {L}),$ $\\mathcal {B}^{\\alpha }_{2,q}({\\mathbf {L}})\\subset \\mathcal {B}^{\\alpha }_{2,q}(\\mathcal {L})\\subset \\mathcal {B}^{\\alpha /2}_{2,q}({\\mathbf {L}}),$ which follow from a much more general results in [25], [11], [16].",
"We would like to stress that subelliptic function spaces are different from the usual (elliptic) spaces.",
"For example, if $W_{2}^{\\alpha }(\\mathbf {L}) $ is the regular Sobolev space than general theory implies the embeddings $W_{2}^{\\alpha }(\\mathcal {L})\\subset W_{2}^{\\alpha /2}(\\mathbf {L}),\\:\\:\\:\\mathcal {B}^{\\alpha }_{2,q}(\\mathcal {L})\\subset \\mathcal {B}^{\\alpha /2}_{2,q}(\\mathbf {L}).$ As the following Lemma shows, these embeddings are generally sharp.",
"Lemma 3.1 For every $\\alpha >0$ and $\\delta >\\alpha /2$ there exists a function that belongs to $W_{2}^{\\alpha }(\\mathcal {L})$ but does not belong to $W_{2}^{\\delta }(\\mathbf {L}).$ For a $\\delta >\\alpha /2>0$ pick any $ \\gamma $ that satisfies the inequalities $-\\frac{1}{2}-\\delta <\\gamma <-\\frac{1}{2}-\\frac{\\alpha }{2}$ Let $c_{n,l}$ be a sequence such that $c_{n,l}=0$ if $n\\ne l$ and $c_{l,l}=(2l+1)^{\\gamma }$ .",
"For a function with such Fourier coefficients the norm (REF ) is finite since $\\sum _{l=0}^{\\infty }(2l+1)^{\\alpha }(2l+1)^{2\\gamma }=\\sum _{l=0}^{\\infty }(2l+1)^{\\alpha +2\\gamma }<\\infty ,\\:\\:\\: \\alpha +2\\gamma <-1,$ but the norm (REF ) is infinite $\\sum _{l=0}^{\\infty }(2l+1)^{2\\delta }(2l+1)^{2\\gamma }=\\sum _{l=0}^{\\infty }(2l+1)^{2(\\delta +\\gamma )},\\:\\:\\:2(\\delta +\\gamma )>-1.$"
],
[
"Sub-Riemannian structure on compact homogeneous manifolds",
"Let $\\mathbf {M}=\\mathbf {G}/\\mathbf {H}$ be a compact homogeneous manifold and ${\\bf X}=\\lbrace X_{1},\\ ...,X_{d}\\rbrace $ be a basis of the Lie algebra $\\mathbf {g}$ , the same as in (REF ).",
"Let ${\\bf Y}=\\lbrace Y_{1},...,Y_{m}\\rbrace $ be a subset of ${\\bf X}=\\lbrace X_{1},\\ ...,X_{d}\\rbrace $ such that $Y_{1},...,Y_{m}$ and all their commutators $Y_{j,k}=[Y_{j}, \\:Y_{k}]=Y_{j}Y_{k}-Y_{k}Y_{j},\\:\\:\\:Y_{j_{1},...,j_{n}}=[Y_{j_{1}},[....[Y_{j_{n-1}}, Y_{j_{n}}]...]],$ of order $n\\le Q$ span the entire algebra $\\mathbf {g}$ .",
"Let $Z_{1}=Y_{1}, Z_{2}=Y_{2}, ... , Z_{m}=Y_{m}, \\:\\:\\:... \\:\\:\\:, Z_{N},$ be an enumeration of all commutators (REF ) up to order $n\\le Q$ .",
"If a $Z_{j}$ corresponds to a commutator of length $n$ we say that $deg(Z_{j})=n$ .",
"Images of vector fields (REF ) under the natural projection $p: \\mathbf {G}\\rightarrow \\mathbf {M}=\\mathbf {G}/\\mathbf {H}$ span the tangent space to $\\mathbf {M}$ at every point and will be denoted by the same letters.",
"Definition 4 A sub-Riemann structure on $\\mathbf {M}=\\mathbf {G}/\\mathbf {H}$ is defined as a set of vectors fields on $\\mathbf {M}$ which are images of the vector fields (REF ) under the projection $p$ .",
"They can also be identified with differential operators in $L_{p}({\\bf M}),\\:1\\le p<\\infty ,$ under the quasi-regular representation of ${\\bf G}$ .",
"One can define a non-isotropic metric $\\mu $ on ${\\bf M}$ associated with the fields $\\lbrace Y_{1},...,Y_{m}\\rbrace $ .",
"Definition 5 [11] Let $C(\\epsilon )$ denote the class of absolutely continuous mappings $\\varphi : [0,1]\\rightarrow {\\bf M}$ which almost everywhere satisfy the differential equation $\\varphi ^{^{\\prime }}(t)=\\sum _{j=1}^{m}b_{j}(t)Z_{j}(\\varphi (t)),$ where $|b_{j}(t)|<\\epsilon ^{deg(Z_{j})}$ .",
"Then we define $\\mu (x,y)$ as the lower bound of all such $\\epsilon >0$ for which there exists $\\varphi \\in C(\\epsilon )$ with $\\varphi (0)=x,\\: \\varphi (1)=y$ .",
"The corresponding family of balls in ${\\bf M}$ is given by $B^{\\mu }(x,\\epsilon )=\\lbrace y\\in {\\bf M} : \\ \\mu (x,y)<\\epsilon \\rbrace .$ These balls reflect the non-isotropic nature of the vector fields $Y_{1},...,Y_{m}$ and their commutators.",
"For a small $\\epsilon >0$ ball $B^{\\mu }(x,\\epsilon )$ is of size $\\epsilon $ in the directions $Y_{1},...,Y_{m}$ , but only of size $\\epsilon ^{n}$ in the directions of commutators of length $n$ .",
"It is known [11] that the following property holds for certain $c=c(Y_{1},...,Y_{m}),\\: C=C(Y_{1},...,Y_{m})$ : $c\\rho (x,y)\\le \\mu (x,y)\\le C\\left(\\rho (x,y)\\right)^{1/Q}$ where $\\rho $ stands for an $\\mathbf {G}$ -invariant Riemannian metric on ${\\bf M}=\\mathbf {G}/\\mathbf {H}$ .",
"We will be interested in the following sub-elliptic operator (sub-Laplacian) $-\\mathcal {L}=Y_{1}^{2}+...+Y_{m}^{2}$ which is hypoelliptic [7] self-adjoint and non-negative in $L_{2}(\\bf {M})$ .",
"Definition 6 The space of $\\omega $ -bandlimited functions $\\mathbf {E}_{\\omega }(\\mathcal {L})$ is defined as the span of all eigenfunctions of $\\mathcal {L}$ whose eigenvalues are not greater than $\\omega .$ Due to the uncertainty principle bandlimited functions in $\\mathbf {E}_{\\omega }(\\mathcal {L})$ are not localized on ${\\bf M}$ in the sense that their supports coincide with ${\\bf M}$ .",
"Using the operator $\\mathcal {L}$ we define non-isotropic Sobolev spaces $W_{p}^{k}(\\mathcal {L}),\\:\\:1\\le p<\\infty , $ and non-isotropic Besov spaces $\\mathcal {B}^{\\alpha }_{p,q}(\\mathcal {L}),\\: \\:1\\le p<\\infty ,\\:1\\le q\\le \\infty ,$ by using formulas (REF ) and (REF ) respectively."
],
[
"Product property for subelliptic Laplace operator",
"The results of this section play a crucial role in our construction of the Parseval frames.",
"In what follows we consider previously defined operators $-\\mathbf {L}=X_{1}^{2}+X_{2}^{2}+\\ ... +X_{d}^{2}, \\ d=dim\\ \\mathbf {G},$ and $-\\mathcal {L}=Y_{1}^{2}+...+Y_{m}^{2}, \\ m<d,$ as differential operators in $L_{2}(\\bf {M})$ .",
"Lemma 5.1 [5] If ${\\bf M}={\\bf G}/{\\bf H}$ is a compact homogeneous manifold then for any $f$ and $g$ in ${\\mathbf {E}}_{\\omega }(\\mathbf {L})$ , their product $fg$ belongs to ${\\mathbf {E}}_{4d\\omega }(\\mathbf {L})$ , where $d$ is the dimension of the group ${\\bf G}$ .",
"For every $X_{j}$ one has $X_{j}^{2}(fg)=f(X_{j}^{2}g)+2(X_{j}f)(X_{j}g)+g(X_{j}^{2}f).$ Thus, the function ${\\mathbf {L}}^{k}\\left(fg\\right)$ is a sum of $(4d)^{k}$ terms of the form $(X_{i_{1}}...X_{i_{m}}f)(X_{j_{1}}...X_{j_{2k-m}}g).$ This implies that $\\left\\Vert \\mathbf {L}^{k}\\left(fg\\right)\\right\\Vert _{\\infty }\\le (4d)^{k}\\sup _{0\\le m\\le 2k}\\sup _{x,y\\in {\\bf M}}\\left|X_{i_{1}}...X_{i_{m}}f(x)\\right|\\left|X_{j_{1}}...X_{j_{2k-m}}g(y)\\right|.$ Let us show that for all $f,g \\in {\\bf E}_{\\omega }({\\mathbf {L}})$ the following inequalities hold: $ \\Vert X_{i_{1}}...X_{i_{m}}f\\Vert _{L_{2}({\\bf M})}\\le \\omega ^{m/2}\\Vert f\\Vert _{L_{2}({\\bf M})}$ and $\\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{L_{2}({\\bf M})}\\le \\omega ^{(2k-m)/2}\\Vert g\\Vert _{L_{2}({\\bf M})}.$ By construction (see (REF )) the operator $-{\\mathbf {L}}=X_{1}^{2}+\\ ...+X_{d}^{2}$ commutes with every $X_{j}$ and the same is true for $(-\\mathbf {L})^{1/2}$ .",
"From here one can obtain the following equality: $\\Vert {\\mathbf {L}}^{s/2}f\\Vert _{L_{2}({\\bf M})}^{2}=\\sum _{1\\le i_{1},...,i_{s}\\le d}\\Vert X_{i_{1}}...X_{i_{s}}f\\Vert _{L_{2}({\\bf M})}^{2},\\ s\\in \\mathbf {N}, $ which implies the estimates (REF ) and (REF ).",
"The formula (REF ) along with the formula $\\Vert {\\mathbf {L}}^{m/2}f\\Vert _{L_{2}({\\bf M})}\\le \\omega ^{m/2}\\Vert f\\Vert _{L_{2}({\\bf M})}.$ imply the estimate $\\Vert {\\mathbf {L}}^{k}(fg)\\Vert _{L_{2}({\\bf M})}\\le (4d)^{k}\\sup _{0\\le m\\le 2k}\\Vert X_{i_{1}}...X_{i_{m}}f\\Vert _{L_{2}({\\bf M})}\\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{\\infty }\\le (4d)^{k}\\omega ^{m/2}\\Vert f\\Vert _{L_{2}({\\bf M})}\\sup _{0\\le m\\le 2k}\\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{\\infty }.$ Using the Sobolev embedding Theorem and the elliptic regularity of $\\mathbf {L}$ , we obtain for every $s>\\frac{dim {\\bf M}}{2}$ $\\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{\\infty }\\le C({\\bf M})\\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{W_{2}^{s}({\\bf M})}\\le C({\\bf M})\\left\\lbrace \\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{L_{2}({\\bf M})}+\\Vert \\mathbf {L}^{s/2}X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{L_{2}({\\bf M})}\\right\\rbrace ,$ where $W_{2}^{s}({\\bf M})$ is the Sobolev space of $s$ -regular functions on ${\\bf M}$ .",
"The estimate (REF ) gives the following inequality: $\\Vert X_{j_{1}}...X_{j_{2k-m}}g\\Vert _{\\infty }\\le C({\\bf M})\\left\\lbrace \\omega ^{k-m/2}\\Vert g\\Vert _{L_{2}({\\bf M})}+\\omega ^{k-m/2+s}\\Vert g\\Vert _{L_{2}({\\bf M})}\\right\\rbrace \\le C({\\bf M})\\omega ^{k-m/2}\\left\\lbrace \\Vert g\\Vert _{L_{2}({\\bf M})}+\\omega ^{s/2}\\Vert g\\Vert _{L_{2}({\\bf M})}\\right\\rbrace =C({\\bf M},g,\\omega ,s)\\omega ^{k-m/2},\\:\\:\\:s>\\frac{dim\\ {\\bf M}}{2}.$ Finally we have the following estimate: $\\Vert \\mathbf {L}^{k}(fg)\\Vert _{L_{2}({\\bf M})}\\le C({\\bf M},f,g,\\omega ,s)(4d\\omega )^{k},\\:\\:\\:s>\\frac{dim \\ {\\bf M}}{2},\\:\\:k\\in \\mathbf {N},$ which leads to our result.",
"Lemma 5.2 There exist positive $c,\\:C$ such that for $\\omega >1$ the following embeddings hold ${\\bf E}_{\\omega }(\\mathcal {L})\\subset {\\bf E}_{c\\omega ^{Q}}(\\mathbf {L}) ,$ ${\\bf E}_{\\omega }(\\mathbf {L}) \\subset {\\bf E}_{C\\omega }(\\mathcal {L}).$ There exists a constant $a=a(\\mathbf {L},\\:\\mathcal {L})$ such that for all $f$ in the Sobolev space $W_{2}^{Q}(\\mathbf {M})$ [11] $\\Vert \\mathbf {L}f\\Vert \\le a\\Vert (I+\\mathcal {L})^{Q}f\\Vert .$ Since $\\mathbf {L}$ belongs to the center of the enveloping algebra of the Lie algebra $\\mathbf {g}$ it commutes with $\\mathcal {L}$ .",
"Thus one has for sufficiently smooth $f$ : $\\Vert \\mathbf {L}^{l}f\\Vert \\le a^{l}\\Vert (I+\\mathcal {L})^{Q l}f\\Vert ,\\:\\: l\\in \\mathbf {R}.$ It implies that if $f\\in {\\bf E}_{\\omega }(\\mathcal {L})$ , then for $\\omega \\ge 1$ $\\Vert \\mathbf {L}^{l}f\\Vert \\le a^{l}\\Vert (I+\\mathcal {L})^{Q l}f\\Vert \\le \\left(a(1+\\omega )^{Q}\\right)^{l}\\Vert f\\Vert \\le \\left(2a\\omega ^{Q}\\right)^{l}\\Vert f\\Vert ,\\:\\: l\\in \\mathbf {R},$ which shows that $f\\in {\\bf E}_{2a\\omega ^{Q}}(\\mathbf {L})$ .",
"Conversely, since for some $b=b(\\mathbf {L},\\:\\mathcal {L})$ $\\Vert \\mathcal {L}f\\Vert \\le b\\Vert (I+\\mathbf {L})f\\Vert ,\\:\\:f\\in W_{2}^{2}(\\mathbf {M}),$ we have $\\Vert \\mathcal {L}^{l}f\\Vert \\le b^{l}\\Vert (I+\\mathbf {L})^{l}f\\Vert ,\\:\\:f\\in W_{2}^{2l}(\\mathbf {M}),$ and for $f\\in {\\bf E}_{\\omega }(\\mathbf {L})$ $\\Vert \\mathcal {L}^{l}f\\Vert \\le b^{l}\\Vert (I+\\mathbf {L})^{l}f\\Vert \\le \\left(b(1+\\omega )\\right)^{l}\\Vert f\\Vert \\le (2b\\omega )^{l}\\Vert f\\Vert ,\\:\\:f\\in W_{2}^{2l}(\\mathbf {M}).$ The product property of bandlimited functions is described in the following Theorem.",
"Theorem 5.3 There exists a constant $C_{0}=C_{0}(\\mathcal {L})>0$ such that for any $f,\\:g\\in {\\bf E}_{\\omega }(\\mathcal {L})$ the product $fg$ belongs to ${\\bf E}_{C_{0}\\omega ^{Q}}(\\mathcal {L})$ .",
"If $f,g\\in {\\bf E}_{\\omega }(\\mathcal {L})$ then $f,g\\in {\\bf E}_{c\\omega ^{Q}}(\\mathbf {L})$ .",
"According to Lemma REF their product $fg$ belongs to ${\\bf E}_{4dc\\omega ^{Q}}(\\mathbf {L})$ which implies that for some $C_{0}=C_{0}(\\mathcal {L})$ the product $fg$ belongs to ${\\bf E}_{C_{0}\\omega ^{Q}}(\\mathcal {L})$ ."
],
[
"Positive cubature formulas on sub-Riemannian manifolds",
"Now we are going to prove existence of cubature formulas which are exact on $\\mathbf {E}_{\\omega }(\\mathcal {L})$ , and have positive coefficients of the right size.",
"Let $\\mathcal {M}_{r}=\\lbrace x_{k}\\rbrace $ be a $r$ -lattice and $\\lbrace B^{\\mu }(x_{k},r)\\rbrace $ be an associated family of balls that satisfy only properties (1) and (2) of Lemma REF .",
"We define $U_{1}=B^{\\mu }(x_{1}, r/2)\\setminus \\cup _{i,\\:i\\ne 1}B^{\\mu }(x_{i}, r/4),$ and $U_{k}=B^{\\mu }(x_{k}, r/2)\\setminus \\left(\\cup _{j<k}U_{j}\\cup _{i,\\:i\\ne k}B^{\\mu }(x_{i}, r/4)\\right).$ One can verify the following properties.",
"Lemma 6.1 The sets $\\left\\lbrace U_{k}\\right\\rbrace $ form a disjoint measurable cover (up to a set of measure zero) of $\\mathbf {M}$ and $B^{\\mu }(x_{k}, r/4)\\subset U_{k}\\subset B^{\\mu }(x_{k}, r/2)$ We have the following Plancherel-Polya inequalities [20], [21].",
"Theorem 6.2 There exist positive constants $a_{1}=a_{1}(\\mathbf {M, Y}), a_{2}=a_{2}(\\mathbf {M, Y})$ , and $a_{0}=a_{0}(\\mathbf {M,Y})$ such that, if for a given $\\omega >0$ one has $0<r<a_{0}\\omega ,$ then for any metric $r$ -lattice $\\mathcal {M}_{r}=\\lbrace x_{k}\\rbrace $ the following inequalities hold $a_{1}\\sum _{k}|U_{k}||f(x_{k})|^{2}\\le \\Vert f\\Vert _{L_{2}(\\mathbf {M})}\\le a_{2}\\sum _{k} |U_{k}||f(x_{k})|^{2}, $ for every $f\\in \\mathbf {E}_{\\omega }(\\mathcal {L}).$ One has $|f(x)|\\le |f(x_{k})|+|f(x)-f(x_{k})|,$ $\\int _{U_{k}}|f(x)|^{2}dx\\le 2\\left(|U_{k}||f(x_{k})|^{2}+\\int _{U_{k}}|f(x)-f(x_{k})|^{2}dx\\right),$ and $\\Vert f\\Vert ^{2}\\le \\sum _{k}\\int _{U_{k}}|f(x)|^{2}dx\\le 2\\left( \\sum _{k}|U_{k}||f(x_{k})|^{2}+ \\sum _{k}\\int _{U_{k}}|f(x)-f(x_{k})|^{2}dx\\right).$ Take an $X\\in \\mathbf {g},\\:\\:|X|=1, $ for which $ \\exp \\ tX\\cdot x_{k} =x$ for some $t\\in \\mathbf {R}$ .",
"Since every such vector field (as a field on $\\mathbf {M}$ ) is a linear combination of the fields $[Y_{i_{1}},...[Y_{i_{l-1}}, Y_{i_{l}}]...], 1\\le l\\le Q, 1\\le i_{j}\\le m$ , the Newton-Leibniz formula applied to a smooth $f$ along the corresponding integral curve joining $x$ and $x_{k}$ gives $|f(x)-f(x_{k})|^{2}\\le Cr^{2}\\sum _{l=1}^{Q} \\sum _{1\\le i_{1},i_{2},... i_{l}\\le m}\\left(\\sup _{y\\in B^{\\mu }(x_{k},r/2)}\\left|Y_{i_{1}}Y_{i_{2}}... Y_{i_{l}}f(y)\\right|\\right)^{2}.$ Applying anisotropic version of the Sobolev inequality [11] we obtain $|f(x)-f(x_{k})|^{2}\\le C r^{2}\\sum _{l=1}^{Q} \\sum _{1\\le i_{1},i_{2},... i_{l}\\le m}\\left(\\sup _{y\\in B^{\\mu }(x_{k},r/2)}\\left|Y_{i_{1}}Y_{i_{2}}... Y_{i_{l}}f(y)\\right|\\right)^{2}\\le $ $C r^{2} \\sum ^{Q}_{l=0}\\sum _{1\\le i_{1},i_{2}, ...,i_{l}\\le m}\\Vert Y_{i_{1}}Y_{i_{2}}...Y_{i_{l}}f\\Vert ^{2}_{H^{Q/2+\\varepsilon }(B^{\\mu }(x_{k}, r/2))},$ where $ x\\in U_{k},\\:\\:\\varepsilon > 0,\\:\\:C=C(\\varepsilon ).$ Next, $\\sum _{k}\\int _{B^{\\mu }(x_{k}, r/2)}|f(x)-f(x_{k})|^{2}dx \\le $ $C r^{n+2} \\sum ^{Q}_{l=0}\\sum _{1\\le i_{1},i_{2}, .. i_{l}\\le m}\\sum _{k}\\Vert Y_{i_{1}}... Y_{i_{l}}f\\Vert ^{2}_{H^{Q/2+\\varepsilon }(B^{\\mu }(x_{k}, r/2))}\\le $ $C r^{n+2}\\sum ^{Q}_{l=0}\\sum _{1\\le i_{1},...,i_{l}\\le m}\\Vert Y_{i_{1}}... Y_{i_{l}}f\\Vert ^{2}_{H^{Q/2+\\varepsilon }(\\mathbf {M})}\\le C r^{n+2} \\left(\\Vert f\\Vert ^{2}+\\Vert \\mathcal {L}^{Q}f\\Vert ^{2}\\right).$ All together we obtain the inequality $\\Vert f\\Vert ^{2}\\le 2 \\sum _{k}|U_{k}||f(x_{k})|^{2}+Cr^{n+2} \\left(\\Vert f\\Vert ^{2}+\\Vert \\mathcal {L}^{Q}f\\Vert ^{2}\\right).$ Note that for $f\\in \\mathbf {E}_{\\omega }(\\mathcal {L})$ $\\Vert \\mathcal {L}^{Q}f\\Vert \\le C\\omega ^{Q}\\Vert f\\Vert .$ Thus, if for a given $\\omega >0$ we pick an $r>0$ a way that $Cr^{n+2}(1+\\omega )^{Q}<1$ then for a certain $C_{1}=C_{1}(M)>0$ one obtains the right-hand side of (REF ) $\\Vert f\\Vert ^{2}\\le C_{1} \\sum _{k}|U_{k}||f(x_{k})|^{2}.$ The left-hand side of (REF ) follows from the Sobolev and Bernstein inequalities.",
"The Plancherel-Polya inequalities (REF ) can be used to prove the so-called sub-elliptic positive cubature formula.",
"The proof goes along the same lines as in [5], [24], (see also [3], [1]).",
"The precise statement is the following.",
"Theorem 6.3 There exists a constant $a=a(\\mathbf {M, Y})>0$ such that for a given $\\omega >0$ if $r=a\\omega ^{-1}$ then for any $r$ -lattice $ \\mathcal {M}_{r}=\\lbrace x_{k}\\rbrace $ there exist strictly positive coefficients $\\lbrace \\alpha _{k}\\rbrace $ , for which the following equality holds for all functions in $ \\mathbf {E}_{\\omega }(\\mathcal {L})$ : $\\int _{\\mathbf {M}}fdx=\\sum _{k}f(x_{k}) \\alpha _{k} .$ Moreover, there exists constants $\\ b_{1}>0, \\ b_{2}>0, $ such that the following inequalities hold: $b_{1}|U_{k}|\\le \\alpha _{k}\\le b_{2}|U_{k}|,$ where the sets $U_{k}$ are defined in (REF )."
],
[
"Space localization of kernels",
"According to the spectral theorem if $F$ is a Schwartz function on the line, then there is a well defined operator $F(\\mathcal {L})$ in the space $L_{2}(\\mathbf {\\mathbf {M}})$ such that for any $f\\in L_{2}(\\mathbf {\\mathbf {M}})$ one has $\\left(F(\\mathcal {L})f\\right)(x)=\\int _{\\mathbf {\\mathbf {M}}}\\mathcal {K}^{F}(x,y)f(y)dy,$ where $dy$ is the invariant normalized measure on $\\mathbf {\\mathbf {M}}$ .",
"If $\\left\\lbrace \\lambda _{j}\\right\\rbrace $ and $\\left\\lbrace u_{j}\\right\\rbrace $ are sets of eigenvalues and eigenfunctions of $\\mathcal {L}$ respectively then $\\mathcal {K}^{F}(x,y)=\\sum _{j=0}^{\\infty }F(\\lambda _{j})u_{j}(x)\\overline{u_{j}}(y).$ We will be especially interested in operators of the form $F(t^{2}\\mathcal {L})$ , where $F$ is a Schwartz function and $t>0$ .",
"The corresponding kernel will be denoted as $\\mathcal {K}_{t}^{F}(x,y)$ and $\\mathcal {K}_{t}^{F}(x,y)=\\sum _{j=0}^{\\infty }F(t^{2}\\lambda _{j})u_{j}(x)\\overline{u_{j}}(y).$ Note, that variable $t$ here is a kind of scaling parameter.",
"The following important estimate was proved in [1] in the setting of the so-called Dirichlet spaces.",
"It is a consequence of the main result in [10] that sub-Riemannin manifolds we consider in our article are the Dirichlet spaces.",
"Theorem 7.1 If $F\\in C_{0}^{\\infty }(\\mathbf {R})$ is even than for every $N>2Q$ there exists a $C_{N}=C_{N}(F,\\mathbf {M, Y})>0$ such that $\\left| \\mathcal {K}_{t}^{F}(x,y)\\right|\\le C_{N}\\left(\\left|B^{\\mu }(x, t)\\right|\\left|B^{\\mu }(y, t)\\right|\\right)^{-1/2}\\left(1+t^{-1}\\mu (x,y)\\right)^{-N},\\:\\:\\:0<t\\le 1.$"
],
[
" Parseval space-frequency localized frames on sub-Riemannian manifolds and proof of Theorem ",
"Let $g\\in C^{\\infty }(\\mathbf {R}_{+})$ be a monotonic function with support in $ [0,\\: 2^{2}], $ and $g(s)=1$ for $s\\in [0,\\:1], \\:0\\le g(s)\\le 1, \\:s>0.$ Setting $G(s)=g(s)-g(2^{2}s)$ implies that $0\\le G(s)\\le 1, \\:\\:s\\in supp\\:G\\subset [2^{-2},\\:2^{2}].$ Clearly, $supp\\:G(2^{-2j}s)\\subset [2^{2j-2}, 2^{2j+2}],\\:j\\ge 1.$ For the functions $F_{0}(s)=\\sqrt{g(s)}, \\:\\:F_{j}(s)=\\sqrt{G(2^{-2j}s)},\\:\\:j\\ge 1, \\:\\:\\:$ one has $\\sum _{j\\ge 0}F_{j}^{2}(s)=1, \\:\\:s\\ge 0$ .",
"Using the spectral theorem for $\\mathcal {L}$ one can define bounded self-adjoint operators $F_{j}(\\mathcal {L})$ as $F_{j}(\\mathcal {L})f(x)=\\int _{\\mathbf {\\mathbf {M}}}\\mathcal {K}^{F}_{2^{-j}}(x,y)f(y)dy,$ where $\\mathcal {K}^{F}_{2^{-j}}(x,y)= \\sum _{\\lambda _{m}\\in [2^{2j-2}, 2^{2j+2}]} F(2^{-2j}\\lambda _m) u_m(x) \\overline{u_m(y)}.$ The same spectral theorem implies $\\sum _{j\\ge 0} F_{j}^2(\\mathcal {L})f = f,\\:\\:f \\in L_{2}(\\mathbf {\\mathbf {M}}),$ and taking inner product with $f$ gives $\\Vert f\\Vert ^2=\\sum _{j\\ge 0}\\left< F_{j}^2(\\mathcal {L})f,f\\right>=\\sum _{j\\ge 0}\\Vert F_{j}(\\mathcal {L})f\\Vert ^2 .$ Moreover, since the function $ F_{j}(s)$ has its support in $[2^{2j-2},\\:\\:2^{2j+2}]$ the functions $ F_{j}(\\mathcal {L})f $ are bandlimited to $[2^{2j-2},\\:\\:2^{2j+2}]$ .",
"Next, consider the sequence $\\omega _{j}=2^{2j+2},\\:j=0, 1, ....\\:$ .",
"By (REF ) the equality $\\Vert f\\Vert ^2=\\sum _{j\\ge 0}\\Vert F_{j}(\\mathcal {L})f\\Vert ^2 $ holds, where every function $ F_{j}(\\mathcal {L})f $ is bandlimited to $[2^{2j-2},\\:\\:2^{2j+2}]$ .",
"Since for every $\\overline{F_{j}( \\mathcal {L})f} \\in \\mathbf {\\mathbf {E}}_{2^{2j+2}}(\\mathcal {L})$ one can use Theorem to conclude that $|F_{j}( \\mathcal {L})f|^2\\in \\mathbf {\\mathbf {E}}_{C_{0}2^{Q(2j+2)}}(\\mathcal {L}).$ According to Theorem REF there exists a constant $a=a(\\mathbf {M, Y})>0$ such that for all natural $j$ if $r_j = b2^{-Q(j+1)},\\:\\:b=aC_{0},$ then for any $r_{j}$ -lattice $\\mathcal {M}_{r_{j}}$ one can find positive coefficients $\\alpha _{j,k}$ with for which the following exact cubature formula holds $\\Vert F_{j}(\\mathcal {L})f\\Vert ^2_2 = \\sum _{k=1}^{K_j}\\alpha _{j,k}\\left|F_{j}(\\mathcal {L})f(x_{j,k})\\right|^2,$ where $x_{j,k} \\in \\mathcal {M}_{r_{j}}$ , $k = 1,\\ldots ,K_j = card\\:(\\mathcal {M}_{r_{j}})$ .",
"Using the kernel $\\mathcal {K}_{2^{-j}}^{F}$ of the operator $F_{j}(\\mathcal {L})$ we define the functions $\\Theta _{j,k}(y) = \\sqrt{\\alpha _{j,k}}\\:\\overline{\\mathcal {K}^{F}_{2^{-j}}}(x_{j,k},y) =\\sqrt{\\alpha _{j,k}} \\sum _{\\lambda _{m}\\in [2^{2j-2}, 2^{2j+2}]} \\overline{F}(2^{-2j}\\lambda _m) \\overline{u}_m(x_{j,k}) u_m(y).$ One can easily see that for every $f \\in L_2(\\mathbf {\\mathbf {M}})$ the equality $ \\Vert f\\Vert ^2_2 = \\sum _{j,k} |\\langle f,\\Theta _{j,k} \\rangle |^2$ holds.",
"Moreover, the first two items of Theorem REF are also satisfied.",
"Thus, Theorem REF is proven.",
"As an application one can obtain description of sub-elliptic Besov spaces $\\mathcal {B}_{p,q}^{\\alpha }(\\mathcal {L}),\\:\\:1\\le p<\\infty ,\\:1\\le q\\le \\infty ,$ in terms of the Fourier coefficients with respect to this frame $\\left\\lbrace \\Theta _{j,k}\\right\\rbrace $ .",
"Consider the quasi-Banach space ${\\bf b}_{p,q}^{\\alpha }$ which consists of sequences $s=\\lbrace s^j_k\\rbrace $ ($j \\ge 0,\\ 1 \\le k \\le {\\mathcal {K}}_j$ ) satisfying $\\Vert s\\Vert _{{\\bf b}_{p,q}^{\\alpha }}=\\left(\\sum _{j \\ge 0}^{\\infty } 2^{j\\alpha q} \\left(\\sum _k\\left| B^{\\mu }(x_{k}^{j},\\:2^{-j})\\right|^{1/p-1/2} |s^j_k|^p\\right)^{q/p}\\right)^{1/q} < \\infty ,$ and introduce the following mappings $\\tau (f) = \\lbrace \\langle f, \\Theta ^{j}_{k}\\rangle \\rbrace ,$ and $\\sigma (\\lbrace s^j_k\\rbrace ) = \\sum _{j\\ge 0}^{\\infty }\\sum _k s^j_k \\Theta ^{j}_{k}.$ It is not difficult to prove the following result (see [5] for the Riemann case).",
"Theorem 8.1 Let $\\Theta ^{j}_{k}$ be the same as above.",
"Then for $1\\le p< \\infty ,\\:\\:0<q\\le \\infty ,\\:\\:\\alpha >0$ the following statements are valid: $\\tau $ in (REF ) is a well defined bounded operator $\\tau : \\mathcal {B}_{p,q}^{\\alpha }(\\mathcal {L}) \\rightarrow {\\bf b}_{p,q}^{\\alpha }$ ; $\\sigma $ in (REF ) is a well defined bounded operator $\\sigma : {\\bf b}_{p,q}^{\\alpha } \\rightarrow \\mathcal {B}_{p,q}^{\\alpha }(\\mathcal {L})$ ; $\\sigma \\circ \\tau = id$ ; Moreover, the following norms are equivalent: $\\Vert f\\Vert _{\\mathcal {B}^{\\alpha }_{p,q}(\\mathcal {L})} \\asymp \\Vert \\tau (f)\\Vert _{{\\bf b}_{p,q}^{\\alpha }},$ where $\\Vert \\tau (f)\\Vert _{{\\bf b}_{p,q}^{\\alpha }}=\\left(\\sum _{j \\ge 0}^{\\infty } 2^{j\\alpha q} \\left(\\sum _k\\left| B^{\\mu }(x_{k}^{j},\\:2^{-j})\\right|^{1/p-1/2} |s^j_k|^p\\right)^{q/p}\\right)^{1/q}.$ The constants in these norm equivalence relations can be estimated uniformly over compact ranges of the parameters $p,q,\\alpha $ .",
"I am thankful to Hartmut Führ and Gerard Kerkyacharian for stimulating discussions."
]
] | 1612.05755 |
[
[
"Quantum harmonic oscillator state control in a squeezed Fock basis"
],
[
"Abstract We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms.",
"We demonstrate that for squeezed Fock states with low $n$ the engineered Hamiltonians reproduce the $\\sqrt{n}$ scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields.",
"Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of $n = 6$ with up to 8.7 dB of squeezing, as well as demonstrating superpositions of these states.",
"These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science."
],
[
"Supplemental Material",
"In figure REF we give the spin population data for the $\\hat{H}_{+}$ and $\\hat{H}_-$ Rabi oscillations.",
"In this supplement we discuss the effects which we think are playing the primary role in the loss of oscillation contrast in these experiments, which we ascribe to time-evolution of the basis states of the system due to a combination of trap frequency offsets and fast fluctuations.",
"This is different to standard Jaynes-Cummings physics for which the basis states (the energy eigenstates) are themselves immune to dephasing (although dephasing of the superposition of the two energy eigenstates during the Rabi oscillations does play a role).",
"The squeezed Fock states are sensitive to this mechanism because their axis of squeezing evolves with a rate given by this detuning.",
"This sensitivity also increases with excitation number $n$ .",
"The data which were used to produce the scaling of the Rabi frequencies in figure 2 of the main text are shown in figure REF .",
"For higher $n$ , it is clear that the loss of coherence does not fit well to either a Gaussian or exponential - the examples in figure REF are for Gaussian decay, fitting a function $P(\\downarrow , t) &=& \\frac{(1 + (-1)^p)}{2} \\nonumber \\\\&+& (-1)^p\\frac{1}{2}\\left(1 + e^{-\\gamma t^2} \\cos (\\Omega t) \\right)$ where $p = 1$ for $\\hat{H}_+$ flopping and $p = 0$ for $\\hat{H}_-$ flopping.",
"Neither the Gaussian or exponential decay models are able to account for the amplitude of oscillations at both long and short times.",
"Rabi oscillations persist at higher amplitudes at long times than would be expected from fits to the data at short times.",
"This effect is particularly apparent in the higher $n$ states.",
"To understand this better, we performed simulations of the population evolution under the influence of an $\\hat{H}_+$ Hamiltonian with a fixed detuning between the trap frequency and the frequency corresponding to half the difference between the red-sideband and blue-sideband drives.",
"This would account for a mis-setting of the drive frequencies, or equivalently that the trap frequency of the ion had shifted.",
"Figure REF shows flopping as a function of time for $\\left| {\\downarrow } \\right> \\left| {\\zeta , 0} \\right>\\leftrightarrow \\left| {\\uparrow } \\right> \\left| {\\zeta , 1} \\right>$ and $\\left| {\\downarrow } \\right> \\left| {\\zeta , 6} \\right>\\leftrightarrow \\left| {\\uparrow } \\right> \\left| {\\zeta , 7} \\right>$ for example curves for a 10 Hz, 20 Hz and 30 Hz detuning.",
"In simulating such systems the energy eigenstate basis is not an efficient choice.",
"Instead, we have found it useful to consider the effect of the detuning in terms of the creation and annihilation operators in the squeezed Fock basis.",
"The Hamiltonian $\\hat{H}_{\\rm det} = \\hbar \\delta {\\,\\hat{a}^{\\dagger }}{\\,\\hat{a}}$ can then be rewritten as $\\hat{H}_{\\rm det} &=& \\hbar \\delta \\left( \\hat{K}^\\dagger \\hat{K} \\cosh (2 r) + \\sinh ^2(r)\\right) \\nonumber \\\\&+& \\hbar \\delta \\frac{\\sinh (2r)}{2} \\left((\\hat{K}^\\dagger )^2 + \\hat{K}^2\\right).",
"$ The first term would imply that the Jaynes-Cummings Hamiltonian does not meet the resonance condition.",
"By analogy with standard Rabi physics (and as we see in simulations) this leads to Rabi oscillations with a higher frequency but with amplitude $<1$ .",
"The final two terms induce transitions to nearby states differing in their quantum number by 2, with some probability during the evolution to come back to the original state.",
"These lead to the collapse and revival effects which can be observed in the simulations shown in figure REF .",
"It is notable that the $n$ dependent terms in the Hamiltonian scale as $n$ for the diagonal terms, and as $\\sqrt{(n + 1)(n + 2)}$ and $\\sqrt{n(n-1)}$ for the off-diagonal terms, which produces the amplified sensitivity for the higher $n$ states.",
"In the limit of large $r$ both terms scale as $\\exp (2r)$ , which is proportional to the quadrature squeezing factor.",
"It is noteworthy that a fixed offset might be hard to measure when close to the bottom of the state ladder, but would only become apparent at higher excitations.",
"For each of the fixed frequency offsets we see partial collapse and revival occurring in both cases.",
"This effect is not dissimilar to what we see in the data, although it is a weak effect.",
"The suspicion of a fixed frequency offset is supported by the presence of data sets in which the $\\hat{H}_+$ data shows different behaviour to those of $\\hat{H}_-$ even though the transitions used are between the same pair of Fock states.",
"Since the two measurements were performed some time apart, this is perhaps due to a drift in the trap frequency between the measurements.",
"We expect that in addition to a possible fixed offset fast fluctuations of our trap frequency add to the decay.",
"As an example we perform a simulation solving the Master equation in Lindblad form, in which we use the detuned versions of $\\hat{H}_+$ and $\\hat{H}_-$ with a fixed detuning of 30 and jump operators for an amplitude reservoir ($\\hat{L}_{\\rm {A1}}=\\sqrt{\\Gamma }_{\\rm {A}} {\\,\\hat{a}^{\\dagger }}$ , $\\hat{L}_{\\rm {A2}}=\\sqrt{\\Gamma }_{\\rm {A}} {\\,\\hat{a}}$ with $\\Gamma _{\\rm {A}}/(2\\pi ) =$ 10.7Hz calibrated by measuring the rate of heating of the motional ground state) and a phase reservoir ($\\hat{L}_{\\rm {Ph}}=\\sqrt{\\Gamma }_{\\rm {Ph}} {\\,\\hat{a}^{\\dagger }}{\\,\\hat{a}}$ with $\\Gamma _{\\rm {Ph}}/(2\\pi ) =$ 5Hz, tuned to match the flopping curves in the squeezed state ladder) [16].",
"We calculate the evolution in the same stepwise fashion as in the experiment with $\\hat{H}_+$ and $\\hat{H}_-$ applied alternating and a final probe pulse of the respective Hamiltonian.",
"Results of the simulation are shown in figure REF , matching the data better than using any effect alone.",
"Figure: Rabi oscillation data for the H ^ + \\hat{H}_+ and H ^ - \\hat{H}_- transitions for each of the transitions a) ↓(↑)ζ,n↔↑(↓)ζ,n+1\\left| {\\downarrow (\\uparrow )} \\right>\\left| {\\zeta , n} \\right> \\leftrightarrow \\left| {\\uparrow (\\downarrow )} \\right> \\left| {\\zeta , n + 1} \\right> for nn from 0 to 6, given as plots i) to vii) respectively.",
"Data shown on the left is for H ^ + \\hat{H}_+ and on the right are the data for H ^ - \\hat{H}_-.",
"The fits are discussed in the text of this section.Figure: Simulated flopping curves on the H ^ + \\hat{H}_+ Hamiltonian for a) ↓ζ,0↔↑ζ,1\\left| {\\downarrow } \\right> \\left| {\\zeta , 0} \\right>\\leftrightarrow \\left| {\\uparrow } \\right> \\left| {\\zeta , 1} \\right> and b) ↓ζ,6↔↑\\left| {\\downarrow } \\right> \\left| {\\zeta , 6} \\right>\\leftrightarrow \\left| {\\uparrow } \\right> for fixed detunings of 10 (green, dashed), 20 (red, dot-dashed) and 30 (blue, solid) using r=1r = 1 and Ω/(2π)=\\Omega /(2 \\pi ) = 4.3.",
"While for the ground state simulation no effect of the detuning is visible and the traces are nearly overlapped, for the excited state the effects of the detuning are clearly visible.",
"In addition to a change in the Rabi oscillation frequency a collapse and revival effect is observed.",
"This is due to the transition terms in the Hamiltonian of equation .whatever Figure: Experimental data overlaid with simulated flopping curves using a Master equation approach on the H ^ + \\hat{H}_+ Hamiltonian for a) ↓ζ,0↔↑ζ,1\\left| {\\downarrow } \\right> \\left| {\\zeta , 0} \\right>\\leftrightarrow \\left| {\\uparrow } \\right> \\left| {\\zeta , 1} \\right> and b) ↓ζ,6↔↑\\left| {\\downarrow } \\right> \\left| {\\zeta , 6} \\right>\\leftrightarrow \\left| {\\uparrow } \\right> with a fixed detuning of 30 combined with Lindblad collapse operators for heating and dephasing noise."
]
] | 1612.05570 |
[
[
"Construction of a new class of quantum Markov fields"
],
[
"Abstract In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs.",
"The construction is based on a specific tessellation on the considered graph, that allows us to express the Markov property for the local structure of the graph.",
"Our main result concerns the existence and uniqueness of quantum Markov field over such graphs."
],
[
"colorlinks=true,linkcolor=red, anchorcolor=green, citecolor=cyan, urlcolor=red, filecolor=magenta, pdftoolbar=true Copyright 2016 by the Tusi Mathematical Research Group."
]
] | 1612.05549 |
[
[
"Computing solutions of linear Mahler equations"
],
[
"Abstract Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable.",
"They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms.",
"Recently, the problem of solving Mahler equations in closed form has occurred in connection with number-theoretic questions.",
"A difficulty in the manipulation of Mahler equations is the exponential blow-up of degrees when applying a Mahler operator to a polynomial.",
"In this work, we present algorithms for solving linear Mahler equations for series, polynomials, and rational functions, and get polynomial-time complexity under a mild assumption.",
"Incidentally, we develop an algorithm for computing the gcrd of a family of linear Mahler operators."
],
[
"Context",
"Our interest in the present work is in computing various classes of solutions to linear Mahler equations of the form $\\ell _r (x) y (x^{b^r}) + \\cdots + \\ell _1 (x) y (x^b) + \\ell _0 (x) y (x) = 0 ,\\qquad \\mathrm {(\\ref {eq:mahler-eqn})}$ where $\\ell _0, \\ldots , \\ell _r$ are given polynomials, $r > 0$ is the order of the equation, and $b \\ge 2$ is a fixed integer.",
"Mahler equations were first studied by Mahler himself in a nonlinear context [17].",
"His aim was to develop a general method to prove the transcendence of values of certain functions.",
"Roughly speaking, the algebraic relations over $\\bar{\\mathbb {Q}}$ between certain of these values come from algebraic relations over $\\bar{\\mathbb {Q}}(x)$ between the functions themselves.",
"This direction was continued by several authors.",
"We refer to Pellarin's introduction [19] for a historical and tutorial presentation, and to the references therein; see also Nishioka [18] for a textbook.",
"Mahler equations are closely linked with automata theory: the generating series of any $b$ -automatic sequence is a Mahler function, that is, a solution of a linear Mahler equation; see [10], [11].",
"Mahler functions also appear in many areas at the interface of mathematics and computer science, including combinatorics of partitions, enumeration of words, and the analysis of divide-and-conquer algorithms.",
"Very recently, functional relations between Mahler functions have been further studied with a bias to effective tests and procedures [4], [5], [6], [12], [21].",
"Such studies motivate the need for algorithms that solve Mahler equations in various classes of functions.",
"For instance, testing transcendence of a Mahler series by the criterion of Bell and Coons [6] requires to compute truncations of Mahler series to suitable orders.",
"So does the algorithm by Adamczewski and Faverjon [4], [5] for the explicit computation of all linear dependence relations over $\\mathbb {Q}$ between evaluations of Mahler functions at algebraic numbers.",
"Besides, Mahler functions being either rational or transcendental—but never algebraic—, solving Mahler equations for their rational functions is another natural approach to testing transcendence, and an alternative to Bell and Coons' (see further comments on this in §REF ).",
"Similarly, the hypertranscendence criterion by Dreyfus, Hardouin, and Roques [12] relies on determining if certain Mahler equations possess ramified rational solutions."
],
[
"Related work",
"Mahler equations are a special case of difference equations, in the sense of functional equations relating iterates of a ring endomorphism $\\sigma $ applied to the unknown function.",
"Algorithms dealing with difference equations have been widely studied.",
"In particular, the computation of rational solutions of linear difference equations with coefficients polynomial in the independent variable $x$ is an important basic brick coming up repeatedly in other algorithms.",
"Algorithms in the cases of the usual shift $\\sigma (x) = x + 1$ and its $q$ -analogue $\\sigma (x) = q x$ have been given by Abramov [2], [3] for equations with polynomial coefficients: in both cases, the strategy is to compute a denominator bound before changing unknown functions and computing the numerator as a polynomial solution of an auxiliary difference equation.",
"Bronstein [8] provides a similar study for difference equations over more general coefficient domains; his denominator bound is however stated under a restriction (unimonomial extensions) that does not allow for the Mahler operator $\\sigma (x) = x^b$ .",
"Mahler equations can also be viewed as difference equations in terms of the usual shift $\\sigma (t) = t + 1$ after performing the change of variables $t = \\log _b \\log _b x$ .",
"This reduction from Mahler to difference equation, however, does not preserve polynomial coefficients, which means that neither Abramov's nor Bronstein's algorithm can be used in this setting.",
"There has been comparatively little interest in algorithmic aspects specific to Mahler equations.",
"To the best of our knowledge, the only systematic study is by Dumas in his PhD thesis [13].",
"In particular, he describes procedures for computing various types of solutions of linear Mahler equations [13].",
"However, beside a few gaps of effectiveness, that work does not take computational complexity issues into account.",
"To a large extent, the results of the present work can be viewed as refinements of it, with a focus on efficiency and complexity analysis.",
"More recently, Bell and Coons [6] give degree bounds that readily translate into algorithms for polynomial and rational solutions based on undetermined coefficients.",
"With regard to series solutions, van der Hoeven [22] suggests an algorithm that applies, under hypotheses, to certain equations of the form (REF ) as well as to certain nonlinear generalizations, and computes the first $n$ terms of a power series solution in $\\operatorname{\\tilde{O}}(n)$ arithmetic operations.",
"At least in the linear case and in analogy to the case of difference equations, this leaves the open question of an algorithm in complexity $\\operatorname{O}(n)$ ."
],
[
"Setting",
"Our goal in this article is to present algorithms that compute complete sets of polynomial solutions, rational function solutions, truncated power series solutions, and truncated Puiseux series solutions of (REF ).",
"More precisely, let $\\mathbb {K}$ be a (computable) subfield of $\\mathbb {C}$ , and suppose $\\ell _0, \\ldots , \\ell _r \\in \\mathbb {K}[x]$ .",
"Denote by $\\mathbb {K}((x^{1/*}))$ the field $\\bigcup _{n=1}^{+\\infty } \\mathbb {K}((x^{1/n}))$ of formal Puiseux series with coefficients in $\\mathbb {K}$ .",
"Let $M$ denote the Mahler operator of radix $b$, that is the automorphism of $\\mathbb {K}((x^{1/*}))$ that substitutes $x^b$ for $x$ and reduces to the identity map on $\\mathbb {K}$ .",
"Writing $x$ again for the operator of multiplication of a series by $x$ , $M$ and $x$ follow the commutation rule $M x = x^b M$ .",
"Equation (REF ) then rewrites as $L y = 0$ where $L = \\ell _r M^r + \\dots + \\ell _0\\qquad \\mathrm {(\\ref {eq:mahler-opr})}$ in the algebra generated by $M$ and $x$ .",
"We are interested in the algebraic complexity of computing the kernel of $L$ in each of $\\mathbb {K}[x]$ , $\\mathbb {K}(x)$ , $\\mathbb {K}[[x]]$ , and $\\mathbb {K}((x^{1/*}))$ .",
"Table: Complexity of the solving algorithms presented in the paper,assuming ℓ 0 ≠0\\ell _0 \\ne 0.We always assume that $\\ell _r$ is nonzero.",
"Except where otherwise noted, we also assume $\\ell _0 \\ne 0$ .",
"From a decidability viewpoint, the latter assumption is no loss of generality thanks to the following result [13].",
"Proposition 1.1 Given a linear Mahler equation of the form (REF ), one can compute an equation of the same form, with $\\ell _0 \\ne 0$ , that has exactly the same formal Laurent series solutions—and therefore, the same polynomial solutions and the same rational-function solutions.",
"Note however that this result does not say anything about the cost of reducing to the case $\\ell _0 \\ne 0$ .",
"We give a complexity bound for this step in §.",
"As it turns out, this bound often dominates our complexity estimates for the actual solving algorithms.",
"Let us therefore stress that all other complexity results are stated under the assumption that $\\ell _0$ is nonzero.",
"For $0 \\le k \\le r$ , we denote by $v_k \\in \\mathbb {N}\\cup \\lbrace +\\infty \\rbrace $ and $d_k \\in \\mathbb {N}\\cup \\lbrace -\\infty \\rbrace $ the valuation and degree of the coefficient $\\ell _k$ .",
"Let $d \\ge \\max _{0\\le k \\le r} d_k$ .",
"Polynomials are implicitly represented in dense form, so that polynomials of degree $d$ in $\\mathbb {K}[x]$ have size $d+1$ .",
"All complexity estimates are given in terms of arithmetical operations in $\\mathbb {K}$ , which we denote “ops”.",
"The complexity of multiplying two polynomials of degree at most $n$ is denoted by $\\operatorname{M}(n)$ ; we make the standard assumptions that $\\operatorname{M}(n) = \\operatorname{O}(n^2)$ and that $n \\mapsto \\operatorname{M}(n)/n$ is nondecreasing.",
"Given two integers or polynomials $a$ and $b$ , we denote their gcd by $a\\wedge b$ and their lcm by $a\\vee b$ ; we use $\\bigwedge $ and $\\bigvee $ for $n$ ary forms.",
"The following identities are used repeatedly in the text.",
"We gather and repeat them here for easier reference: $\\ell _r (x) y (x^{b^r}) + \\cdots + \\ell _1 (x) y (x^b) + \\ell _0 (x) y (x) = 0 , \\\\L = \\ell _r M^r + \\dots + \\ell _0 , \\\\\\nu = \\max _{k \\ge 1} \\frac{v_0 - v_k}{b^k - 1} ,\\qquad \\mu = v_0 + \\nu .\\qquad \\mathrm {(\\text{\\sc eqn},\\text{\\sc opr},\\text{\\sc mu-nu})}$"
],
[
"General strategy and outline",
"The article is organized as follows.",
"In §, we develop algorithms to compute truncated series solutions of equations of the form (REF ).",
"We start with an example that illustrates the structure of the solution space and some of the main ideas behind our algorithms (§REF ).",
"Then, we introduce a notion of Newton polygons, and use it to prove that the possible valuations (resp.",
"degrees) of the solutions of (REF ) in $\\mathbb {K}((x^{1/*}))$ (resp.",
"$\\mathbb {K}[x]$ ) belong to a finite set that we make explicit (§REF ).",
"We compute a suitable number of initial coefficients by solving a linear system (§REF ), then prove that the following ones can be obtained iteratively in linear time, and apply these results to give a procedure that computes a complete set of truncated series solutions (§REF ).",
"Finally, we extend the same ideas to the case of solutions in $\\mathbb {K}[x]$ (§REF ) and in $\\mathbb {K}((x^{1/*}))$ (§REF ).",
"The next section, §, deals with solutions in $\\mathbb {K}(x)$ .",
"The general idea is to first obtain a denominator bound, that is a polynomial $q$ such that $Lu=0$ with $u\\in \\mathbb {K}(x)$ implies $qu\\in \\mathbb {K}[x]$ (§REF ).",
"Based on elementary properties of the action of $M$ on elements of $\\mathbb {K}[x]$ (§REF ), we give several algorithms for computing such bounds (§REF –§REF ).",
"This reduces the problem to computing a set of polynomial solutions with certain degree constraints, which can be solved efficiently using the primitives developed in §, leading to an algorithm for solving linear Mahler equations in $\\mathbb {K}(x)$ (§REF ).",
"We briefly comment on a comparison, in terms of complexity, of Bell and Coons' transcendence test and the approach by solving the Mahler equation for rational functions (§REF ).",
"The net result is that the new approach is faster.",
"Finally, in §, we generalize our study to the situation where the coefficient $\\ell _0$ in (REF ) is zero.",
"This makes us develop an unexpected algorithm for computing the gcrd of a family of operators, which we analyze and compare to the more traditional approach via Sylvester matrices and subresultants."
],
[
"Acknowledgment",
"The authors are indebted to Alin Bostan for helpful discussions and for pointing us to the work of Grigor'ev [14]."
],
[
"A worked example",
"The aim of this section is to illustrate our solving strategy in $\\mathbb {K}[[x]]$ and $\\mathbb {K}((x^{1/*}))$ on an example that we treat straightforwardly.",
"In radix $b = 3$ , consider the equation $L y = 0$ where $L = x^3(1-x^3+x^6)(1-x^7-x^{10}) \\, M^2 \\\\- (1 - x^{28} - x^{31} - x^{37} - x^{40}) \\, M+ x^6(1+x)(1-x^{21}-x^{30}).$ Assume that $y\\in \\mathbb {K}((x^{1/*}))$ is a solution whose valuation is a rational number $v$ .",
"The valuations of $\\ell _k M^ky$ , for $k = 0,1,2$ , are respectively equal to $6+v, 3v, 3+9v$ .",
"If one of these rational numbers was less than the other two, then the valuation of the sum $\\sum _{k=0}^2 \\ell _k M^ky$ would be this smaller number, and $L y$ could not be zero.",
"Consequently, at least two of the three rational numbers $6+v, 3v, 3+9v$ have to be equal to their minimum.",
"After solving, we find $v\\in \\lbrace -1/2,3 \\rbrace $ .",
"First consider the case $v=3$ , and write $y=\\sum _{n\\ge 3} y_nx^n$ .",
"For $m$ from 10 to 15, extracting the coefficients of $x^m$ from both sides of $0 = \\ell _0y+\\ell _1My+\\ell _2M^2y$ , we find that $y_3,\\dots ,y_9$ satisfy $\\begin{array}{c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c}0 &= &y_3 &{}+y_4 , &&&&& \\\\0 &= &&\\phantom{{}+{}}y_4 &{}+y_5 , &&&& \\\\0 &= &&{}-y_4 &{}+y_5 &{}+y_6 , &&& \\\\0 &= &&&&\\phantom{{}+{}}y_6 &{}+y_7 , && \\\\0 &= &&&&&\\phantom{{}+{}}y_7 &{}+y_8 , & \\\\0 &= &&&{}-y_5&&&{}+y_8 &{}+y_9 .\\end{array}$ More generally, extracting the coefficient of $x^m$ yields the relation $\\bigl ( y_{m-6} + y_{m-7} - y_{m-27} - y_{m-28} - y_{m-36} - y_{m-37} \\bigr ) \\\\- \\bigl ( y_{\\frac{m}{3}} - y_{\\frac{m-28}{3}} - y_{\\frac{m-31}{3}} - y_{\\frac{m-37}{3}} - y _{\\frac{m-40}{3}} \\bigr ) \\\\+ \\bigl ( y_{\\frac{m-3}{9}} - y_{\\frac{m-6}{9}} + y_{\\frac{m-9}{9}} - y_{\\frac{m-10}{9}} - y_{\\frac{m-19}{9}} \\bigr ) = 0 ,$ where $y_s$ is understood to be zero if the rational number $s$ is not a nonnegative integer.",
"This equation takes different forms, depending on the residue of $m$ modulo 9: for example, for $m = 20$ and $m = 42$ , it reduces to, respectively, $y_{14} + y_{13} = 0, \\qquad y_{36}+y_{35}-y_{15}-2y_{14}-y_6-y_5-y_4 = 0.$ Despite these variations, for any $m\\ge 10$ the index $n = m-6$ is the largest integer index occurring in (REF ).",
"It follows that for successive $m\\ge 10$ , we can iteratively obtain $y_n$ from (REF ) in terms of already known coefficients of the series.",
"Conversely, any sequence $(y_n)_{n\\ge 3}$ that satisfies (REF ) gives a solution $y=\\sum _{n\\ge 3} y_nx^n$ of (REF ).",
"As a consequence, the power series solution is entirely determined by the choice of $y_3$ and the space of solutions of (REF ) in $\\mathbb {K}[[x]]$ has dimension one.",
"A basis consists of the single series $x^3-x^{4}+x^5-2x^6+2x^7-2x^8+3x^9-3x^{10}+3x^{11}-5x^{12}+ \\cdots .$ The other possible valuation, $v=-1/2$ , is not a natural number.",
"To revert to the simpler situation of the previous case, we perform the change of variables $x = t^2$ followed by the change of unknowns $y(t) = \\tilde{y}(t) / t$ .",
"The equation becomes $\\tilde{L} \\tilde{y} = 0$ with $\\tilde{L} =(1 - t^6 + t^{12}) (1 - t^{14} - t^{20}) \\, M^2 \\\\- (1 - t^{56} - t^{62} - t^{74} - t^{80}) \\, M+ t^{14}(1 + t^2)(1 - t^{42} - t^{60}) .$ To understand this calculation, remember that $M$ was defined on $\\mathbb {K}((x^{1/*}))$ , so that $M(t) = M(x^{1/2}) = x^{3/2} = t^3$ .",
"We now expect $\\tilde{L}$ to have solutions $\\tilde{y} = \\sum _{n\\ge 0} \\tilde{y}_n t^n$ of valuation 0 and 7 with respect to $t$ , and the solutions of $\\tilde{L}$ with valuation 0 to correspond to the solutions of $L$ with valuation $-1/2$ .",
"Extracting the coefficients of $x^m$ for $m$ from 0 to 24 from both sides of $\\tilde{L} \\tilde{y} = 0$ and skipping tautologies, we find that $\\tilde{y}_0,\\dots ,\\tilde{y}_{10}$ satisfy $\\begin{array}{c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c@{\\ }c}0 &= &&{}-\\tilde{y}_1, &&&&&&&&& \\\\0 &= &{}-\\tilde{y}_0&&{}-\\tilde{y}_2, &&&&&&&& \\\\0 &= &&\\phantom{{}+{}}\\tilde{y}_1&&{}-\\tilde{y}_3, &&&&&&& \\\\0 &= &\\phantom{{}+{}}\\tilde{y}_0&&&&{}-\\tilde{y}_4, &&&&&& \\\\0 &= &&&&&&{}-\\tilde{y}_5, &&&&& \\\\0 &= &\\phantom{{}+{}}\\tilde{y}_0&&{}+\\tilde{y}_2, &&&&&&&& \\\\0 &= &&\\phantom{{}+{}}\\tilde{y}_1&&{}+\\tilde{y}_3, &&&&&&& \\\\0 &= &&&\\phantom{\\,\\,\\,}2\\tilde{y}_2&&{}+\\tilde{y}_4&&{}-\\tilde{y}_6, &&&& \\\\0 &= &&&&\\phantom{{}+{}}\\tilde{y}_3&&{}+\\tilde{y}_5, &&&&& \\\\0 &= &&&&&\\phantom{{}+{}}\\tilde{y}_4&&{}+\\tilde{y}_6, && \\\\0 &= &&\\phantom{{}+{}}\\tilde{y}_1&&&&{}+\\tilde{y}_5, &&&&& \\\\0 &= &&&&&&&\\phantom{{}+{}}\\tilde{y}_6&&{}+\\tilde{y}_8, && \\\\0 &= &&{}-\\tilde{y}_1&&&&&&{}+\\tilde{y}_7&&{}+\\tilde{y}_9, & \\\\0 &=&&&{}-\\tilde{y}_2&&&&&&&&{}+\\tilde{y}_{10}.\\end{array}$ Reasoning as above, we derive that, given $\\tilde{y}_0$ while enforcing $\\tilde{y}_7=0$ , there is exactly one power series solution to $\\tilde{L}$ .",
"More specifically when $\\tilde{y}_0 = 1$ and $\\tilde{y}_7=0$ , we find the series $1-t^2+t^4-t^6+t^8-t^{10}+t^{12} + \\cdots .$ Hence, there is a 2-dimensional solution space in $\\mathbb {K}((x^{1/*}))$ for the original equation (REF ), with a basis consisting of the power series (REF ) and the additional Puiseux series $x^{-1/2}-x^{1/2}+x^{3/2}-x^{5/2}+x^{7/2}-x^{9/2}+x^{11/2} + \\cdots .$"
],
[
"Valuations and degrees",
"Let us assume that $y \\in \\mathbb {K}((x^{1/*}))$ is a solution of (REF ), whose valuation is a rational number $v$ .",
"The valuation of the term $\\ell _k M^k y$ is then $v_k + b^k v$ .",
"Among those expressions, at least two must be minimal to permit the left-hand side of (REF ) to be 0: therefore, there exist distinct indices $k_1,k_2$ between 0 and $r$ such that $v_{k_1} + b^{k_1} v= v_{k_2} + b^{k_2} v= \\min _{0 \\le k \\le r} v_k + b^k v.$ This necessary condition for $L y = 0$ can be interpreted using a Newton polygon analogous to that of algebraic equations [23]: to each monomial $x^j M^k$ in $L$ , we associate the point $(b^k,j)$ in the first quadrant of the Cartesian plane endowed with coordinates $U$ and $V$ (see Fig.",
"REF ).",
"We call the collection of these points the Newton diagram of $L$ , and the lower (resp.",
"upper) boundary of its convex hull the lower (resp.",
"upper) Newton polygon of $L$ .",
"That two integers $k_1, k_2$ satisfy (REF ) exactly means that $(b^{k_1}, v_{k_1})$ and $(b^{k_2}, v_{k_2})$ belong to an edge $E$ of slope $-v$ of the corresponding lower Newton polygon.",
"Given an edge $E$ as above, an arithmetic necessary condition holds in addition to the geometric one just mentioned: the coefficients of the monomials of $L$ associated to points of $E$ must add up to zero.",
"We call an edge with this property admissible.",
"Example 2.1 The lower Newton polygon of the operator (REF ) appears in dashed lines in Figure REF .",
"It contains two admissible edges, corresponding to the valuations 3 and $-1/2$ .",
"We get the following criterion, already stated in [13] with a slightly different proof.",
"Lemma 2.2 Let $L$ be defined as in ().",
"The valuation $v$ of any formal Puiseux series solution of (REF ) is the opposite of the slope of an admissible edge of the lower Newton polygon of $L$ .",
"It satisfies $-\\frac{v_r}{b^{r-1}(b-1)}\\le v= - \\frac{v_{k_1} - v_{k_2}}{b^{k_1} - b^{k_2}}\\le \\frac{v_0}{b-1},$ where $(b^{k_1}, v_{k_1})$ and $(b^{k_2}, v_{k_2})$ are the endpoints of the implied edge.",
"The fact that $v$ is the opposite of a slope together with its explicit form follow from (REF ) and the discussion above.",
"There remains to prove the upper and lower bounds.",
"The leftmost edge of the lower Newton polygon of $L$ provides the largest valuation and its slope $(v_k - v_0)/(b^k - 1)$ for some $k \\ge 1$ is bounded below by $- v_0/(b - 1)$ .",
"In the same way, the rightmost edge provides the smallest valuation and its slope, of the form $(v_r - v_k)/(b^r - b^k)$ for some $k < r$ , is bounded above by $v_r/(b^r - b^{r-1})$ .",
"Figure: The Newton diagram of the equation treated in §for radix b=3b = 3,with corresponding lower Newton polygon (dashed line)and upper Newton polygon (dotted line).Figure: The infinite matrix RR correspondingto the example treated in §:solid circles denote nonzero entries,hollow circles denote recombinations to zero.Proposition 2.3 The dimension of the space of solutions of the homogeneous equation $L y = 0$ in $\\mathbb {K}((x^{1/*}))$ is bounded by the order $r$ of $L$ .",
"The space of solutions admits a basis consisting of Puiseux series with pairwise distinct valuations.",
"The number of possible valuations is bounded by the edge count of the lower Newton polygon of $L$ , which is at most $r$ .",
"Remark 2.4 As we will see, the dimension of the solutions in $\\mathbb {K}((x^{1/*}))$ can be strictly less than $r$ .",
"It is natural to ask how to construct a “full” system of $r$ linearly independent formal solutions in some larger extension of $\\mathbb {K}(x)$ .",
"We will not pursue this question here and point to Roques's work for an answer; see [21] and [20].",
"See also Remark REF below.",
"In analogy with the previous discussion on valuations of solutions, if a Puiseux series solution of (REF ) involves monomials with maximal exponent $\\delta $ , then the expression $d_k + b^k \\delta $ must reach its maximum at least twice as $k$ ranges from 0 to $r$ .",
"As we see by the same reasoning as above (or by changing $x$ to $1/x$ , which exchanges the lower and upper Newton polygons), $-\\delta $ is then one of the slopes of the upper Newton polygon of $L$ .",
"The largest possible value corresponds to the rightmost edge.",
"Lemma 2.5 The maximum exponent $\\delta $ of a monomial in a finite Puiseux series solution, and in particular the degree of a polynomial solution, is the opposite of the slope of an admissible edge of the upper Newton polygon.",
"It satisfies $\\delta = - \\frac{d_{k_1} - d_{k_2}}{b^{k_1} - b^{k_2}} \\le \\frac{d}{b^{r-1} (b-1)},$ for some ${k_1} \\ne {k_2}$ .",
"The admissibility of an edge of the upper Newton polygon is defined in analogy with admissibility in the lower Newton polygon."
],
[
"The nonhomogeneous case",
"One of the proofs of results about Puiseux series solutions in §REF makes use of extended Newton diagrams that take into account the right-hand side of nonhomogeneous equations.",
"For $L$ as in () and a Puiseux series $\\ell _{-\\infty }$ of valuation $v_{-\\infty } \\in \\mathbb {Q}\\cup \\lbrace +\\infty \\rbrace $ , consider the nonhomogeneous equation $\\ell _r (x) y (x^{b^r}) + \\cdots + \\ell _1 (x) y (x^b) + \\ell _0 (x) y (x) =\\ell _{-\\infty }(x) .$ Given a Puiseux series solution $y \\in \\mathbb {K}((x^{1/*}))$ of this equation, with valuation $v \\in \\mathbb {Q}$ , we define the Newton diagram of $(L, \\ell _{-\\infty })$ as the Newton diagram of $L$ , augmented with all points $(0, \\alpha )$ for which $x^\\alpha $ appears with nonzero coefficient in $\\ell _{-\\infty }$ .",
"The notion of lower Newton polygon extends correspondingly.",
"As in §REF , these definitions are motivated by analyzing the minimum of the valuations $v_k + b^k v$ of the terms of the left-hand side of (REF ): either this minimum is equal to $v_{-\\infty }$ , or it is less than $v_{-\\infty }$ and must be reached as least twice on the left-hand side.",
"In both cases, making the convention that $b^{-\\infty } = 0$ , there exist distinct indices $k_1, k_2$ , now in $\\lbrace -\\infty , 0, 1, \\dots , r \\rbrace $ , such that the analogue $v_{k_1} + b^{k_1} v= v_{k_2} + b^{k_2} v= \\min _{k \\in \\lbrace -\\infty , 0, 1, \\dots , r\\rbrace } v_k + b^k v$ of (REF ) holds.",
"Again, this exactly means that $(b^{k_1}, v_{k_1})$ and $(b^{k_2}, v_{k_2})$ belong to an edge $E$ of slope $-v$ of the lower Newton polygon, now of $(L, \\ell _{-\\infty })$ .",
"Depending on $v_{-\\infty }$ and $\\hat{v} = \\min _{0 \\le k \\le r} (v_k + b^k v)$ , the lower Newton polygon of $(L, \\ell _{-\\infty })$ can: be equal to that of $L$ , if $\\ell _{-\\infty } = 0$ ; add an edge to its left, if $v_{-\\infty } > \\hat{v}$ ; prolong its leftmost edge, if $v_{-\\infty } = \\hat{v}$ ; or replace some of its leftmost edges, if $v_{-\\infty } < \\hat{v}$ .",
"We defined the admissibility of an edge $E$ of the lower Newton polygon of $L$ in terms of the coefficients of those monomials $x^{v_k} M^k$ in $L$ associated to points on $E$ .",
"We extend the definition to edges of the lower Newton polygon of $(L, \\ell _{-\\infty })$ by the convention that, if a point has to be considered for $k = -\\infty $ , the corresponding coefficient is the opposite of the coefficient of $x^{v_{-\\infty }}$ in $\\ell _{-\\infty }$ .",
"Admissibility is again a necessary condition for $v$ to be a possible valuation of a solution of (REF )."
],
[
"Approximate series solutions",
"We now concentrate on the search for power series solutions $y(x) = y_0 + y_1 x + \\cdots \\in \\mathbb {K}[[x]]$ of (REF ).",
"Extracting the coefficient of $x^m$ in both sides of it yields a linear equation for the coefficients $y_n$ .",
"This linear equation can be viewed as a row, denoted $R_m$ , of an infinite matrix $R= R(L)$ .",
"The matrix $R$ consists of overlapping strips with different slopes.",
"We view its row and column indices, starting at 0, as continuous variables $Y$ and $X$ with the $Y$ -axis oriented downwards.",
"Each nonzero term $\\ell _k(x)M^k$ then corresponds to matrix entries in the strip $b^k X + v_k \\le Y \\le b^k X + d_k$ .",
"By definition of $v_k$ and $d_k$ , the entries lying on the lines $Y = b^k X + d_k$ and $Y = b^k X + v_k$ that delimit the strip are nonzero, except maybe at intersection points of such lines (obtained for different $k$ ).",
"Because of our assumption that $\\ell _0$ is nonzero, the smallest slope is 1, obtained for $k = 0$ .",
"For large $Y$ , the line $Y = X + v_0$ becomes the topmost one, and each row $R_m$ determines a new coefficient $y_n$ uniquely, for $n=m-v_0$ .",
"Thus, the power series solutions are characterized by a finite subsystem of $R$ .",
"In order to state this fact more precisely in Proposition REF below, define $\\nu = \\max _{k \\ge 1} \\frac{v_0 - v_k}{b^k - 1} ,\\qquad \\mu = v_0 + \\nu .\\qquad \\mathrm {(\\ref {eq:def-sizes})}$ In terms of the Newton diagram, $\\nu $ and $\\mu $ are, respectively, the opposite of the slope and the $V$ -intercept of the leftmost edge of the lower Newton polygon.",
"Note that, as we can deduce from the proof of Lemma REF , there is no nonzero power series solution when $\\nu < 0$ , which happens if and only if $v_0$ is a strict minimum of all the $v_k$ over $0 \\le k \\le r$ .",
"Proposition 2.6 Assume that $\\nu \\ge 0$ .",
"A vector $(y_0, \\dots , y_{\\lfloor \\nu \\rfloor })$ is a vector of initial coefficients of a formal power series solution $ y = y_0 + \\dots + y_{\\lfloor \\nu \\rfloor } x^{\\lfloor \\nu \\rfloor }+ y_{\\lfloor \\nu \\rfloor + 1} x^{\\lfloor \\nu \\rfloor + 1} + \\cdots $ of (REF ) if and only if it satisfies the linear system given by the upper left $(\\lfloor \\mu \\rfloor + 1) \\times (\\lfloor \\nu \\rfloor + 1)$ submatrix of $R$ .",
"The power series solution (REF ) extending $(y_0, \\dots , y_{\\lfloor \\nu \\rfloor })$ is then unique.",
"A series $y = y_0 + y_1 x + \\cdots $ is a solution if and only if its coefficients satisfy the system $(R_m)_{m \\ge 0}$ .",
"Whenever $v_0 + n < v_1 + b^1 n, \\quad \\dots , \\quad v_0 + n < v_r + b^r n,$ the row $R_{v_0 + n}$ of $R$ is the first one with a nonzero entry of index $n$ .",
"It then determines $y_n$ in terms of $y_0, \\dots , y_{n-1}$ .",
"Condition (REF ) is equivalent to $n > \\nu $ , hence, for any given $(y_n)_{0 \\le n \\le \\nu }$ , there is a unique choice of $(y_n)_{n > \\nu }$ satisfying all the equations $R_m$ for $m > v_0 + \\nu = \\mu $ .",
"As, when (REF ) holds for a given $n$ , the entries of index $n$ of $R_{m}$ with $m < v_0 + n$ are zero, the remaining equations $(R_m)_{0 \\le m \\le \\mu }$ only involve the unknowns $(y_n)_{0 \\le n \\le \\nu }$ .",
"We note in passing the following corollary, which is the essential argument in the proof of [21].",
"Corollary 2.7 In case the leftmost edge of the lower Newton polygon of $L$ lies on the axis of abscissas and is admissible, Equation (REF ) admits a power series solution of valuation 0.",
"We then have $\\nu = \\mu = 0$ , so the only condition to check is that the first entry of $R_0$ is zero.",
"This is equivalent to the edge being admissible.",
"The geometric interpretation of the quantities $\\mu $ and $\\nu $ defined by () is a special case of a general correspondence between the structure of the matrix $R$ and the Newton diagram of $L$ via the point-line duality of plane projective geometry.",
"The correspondence stems from the fact that a monomial $x^j M^k$ of $L$ is associated both to a point $(b^k,j)$ in the Newton diagram and, by considering its action on powers of $x$ , to the entries of $R$ lying on the line $Y = b^k X + j$ .",
"More generally, under projective duality, each point $(U,V)$ in the plane of the Newton diagram corresponds to a line $Y = U X + V$ in the plane of the matrix $R$ , while, conversely, the dual of a point $(X,Y)$ is the line $V = -XU + Y$ .",
"A line through two points $(U_1, V_1)$ and $(U_2, V_2)$ corresponds to the intersection of their duals.",
"In particular, the point $P_0 = (1, v_0)$ corresponds to the right boundary $\\Delta : Y = X + v_0$ of the strip of entries of slope 1 in the matrix $R$ (see Figures REF and REF ).",
"In the $(U,V)$ -plane, the line containing the leftmost edge of the lower Newton polygon passes through that point $P_0 = \\Delta ^*$ .",
"This line is $\\Lambda : V = -\\nu U + \\mu $ and corresponds to the bottommost intersection $\\Lambda ^* = (\\nu , \\mu )$ of $\\Delta $ with the right boundary of another strip.",
"Below this intersection, the entries of $R$ lying on $\\Delta $ are the topmost nonzero entries of their respective columns, and, at the same time, the rightmost nonzero entries of their respective rows: as already observed, each row $R_m$ then determines a new $y_n$ .",
"Example 2.8 For the operator $L$ of §REF , the right boundaries of the strips associated to the three terms of $L$ have equations $Y = X + 6$ , $Y = 3 X$ , and $Y = 9 X + 3$ respectively (dotted lines in Fig.",
"REF ).",
"The first two of them meet at $\\Lambda ^* = (3, 9)$ (Fig.",
"REF , hollow circle at the bottom right corner of the gray rectangle), and the line $\\Delta : Y = X + 6$ becomes the rightmost line for $Y > 9$ .",
"For $m \\ge 10$ , the row $R_m$ reflects the relation (REF ).",
"In particular, the existence of a power series solution is entirely determined by the small linear system that uses the rows $R_0$ to $R_9$ and the unknowns $y_0$ to $y_3$ (gray rectangle on Figure REF ).",
"Solving the system yields $y_0 = y_1 = y_2 = 0$ and $y_3$ arbitrary.",
"We then recover the results of §REF : the space of solutions of (REF ) in $\\mathbb {K}[[x]]$ has dimension one and a basis consists of the single series (REF ).",
"The $V$ -intercept of the leftmost edge of the lower Newton polygon is $\\mu =9$ , and the corresponding slope is $-\\nu = -3$ .",
"In this case, it is both the column dimension of the small system and the valuation of the solution.",
"Observe how the bottom right sector depicted in light gray corresponds to the system starting with equations (REF ): as the top left rectangle imposes $y_0=y_1=y_2=0$ , the dots on the left of the sector in light gray play no role in the equations.",
"As we will see, in the situation of Proposition REF , the coefficients $y_{\\lfloor \\nu \\rfloor + 1}$ to $y_{\\lfloor \\nu \\rfloor + n}$ of $y$ can be computed from $y_0, \\dots , y_{\\lfloor \\nu \\rfloor }$ in $\\operatorname{O}(n)$ ops for fixed $L$ .",
"This motivates to call the truncation to order $\\operatorname{O}(x^{\\lfloor \\nu \\rfloor + 1})$ of a series solution an approximate series solution of (REF )."
],
[
"Power series solutions",
"Our goal at this point is to describe an algorithm that computes the formal power series solutions of (REF ), truncated to any specified order.",
"We first explain how to compute the entries of the matrix $R$ .",
"It is convenient, for expository reasons, to frame this computation as an individual step that returns a sparse representation of a submatrix of $R$ corresponding to a subset of the rows.",
"Indeed, in our complexity model dense matrices could not lead to good bounds.",
"We therefore define a matrix representation to be row-sparse if iterating over the nonzero entries of any given row does not require any zero test in $\\mathbb {K}$ .",
"Then, the algorithm essentially amounts to an explicit expression for the coefficients of recurrences similar (REF ), which can as well be computed on the fly.",
"In view of the computation of ramified solutions (§REF ), Algorithms REF and REF accept as input a $\\mathbb {K}$ -linear transformation $\\phi $ to be applied to the operator $L$ .",
"In general, $\\phi $ will take the form $\\phi (x^j M^k)= x^{\\alpha b^k + \\beta j - \\gamma } M^k ,\\qquad \\alpha , \\gamma \\in \\mathbb {Z}, \\quad \\beta \\in \\mathbb {N}_{> 0}, \\quad \\beta \\wedge b = 1,$ with $\\alpha , \\beta , \\gamma $ chosen such that $\\phi (L)$ has plain (as opposed to Laurent) polynomial coefficients.",
"The reader only interested in polynomial, rational, and power series solutions of $L$ may safely assume $\\phi = \\operatorname{id}$ , i.e., $\\alpha =\\gamma =0,\\beta =1$ .",
"Lemma 2.9 Algorithm REF computes the submatrix $R_E$ obtained by taking the first $w$ entries of the rows of $R(\\phi (L))$ with index $m \\in E$ in $\\operatorname{O}\\bigl ((r + d) |E|\\bigr )$ ops.",
"Each row of $R_E$ has at most $r + 2 d$ nonzero entries.",
"Write $\\tilde{L} = \\sum _{k=0}^r \\tilde{\\ell }_k(x) M^k = \\phi (L)$ .",
"Recall that the row $R_m$ is obtained by extracting the coefficient of $x^m$ in the equality $\\tilde{L} y = 0$ , where $y = \\sum _{n \\ge 0} y_n x^n$ .",
"More precisely, $R_{m,n}$ is the coefficient of $y_n x^m$ in the series $\\tilde{L} y= \\sum _{k=0}^r \\sum _{j=0}^{d} \\tilde{\\ell }_{k,j} x^j\\sum _{n=0}^{\\infty } y_n x^{b^k n}= \\sum _{m=0}^{\\infty } \\sum _{n=0}^{\\infty }\\Bigl ( \\sum _{j+b^kn = m} \\tilde{\\ell }_{k,j} \\Bigr ) y_n x^m.$ The definition of $\\phi $ translates into $\\tilde{\\ell }_{k, j} = 0$ when $j \\lnot \\equiv \\alpha b^k - \\gamma \\pmod {\\beta }$ , and otherwise $\\tilde{\\ell }_{k, j} = \\ell _{k, j^{\\prime }}$ for $j = \\alpha b^k + \\beta j^{\\prime } - \\gamma $ .",
"Therefore, $R_{m,n}$ is equal to the sum of $\\ell _{k,j^{\\prime }}$ for $(k,j^{\\prime })$ satisfying $\\alpha b^k + \\beta j^{\\prime } - \\gamma = m - n b^k$ .",
"For fixed $m$ and $k$ , the coefficient $\\ell _{k,j^{\\prime }}$ only contributes when $\\beta j^{\\prime } \\equiv m + \\gamma \\pmod {b^k}$ .",
"Its contribution is then to $R_{m,n}$ with $n = b^{-k}(B - \\beta j^{\\prime })$ where $B = m + \\gamma - \\alpha b^k$ , and we are only interested in $0 \\le n < w$ , i.e., $B - b^k w < \\beta j^{\\prime } \\le B$ .",
"Using the assumption that $\\beta $ is coprime with $b$ , the condition on $\\beta j^{\\prime } \\pmod {b^k}$ rewrites as $j^{\\prime } \\equiv j^{\\prime }_0 \\pmod {b^k}$ , where $j^{\\prime }_0$ is the integer computed at step REF .",
"Therefore, the loop REF correctly computes the contribution of $\\tilde{\\ell }_k$ to the entries of index less than $w$ of the row $R_{m_i}$ , and hence the algorithm works as stated.",
"The only operations in $\\mathbb {K}$ performed by the algorithm are one addition and possibly one comparison (to update the sparse structure) at each loop pass over step REF .",
"The total number of iterations of the innermost loop for a given $i$ is at most $\\sum _{k=0}^r \\left\\lceil \\frac{d_k}{b^k} \\right\\rceil \\le r + \\frac{b}{b-1} d\\le r + 2 d$ and bounds the number of nonzero entries in the row of index $m_i$ .",
"The complexity in ops follows by summing over $i$ .",
"According to Proposition REF , the number of linearly independent power series solutions and their valuations are determined by a small upper upper left submatrix of $R$ .",
"As a direct attempt at solving the corresponding linear system would have too high a complexity (see Remark REF ), our approach is to first find a set of candidate solutions, spanning a low-dimensional vector space that contains the approximate series solutions, and to refine the solving in a second step.",
"Geometrically, the idea to obtain a candidate solution $g = g_0 + g_1 x + \\cdots $ is to follow the “profile” of $R$ (more precisely, the right boundary of the overlapping strips described in the previous section), using a single equation $R_m$ to try and compute each coefficient $g_n$ from $g_0, \\dots , g_{n-1}$ .",
"(That is, for each $n$ , we resolutely skip all but one equations susceptible to determine $g_n$ .)",
"By duality, this corresponds to keeping a varying line of increasing integer slope in contact with the lower Newton polygon, and having it “pivot” around it.",
"In this process, the only case that potentially leaves a degree of freedom in the choice of $g_n$ is when column $n$ contains a “corner” of the profile, corresponding to an edge of the Newton polygon.",
"As a consequence, it is enough to construct at most $r$ independent candidates solutions.",
"The second step then consists in recombining the candidates in such a way that the equations $R_m$ that were skipped in the first phase be satisfied.",
"This strategy is made more precise in Algorithm REF , which will then be specialized to power series solutions (and later to other types of solutions) by a suitable choice of $E$ , $h$ and $w$ .",
"By construction, Algorithm REF outputs polynomials of degree less than $w$ that are solutions of a subsystem of the linear system induced by $L$ .",
"These polynomials need not a priori prolong into actual solutions.",
"Lemma 2.10 Algorithm REF runs in $\\operatorname{O}(r w d + r^2 w + r^2\\operatorname{M}(h))$ ops, and returns a basis of the kernel of the linear map induced by $\\phi (L)$ from $\\mathbb {K}[x]_{<w}$ to $\\mathbb {K}[x]/(x^h)$ .",
"When $S_E$ is lower, respectively upper, triangular it is possible at step REF to compute $G$ by forward, respectively backward, substitution, in such a way that $S_E G = 0$ .",
"By interpreting the $h \\times w$ upper left submatrix $S$ of $R$ as the matrix of a restriction of $L$ to suitable monomial bases, it follows from the definition of $S^{\\prime }$ that $S^{\\prime } = S G$ .",
"Step REF computes $K$ such that $S^{\\prime } K = 0$ .",
"The columns of $F$ , computed as $GK$ at step REF , span the kernel of $S$ : Indeed, assume $S f = 0$ , so that by selecting rows $S_E f = 0$ , and $f$ can be written as $G \\gamma $ for some $\\gamma $ .",
"Then, $S^{\\prime } \\gamma = S G \\gamma = S f = 0$ .",
"But this means that $\\gamma = K \\eta $ for some $\\eta $ , so that $f = G K \\eta = F \\eta $ .",
"Conversely, we have $S F = S G K = S^{\\prime } K = 0$ , so that any vector of the form $F \\eta $ belongs to $\\ker S$ .",
"Additionally, since the columns of $G$ , respectively those of $K$ , are linearly independent, $G K \\eta = 0$ implies $K \\eta = 0$ , which implies $\\eta = 0$ .",
"The columns of $F = G K$ hence form a basis of $\\ker S$ .",
"By Lemma REF , step REF takes $\\operatorname{O}(w (r + d))$ ops.",
"The number of nonzero entries in each row of $S_E$ is bounded by $r + 2d$ by Lemma REF , hence the cost of computing $\\rho $ linearly independent solutions by substitution at step REF is $\\operatorname{O}(\\rho w (r + d))$ .",
"As no more than $r$ of the diagonal entries of $S_E$ are zero, $\\rho $ is at most $r$ .",
"The computation of each column of $S^{\\prime }$ at step REF amounts to adding ${r+1}$ products of the $\\ell _k$ by the $M^k S_i$ , truncated to order $h$ , for a total of $\\operatorname{O}(r^2 \\operatorname{M}(h))$ ops.",
"As $\\rho \\le r$ , computing the kernel of $S^{\\prime }$ at step REF via an LSP decomposition (a generalization of the LUP decomposition) requires $\\operatorname{O}(hr^{\\omega -1}) = o(r^2 \\operatorname{M}(h))$ ops [16].",
"Finally, the recombination at step REF takes $\\operatorname{O}(wr^{\\omega -1}) = o(r^2 w)$ ops as $\\sigma \\le \\rho \\le r$ .",
"Remark 2.11 Note that a direct attempt to solve $S$ , when, say, $\\phi = \\operatorname{id}$ and $w = \\operatorname{O}(d)$ , would result in a complexity $\\operatorname{O}(d^\\omega )$ (e.g., using the LSP decomposition), as opposed to $\\operatorname{O}(d^2)$ when using Algorithm REF and disregarding the dependency in $r$ .",
"Let $\\tilde{v}_k$ be the valuation of the coefficient $\\tilde{\\ell }_k$ of $\\phi (L) = \\sum _k \\tilde{\\ell }_k(x) M^k$ .",
"In analogy with (), define $ \\tilde{\\nu }= \\max _{k \\ge 1} \\frac{\\tilde{v}_0 - \\tilde{v}_k}{b^k - 1} ,\\qquad \\tilde{\\mu }= \\tilde{v}_0 + \\tilde{\\nu }.$ We now specialize the generic solver to the computation of approximate series solutions (in the sense of the previous subsection) of $\\phi (L)$ .",
"The case $\\phi = \\operatorname{id}$ is formalized as Algorithm REF on page REF .",
"Proposition 2.12 Assume $\\tilde{\\nu }\\ge 0$ .",
"Algorithm REF , called with $h = \\lfloor \\tilde{\\mu }\\rfloor + 1,\\quad w = \\lfloor \\tilde{\\nu }\\rfloor + 1,\\quad E = \\bigl ( \\min _k (\\tilde{v}_k + n b^k) \\bigr )_{0 \\le n < w},$ runs in $\\operatorname{O}(r d \\tilde{v}_0 + r^2 \\operatorname{M}(\\tilde{v}_0))$ ops and returns a basis of approximate series solutions of the equation $\\phi (L) \\, y = 0$ .",
"First of all, when $m = m_i \\in E$ , none of the terms $\\tilde{\\ell }_k M^k$ of $\\phi (L)$ contributes to the entries of $S$ located above $S_{m, n}$ .",
"The matrix $S_E$ is thus lower triangular.",
"In addition, $R_{m,n}$ is zero (if and) only if $-n$ is an (admissible) slope of the lower Newton polygon, so that no more than $r$ of the diagonal entries of $S_E$ are zero.",
"Both preconditions of Algorithm REF are therefore satisfied.",
"By Proposition REF and Lemma REF , it follows from the choice of $h$ and $w$ that the $f_j$ form a basis of approximate series solutions.",
"Using the inequalities $h \\le b v_0/(b-1) + 1 = \\operatorname{O}(\\tilde{v}_0)$ and $w \\le v_0/(b-1) + 1 = \\operatorname{O}(\\tilde{v}_0)$ in the formula of Lemma REF , the total complexity is as announced.",
"Given an approximate series solution, the next terms of the corresponding series solutions can be computed efficiently one by one using simple recurrence formulae.",
"Proposition 2.13 Given an approximate series solution $\\hat{y} = y_0+\\dots + y_{\\lfloor \\tilde{\\nu }\\rfloor }x^{\\lfloor \\tilde{\\nu }\\rfloor }$ of (REF ), Algorithm REF computes the truncation to the order $\\operatorname{O}(x^{\\lfloor \\tilde{\\nu }\\rfloor + n})$ of the unique solution $y$ of (REF ) of the form $y = \\hat{y} + \\operatorname{O}(x^{\\lfloor \\tilde{\\nu }\\rfloor + 1})$ in $\\operatorname{O}((r + d) \\, n)$ ops.",
"By Proposition REF , the system to be solved at step REF is compatible.",
"According to the description of $R$ provided above, the submatrix $(R_{m,n})_{m > \\lfloor \\tilde{\\mu }\\rfloor , n > \\lfloor \\tilde{\\nu }\\rfloor }$ is lower triangular, with nonzero diagonal coefficients, so that the system can be solved by forward substitution.",
"As explained in §REF , the output is a truncation of a solution of $\\phi (L)$ .",
"By Lemma REF , the cost in ops of step REF is $\\operatorname{O}((r + d) \\, n)$ , and each row of $S$ contains at most $r + 2 d$ nonzero entries.",
"Therefore, step REF costs $\\operatorname{O}((r + d) \\, n)$ ops."
],
[
"Polynomial solutions",
"Our goal in this subsection is Algorithm REF , which computes a basis of all polynomial solutions.",
"Lemma REF provides us with an upper bound $d/(b^r - b^{r-1}) + 1 = \\operatorname{O}(d/b^r)$ for the degree of any polynomial solution.",
"Before we take this into account, we provide an algorithm to compute polynomial solutions with degree bounded by $w \\ge 0$ , which runs in a complexity that is sensitive to $w$ .",
"In the same way as in Proposition REF , to obtain candidate polynomial solutions $f = f_0 + \\dots + f_{w-1} x^{w-1}$ , we set $f_n = 0$ for $n \\ge w$ and then compute $f_n$ for decreasing $n$ by “following” the “left profile” of the matrix $R$ (or, dually, the upper Newton polygon).",
"The corresponding specialization of Algorithm REF is formalized as Algorithm REF .",
"Proposition 2.14 Assume $\\nu \\ge 0$ .",
"Algorithm REF , called with $\\phi = \\operatorname{id}$ and $h = d + (w-1) b^r + 1,\\qquad E = \\bigl ( \\max _k (d_k + n b^k) \\bigr )_{0 \\le n \\le w},$ returns a basis of the space of polynomial solutions of (REF ) of degree less than $w$ .",
"For $w = \\operatorname{O}(d/b^r)$ , the algorithm runs in $\\operatorname{\\tilde{O}}(w d + \\operatorname{M}(d))$ ops.",
"The proof is similar to that of Proposition REF : the extracted submatrix of $R$ is now upper triangular; the zeros on its diagonal correspond to the admissible nonpositive integer slopes of the upper Newton polygon; the number of such zeros is not more than $r$ .",
"Both preconditions of Algorithm REF are therefore satisfied and Lemma REF applies.",
"Additionally, the choice of $h$ in terms of $w$ is such that $\\deg (Ly) < h$ whenever $\\deg y < w$ for a polynomial $y$ .",
"So, the basis returned is that of the kernel of the map induced by $L$ from $\\mathbb {K}[x]_{<w}$ to $\\mathbb {K}[x]$ , as announced.",
"For the complexity result, the hypothesis on $w$ implies $h = \\operatorname{O}(d)$ and $r = \\operatorname{O}(\\log _b d)$ , so that the conclusion of Lemma REF specializes to $\\operatorname{\\tilde{O}}(w d + \\operatorname{M}(d))$ ops.",
"Remark 2.15 The loose bound on $w$ , namely $w = \\operatorname{O}(d/b^r)$ , permits in particular to obtain a result when $d$ is not the maximal degree of the $\\ell _k$ , but only bounds them up to a multiplicative constant.",
"In this case, the complexity announced by Proposition REF specializes to the same complexity as in Corollary REF .",
"This will be used for the numerators of rational-function solutions in §REF .",
"By Lemma REF , the degree of any polynomial solution is bounded above by $\\delta _0 = d/(b^r - b^{r-1}) + 1$ .",
"Specializing Proposition REF to $w = \\lfloor \\delta _0 \\rfloor $ , we obtain a bound for the complexity of computing the whole space of polynomial solutions.",
"Corollary 2.16 Assuming $\\nu \\ge 0$ , Algorithm REF , called with $\\phi = \\operatorname{id}$ , $h = 3 d + 1,\\qquad w = \\Bigl \\lfloor \\frac{d}{b^{r-1} (b-1)} \\Bigr \\rfloor + 1,\\qquad E = \\bigl ( \\max _k (d_k + n b^k) \\bigr )_{0 \\le n \\le w},$ computes a basis of the polynomial solutions of (REF ) in $\\operatorname{\\tilde{O}}(d^2/b^r + \\operatorname{M}(d))$ ops.",
"Observe that the choice for $w$ induces that $h$ , as defined in Algorithm REF , satisfies $h \\le 3d + 1$ .",
"The result follows from this fact and $w = \\operatorname{O}(d/b^r)$ ."
],
[
"Puiseux series solutions",
"We now discuss the computation of solutions of (REF ) in $\\mathbb {K}((x^{1/*}))$ .",
"Even though Proposition REF does not apply, we still assume that the coefficient $\\ell _0$ of $L$ is nonzero.",
"There is no loss of generality in doing so: if $L = L_1 M^w$ for some $w \\in \\mathbb {N}$ , then the Puiseux series solutions of $L$ are exactly the $y(x^{b^{-w}})$ where $y$ ranges over the Puiseux series solutions of $L_1$ .",
"Additionally, the order of $L_1$ is bounded by that of $L$ , so that the complexity estimates depending on it will still hold (and equations of order zero that result from the transformation when $r = w$ have no nontrivial solutions).",
"The computation of solutions $y \\in \\mathbb {K}((x^{1/N}))$ with a given ramification index $N$ is similar to that of power series solutions.",
"In order to compute a full basis of solutions in $\\mathbb {K}((x^{1/*}))$ , however, we need a bound on the ramification index necessary to express them all.",
"Lemma REF , communicated to us by Dreyfus and Roques, and Proposition REF below provide constraints on the possible ramification indices.",
"Lemma 2.17 If $y \\in \\mathbb {K}((x^{1/*}))$ is a Puiseux series such that $L y \\in \\mathbb {K}((x^{1/q^{\\prime }}))$ where $q^{\\prime }$ is coprime with $b$ , then $y \\in \\mathbb {K}((x^{1/q}))$ for some $q$ coprime with $b$ .",
"Let $q_0$ be the smallest positive integer such that $y \\in \\mathbb {K}((x^{1/q_0}))$ .",
"Set $g = q_0 \\wedge b$ and $q^{\\prime \\prime } = q_0 / g$ , so that $M y \\in \\mathbb {K}((x^{b/q_0})) \\subset \\mathbb {K}((x^{1/q^{\\prime \\prime }}))$ .",
"The expression $y = \\ell _0^{-1} \\, \\bigl (L y - (\\ell _1 + \\cdots + \\ell _r M^{r-1}) M y \\bigr )$ shows that $y \\in \\mathbb {K}((x^{1/q_1}))$ where $q_1 = q^{\\prime } q^{\\prime \\prime }$ .",
"By minimality of $q_0$ , we have $q_1 = k q_0$ for some $k \\in \\mathbb {N}$ , which simplifies to $q^{\\prime } = k g$ .",
"Since $q^{\\prime }$ was assumed to be coprime with $b$ , this implies $g=1$ .",
"Remark 2.18 Some non-Puiseux formal series solutions of Mahler equations with $\\ell _0 \\ne 0$ do involve ramifications of order divisible by $b$ : perhaps the simplest example, akin to [9] (see also [1]), is $y = x^{1/b} + x^{1/b^2} + x^{1/b^3} + \\cdots $ , which satisfies $(M-x^{b-1}) (M - 1) \\, y = 0$ .",
"The following proposition formalizes, as a consequence of Lemma REF and the properties of Newton polygons discussed in §REF , that no ramification is needed beyond those present in the candidate leading terms given by the Newton polygon.",
"Call $\\mathcal {N}$ the lower Newton polygon of $L$ , and let $Q$ denote the set of denominators $q$ of slopes (written in lowest terms) of admissible edges of $\\mathcal {N}$ such that $q \\wedge b = 1$ .",
"Proposition 2.19 Any Puiseux-series solution $y$ of $L y = 0$ belongs to $\\mathcal {V} = \\sum _{q \\in Q} \\mathbb {K}((x^{1/q}))$ .",
"In particular, the space of solutions of $L$ in $\\mathbb {K}((x^{1/*}))$ is contained in $\\mathbb {K}((x^{1/N}))$ , where $N\\le b^r - 1$ denotes the lcm of the elements of $Q$ .",
"Let $y \\in \\mathbb {K}((x^{1/*}))$ satisfy $L y = 0$ , and suppose by contradiction that $y$ contains a nonzero term of exponent $p_1/q_1$ where $p_1 \\wedge q_1 = 1$ and $q_1$ does not divide any element of $Q$ .",
"Choose $p_1/q_1$ minimal with these properties.",
"Write $y = y_0 + y_1$ where $y_0$ consists of the terms of $y$ with exponent strictly less than $p_1/q_1$ , so that $y_0 \\in \\mathcal {V}$ and $y_1$ has valuation $p_1/q_1$ .",
"Then $g = L y_0$ belongs to $\\mathcal {V}$ , so that there exists $q^{\\prime } \\in \\mathbb {N}$ for which $q^{\\prime } \\wedge b = 1$ and $g \\in \\mathbb {K}((x^{1/q^{\\prime }}))$ .",
"Since $L y_1 = -g$ , Lemma REF implies that $y_1 \\in \\mathbb {K}((x^{1/q}))$ for some $q$ coprime with $b$ .",
"In particular, $q_1$ is coprime with $b$ .",
"Since $p_1/q_1$ is the valuation of a solution of the equation $L z = -g$ , its opposite $s=-p_1/q_1$ is the slope of an admissible edge $\\mathcal {E}$ of the lower Newton polygon $\\mathcal {N}_g$ of $(L, -g)$ (see §REF ).",
"On the other hand, because of the definition of $Q$ and the properties $q_1 \\wedge b = 1$ and $q_1 \\notin Q$ , the edge $\\mathcal {E}$ cannot be an edge of $\\mathcal {N}$ .",
"Therefore, by the description in §REF , $g$ must be nonzero and the edge $\\mathcal {E}$ must be the leftmost edge of $\\mathcal {N}_g$ .",
"The valuation of $g \\in \\mathcal {V}$ is thus a rational number $p_0/q_0$ (not necessarily in lowest terms) with $q_0 \\in Q$ , so that in particular $q_0 \\wedge b = 1$ .",
"As $s$ is the slope of $\\mathcal {E}$ in $\\mathcal {N}_g$ , it is of the form $(q_0 v_k - p_0)/(q_0 b^k)$ for some $k \\in \\lbrace 0, \\dots , r\\rbrace $ .",
"Then, $q_1$ divides $q_0 b^k$ .",
"As it is coprime with $b$ , this implies that $q_1$ divides $q_0 \\in Q$ , a contradiction.",
"We have proved that $y$ belongs to $\\mathcal {V}$ .",
"Next, it is clear that $\\mathcal {V}$ is contained in $\\mathbb {K}((x^{1/N}))$ .",
"Finally, letting $(b^{k_i}, v_i)$ denote the vertices of $\\mathcal {N}$ (sorted from left to right as $i$ increases), the lcm $N$ satisfies $N \\le \\prod _i (b^{k_{i+1} - k_i} - 1) < b^r$ , as claimed.",
"Remark 2.20 The bound $N < b^r$ is tight, as shown by the example of $M^r - x$ , which admits the solution $x^{1/(b^r - 1)}$ .",
"In order to obtain an algorithm that computes a basis of the space of Puiseux series solutions, there remains to generalize the results of §REF –REF to the case of solutions lying in $\\mathbb {K}((x^{1/N}))$ where $N$ is given.",
"Motivated by the structure of the space $\\mathcal {V}$ described in Proposition REF , we do not require here that $N$ be equal to the lcm of all elements of $Q$ : setting it to the lcm of any subset of these elements also makes sense.",
"For the most part, the algorithms searching for power series solutions apply mutatis mutandis when the indices $m$ and $n$ are allowed to take negative and noninteger rational values.",
"Nevertheless, some care is needed in the complexity analysis, so we explicitly describe a way to reduce the computation of ramified solutions of $L$ to that of power series solutions of an operator $\\tilde{L}$ .",
"Denote $x = t^\\beta $ , and consider the change of unknown functions $y(x) = t^\\alpha z(t)$ , for $\\alpha \\in \\mathbb {Z}$ and $\\beta \\in \\mathbb {N}_{>0}$ to be determined.",
"Observe that $M t = t^b$ .",
"If $y(x)$ is a solution of $Ly = 0$ , then $z(t)$ is annihilated by $\\tilde{L} = t^{-\\gamma } L \\, t^\\alpha = t^{-\\gamma } \\sum _{k=0}^r t^{\\alpha b^k} \\ell _k(t^\\beta ) M^k= \\sum _{k=0}^r \\tilde{\\ell }_k(t) M^k$ where $\\gamma \\in \\mathbb {Z}$ can be adjusted so that the $\\tilde{\\ell }_k$ belong to $\\mathbb {K}[t]$ .",
"We then have $\\tilde{L} = \\phi (L)$ where $\\phi $ is the $\\mathbb {K}$ -linear map, already introduced in §REF , that sends $x^j M^k$ to $\\phi (x^j M^k)= t^{-\\gamma } t^{\\beta j} M^k t^\\alpha = t^{-\\gamma + \\beta j + \\alpha b^k} M^k .$ Viewing monomials $x^j M^k$ as points in the plane of the Newton diagram, the map $\\phi $ induces an affine shearing $[\\phi ]: \\begin{pmatrix}b^k \\\\ j\\end{pmatrix} \\mapsto \\begin{pmatrix}1 & 0 \\\\ \\alpha & \\beta \\end{pmatrix}\\begin{pmatrix}b^k \\\\ j\\end{pmatrix} +\\begin{pmatrix}0 \\\\ -\\gamma \\end{pmatrix}.$ As in §REF , denote by $\\tilde{v}_k$ and $\\tilde{d}_k$ the valuations and degrees of the coefficients of $\\tilde{L}$ , and by $\\tilde{\\mu }$ and $\\tilde{\\nu }$ the quantities defined by () with $v_k$ replaced by $\\tilde{v}_k$ .",
"Figure: The transformation in Example puts the edge with slope 1/21/2 of the lower Newton polygon of LL (left)onto the UU-axis (Newton polygon of L ˜\\tilde{L}, right).Lemma 2.21 Fix an edge $S_0$ of the lower Newton polygon of $L$ , of slope $-p/q$ for (not necessarily coprime) $p\\in \\mathbb {Z}$ and $q\\in \\mathbb {N}$ .",
"Let $c$ be the $V$ -intercept of the line supporting $S_0$ .",
"Set $\\alpha = p$ , $\\beta = q$ , and $\\gamma = q c$ in (REF ).",
"Then: the operator $\\tilde{L} = \\phi (L)$ has polynomial coefficients; its Newton diagram is the image of that of $\\tilde{L}$ by $[\\phi ]$ , with the edge $S_0$ being mapped to a segment of the $U$ -axis; in terms of those of $L$ , the parameters associated to $\\tilde{L}$ satisfy $\\tilde{d}_k = -q c + p b^k + q d_k \\ge \\tilde{v}_k = -q c + p b^k + q v_k \\ge 0, \\\\\\tilde{\\nu }= q \\nu -p \\ge 0,\\qquad \\tilde{\\mu }= q (\\mu - c) \\ge 0.$ Observe that $q c$ is equal to the common value on $S_0$ of $p U + qV$ .",
"Since the endpoints of $S_0$ have integer coordinates, this value is an integer, and hence the coefficients of $\\tilde{L}$ are Laurent polynomials.",
"The transformation $[\\phi ]$ of the Newton plane maps segments of slope $s$ to segments of slope $(\\alpha + \\beta s)/(1 + 0 \\cdot s) = p + q s$ , and in particular maps $S_0$ to a horizontal segment.",
"By the choice of $c$ , that segment lies on the $U$ -axis.",
"Since $q > 0$ , images by $[\\phi ]$ of points above $S_0$ lie above $[\\phi ](S_0)$ .",
"As monomials of $L$ correspond to points lying on or above $S_0$ , their images by $\\phi $ are monomials of nonnegative degree.",
"This proves assertion REF .",
"It follows that $\\tilde{L}$ has a Newton diagram in the sense of our definition, and it is then clear this Newton diagram is as stated by REF .",
"The expressions of $\\tilde{v}_k$ and $\\tilde{d}_k$ in REF are a consequence of (REF ), using again the positivity of $\\beta $ .",
"Those of $\\tilde{\\nu }$ and $\\tilde{\\mu }$ follow.",
"We already observed that $\\tilde{v}_k \\ge 0$ .",
"Finally, $-\\tilde{\\nu }$ and $\\tilde{\\mu }$ are, respectively, the slope and $V$ -intercept of the leftmost edge of the lower Newton polygon $\\tilde{\\mathcal {N}}$ of $\\tilde{L}$ .",
"Since $\\tilde{\\mathcal {N}}$ has a horizontal edge, $\\tilde{\\nu }$ and $\\tilde{\\mu }$ are nonnegative.",
"Example 2.22 Consider again the Mahler operator $L$ in (REF ) treated for $b = 3$ in §REF .",
"We already observed that the slopes of the Newton polygon of $L$ are $-3$ and $1/2$ and that they are admissible, and, in §REF , we performed the transformation (REF ) for the parameters $\\alpha = -1$ , $\\beta = 2$ , and $\\gamma = -3$ , to obtain the operator $\\tilde{L}$ in (REF ).",
"The slopes of the Newton polygon of $\\tilde{L}$ are $-7$ and 0 and are both admissible.",
"Theorem 2.23 Algorithm REF runs in $ \\operatorname{O}(r^2 \\operatorname{M}(N d) + r N (d^2 + (r+d) \\, n))= \\operatorname{\\tilde{O}}(r^2 N d \\, (d + n))~\\text{ops} $ (assuming a softly linear-time polynomial multiplication) and computes the truncation to order $\\operatorname{O}(x^{n+1})$ of a basis of solutions of (REF ) in $\\mathbb {K}((x^{1/N}))$ .",
"The discussion at the beginning of this section shows that $z(x) \\in \\mathbb {K}((x^{1/*}))$ is a solution of the operator $\\tilde{L}$ computed at step REF if and only if $y(x) = x^{-s} z(x^{1/N})$ is a solution of $L$ .",
"By Lemma REF and the choice of $s$ , solutions of $L$ in $\\mathbb {K}((x^{1/N}))$ have valuation at least $-s$ , and hence correspond to solutions of $\\tilde{L}$ lying in $\\mathbb {K}[[x]]$ .",
"Since the mapping $z \\mapsto y$ is linear and invertible, a basis of solutions of $\\tilde{L}$ in $\\mathbb {K}[[x]]$ provides a basis of solutions of $L$ in $\\mathbb {K}((x^{1/N}))$ .",
"Let $S_0$ be the edge of the Newton polygon of $L$ considered at step REF , so that the notation of the algorithm agrees with that of Lemma REF .",
"Lemma REFREF then provides expressions various parameters associated to $\\tilde{L}$ in terms of $s$ , $c$ , and quantities that can be read off $L$ .",
"Since $\\tilde{\\nu }$ is nonnegative, Proposition REF applies and shows that step REF computes a basis $(f_1,\\dots ,f_{\\sigma })$ of the space of approximate solutions of $\\tilde{L}$ in $\\mathbb {K}[[x]]$ in $\\operatorname{O}(r d \\tilde{v}_0 + r^2 \\operatorname{M}(\\tilde{v}_0))$ ops.",
"Denote by $(z_1, \\dots , z_\\sigma )$ the basis of power series solutions of $\\tilde{L}$ such that each $z_i$ extends $f_i$ .",
"Then, according to Proposition REF , the series $\\hat{z}_i$ computed at step REF satisfy $z_i = \\hat{z}_i + \\operatorname{O}(x^{N(s+n)+1})$ , and their computation takes $\\operatorname{O}(\\sigma (r + d) \\tilde{n})$ ops.",
"Finally, the truncated Puiseux series returned by the algorithm satisfy $\\hat{y}_i = x^{-s} \\hat{z}_i(x^{1/N})$ , hence are truncations of elements of a basis of solutions of $\\tilde{L}$ in $\\mathbb {K}((x^{1/N}))$ .",
"Steps other than REF and REF do not perform any operation in $\\mathbb {K}$ , so that the cost in ops of the algorithm is concentrated in those two steps.",
"Let $(b^{k_1}, v_{k_1})$ and $(b^{k_2}, v_{k_2})$ with $k_1 < k_2$ be the endpoints of $S_0$ , so that $ q c = p b^{k_1} + q v_{k_1} = p b^{k_2} + q v_{k_2}.$ Lemma REFREF gives $\\tilde{v}_0 = q v_0 + p - q c$ .",
"If $p \\ge 0$ , then (REF ) implies $q c \\ge p$ and hence $\\tilde{v}_0 \\le q v_0 \\le N d$ .",
"If, now, $p < 0$ , first observe that since $b^{k_2} \\ge 2 b^{k_1}$ , we have $-p b^{k_1} \\le -p (b^{k_2} - b^{k_1}) = q (v_{k_2} - v_{k_1})$ .",
"It follows that $-q c = - p b^{k_1} - q v_{k_1} \\le q v_{k_2}$ , whence $\\tilde{v}_0 \\le q (v_0 + v_{k_2}) \\le 2 N d$ .",
"In both cases, we have proved that $\\tilde{v}_0 = \\operatorname{O}(N d)$ .",
"The complexity estimate for step REF thus rewrites as $\\operatorname{O}(r N d^2 + r^2 \\operatorname{M}(N d))$ ops.",
"As $s \\le d$ (because all slopes of the Newton polygon are bounded by $d$ in absolute value) and $\\sigma \\le r$ , that of step REF becomes $\\operatorname{O}(r N (r + d) (d + n))$ ops.",
"The total running time is therefore $\\operatorname{O}(r^2 \\operatorname{M}(N d) + r N (d^2 + (r+d) \\, n))$ ops.",
"Recall that $Q$ denotes the set of denominators $q$ of slopes, written in lowest terms, of admissible edges of $\\mathcal {N}$ such that $q \\wedge b = 1$ .",
"Corollary 2.24 Algorithm REF with $N$ set to the lcm of elements in $Q$ , returns the truncation to order $\\operatorname{O}(x^{n+1})$ of a basis of solutions of (REF ) in $\\mathbb {K}((x^{1/*}))$ in $\\operatorname{\\tilde{O}}(r^2b^r d (d + n))$ ops, assuming $\\operatorname{M}(k) = \\operatorname{\\tilde{O}}(k)$ .",
"This follows by combining Proposition REF with Theorem REF .",
"Example 2.25 With $b = 3$ , let us consider the order $r = 11$ Mahler operator $L =x^{568}- (x^{1218} + x^{1705}) M+ x^{3655} M^2- (x^{162} - x^{10962}) M^3 \\\\+ (1+x^{487}-x^{4104}-x^{4536}-x^{32887}) M^{4}- (x - x^{11826} - x^{12313} - x^{13122} - x^{13609}) M^5 \\\\- (1 + x^{35479} + x^{39367}) M^6+ (x+x^{95634}-x^{106434}-x^{118098}) M^7 \\\\- (x^{286416} + x^{286903} - x^{319303} - x^{354295}) M^8+ x^{859249} M^9 \\\\+ x^{2577744} M^{10}- x^{7733233} M^{11}.$ Its associated parameters are $w = 0$ , $v_0 = 568$ , and a Newton polygon made from five segments, all admissible, with slopes $-203/13$ , $-3$ , 0, $1/1458$ , and $221/5$ .",
"Except for $1458 = 2 \\cdot 3^6$ , the denominators are coprime with $b = 3$ and their lcm is $N = 65$ .",
"The rightmost slope is $s = 221/5$ and we perform the change of variables of Algorithm REF with $\\alpha = -2873$ , $\\beta = 65$ , hence $\\gamma = -6283186$ and this provides us with the new operator $\\tilde{L} =t^{6317233}- (t^{6353737} + t^{6385392}) {M}+ t^{6494904} {M}^2- (t^{6216145} - t^{6918145}) {M}^3 \\\\+ (t^{6050473}+t^{6082128}-t^{6317233}-t^{6345313}-t^{8188128}) {M}^{4} \\\\- (t^{5585112} - t^{6353737} - t^{6385392} - t^{6437977} - t^{6469632}) {M}^5 \\\\- (t^{4188769} -t^{6494904}-t^{6747624}) {M}^6+ (1+t^{6216145}-t^{6918145}-t^{7676305}) {M}^7 \\\\- (t^{6050473} + t^{6082128} - t^{8188128} - t^{10462608}) {M}^8+ t^{5585112} {M}^9+ t^{4188769} {M}^{10}- {M}^{11}.$ We want to find a basis of Puiseux solutions for $L$ with a precision $\\operatorname{O}(x^{n})$ where $n = 10^6$ .",
"According to Algorithm REF , this leads us to compute a basis of formal series solutions for $\\tilde{L}$ with a precision $\\operatorname{O}(x^{\\tilde{n}})$ where $\\tilde{n} = 65002873$ .",
"We first apply Algorithm REF with $\\tilde{\\nu }= 3888$ , $\\tilde{\\mu }= 6321121$ .",
"The computation shows that the space of solutions has dimension 2.",
"We extend the solutions to the requested precision by Algorithm REF and we obtain a basis of formal series solutions $\\tilde{f}_1(t) =1+{t}^{28080}+{t}^{657072}+{t}^{2274480}+{t}^{2302560}+{t}^{17639856}+{t}^{53222832}\\\\+{t}^{53250912}+{t}^{62068032}+\\operatorname{O}\\left( {t}^{65002873}\\right),$ $\\tilde{f}_2 (t) ={t}^{3888}+{t}^{314928}+{t}^{343008}+{t}^{9160128}+{t}^{25509168}+{t}^{25537248}\\\\+{t}^{27783648}+{t}^{27811728}+\\operatorname{O}\\left( {t}^{65002873}\\right).$ Reversing the change of variable, we find the basis $f_1(x) =x^{-{\\frac{221}{5}}}+x^{{\\frac{1939}{5}}}+x^{{\\frac{50323}{5}}}+x^{{\\frac{174739}{5}}}+x^{{\\frac{176899}{5}}}+x^{{\\frac{1356691}{5}}}+x^{{\\frac{4093843}{5}}}\\\\+x^{{\\frac{4096003}{5}}}+x^{{\\frac{4774243}{5}}}+\\operatorname{O}\\left( x^{1000000} \\right),$ $f_2(x) =x^{{\\frac{203}{13}}}+x^{{\\frac{62411}{13}}}+x^{{\\frac{68027}{13}}}+x^{{\\frac{1831451}{13}}}+x^{{\\frac{5101259}{13}}}+x^{{\\frac{5106875}{13}}}\\\\+x^{{\\frac{5556155}{13}}}+x^{{\\frac{5561771}{13}}}+\\operatorname{O}\\left( x^{1000000} \\right).$ These truncated series satisfy $Lf_1 = \\operatorname{O}(x^{e})$ , $Lf_2 = \\operatorname{O}(x^{e})$ with $e = v_0 + n = 1000568$ ."
],
[
"Rational solutions",
"We now turn to the computation of rational function solutions of Mahler equations of the form (REF ).",
"Our algorithm follows a classical pattern: it first computes a denominator bound, that is, a polynomial that the denominator of any (irreducible) rational solution must divide.",
"Then it makes a change of unknown functions and computes the possible numerators using the algorithm of §REF .",
"As is usual with other functional equations, the denominator bound is obtained by analyzing the action of the operator $L$ on zeros and poles of the functions it is applied to."
],
[
"Denominator bounds: setting",
"We will call a rational function $p/(x^{\\bar{v}} q)$ in lowest terms if it satisfies the following conditions: $\\bar{v} \\ge 0$ ; $p,q\\in \\mathbb {K}[x]$ are coprime polynomials; $q(0) \\ne 0$ ; and $p(0)$ can be zero only if $\\bar{v} = 0$ .",
"Consider a rational solution $p/(x^{\\bar{v}} q)$ of (REF ), written in lowest terms.",
"We already know from Lemma REF that $\\bar{v} \\le v_r/(b^r-b^{r-1})$ , so we are left with the problem of finding a multiple of $q$ .",
"Write $T a = \\bigvee _{i=0}^{r-1} M^i a$ .",
"We will freely use the fact that $T(ab) \\mid (T a) \\, (T b)$ for all $a$ and $b$ .",
"For any $j$ between 0 and $r$ , multiplying the equation $\\ell _r(x) M^r y + \\cdots + \\ell _1(x) \\, M y + \\ell _0(x) y = 0,$ by $(M^r x^{\\bar{v}}) \\, (M^j q) \\, \\bigvee _{i\\ne j} M^i q$ and reducing modulo $M^j q$ yields $ M^j q\\mid x^{(b^r - b^j) {\\bar{v}}} \\ell _j \\, (M^j p) \\, \\bigvee _{i \\ne j} M^i q.$ As $q$ is coprime with $p$ and $q(0) \\ne 0$ , Equation (REF ) with $j=r$ implies $ M^r q \\mid \\ell _r \\, T q.$ This relation is our starting point for computing a polynomial $q^\\star $ , depending only on $\\ell _r$ , such that $q \\mid q^\\star $ .",
"The algorithm for this task, presented in §REF , operates with polynomials over $\\mathbb {K}$ , but it may be helpful in order to get an intuition to first consider the case $\\mathbb {K}= \\mathbb {C}$ .",
"Assume for simplicity that $q$ is squarefree.",
"Equation (REF ) then says that, if $\\alpha $ is a zero of $q$ , each of its $b^r$ th roots is either a $b^k$ th root with $k < r$ of some zero of $q$ or a zero of $\\ell _r$ .",
"Thus, when $\\alpha $ is not a root of unity, its $b^r$ th roots are either zeros of $\\ell _r$ or roots of lower order of some other zero of $q$ , whose $b^r$ th roots then satisfy the same property.",
"(Compare Lemma REF below.)",
"As $q$ has finitely many zeros, this cannot continue indefinitely, so, in this case, we will eventually find a zero $\\alpha $ whose $b^r$ th roots are zeros of $\\ell _r$ .",
"A difficulty arises when $\\alpha $ is a root of unity, but then at most one of its $b$ th roots can be part of a cycle of the map $\\zeta \\mapsto \\zeta ^b$ (cf.",
"Lemma REF ), and a closer examination shows that the $b-1$ other roots behave essentially like non-roots of unity."
],
[
"Properties of the Mahler and Gräffe operators",
"Going back to the general case, and before making the reasoning sketched above more precise, let us state a few properties of the action of $M$ on polynomials.",
"Besides $M$ , we consider the Gräffe operator defined by $G : \\mathbb {K}[x] \\rightarrow \\mathbb {K}[x], \\hspace{10.0pt} p \\mapsto \\operatorname{Res}_y (y^b - x, p (y)).$ In other words, $Gp$ is the product $p(x^{1/b}) p(\\zeta x^{1/b}) \\cdots p(\\zeta ^{b-1} x^{1/b})$ for any primitive $b$ th root of unity $\\zeta $ .",
"While $M$ maps a polynomial $p$ to a polynomial whose complex zeros are the $b$ th roots of the zeros of $p$ , the zeros of $Gp$ are the $b$ th powers of the zeros of $p$ .",
"As a direct consequence of the definitions, $M$ and $G$ act on degrees by: $ \\deg Mp = b \\deg p, \\qquad \\deg Gp = \\deg p .",
"$ Some other elementary properties that will be useful in the sequel are as follows.",
"Lemma 3.1 For any nonzero $i \\in \\mathbb {N}$ , the following relations between $M$ and $G$ hold for all $p, q \\in \\mathbb {K}[x]$ : $G^i M^i p = p^{b^i}$ , $p \\mid M^iG^ip$ , $p \\mid q \\Longleftrightarrow M^i p \\mid M^i q$ .",
"The case $i>1$ reduces to the case $i=1$ by changing the radix, since $M^i$ (resp.",
"$G^i$ ) is nothing but the Mahler (resp.",
"Gräffe) operator of radix $b^i$ ; so we set $i=1$ .",
"The assertions REF and REF are direct consequences of the definition of $G$ as a resultant.",
"The direct implication in REF is clear.",
"For the converse, write the Euclidean division $q = up + v$ .",
"If $M q = sM p$ for some $s \\in \\mathbb {K}[x]$ , then $(M u) \\, (M p) + (M v) = sM p$ , whence $M v =0$ since $\\deg Mv < \\deg Mp$ .",
"Lemma 3.2 If $p \\in \\mathbb {K}[x]$ is monic irreducible and $i \\in \\mathbb {N}$ , then $G^i p = q^e$ for some monic irreducible $q \\in \\mathbb {K}[x]$ and $e \\in \\mathbb {N}$ .",
"Furthermore, $G^i p = p$ if and only if $p$ divides $M^i p$ .",
"If this holds for $i>0$ , $G^j p$ is monic irreducible for any $j \\in \\mathbb {N}$ .",
"To prove the first point, consider the factorization $G^{i} p = c q_1^{e_1} \\cdots q_s^{e_{s}}$ of $G^{i}p$ for monic irreducible and pairwise coprime $q_j$ and a nonzero $c \\in \\mathbb {K}$ .",
"Because of Lemma REFREF , the polynomials $M^{i} q_1^{e_1}, \\dots , M^{i} q_s^{e_{s}}$ are pairwise coprime.",
"We have $ M^iG^i p = c \\, (M^i q_1^{e_1}) \\cdots (M^i q_s^{e_s}), $ and, by Lemma REFREF , $p \\mid M^{i}q_j^{e_{j}}$ for some $j$ .",
"It follows that $G^{i} p \\mid G^{i}M^{i} q_j^{e_{j}} = q_j^{e_{j}b^{i}}$ by Lemma REFREF , proving the first point.",
"Now if $p \\mid M^i p$ , then $G^i p \\mid p^{b^i}$ , and necessarily there is $e \\in \\mathbb {N}$ such that $G^i p = p^e$ .",
"In fact, $e = 1$ and $G^i p = p$ as $G^i p$ and $p$ have the same degree and $p$ is irreducible.",
"Conversely, if $G^i p = p$ , then $p$ divides $M^i p$ by Lemma REFREF .",
"Assume $G^i p = p$ for some $i>0$ .",
"Let $j \\in \\mathbb {N}$ and $m \\in \\mathbb {N}$ such that $mi \\ge j$ .",
"Then $p = G^{mi} p = G^{mi-j} (G^j p)$ is monic irreducible, so that $G^j p$ is monic irreducible too.",
"Lemma 3.3 Let $f\\in \\mathbb {K}[x]$ be a nonconstant polynomial with $f(0) \\ne 0$ .",
"If $f$ and its derivative $f^{\\prime }$ are coprime, so are $Mf$ and $(Mf)^{\\prime }$ .",
"Assume $f \\wedge f^{\\prime }=1$ .",
"Applying $M$ to a Bézout relation shows that $Mf \\wedge M(f^{\\prime })=1$ .",
"Now, $(Mf)^{\\prime } = bx^{b-1} M(f^{\\prime })$ , so a common factor $s$ of $Mf$ and $(Mf)^{\\prime }$ must divide $x$ .",
"As $x$ cannot divide $Mf$ because $x\\nmid f$ , the only possibility is that $s$ be a constant.",
"The following lemma generalizes the fact that the iterated $b$ th roots of a complex number $\\alpha \\ne 0$ are all distinct, except in some cases where $\\alpha $ is a root of unity.",
"Lemma 3.4 Let $p \\in \\mathbb {K}[x]$ be monic and irreducible.",
"For general $\\mathbb {K}$ , $M^i p$ and $M^j p$ are coprime for all $i > j \\ge 0$ if none of the $G^i p$ for $i \\ge 1$ is equal to $p$ .",
"When $\\mathbb {K}=\\mathbb {Q}$ , the same conclusion holds if $G p$ is not equal to $p$ .",
"We proceed by contraposition, assuming the negation of the common conclusion: for monic irreducible $p$ , assume $M^i p \\wedge M^j p \\ne 1$ for some $i > j \\ge 0$ .",
"Set $k = i - j \\ge 1$ .",
"Lemma REFREF implies that $M^k p$ and $p$ are not coprime.",
"Then $p$ divides $M^k p$ and Lemma REF implies that $G^k p = p$ .",
"This proves the result for general $\\mathbb {K}$ .",
"For $\\mathbb {K}=\\mathbb {Q}$ , a further consequence is that the map $\\alpha \\mapsto \\alpha ^{b^k}$ is a permutation of the roots of $p$ in $\\bar{\\mathbb {Q}}$ .",
"Hence, all roots of $p$ satisfy $\\alpha ^B = \\alpha $ for some power $B = b^e$ of $b$ , with $e>0$ .",
"This means that $p$ divides $x^B - x$ .",
"If $p = x$ , $Gp = p$ ; otherwise, $p$ is a cyclotomic polynomial $\\Phi _a$ with $a \\mid b^e-1$ , so $a \\wedge b = 1$ .",
"Applying the formula in [13] yields $M \\Phi _a = \\prod _{b^{\\prime } \\mid b} \\Phi _{a b^{\\prime }}$ , so that $p$ divides $M p$ .",
"Lemma REF now implies $G p = p$ again, completing the proof.",
"Remark 3.5 Over a general subfield $\\mathbb {K}\\subset \\mathbb {C}$ , the cyclotomic polynomial $\\Phi _a$ factors as $\\Phi _a = \\Psi _1 \\cdots \\Psi _s$ and $G$ acts as a cyclic permutation of the $\\Psi _i$ .",
"See also [13] for a detailed description of the case $a \\wedge b \\ne 1$ .",
"Lemma REF states a result for polynomials $p$ that are not part of a cycle of the map $G$ .",
"As a matter of fact, a related graph whose structure plays a crucial role in what follows is that of the map $\\sqrt{G}$ that maps a monic irreducible $p$ to the unique monic irreducible $q$ such that $G p$ is some power of $q$ : we call this map the radical of $G$ , as it ignores the exponent generally introduced by $G$ .",
"An immediate degree argument shows that the cycles of $G$ are exactly the cycles of $\\sqrt{G}$ , and consist of monic irreducible polynomials only.",
"To find a kind of generalization of Lemma REF that applies to polynomials on cycles of $\\sqrt{G}$ , we can always reduce to its hypothesis $G^i p \\ne p$ for nonzero $i$ , by “stepping back one step” in the graph of $\\sqrt{G}$ , thus leaving the cycle.",
"Lemma 3.6 Let $f\\in \\mathbb {K}[x]$ be a nonconstant polynomial with $f(0) \\ne 0$ .",
"There exists a monic irreducible factor $q\\in \\mathbb {K}[x]$ of $Mf$ such that $G^kq\\ne q$ for all nonzero $k\\in \\mathbb {N}$ .",
"Choose a monic irreducible factor $p$ of $f$ and write $Mp = q_1\\cdots q_s$ for monic irreducible $q_i$ .",
"By contradiction, assume that for each $i$ , there is some nonzero $k_i$ for which $G^{k_i}q_i = q_i$ .",
"It follows that for $k=k_1\\cdots k_s$ and all $i$ , $G^kq_i = q_i$ .",
"Lemma REFREF implies $p^b = (G q_1) \\cdots (G q_s)$ , and because of Lemma REF , for all $i$ , $G q_i$ is irreducible.",
"Hence, there exist nonzero $e_i\\in \\mathbb {N}$ such that $G q_i = p^{e_i}$ , with $b=e_1+\\dots +e_s$ .",
"Therefore, for each $i$ , $q_i = G^{k-1}p^{e_i}$ , so that, as $q_i$ is irreducible, $e_i=1$ , and thus all $q_i$ are equal to some same monic irreducible $\\tilde{q}$ .",
"It follows that $Mp=\\tilde{q}^b$ .",
"As $p$ is irreducible, Lemma REF applies to show that $Mp \\wedge (Mp)^{\\prime } = 1$ , which is impossible.",
"The result follows by setting $q=q_i$ for a suitable $i$ .",
"Figure: Graph of the radical G\\sqrt{G} of the Gräffe operatorfor b=6b = 6 in ℚ[x]\\mathbb {Q}[x].Here, aa is a positive integer, coprime to bb.In general, the graph of G\\sqrt{G} consists ofa loop rooted at xx (top left),bi-infinite trees (bottom),and cycles between cyclotomic polynomialswith infinite trees rooted at them (top right).Example 3.7 To suggest the graph structures induced by the Mahler and Gräffe operators, we depict on Figure REF the graph of the radical $\\sqrt{G}$ .",
"Applying $M$ to some vertex $p$ in the graph results in the product of all antecedents under the map.",
"For example, $M (x-2^6) = (x-2)(x+2)(x^2-2x+4)(x^2+2x+4)$ , and $M \\Phi _a = \\Phi _a \\Phi _{2a} \\Phi _{3a} \\Phi _{6a}$ .",
"In the second example, $\\Phi _a$ appears to the right as a consequence of it being mapped to itself by $G$ .",
"The depicted case, $b = 6$ , is typical for $\\mathbb {Q}[x]$ .",
"In particular, all cycles have length 1 as a consequence of the second part of Lemma REF ."
],
[
"Denominator bounds: algorithm",
"Armed with the previous lemmas, we can now prove the key result that leads to our main denominator bound.",
"Still, to avoid repetitions in the proof of Proposition REF below, we first state two intermediate lemmas.",
"The following lemma can be expressed more intuitively as follows: for any $\\tilde{f}$ that is not on a cycle of $\\sqrt{G}$ , any $g$ that appears on the tree rooted at $\\tilde{f}$ of antecedents under $\\sqrt{G}$ is also not on a cycle.",
"Lemma 3.8 Let $\\tilde{f} \\in \\mathbb {K}[x]$ be monic irreducible and satisfy $G^i \\tilde{f} \\ne \\tilde{f}$ for all $i > 0$ .",
"Further, let $g \\in \\mathbb {K}[x]$ be monic irreducible and divide $M^j \\tilde{f}$ for some $j \\ge 0$ .",
"Then $G^i g \\ne g$ for all $i > 0$ .",
"Suppose $G^i g = g$ for some $i \\ge 1$ .",
"By Lemma REF , $G^j g$ is monic irreducible, and since $G^j g \\mid G^j M^j \\tilde{f} = \\tilde{f}^{b^j}$ , it must be $\\tilde{f}$ .",
"Thus, $G^i \\tilde{f} = G^{i+j} g = G^j g = \\tilde{f}$ , in contradiction with the definition of $\\tilde{f}$ .",
"Lemma 3.9 Let $s \\ge r-1$ and $m \\ge 1$ be integers, and let $f \\in \\mathbb {K}[x]$ be monic irreducible, $q \\in \\mathbb {K}[x]$ be nonconstant, and $\\ell \\in \\mathbb {K}[x]$ be nonzero, and such that $x \\nmid q$ , $M^s f^m \\mid M^r q \\mid \\ell \\, T q$ , and $M^{s-i} f \\wedge q = 1$ whenever $0 \\le i < r$ .",
"Then $M^s f^m$ divides $\\ell $ .",
"Let $h^k \\mid M^s f^m$ for a monic irreducible $h \\in \\mathbb {K}[x]$ and $k > 0$ , so that $h^k \\mid \\ell \\, T q$ .",
"We prove by contradiction that $h$ is coprime with $T q$ : suppose there exists some $i$ satisfying $0 \\le i <r$ such that $h$ divides $M^i q$ .",
"Then, $G^i h$ divides both $G^i M^i q$ and $G^i M^s f$ , which, upon applying Lemma REFREF , are equal to powers of $q$ and $M^{s-i} f$ , respectively.",
"This contradicts the coprimality of $q$ and $M^{s-i} f$ .",
"We conclude that $h^k \\mid \\ell $ , and the conclusion follows upon considering all $h^k \\mid M^s f^m$ .",
"The following proposition will be used implicitly as a termination test in Algorithm REF : as long as there exists a nonpolynomial rational solution $p/q$ , the nonconstant polynomial $u$ proved to exist contains (potential) factors of $q$ and can be used to change unknowns in a way that lessens the degree of $\\ell _r$ .",
"An interpretation of the structure of the proof is as follows: If some factor of $q$ appears out of all cycles of $\\sqrt{G}$ , there exists such a factor $u$ with no other factor of $q$ in the tree rooted at $u$ , and this $u$ satisfies $M^r u \\mid \\ell $ .",
"Otherwise, each factor $f$ of $q$ is on a cycle and leads to some antecedent $\\tilde{f}$ under $\\sqrt{G}$ that is on no cycle, for which $f$ divides $G \\tilde{f}$ .",
"Considering all possible $f$ and taking multiplicities into account, we construct a polynomial $u$ such that $M^{r-1} u \\mid \\ell $ and $q \\mid G u$ .",
"Proposition 3.10 Let $\\ell \\in \\mathbb {K}[x]$ be a nonzero polynomial and $q \\in \\mathbb {K}[x]$ be a nonconstant polynomial such that $x \\nmid q$ and $M^r q \\mid \\ell \\, T q$ .",
"Then there exists a nonconstant $u \\in \\mathbb {K}[x]$ such that: either $M^r u \\mid \\ell $ , or $M^{r-1} u \\mid \\ell $ and $q \\mid G u$ .",
"We consider two cases, the first one being when there exists a monic irreducible $f$ dividing $q$ such that $G^if \\ne f$ for all $i>0$ .",
"In this case, we first prove that we can also assume without loss of generality that $M^j f \\wedge q = 1$ for all $j > 0$ .",
"Assume the contrary: that the gcd is nontrivial for at least one $j > 0$ .",
"By Lemma REF , the $M^j f$ for $j\\in \\mathbb {N}$ are pairwise coprime, and since $q$ has finitely many factors, $M^j f \\wedge q \\ne 1$ for at most finitely many $j$ .",
"Set $j$ to the maximal possible value and $g$ to a monic irreducible factor of $M^j f \\wedge q$ .",
"Lemma REF applied to $g$ and $\\tilde{f} = f$ implies that $G^i g \\ne g$ for all $i > 0$ , and $g$ can replace $f$ with the added property on the $M^j g$ .",
"At this point, Lemma REF applies with $s = r$ and $m = 1$ , proving that $M^r f$ divides $\\ell $ .",
"The proposition is proved in this case by choosing $u = f$ .",
"In the second case, let $q = c \\prod _k f_k^{m_k}$ be the irreducible factorization of $q$ , for a nonzero constant $c$ and two-by-two distinct monic irreducible $f_k$ , and with, for each $k$ , some $i_k>0$ satisfying $G^{i_k} f_k = f_k$ .",
"Fix any $k$ .",
"Lemma REF provides a monic irreducible factor $\\tilde{f}_k\\in \\mathbb {K}[x]$ of $M f_k$ such that $G^i \\tilde{f}_k \\ne \\tilde{f}_k$ for all $i > 0$ .",
"If $M^i \\tilde{f}_k \\wedge q$ was nontrivial for some $i \\in \\mathbb {N}$ , this gcd would contain some monic irreducible factor $g$ , necessarily equal to some $f_{k^{\\prime }}$ , and Lemma REF would contradict the existence of $i_{k^{\\prime }}$ .",
"So the polynomials $M^j \\tilde{f}_k$ are coprime with $q$ for all $j \\in \\mathbb {N}$ .",
"Upon setting $s = r - 1$ , $m = m_k$ , and $g = \\tilde{f}_k$ , $M^s g^m = M^{r-1} \\tilde{f}_k^{m_k} \\mid M^r f_k^{m_k} \\mid M^r q$ , and $M^{s-i} g = M^{r-1-i} \\tilde{f}_k$ is coprime with $q$ for all $i$ satisfying $0 \\le i < r$ , so that Lemma REF proves that $M^{r-1} \\tilde{f}_k^{m_k} = M^s g^m$ divides $\\ell $ .",
"Additionally, $G g = G \\tilde{f}_k \\mid G M f_k = f_k^b$ , so that $G g$ is a power of $f_k$ , hence $f_k \\mid G g = G \\tilde{f}_k$ , and next $f_k^{m_k} \\mid G \\tilde{f}_k^{m_k}$ .",
"Gathering the results over all $k$ , the $\\tilde{f}_k$ are pairwise coprime because the $f_k$ are; it follows that all $M^{r-1} \\tilde{f}_k^{m_k}$ divide $\\ell $ and are pairwise coprime, so that, finally, the product $u = \\prod _k \\tilde{f}_k^{m_k}$ satisfies $M^{r-1} u \\mid \\ell $ and $q \\mid G u$ .",
"Remark 3.11 In the first case of the proof, which builds $u$ satisfying $M^r u \\mid \\ell $ , it is of interest to compare the construction with that in the case of usual recurrences [2].",
"The obtained $u$ is extremal, in the sense that no other factor of $q$ can be found in the tree rooted at it, that is to say by iterating $\\sqrt{G}$ backward from it; this is used to compute $u$ from the leading coefficient $\\ell $ of the Mahler operator.",
"In the case of usual recurrences, the shift operator $S$ (with respect to the variable $n$ ) and its inverse $S^{-1}$ play roles similar to $M$ and $\\sqrt{G}$ , respectively.",
"In Abramov's algorithm for denominator bounds, poles are searched for by considering poles that are extremal in a class $\\alpha + \\mathbb {Z}$ : in particular, a pole $\\beta \\in \\alpha + \\mathbb {Z}$ with minimal real part corresponds to a monic irreducible factor $u = n - \\beta $ such that $S^r u$ divides the leading coefficient $\\ell $ of the recurrence operator.",
"Corollary 3.12 When $d < b^{r-1}$ , Eq.",
"(REF ) has no nonconstant rational solution.",
"With the notation above, Lemma REF implies $\\bar{v} = 0$ .",
"If a nonconstant $q$ could satisfy Eq.",
"(REF ), Proposition REF would apply, inducing the contradiction $b^{r-1} \\le \\deg \\ell _r \\le d$ .",
"So $q$ is constant, and Lemma REF applies and proves $p$ is constant.",
"Proposition REF forms the basis of Algorithm REF , which repeatedly searches for factors of the form $M^r u$ to “be removed” from $\\ell _r$ (while “adding back” other factors of strictly smaller degree) and accumulates the corresponding $u$ into the denominator bound.",
"The update of $\\ell $ at step REF of each loop iteration can be viewed as a change of unknown functions of the form $y =\\tilde{y}/u_k$ in (REF ).",
"The search for factors of the form $M^r u$ , respectively $M^{r-1} \\tilde{u}$ , uses the following property (for radix $b^r$ , resp.",
"$b^{r-1}$ ).",
"Lemma 3.13 Let $f_0, \\dots , f_{b-1},u \\in \\mathbb {K}[x]$ .",
"The polynomial $\\ell = Mf_0 + x \\, Mf_1 + \\dots + x^{b-1} \\, Mf_{b-1}$ is divisible by $M u$ if and only if $f_0, \\dots , f_{b-1}$ are all divisible by $u$ .",
"The “if” part is clear.",
"Conversely, fix $i < b$ , and assume that $Mu \\mid \\ell $ .",
"Let $\\omega $ be a primitive $b$ th root of unity.",
"Then, $Mu = (Mu)(\\omega ^j x) \\mid \\ell (\\omega ^j x)$ for all $j$ , hence $Mu$ divides $\\sum _{j=0}^{b-1} \\omega ^{-ij} \\ell (\\omega ^j x)= b \\, x^i M f_i.$ As $Mu \\in \\mathbb {K}[x^b]$ and $i < b$ , this implies $Mu \\mid Mf_i$ , and $u \\mid f_i$ by Lemma REFREF .",
"Figure: Portion of the graph of the radical G\\sqrt{G}of the Gräffe operator used for the resolution in Example .Example 3.14 In this example, we let $b = 3$ and use Algorithm REF to analyze the potential poles in rational-function solutions of an operator $ L = \\bigl (p_1(x) \\cdots p_6(x)\\bigr ) \\, M^2 + \\cdots , $ where the $p_i$ are polynomials to be found in Figure REF and the coefficients of $M^1$ and $M^0$ will be disclosed below.",
"In the figure and this example, polynomials of large size are truncated to their first few monomials, and in most cases, we write them in factored form, although polynomials are manipulated in expanded form in the actual algorithm.",
"Following Algorithm REF , we set $\\ell = p_1 \\cdots p_6$ .",
"Step REF is motivated by the first case in Proposition REF : it strives to solve (REF ) by finding a factor $u$ of $q$ such that $M^2 u \\mid \\ell $ .",
"For each $i$ , the only monic irreducible candidate factor of $u$ that can “cover” $p_i$ upon application of $M^2$ is the polynomial $p_{i,2}$ in the figure.",
"However, $M^2 p_{i,2}$ consists of all factors on the level of $p_i$ with same ancestor $p_{i,2}$ .",
"So, for example, $M_2 p_{1,2} = p_1$ and $p_{1,2}$ can be part of $u$ , whereas $M_2 p_{6,2}$ is a strict multiple of $p_6$ so that $p_{6,2}$ cannot be made part of $u$ .",
"As a matter of fact, for $k=1$ in the loop, the algorithm finds $u_1 = p_{1,2} p_{2,2} p_{4,2} p_{5,2}$ at step REF , after rewriting $\\ell $ in the form $\\ell =- M^2\\bigl ( (2x-1)(x^2-x+1)(x^2-x-1)(x^6-x^3-1)(9x^5-133x^4+\\cdots ) \\bigr ) + \\cdots + \\\\x^8 M^2\\bigl ( 2(2x-1)(x^2-x+1)(x^2-x-1)(x^6-x^3-1)(5x^4-74x^3+\\cdots ) \\bigr )$ at step REF .",
"Step REF resets $\\ell $ to a polynomial that factors into $ p_{1,1} \\, p_{1,2} \\, p_{2,1} \\, p_{2,2} \\, p_3 \\, p_4 \\, p_{5,2} \\, p_{4,2} \\, p_6.",
"$ Following the same approach for $k=2$ , a new phenomenon occurs because of the loops in the graph: the candidate factor $p_{2,3}$ that would “cover” $p_{2,1}$ appears in its own tree on the same level as $p_{2,1}$ , and thus has to be rejected.",
"It follows that the algorithm finds $u_2 = p_{3,2} p_{4,2}$ at step REF , after rewriting $\\ell $ in the form $\\ell = - M^2\\bigl ( (x^2-4x-1)(x^2-x-1)(248x^5-5615x^4+\\cdots ) \\bigr ) + \\cdots + \\\\x^8 M^2\\bigl ( (x^2-4x-1)(x^2-x-1)(532x^4-6211x^3+\\cdots ) \\bigr )$ at step REF .",
"Step REF resets $\\ell $ to a polynomial that factors into $ p_{1,1} \\, p_{1,2} \\, p_{2,1} \\, p_{2,2} \\, p_{3,1} \\, p_{5,2} \\, p_{4,2} \\, p_{3,2} \\, p_6.",
"$ Following the same approach for $k=3$ leads to $u_3 = 1$ : no further factor $u$ of $q$ exists and helps solving Eq.",
"(REF ) by ensuring $M^2 u \\mid \\ell $ .",
"This leads to step REF , which is motivated by the second case in Proposition REF : Eq.",
"(REF ) now implies $M^2 q \\mid q \\wedge M q$ , which is solved by finding $\\tilde{u}$ such that $M \\tilde{u} \\mid \\ell $ .",
"A difference to step REF is that at step REF , candidates are looked for just $2 - 1 = 1$ level above the factors to be “covered”.",
"A similar calculation as previously explains that the algorithm finds $\\tilde{u} = p_{1,2} p_{2,2} p_{3,2} p_{4,2}$ , after rewriting $\\ell $ in the form $\\ell = M^2\\bigl ( (2x-1)(x^2-x+1)(x^2-4x-1)(x^2-x-1)(181x^{13}-1198x^{12}+\\cdots ) \\bigr ) \\\\- x M^2\\bigl ( (2x-1)(x^2-x+1)(x^2-4x-1)(x^2-x-1)(44x^{13}-623x^{12}+\\cdots ) \\bigr ) \\\\+ x^2 M^2\\bigl ( (2x-1)(x^2-x+1)(x^2-4x-1)(x^2-x-1)(4x^{13}-382x^{12}+\\cdots ) \\bigr ) .$ From these factors, only $p_{2,2}$ is cyclotomic.",
"But as the algorithm does not factor polynomials, the other factors cannot be discarded.",
"At step REF , the algorithm returns the bound $q^\\star = u_1 u_2 G \\tilde{u} ={p_{1,2}}^{1+1} {p_{2,2}}^2 {p_{3,2}}^{1+1} {p_{4,2}}^{2+1} p_{5,2} ,$ where the “$+1$ ” indicate factors that could have been saved if a cyclotomic test had been available.",
"The operator $L$ was indeed constructed so as to admit the two explicit rational solutions $\\frac{2x}{(2x-1)(x^2-x-1)}\\qquad \\text{and}\\qquad \\frac{x-3}{(x^2-x+1)(x^2-4x-1)(x^6-x^3-1)} ,$ whose denominators are effectively “covered” by $q^\\ast $ .",
"We remark that, during the steps of the algorithm, the degree of $\\ell $ has dropped from its initial value 145 down to 84, then to 62.",
"Example 3.15 Let $b=3$ and let us consider the Mahler equation $L = (2x^4-x^3-x+3)(2x^9-1)(x^{18}-x^9-1) \\, M^2 \\\\-(x^2+1)(2x^3-1)(x^4+1)(x^6-x^3-1)(2x^{10}-x^9-x+3) \\, M \\\\+x^2(2x-1)(x^2+x+1)(x^2-x+1)(x^2-x-1)(2x^{12}-x^9-x^3+3) .$ Following Algorithm REF , we expand $(2x^4-x^3-x+3)(2x^9-1)(x^{18}-x^9-1)$ to get $\\ell $ , which step REF rewrites $\\ell = M^2(6x^3-9x^2-3x+3) + x M^2(-2x^3+3x^2+x-1) + {} \\\\x^3 M^2(-2x^3+3x^2+x-1) + x^4 M^2(4x^3-6x^2-2x+2) .$ (That is, $f_2 = f_5 = f_6 = f_7 = f_8 = 0$ .)",
"We get $u_1 = 2x^3-3x^2-x+1$ , which factors into $(2x-1)(x^2-x-1)$ .",
"Step REF resets $\\ell $ to a polynomial that factors into $(2x-1)(x^2-x-1)(2x^3-1)(2x^4-x^3-x+3)(x^6-x^3-1)$ .",
"Expanding $\\ell $ as in step REF , we now find $\\ell =M^2(3-10x) + x M^2(-4-15x) + x^2 M^2(-8-19x) + {} \\\\x^3 M^2(5+40x) + x^4 M^2(5-10x) + x^5 M^2(2x+9) + {} \\\\x^6 M^2(-25-16x) + x^7 M^2(15+8x) + x^8 M^2(23) ,$ so that $u_2=1$ .",
"We pass to step REF , which expands $\\ell $ in the form $\\ell = M(-16x^5+40x^4-10x^3-25x^2+5x+3) + {} \\\\x M(8x^5-10x^4-15x^3+15x^2+5x-4) + {} \\\\x^2 M(2x^4-19x^3+23x^2+9x-8) ,$ and $\\tilde{u} = (2x-1)(x^2-x-1)$ .",
"So, $q^\\star = u_1 \\, G \\tilde{u} = (2x-1)(x^2-x-1)(8x-1)(x^2-4x-1)$ .",
"This means that if $y=p/(x^{\\bar{v}} q)$ is solution of $Ly=0$ , where $\\bar{v} \\ge 0$ and $p,q\\in \\mathbb {K}[x]$ satisfy $x\\wedge q=p\\wedge q=p\\wedge x^{\\bar{v}}=1$ , then $q$ divides $q^\\star $ .",
"Using the results of §REF , we find that 0 could not be a pole of a solution in $\\mathbb {K}(x)$ and therefore $\\bar{v}=0$ .",
"Consequently, $q^\\star $ is a denominator bound.",
"Proposition 3.16 Algorithm REF runs in $\\operatorname{O}((\\deg \\ell _r) \\, \\operatorname{M}(d) \\log d)$ ops if $b = 2$ , resp.",
"in $\\operatorname{O}(b^{-r} \\, (\\deg \\ell _r) \\, \\operatorname{M}(d) \\log d)$ ops if $b \\ge 3$ , and computes a polynomial $q^{\\star }$ of degree at most $\\deg \\ell _r$ if $b = 2$ , resp.",
"at most $(\\deg \\ell _r)/b^{r-1}$ if $b \\ge 3$ , such that any rational function solution $y$ of (REF ) can be written in the form $y = p/(x^{\\bar{v}} q^\\star )$ for some $p \\in \\mathbb {K}[x]$ and ${\\bar{v}} \\in \\mathbb {N}$ .",
"For each $k \\ge 1$ reached by the loop REF , let $\\tilde{\\ell }_k$ denote the value of $\\ell $ considered at step REF , so that the value assigned at step REF is $\\tilde{\\ell }_{k+1}$ .",
"(In particular, $\\tilde{\\ell }_1 = \\ell _r$ .)",
"First, observe that, after step REF in each loop iteration, $u_k$ is by Lemma REF a polynomial of maximal degree such that $M^r u_k \\mid \\tilde{\\ell }_k$ .",
"In particular, the next value, $\\tilde{\\ell }_{k+1}$ , computed at step REF , is a polynomial.",
"Set $\\rho = b^r - \\frac{b^r - 1}{b - 1}$ , which is at least 1.",
"Step REF decreases the degree of $\\ell $ by $\\deg \\tilde{\\ell }_k - \\deg \\tilde{\\ell }_{k+1}\\ge \\deg M^r u_k - \\deg T u_k\\ge \\rho \\deg u_k\\ge \\rho .$ In particular, the loop terminates after at most $\\rho ^{-1} (1 + \\deg \\ell _r)$ iterations, and therefore the whole algorithm terminates as well.",
"Second, after step REF , $\\tilde{u}$ is similarly a polynomial of maximal degree such that $M^{r-1} \\tilde{u} \\mid \\tilde{\\ell }_{t+1}$ .",
"Therefore, $b^{r-1} \\deg \\tilde{u}$ is bounded above by the degree of $\\tilde{\\ell }_{t+1}$ , so that $\\biggl ( \\sum _{k=1}^t \\rho \\deg u_k \\biggr ) + b^{r-1} \\deg \\tilde{u}\\le \\biggl ( \\sum _{k=1}^t \\deg \\tilde{\\ell }_k - \\deg \\tilde{\\ell }_{k+1} \\biggr )+ \\deg \\tilde{\\ell }_{t+1}\\le \\deg \\ell _r ,$ where $t$ denotes, as in Algorithm REF , the last value of $k$ for which $\\deg u_k > 0$ .",
"The output from the algorithm is $q^\\star = u_1\\cdots u_t \\, (G \\tilde{u})$ .",
"If $b = 2$ , then $\\rho = 1$ and $\\deg q^\\star = \\bigl (\\sum _k \\deg u_k\\bigr ) + \\deg \\tilde{u}$ is bounded by $\\deg \\ell _r$ ; if $b \\ge 3$ , then $ \\rho = b^{r-1} \\left(b-2 + \\frac{b-2}{b-1}\\right) + \\frac{1}{b-1} \\ge b^{r-1} $ and $\\deg q^\\star $ is bounded by $b^{-(r-1)} \\deg \\ell _r$ .",
"Assume that $p/(x^{\\bar{v}} q)$ is a solution written in lowest terms.",
"Set $\\tilde{q}_0 = q$ and, for $k$ between 1 and $t$ , define the polynomials $\\tilde{q}_k = \\tilde{q}_{k-1} / (u_k \\wedge \\tilde{q}_{k-1})$ .",
"Let us prove by an induction on $k$ that, for $1 \\le k \\le t+1$ : (i) $x \\nmid \\tilde{q}_{k-1}$ ; (ii) $M^r \\tilde{q}_{k-1} \\mid \\tilde{\\ell }_k \\, T \\tilde{q}_{k-1}$ ; and (iii) $q \\mid u_1 \\cdots u_{k-1} \\tilde{q}_{k-1}$ .",
"Initially when $k = 1$ , we have $\\tilde{q}_0 = q$ and $\\tilde{\\ell }_1 = \\ell _r$ , so the three properties hold by our assumption on a solution and Equation (REF ).",
"Assume now that $x \\nmid \\tilde{q}_{k-1}$ , $M^r \\tilde{q}_{k-1} \\mid \\tilde{\\ell }_k \\, T \\tilde{q}_{k-1}$ and $q \\mid u_1 \\cdots u_{k-1} \\tilde{q}_{k-1}$ .",
"It follows from $\\tilde{q}_{k-1} = (u_k \\wedge \\tilde{q}_{k-1}) \\, \\tilde{q}_k \\mid u_k \\tilde{q}_k$ that $x \\nmid \\tilde{q}_k$ and $T \\tilde{q}_{k-1} \\mid (T u_k) \\, (T \\tilde{q}_k)$ .",
"Furthermore, $(M^r (u_k \\wedge \\tilde{q}_{k-1})) \\, (M^r \\tilde{q}_k)= M^r \\tilde{q}_{k-1}\\mid \\tilde{\\ell }_k \\, T \\tilde{q}_{k-1}\\mid \\tilde{\\ell }_k \\, (T u_k) \\, (T \\tilde{q}_k) .$ Write $M^r u_k = a_k \\, M^r (\\tilde{q}_{k-1} \\wedge u_k)$ and $\\tilde{\\ell }_k = b_k \\, M^r u_k$ , for suitable polynomials $a_k$ and $b_k$ .",
"Upon division by $M^r (u_k \\wedge \\tilde{q}_{k-1})$ , Equation (REF ) becomes $M^r \\tilde{q}_k \\mid a_kb_k \\, (T u_k) \\, (T \\tilde{q}_k) .$ By construction, $a_k$ and $M^r \\tilde{q}_k$ are coprime, as they are the cofactors of $M^r(u_k \\wedge \\tilde{q}_{k-1})$ in, respectively, $M^r u_k$ and $M^r \\tilde{q}_{k-1}$ , so Equation (REF ) finally becomes $M^r \\tilde{q}_k \\mid \\frac{\\tilde{\\ell }_k}{M^r u_k} \\, (T u_k) \\, (T \\tilde{q}_k) =\\tilde{\\ell }_{k+1} \\, T \\tilde{q}_k .$ By the divisibility assumption on $q$ and the definition of $\\tilde{q}_k$ , $q \\mid u_1 \\cdots u_{k-1} \\tilde{q}_{k-1}= u_1 \\cdots u_{k-1} \\, (u_k \\wedge \\tilde{q}_{k-1}) \\, \\tilde{q}_k\\mid u_1 \\cdots u_k \\tilde{q}_k ,$ completing the proof by induction.",
"The loop terminates when $\\ell $ no longer has any nonconstant factor of the form $M^r u$ , with $\\ell = \\tilde{\\ell }_{t+1}$ .",
"At this point, $M^r \\tilde{q}_t \\mid \\tilde{\\ell }_{t+1} \\, T \\tilde{q}_t$ and $q \\mid u_1 \\cdots u_t \\tilde{q}_t$ .",
"If $\\tilde{q}_t$ is constant, then $q \\mid u_1 \\cdots u_t \\mid q^\\star $ .",
"On the other hand, if $\\tilde{q}_t$ is not constant, Proposition REF applies, as $x \\nmid \\tilde{q}_t$ , which implies that $\\tilde{\\ell }_{t+1}$ admits a factor of the form $M^{r-1} u$ such that $\\tilde{q}_t \\mid Gu$ .",
"By Lemma REF , step REF computes a polynomial $\\tilde{u}$ such that $M^{r-1} u \\mid M^{r-1} \\tilde{u}$ .",
"It follows by Lemma REFREF that $u \\mid \\tilde{u}$ , next that $\\tilde{q}_t \\mid G \\tilde{u}$ , so that $q$ divides $q^\\star $ , again.",
"Let us turn to the complexity analysis.",
"Applying $M$ to a polynomial requires no arithmetic operation.",
"Each execution of step REF amounts to $b^r-1$ gcds of polynomials of degree less than or equal to $d/b^r$ , for a total cost of $\\operatorname{O}(\\operatorname{M}(d) \\log d)$ ops.",
"The same argument applies to step REF .",
"Similarly, the chain of lcms at step REF requires $\\operatorname{O}\\biggl (\\sum _{i=0}^{r-1} \\operatorname{M}(b^i \\deg u_k) \\log (b^i \\deg u_k)\\biggr )= \\operatorname{O}(\\operatorname{M}(d) \\log d)~\\text{ops} ,$ as $(\\sum _{i=0}^{r-1} b^i) \\deg u_k = \\operatorname{O}(d)$ .",
"Since there are at most $\\rho ^{-1} (1+\\deg \\ell _r)$ iterations of steps REF and REF , the cost of step REF is $\\operatorname{O}(\\rho ^{-1} \\, (\\deg \\ell _r) \\, \\operatorname{M}(d) \\log d)$ .",
"If $b = 2$ , then $\\rho = 1$ and the cost of step REF is $\\operatorname{O}((\\deg \\ell _r) \\, \\operatorname{M}(d) \\log d)$ .",
"If $b\\ge 3$ , then $\\rho \\ge b^{r-1}$ and the cost is $\\operatorname{O}(b^{-r} \\, (\\deg \\ell _r) \\, \\operatorname{M}(d) \\log d)$ .",
"The computation of $G \\tilde{u}$ from $\\tilde{u}$ at step REF can be performed in $\\operatorname{O}(\\operatorname{M}(bd))$ ops [7], [15] and the final product can be computed in $\\operatorname{O}(\\operatorname{M}(d) \\log d)$ ops using a product tree.",
"Proposition REF implicitly provides a bound on $\\deg q$ that essentially (when $\\bar{v} = 0$ and $\\tilde{u} = 1$ , exactly) matches that of Bell and Coons [6].",
"However, a tighter bound holds, especially for $b=2$ .",
"Proposition 3.17 With the notation above, $q$ has degree at most $3 \\deg \\ell _r /b^r$ .",
"Let $g = M^r q \\wedge Tq$ .",
"On the one hand, (REF ) implies $M^r q \\mid \\ell _r g$ , so that $b^r \\deg q \\le \\deg \\ell _r + \\deg g$ .",
"On the other hand, $Mg$ divides $h = M^r q \\vee Tq$ by definition of $T$ , hence $g \\, Mg$ divides $gh = (M^r q) \\, (Tq)$ , whence $ (b + 1) \\deg g \\le \\frac{b^{r+1} - 1}{b-1} \\deg q.",
"$ Comparing the two inequalities leads to $\\deg q\\le \\frac{(b^2 - 1) \\deg \\ell _r}{b^{r+2} - b^{r+1} - b^r + 1}\\le \\frac{(b^2 -1) \\deg \\ell _r}{b^r (b^2 - b - 1)}\\le \\frac{3 \\deg \\ell _r}{b^r}$ since $(b^2-1)/(b^2-b-1) \\le 3$ for $b \\ge 2$ .",
"Remark 3.18 The previous discussion to find $q^\\star $ is entirely based on (REF ) in the case $j = r$ and on expressing the solution $y$ with a minimal denominator $x^{\\bar{v}} q$ .",
"Noting that (REF ) actually holds also for $j \\ne r$ and even if $p \\wedge q \\ne 1$ , we may apply it with $0 \\le j \\le r-1$ to a potential solution written in the form $p/(x^{\\bar{v}} q^\\star )$ to get additional constraints involving $\\ell _0, \\dots , \\ell _{r-1}$ that can be used to remove some factors from $q^\\star $ ."
],
[
"An alternative bound",
"We now describe an alternative method for computing denominator bounds.",
"While it yields coarser bounds, our estimate for its computational cost is better, so that it may be a superior choice in some cases.",
"The results of this subsection are not used in the sequel.",
"Proposition 3.19 If $x^{\\bar{v}} q \\in \\mathbb {K}[x]$ is the denominator of a rational solution of (REF ) written in lowest terms, then it holds that $q \\mid (G^r \\ell _r) \\, (G^{r+1} \\ell _r) \\cdots (G^{r + K} \\ell _r),\\qquad K = \\lfloor \\log _b (3 \\deg \\ell _r) \\rfloor - r.$ Suppose $f$ is monic irreducible and $m$ is positive such that $f^m \\mid q$ , and consider the condition $ M^{r+j} f \\mid \\bigvee _{i=0}^r M^i q.$ Clearly, (REF ) is satisfied for $j=0$ , while it requires $ b^{r+j} \\deg f \\le \\frac{b^{r+1} - 1}{b - 1} \\deg q, $ which in turn implies $j \\le \\log _b \\deg q$ .",
"Plugging in the bound from Proposition REF , we obtain $j \\le \\log _b (3 \\deg \\ell _r) - r$ .",
"Choose $j$ maximal such that (REF ) holds.",
"Then $M^{r+j} f$ cannot divide $Tq$ , and by Lemma REF , $M^{r+j} f$ is squarefree.",
"Let $h$ be a monic irreducible factor of $M^{r+j} f$ not dividing $Tq$ .",
"In the rest of the proof, we write $\\operatorname{sqrfree}p$ for the squarefree part of any polynomial $p$ .",
"For all $k \\ge 0$ , set $h_k = \\operatorname{sqrfree}(G^k h)$ , and denote by $m_k$ the multiplicity of $h_k$ as a factor of $Tq$ .",
"Thus $m_0$ is zero by definition of $h$ .",
"Continuing with $k \\ge 0$ , Lemma REFREF implies $G^k h \\mid MG^{k+1} h$ , so that $h_k \\mid \\operatorname{sqrfree}(MG^{k+1} h) \\mid M \\operatorname{sqrfree}(G^{k+1} h) = M h_{k+1}$ .",
"As $h_k^{m_k} \\mid Tq$ , we deduce that $h_k^{m_{k+1}} \\mid M h_{k+1}^{m_{k+1}} \\mid M Tq \\mid Tq \\wedge M^r q$ , then, by using (REF ), $h_k^{m_{k+1}} \\mid \\ell _r \\, Tq$ .",
"The definition of $m_k$ then yields $h_k^{\\delta _k} \\mid \\ell _r$ for $\\delta _k = \\max (m_{k+1} - m_k, 0)$ .",
"Now, restrict $k$ to the interval $j < k \\le r+j$ .",
"Then, by Lemma REFREF , $h_k \\mid G^k h \\mid G^k M^{r+j} f = (M^{r+j-k} f)^{b^k} ,$ and as $h_k$ is squarefree, $h_k$ divides $M^{r+j-k} f$ .",
"Since $f^m \\mid q$ and $0 \\le r+j-k < r$ , $h_k^m$ divides $T q$ , implying $m_k \\ge m$ .",
"By Lemma REFREF and Equation (REF ), $G^{r+j-k} h_k$ is $f^{b^{r+j}}$ , so that $f$ divides the former.",
"Then, $ f^{\\delta _k} \\mid G^{r+j-k} h_k^{\\delta _k} \\mid G^{r+j-k} \\ell _r .",
"$ Forming the product of these bounds for $k$ ranging from 0 to $j$ , we get $f^m \\mid \\prod _{k=0}^j G^k \\ell _r$ , as $m \\le m_{j+1}$ and $m_0 = 0$ .",
"The result follows by considering all possible $(f,m)$ such that $f^m \\mid q$ .",
"Proposition 3.20 One can compute a polynomial $q^\\ast \\in \\mathbb {K}[x]$ of degree at most ${d \\, (\\log _b d - r + 2)}$ and such that $q \\mid q^\\ast $ in $\\operatorname{O}(\\operatorname{M}(d \\log d) \\log d)$ ops.",
"If $\\deg \\ell _r < b^{r-1}$ , return 1.",
"This is a valid bound by Corollary REF .",
"Otherwise, return the bound from Proposition REF .",
"As with the previous bound, the $G^k \\ell _r$ up to $k = r+K = \\operatorname{O}(\\log d)$ can be computed for a total of $\\operatorname{O}(\\operatorname{M}(b d) \\log d)$ ops [7], [15].",
"The product then takes $\\operatorname{O}(\\operatorname{M}(d \\log d) \\log d)$ ops."
],
[
"Computing numerators",
"In order to obtain a basis of rational solutions $y$ of (REF ), it suffices to obtain a bound $x^{\\bar{v}} q^\\star $ on denominators as in §REF , to construct an auxiliary equation corresponding to the change of unknown functions $y = \\tilde{y} / (x^{\\bar{v}} q^\\star )$ , and to search for its polynomial solutions $\\tilde{y}$ .",
"We first note the following consequence of Lemma REF , already proved by Bell and Coons [6].",
"Proposition 3.21 If $p, q \\in \\mathbb {K}[x]$ , not necessarily coprime, satisfy $L (p/q) = 0$ , then $\\deg p$ is at most $\\deg q + \\lfloor d/(b^r - b^{r-1}) \\rfloor $ .",
"The procedure to obtain rational solutions is summarized in Algorithm REF .",
"Proposition 3.22 Algorithm REF computes a basis of rational solutions of its input equation.",
"Assuming $d \\ge b^{r-1}$ , it runs in $\\operatorname{\\tilde{O}}(d \\operatorname{M}(d) + 2^r d^2 + \\operatorname{M}(2^r d))$ ops when $b = 2$ and $\\operatorname{\\tilde{O}}(b^{-r} d \\operatorname{M}(d))$ ops when $b \\ge 3$ .",
"Assuming further $\\operatorname{M}(n) = \\operatorname{\\tilde{O}}(n)$ , it runs in $\\operatorname{\\tilde{O}}(2^r d^2) = \\operatorname{\\tilde{O}}(d^3)$ ops when $b = 2$ and in $\\operatorname{\\tilde{O}}(b^{-r}d^2)$ ops when $b \\ge 3$ .",
"Define $\\delta $ as in step REF , so that $\\delta \\le d$ .",
"If $\\delta < b^{r-1}$ , the algorithm will stop after step REF .",
"In this case, Corollary REF states that there are no nonconstant rational solution.",
"Therefore, the vector space of rational solutions is $\\mathbb {K}$ when $L(1) = 0$ and $\\lbrace 0\\rbrace $ otherwise.",
"Otherwise, the algorithm continues with $d \\ge b^{r-1}$ .",
"Assume that $y \\in \\mathbb {K}(x)$ is a rational solution of $L y = 0$ , and let $p = x^{\\bar{v}} q^\\star y$ for $q^\\star $ and $\\bar{v}$ computed as in step REF .",
"By Proposition REF combined with Lemma REF , $p$ is a polynomial.",
"By Proposition REF combined with Lemma REF , it has degree at most $\\deg (x^{\\bar{v}} q^\\star ) + \\bar{v} = \\deg q^\\star + 2 \\bar{v}$ .",
"Plugging $y = p/(x^{\\bar{v}} q^\\star )$ into $L y = 0$ and multiplying the resulting equation by the polynomial $x^{\\lfloor b \\delta /(b-1) \\rfloor } \\prod _{i=0}^r M^i q^\\star $ , we see that $p$ satisfies $\\tilde{L} p = 0$ , where $\\tilde{L}$ is defined as in step REF .",
"As $b^k \\bar{v} \\le b \\delta /(b-1)$ for $k \\le r$ , the $e_k$ are nonnegative and the $\\tilde{\\ell }_k$ are polynomials.",
"Thus Algorithm REF applies and, by Proposition REF , $p$ belongs to the span of the $p_k$ computed at step REF of Algorithm REF .",
"Conversely, for all $k$ , the fraction $p_k/(x^{\\bar{v}} p^\\star )$ is a solution of $L y = 0$ .",
"After step REF , we have $b^r = \\operatorname{O}(d)$ , that is, $r = \\operatorname{\\tilde{O}}(1)$ .",
"By Proposition REF , the cost of step REF is $\\operatorname{\\tilde{O}}(d \\operatorname{M}(d))$ ops when $b = 2$ and $\\operatorname{\\tilde{O}}(b^{-r} d \\operatorname{M}(d))$ ops when $b \\ge 3$ .",
"Define $\\tilde{d}= \\frac{2b-1}{b-1} d + \\frac{b^{r+1}-1}{b-1} \\deg q^\\star = {\\left\\lbrace \\begin{array}{ll}\\operatorname{O}(2^r d), & b = 2 , \\\\\\operatorname{O}(d), & b \\ge 3 ,\\end{array}\\right.",
"}$ where the asymptotic bounds follow from Proposition REF .",
"Each polynomial $\\tilde{\\ell }_k$ defined at step REF then satisfies $\\deg \\tilde{\\ell }_k\\le e_k + \\delta + \\frac{b^{r+1} - 1}{b-1} \\deg q^\\star \\le \\tilde{d} ,$ so its computation as a product of $r+1$ factors can be done in $\\operatorname{O}( r \\operatorname{M}(\\tilde{d}) )$ ops.",
"This makes a total of $\\operatorname{O}( r^2 \\operatorname{M}(\\tilde{d}) ) = \\operatorname{\\tilde{O}}( \\operatorname{M}(\\tilde{d}) )$ ops to compute the $\\ell _k$ 's.",
"Observe as well that $1 \\le w = \\operatorname{O}(\\tilde{d} / b^r)$ .",
"According to Proposition REF , step REF thus requires $\\operatorname{\\tilde{O}}(b^{-r} \\tilde{d}^2 + \\operatorname{M}(\\tilde{d}))$ ops, which dominates the cost of step REF .",
"Taking the bounds (REF ) into account, we get that step REF is dominated by step REF when $b\\ge 3$ , so that the total cost is $\\operatorname{\\tilde{O}}(d \\operatorname{M}(d) + 2^r d^2 + \\operatorname{M}(2^r d))$ ops when $b = 2$ and $\\operatorname{\\tilde{O}}(b^{-r} d \\operatorname{M}(d))$ ops when $b \\ge 3$ .",
"With fast multiplication, $\\operatorname{M}(n) = \\operatorname{\\tilde{O}}(n)$ , this simplifies to the announced complexity estimates.",
"Example 3.23 We continue Example REF .",
"We have seen that the denominator bound is $q^\\star = (2x-1)(x^2-x-1)(8x-1)(x^2-4x-1)$ .",
"We set $\\tilde{y} = q^\\star y$ , so that $Ly=0$ if and only if $\\tilde{L} \\tilde{y} = 0$ , where $\\tilde{L}=\\tilde{\\ell _2}M^2+\\tilde{\\ell _1}M^{1}+\\tilde{\\ell _0}$ for $\\tilde{\\ell }_2 &= (2x-1)(8x-1)(x^2-x-1)(x^2-4x-1) \\times {} \\\\&\\qquad (4x^2+2x+1)(2x^4-x^3-x+3)(x^4+x^3+2x^2-x+1) ,[0]\\\\\\tilde{\\ell }_1 &= -(8x-1)(x^2+1)(x^2-4x-1)(2x^3-1)(x^4+1) \\times {} \\\\&\\qquad (x^6-x^3-1)(4x^6+2x^3+1)(2x^{10}-x^9-x+3)(x^{12}+x^9+2x^6-x^3+1) ,[0]\\\\\\tilde{\\ell }_0 &= x^2(2x-1)(x^2+x+1)(x^2-x+1)(x^2-x-1) \\times {} \\\\&\\qquad (4x^2+2x+1)(2x^3-1)(x^4+x^3+2x^2-x+1) \\times {} \\\\&\\qquad (x^6-x^3-1)(4x^6+2x^3+1)(x^{12}+x^9+2x^6-x^3+1)(2x^{12}-x^9-x^3+3) .$ We have to compute the complete set of polynomial solutions of $\\tilde{L}\\tilde{y}=0$ .",
"The degree of $\\tilde{\\ell _2},\\tilde{\\ell _1},\\tilde{\\ell _0}$ are respectively 16, 46, 54.",
"Using Lemma REF , we find that the degree of a nonzero polynomial solution is necessarily 4 or 5.",
"Following Algorithm REF , we equate the coefficients on both sides of $\\tilde{L}\\tilde{y}=0$ up to degree 54, and we obtain that $\\tilde{y} = \\tilde{y}_0 + \\dots + x^5 \\tilde{y}_5$ is solution of $\\tilde{L}\\tilde{y}=0$ if and only if the vector $(\\tilde{y}_0,\\dots ,\\tilde{y}_5)$ is solution of a system of $h=163$ equations.",
"A basis of solutions turns out to consist of $(2x-1)(8x-1)(x^2-4x-1)$ and $(x^2-x-1)(8x-1)(x^2-4x-1)$ .",
"Consequently, a basis of rational-function solutions of $Ly=0$ consists of $ \\frac{1}{2x-1} \\hbox{ and } \\frac{1}{x^2-x-1}.",
"$ Remark 3.24 When Mahler equations are considered in difference Galois theory [12], [21], the interest tends to be in base fields on which $M$ acts as an automorphism, such as $\\mathbb {K}((x^{1/*}))$ and $\\mathbb {K}(x^{1/*})=\\bigcup _{n=1}^{+\\infty } \\mathbb {K}(x^{1/n})$ .",
"By combining the strategy of Algorithm REF with Proposition REF about possible ramifications, we obtain an algorithm that computes a basis of solutions of (REF ) in $\\mathbb {K}(x^{1/*})$ .",
"Assuming $\\operatorname{M}(n) = \\operatorname{\\tilde{O}}(n)$ , it runs in $\\operatorname{\\tilde{O}}(2^{3r} d^3)$ ops when $b = 2$ and in $\\operatorname{\\tilde{O}}(b^{r}d^2)$ ops when $b \\ge 3$ .",
"Note that, as in §REF , these complexity bounds hold even if $\\ell _0$ is zero."
],
[
"Testing transcendence",
"As was announced in the introduction, solving Mahler equations relates to testing the transcendence of Mahler functions.",
"In particular, when computing the rational solutions of a Mahler equation (REF ) shows that there are no nonzero rational solutions, this is a proof that all solutions to (REF ) are transcendental.",
"We compare in this section the complexity of the transcendence test by Bell and Coons [6] with that of a test by our rational solving.",
"To this end, we briefly sketch Bell and Coons' “universal” transcendence test [6] and do a complexity analysis of their approach, using our notation and the same level of sophistication with regard to algorithms for subtasks.",
"Define $\\kappa _1 = \\left\\lfloor \\frac{(b-1) \\, d}{b^{r+1}-2b^r+1} \\right\\rfloor , \\ \\kappa _2 = \\left\\lfloor \\frac{d / (b-1)}{b^{r-1}} \\right\\rfloor , \\ \\kappa = \\kappa _1 + \\kappa _2 + 1 , \\ B = d + \\kappa \\frac{b^{r+1}-1}{b-1} .$ Bell and Coons [6] show that any rational solution $p/q$ to (REF ) without pole at 0 satisfies $\\deg q \\le \\kappa _1$ , $\\deg p \\le \\kappa _1 + \\kappa _2$ , and that if a series $y \\in \\mathbb {K}[[x]]$ solves (REF ), then either $y - p/q \\ne \\operatorname{O}(x^{B+1})$ or $y = p/q$ as series.",
"Then, given $y = y_0 + y_1x + \\cdots $ , Bell and Coons consider the matrix $M = (y_{i+j})_{0\\le i\\le \\kappa , \\ 0\\le j\\le B}$ , whose $i$ th row represents the truncation up to $\\operatorname{O}(x^{B+1})$ of the non-singular part of $y/x^i$ .",
"To any nonzero $\\tilde{q}$ in the left kernel of $M$ , they associate the polynomial $q = \\tilde{q}_\\kappa + \\dots + \\tilde{q}_0 x^\\kappa $ and find a polynomial $p$ of degree at most $\\kappa $ such that $y - p/q = \\operatorname{O}(x^{B+1})$ , therefore such that $y = p/q$ .",
"This leads to the equivalence that $M$ is full rank if and only if $y$ is transcendental.",
"Bell and Coons' test therefore consists in computing the truncation of $y$ up to $\\operatorname{O}(x^{B+\\kappa +1})$ , in forming the matrix $M$ , and in determining if $M$ is of full rank, $\\kappa +1$ .",
"Only considering the linear-algebra task, which will dominate the complexity, Bell and Coons' approach takes $\\operatorname{O}(B\\kappa ^{\\omega -1})$ ops, by the algorithm of Ibarra, Moran and Hui [16].",
"When $b = 2$ , we get $\\kappa = \\operatorname{O}(d)$ , $B = \\operatorname{O}(2^r d)$ , and a complexity $\\operatorname{O}(2^r d^\\omega )$ ; for $b \\ge 3$ , we get $\\kappa = \\operatorname{O}(d/b^r)$ , $B = \\operatorname{O}(d)$ , and a complexity $\\operatorname{O}(d^\\omega / b^{(\\omega -1)r})$ .",
"In either case, the dependency in $d$ is in $\\operatorname{O}(d^\\omega )$ , being not as good as the $\\operatorname{\\tilde{O}}(d^2)$ that can be obtained by Algorithm REF , as Proposition REF justifies.",
"In situations where (REF ) has nonzero rational solutions, a given series solution $y \\in \\mathbb {K}[[x]]$ can easily be tested to be one of them, in $\\operatorname{O}(r^\\omega d) + \\operatorname{\\tilde{O}}(rd)$ ops, because only $\\lfloor \\nu \\rfloor +1 = \\operatorname{O}(d)$ initial coefficients of solutions identify them (see §REF ).",
"So in all cases our Algorithm REF induces a transcendence test in better complexity with respect to $d$ than with the approach of [6]."
],
[
"The case $\\ell _0=0$ and an algorithm for computing gcrd's",
"In this section, we drop the assumption $\\ell _0\\ne 0$ .",
"More precisely, we consider a linear Mahler equation of the form (REF ), with $\\ell _0=\\dots =\\ell _{w-1}=0$ and $\\ell _r\\ell _w\\ne 0$ .",
"We call the integer $w$ the $M$ -valuation of (REF ) and $d=\\max _{k=w,\\dots ,r}\\deg \\ell _k$ its degree.",
"We define the $M$ -valuation and the degree of the corresponding operator () similarly.",
"The goal of this section is to compute a linear Mahler equation with $M$ -valuation equal to 0, such that the new equation and (REF ) have the same set of series solutions in $\\mathbb {K}((x))$ .",
"The algorithm proposed here, Algorithm REF , can be seen as an improvement over an algorithm given by Dumas in his thesis [13].",
"In particular, Algorithm REF , borrowed from [13], performs the subtask of splitting an operator of positive $M$ -valuation into a system of operators of zero $M$ -valuation while preserving the solution set in $\\mathbb {K}((x))$ .",
"Dumas's algorithm next makes use of the right Euclidean structure of the algebra $\\mathcal {M}(\\mathbb {K})$ of linear Mahler operators with coefficients in $\\mathbb {K}(x)$ , and transforms the system into a single, equivalent equation by computing a gcrd (greatest common right divisor) via Euclidean divisions.",
"The problem of this approach is that the degree of the obtained equation explodes in the process.",
"To avoid this, we change the second step of algorithm in [13] so as to reuse Algorithm REF and cancellations of trailing instead of leading coefficients.",
"The splitting process of Algorithm REF is explained in terms of section maps $S_i$, each of which maps a polynomial in $x$ and $M$ to a polynomial in $x$ and $M$ , and whose collection plays the role of a partial inverse for $M$ : for $0 \\le i < b$ , let $S_i$ be the $\\mathbb {K}$ -linear map that sends $x^jM^{k+1}$ to $x^{(j-i)/b}M^k$ if $(j-i)/b$ is an integer and to 0 otherwise.",
"Lemma 4.1 Let $L$ be a linear Mahler operator $L$ of the form () and have degree $d$ and positive $M$ -valuation.",
"Then, whenever $0 \\le i < b$ , the section $S_i(L)$ has degree at most $d/b$ .",
"Additionally, $L$ can be reconstructed from its sections by $L = \\sum _{i=0}^{b-1} x^i M \\, S_i(L) .$ The degree bound and relation (REF ) are shown by immediate calculations.",
"Lemma 4.2 Let $L$ be a linear Mahler operator of the form (), with order $r$ , $M$ -valuation $w$ , and degree $d$ .",
"Then, Algorithm REF returns a set of nonzero linear Mahler operators of order at most $r-w$ , $M$ -valuation 0, and degree at most $ db^{-w}$ .",
"This is shown by a straightforward induction on $w$ .",
"Instead of considering usual Euclidean divisions according to decreasing powers, which would compute a gcrd as in [13], we use in Algorithm REF linear combinations that kill constant terms: given two nonzero Mahler operators $L_1$ and $L_2$ with coefficients in $\\mathbb {K}[x]$ , $M$ -valuation zero, and coefficient of $M^0$ respectively $c_1$ and $c_2$ , we write $R(L_1, L_2)$ for the operator $c_2 L_1 - c_1 L_2$ , whose coefficient of $M^0$ is zero.",
"We call this operator the interreduction of $L_1$ and $L_2$ and a step of the algorithm that replaces an operator $L_1$ by an interreduction $R(L_1, L_2)$ a reduction step.",
"Lemma 4.3 Let $\\mathcal {L}$ be a system of Mahler operators.",
"Replacing an element $L$ of $\\mathcal {L}$ by its sections $S_0(L), \\dots , S_{b-1}(L)$ does not change the set of solutions of $\\mathcal {L}$ in $\\mathbb {K}((x))$ .",
"Nor does replacing $L_1$ by the interreduction $R(L_1, L_2)$ where $L_1, L_2$ are distinct elements of $\\mathcal {L}$ .",
"The second claim is obvious.",
"Regarding the first one (already in [13]), the decomposition (REF ) shows that any common solution of the $S_i(L)$ is a solution of $L$ .",
"If, conversely, $y$ is an unramified solution of $L$ , then the $x^i M S_i(L) \\, y$ , $0 \\le i < b$ , have disjoint support, hence $S_i(L) \\, y = 0$ for all $i$ .",
"Here, the degree of $R(L_1, L_2)$ may well be the sum of the degrees of $L_1$ and $L_2$ , but having generated a multiple of $M$ makes it possible to apply splitting and keep degrees under control.",
"This leads to Algorithm REF , whose correctness and complexity are given in the following proposition.",
"It is worth mentioning that, in general, the equation $\\tilde{L}(y)=0$ returned by Algorithm REF does not have the same set of solutions in $\\mathbb {K}((x^{1/*}))$ as the equation $L(y)=0$ .",
"As an example, let $b=2$ and consider $L=M^2-xM$ .",
"We have $\\tilde{L}=1$ , and the solution space in $\\mathbb {K}((x^{1/*}))$ of $\\tilde{L}(y)=0$ is $\\lbrace 0\\rbrace $ .",
"On the other hand, the solution space in $\\mathbb {K}((x^{1/*}))$ of $L(y)=0$ is the $\\mathbb {K}$ -vector space spanned by $x^{1/2}$ .",
"Proposition 4.4 The operator $L$ has the same set of solutions in $\\mathbb {K}((x))$ as the operator $\\tilde{L}$ returned by Algorithm REF .",
"This operator has order $\\tilde{r}\\le r-w$ , $M$ -valuation 0, and degree $\\tilde{d}\\le db^{-w}$ .",
"Furthermore, Algorithm REF runs in $\\operatorname{O}(r b^r \\operatorname{M}(d/b^w))$ ops.",
"Because $L\\ne 0$ and by construction of Algorithm REF , the initial set $\\mathcal {L}$ is nonempty.",
"Next, by construction of Algorithm REF , at any time of a run, $\\mathcal {L}$ is nonempty and contains only elements of outputs from Algorithm REF , so that, by Lemma REF , if Algorithm REF terminates, its output must be nonzero and of $M$ -valuation zero.",
"Lemma REF implies that the original operator $L$ , the system $\\mathcal {L}$ at any time of the run, and therefore the final operator $\\tilde{L}$ , all share the same set of solutions in $\\mathbb {K}((x))$ .",
"Let us prove the bound on the order and the degree of $\\tilde{L}$ .",
"By Lemma REF , the set $\\mathcal {L}$ computed at step REF consists of Mahler operators with orders bounded by $r-w$ and degrees bounded by $db^{-w}$ .",
"These bounds keep on holding after each run of the loop body at step REF : As the operators $L_1$ and $L_2$ chosen at step REF satisfy the property, their combination $R(L_1, L_2)$ (including the case it is zero) has order bounded by $r-w$ , degree bounded by $2db^{-w}$ , and positive valuation.",
"By Lemma REF , the set $\\mathcal {L}^{\\prime }$ computed at step REF consists of Mahler operators with orders bounded by $r-w-1$ and degrees bounded by $2db^{-(w+1)}$ .",
"As $2/b \\le 1$ , the set $\\mathcal {L}$ retains the property after the update at step REF .",
"Therefore, if the algorithm terminates, it returns at step REF an element of $\\mathcal {L}$ , therefore with the announced order and degree bounds.",
"We finally prove termination and complexity by a joint argument.",
"To this end, we represent the process of Algorithm REF by an oriented tree labeled by operators $L_w^n$ , for integers $n$ and words $w$ on the alphabet $\\lbrace 0,\\dots ,b-1\\rbrace $ .",
"These operators $L_w^n$ will be the operators considered during the execution of the algorithm.",
"This tree is rooted at the node labeled $L_\\epsilon ^0 = L$ , and evolves by following the execution of Algorithm REF .",
"Each time a section of an operator $L_w^n$ is computed by the subtask of Algorithm REF , whether it be at step REF or at step REF , the tree is augmented by new edges from $L_w^n$ to its subsection $L_{wj}^n = S_j(L_w^n)$ .",
"For each choice of $L_1 = L_w^n$ and $L_2 = L_{w^{\\prime }}^{n^{\\prime }}$ at step REF , the tree is augmented by a new edge labeled $L_{w^{\\prime }}^{n^{\\prime }}$ , from $L_w^n$ to $L_\\epsilon ^{m+1}$ , if $m$ is the larger upper index in the tree before reduction.",
"Thus, one obtains that at each stage of the execution, the set $\\mathcal {L}$ is equal to the collection of nonzero leaves of the current tree.",
"Now, by construction of the tree and by design of the algorithm, a reduction step results either in a zero operator or in an operator with positive $M$ -valuation that is immediately split to its sections.",
"Therefore, following a path from the root to a leaf, two reduction edges can only appear if separated by at least one section edge.",
"As section edges reduce orders by at least 1, while reduction edges do not increase orders, the tree has to be finite and the algorithm terminates.",
"The only arithmetic operations of the algorithm are the polynomial products involved in the computation of the $R(L_1, L_2)$ at step REF .",
"It was proved above that any operator of $\\mathcal {L}$ has degree bounded by $d/b^w$ .",
"Because operators all have order at most $r$ and as the size of the tree bounds the number of reductions, the algorithm has total complexity $\\operatorname{O}(r b^r \\operatorname{M}(d/b^w))$ .",
"Remark 4.5 A slightly better complexity can be obtained by a variant of Algorithm REF , in which the $L_1$ at step REF is not chosen as having maximal order, but according to a notion of depth in the tree introduced for the proof of Proposition REF .",
"Doing so guarantees a better behavior of degrees, with a geometric decrease with depth, as opposed to the uniform bound $d/b^w$ used in the proof above.",
"Define the depth $\\beta $ of a node $L_w^n$ in the tree as the number of section edges from the root $L_\\epsilon ^0$ to $L_w^n$ , and change the strategy at step REF to choose $L_1$ among the elements of $\\mathcal {L}$ of lowest depth.",
"By another induction, $L_w^n$ has order not more than $r - \\beta $ , as in the proof above, but its degree is not more than $d/b^w$ if $\\beta \\le w$ , and not more than $(2/b)^{\\beta -w} (d/b^w)$ if $\\beta > w$ .",
"A bound on the complexity becomes $\\sum _{\\beta =w}^r (r+1) b^\\beta \\operatorname{M}\\left( \\frac{2^{\\beta -w} d}{b^\\beta } \\right)\\le \\operatorname{O}\\left( r \\operatorname{M}(2^r d/b^w) \\right) .$ This bound is better than the original complexity $\\operatorname{O}(r b^r \\operatorname{M}(d/b^w))$ when $b \\ge 3$ .",
"For $b=2$ , the new bound is not tight and the variant algorithm has the same complexity bound as Algorithm REF .",
"Example 4.6 Figure: Execution of Algorithm on the operator of Example .Each nonzero operator is given with a corresponding pair (order, degree).Operators are generated in the following order:L ϵ 0 =LL_\\epsilon ^0 = L, L 0 0 L_0^0, L 1 0 L_1^0, L 2 0 L_2^0, L 20 0 L_{20}^0, L 21 0 L_{21}^0, L 22 0 L_{22}^0, L ϵ 1 L_\\epsilon ^1, L 0 1 L_0^1, L 1 1 L_1^1, L 2 1 L_2^1, L ϵ 2 L_\\epsilon ^2, L ϵ 3 L_\\epsilon ^3, L ϵ 4 L_\\epsilon ^4, L ϵ 5 L_\\epsilon ^5, L ϵ 6 L_\\epsilon ^6.Blue and red arrows respectively represent section and reduction steps.Labels on (red) arrows provide the auxiliary operators used for reduction.The process starts with L ϵ 0 =LL_\\epsilon ^0 = L and ends with L 1 1 L_1^1.Observe the strict decrease of orders along blue edgesand large decrease along red edges.Also observe that degrees are divided by at least 3 on blue edgesand, for the only nontrivial red edge of this example,how the reduction of L 0 0 L_0^0 by L 21 0 L_{21}^0 induces an increase of the degreefrom 49 to 58, which is not more than 49+1249 + 12.We apply Algorithm REF with $b = 3$ and the operator $L = \\ell _1 M + \\ell _2 M^2 + \\ell _3 M^3 + \\ell _4 M^4$ with $\\ell _1 & = x^9 (1-x^{15}+x^{51}+x^{54}-x^{87}+x^{108}) (1-x^{12}+x^{24}) ,[0]\\\\\\ell _2 & =- x^3 \\left(1+x^6-x^{20}-x^{21}+x^{30}+x^{32}+x^{33}+x^{36}-x^{44}-x^{45}+x^{54}+x^{56} \\right.",
"\\\\& \\left.",
"\\rule {4em}{0ex} +x^{57}+x^{60}-x^{68}-x^{69}+x^{80}+x^{81}+x^{84}+x^{90}-x^{92}-x^{93}+x^{104} \\right.",
"\\\\& \\left.",
"\\rule {4em}{0ex} +x^{105}+x^{108}+x^{114}-x^{116}-x^{117}+x^{138}+x^{144} \\right) ,[0]\\\\\\ell _3 & =\\left(1+x^3-x^5+x^{17}+x^{18}+x^{21}-x^{23}-x^{29}+x^{35}+x^{36}+x^{39}-x^{47}+x^{54} \\right.",
"\\\\& \\left.",
"\\rule {2em}{0ex} +x^{57}+x^{72}+x^{75}+x^{90}+x^{93}-x^{95}+x^{107}+x^{108}+x^{111}-x^{113}-x^{119}\\right.",
"[0]\\\\& \\left.",
"\\rule {2em}{0ex} +x^{125}+x^{126}+x^{129}-x^{137}+x^{144}+x^{147} \\right) ,\\\\\\ell _4 & = - (1+x^{27}+x^{54}) (1-x^{27}+x^{54}) (1-x^5+x^{17}+x^{18}-x^{29}+x^{36}).$ Starting from $L_\\epsilon ^0 = L$ , we compute its sections (see Fig.",
"REF , blue edges): first, $L_0^0 = S_0(L_\\epsilon ^0)$ , which has $M$ -valuation 0 so that the process of splitting stops for it; next, $L_1^0 = S_1(L_\\epsilon ^0)$ , which is zero and is dropped; last, $L_2^0 = S_2(L_\\epsilon ^0)$ , which has $M$ -valuation 1.",
"Splitting continues for the latter and provides $L_{20}^0 = S_0(L_0^0)$ , $L_{21}^0 = S_1(L_2^0)$ , $L_{22}^0 = S_2(L_0^0)$ , all with $M$ -valuation 0.",
"Note that during this splitting, the operators $L_\\epsilon ^0$ , $L_1^0 = 0$ , and $L_2^0$ disappear.",
"A reduction is made (see Fig.",
"REF , red edges) where $R(L_0^0,L_{21}^0) = L_\\epsilon ^6$ replaces $L_0^0$ .",
"The process continues and, at the end, there only remains $L_1^1 & = x^5 (1+x+x^2) (1-x+x^2) (1-x^{4}+x^8) \\\\& - x^3 (1+x+x^2) (1-x+x^2) (1-x^2+x^{4}-x^6+x^8) (1+2\\,x^2+x^{4}) M \\\\& + x^3 (1+x+x^2) (1-x+x^2) (1+x^3+x^6) (1-x^{3}+x^6) M^2.$ It is worth noting that $L_1^1$ has a content $c = x^3 (1+x+x^2) (1-x+x^2)$ , so that we can write $L_1^1 = c \\bar{L}_1^1$ where $\\bar{L}_1^1$ is a primitive polynomial (with respect to $M$ ).",
"The computation shows that $L$ is in the left ideal generated by $\\bar{L}_1^1$ in the algebra $\\mathcal {M}(\\mathbb {Q})$ .",
"This and exhibiting the $M$ -valuation $w = 1$ of $L$ provides factorizations $L = L^{\\prime } M = L^{\\prime \\prime } M \\bar{L}_1^1$ .",
"We can say that $M^w$ has been pushed as much as possible to the left.",
"Using Algorithm REF , we find that a basis of solutions of $L_1^1$ in $\\mathbb {K}(x)$ is given by 1 and $\\frac{x}{x^2-1}$ .",
"Since $L_1^1$ has order two, this also forms a basis of solutions of $L_1^1$ in $\\mathbb {K}((x))$ , as a consequence of Proposition REF , and by Proposition REF , a basis of solutions of $L$ in $\\mathbb {K}((x))$ .",
"We now proceed to prove that Algorithm REF indeed computes a gcrd with controlled degree.",
"This is proved in Theorem REF below, using the following lemmas.",
"Lemma 4.7 For any operators $P_1$ , $P_2$ , and any integer $i$ such that $0\\le i < b$ , $S_i(P_1 M P_2) = S_i(P_1 M) P_2$ .",
"By linearity, it is sufficient to consider $P_1 = x^{j_1} M^{k_1}$ and $P_2 = x^{j_2} M^{k_2}$ .",
"Then, $P_1 M P_2 = x^{j_1 + b^{k_1+1} j_2} M^{k_1+k_2+1}$ .",
"Either $b$ divides $j_1-i$ and $S_i(P_1 M P_2) = x^{(j_1-i)/b + b^{k_1} j_2} M^{k_1+k_2}= x^{(j_1-i)/b} M^{k_1} x^{j_2} M^{k_2}= S_i(P_1 M) P_2 ,$ or $b$ does not divide $j_1-i$ and both extreme terms are zero, thus equal again.",
"Lemma 4.8 For any operators $P_1$ , $P_2$ , and $P$ , all of $M$ -valuation 0, let $c$ be the coefficient of $M^0$ in $P$ .",
"Then, $R(P_1 P, P_2 P) = c R(P_1, P_2) P$ .",
"The property holds, as obviously the coefficient of $M^0$ in a product is the product of the coefficients of $M^0$ in the factors.",
"Theorem 4.9 Steps REF and REF of Algorithm REF compute a gcrd of the elements of the split $\\mathcal {L}$ of $L$ obtained at step REF .",
"The degree of this particular gcrd is bounded by the maximal degree of the elements of $\\mathcal {L}$ .",
"Let $I$ denote the left ideal $\\mathcal {M}(\\mathbb {K}) \\mathcal {L}$ generated by $\\mathcal {L}$ at any time in the run of the algorithm.",
"Call $G$ the monic gcrd of the elements of the set $\\mathcal {L}$ as obtained from $L$ at the end of step REF .",
"By (REF ), $G$ is a right factor of $L$ .",
"By the definition of $R(\\cdot ,\\cdot )$ and because of (REF ) again, the ideal $I$ can only increase during the run of the algorithm, so that during step REF , $\\mathcal {M}(\\mathbb {K}) L \\subset \\mathcal {M}(\\mathbb {K}) G \\subset I$ .",
"We show by induction that $G$ is a right factor of all elements of $\\mathcal {L}$ at any time in step REF , in other words, that $I \\subset \\mathcal {M}(\\mathbb {K}) G$ .",
"This is true by the definition of $G$ when entering the loop.",
"The set $\\mathcal {L}$ contains only elements with $M$ -valuation 0, and it cannot be empty when entering the loop, so $G$ has $M$ -valuation 0 as well.",
"At any step REF , divisibility on the right by $G$ is preserved for $R(L_1, L_2)$ , by Lemma REF .",
"As $R(L_1, L_2)$ has positive $M$ -valuation, one can choose $P_2 = G$ and find $P_1$ so as to write $R(L_1, L_2) = P_1 M P_2$ .",
"By Lemma REF , it follows that divisibility on the right by $G$ is also preserved for each element of $\\mathcal {L}^{\\prime }$ , then for each element of the next value of $\\mathcal {L}$ .",
"As a consequence, during step REF , $I$ constantly equals $\\mathcal {M}(\\mathbb {K}) G$ .",
"In particular, the final operator $\\tilde{L}$ is proportional to $G$ .",
"The degree bound was proved as part of Proposition REF .",
"Note that the origin of the initial $\\mathcal {L}$ as a split of $L$ , at step REF of Algorithm REF plays no role in the proof of Theorem REF .",
"Thus, Algorithm REF implicitly contains an algorithm for computing the gcrd of any family $\\mathcal {L}$ of operators of $M$ -valuation zero.",
"Remark 4.10 We developed Algorithm REF without targeting a gcrd and realized Theorem REF only a posteriori.",
"As Algorithm REF indeed works by computing a gcrd as the original algorithm in [13], it is now instructive to compare the result of Proposition REF with bounds on the size of gcrds of Mahler operators given by existing methods.",
"Such a bound can be computed using a variant of the subresultant argument given by Grigor'ev [14] in the differential case.",
"Let $L_1, \\dots , L_n$ be operators of respective order $r_1 \\ge r_2 \\ge \\dots \\ge r_n \\ge 1$ and degree $d_1, \\dots , d_n \\le \\delta $ .",
"Let $G = U_1 L_1 + \\dots + U_n L_n$ be their greatest common right divisor.",
"We can assume that the order of each term $U_i L_i$ is less than $t = r_1 + r_n$ .",
"Indeed, for all $i,j$ the linear equation $V_{i,j} L_i = V_{j,i} L_j$ with $V_{i,j}$ , resp.",
"$V_{j,i}$ , constrained to have degree at most $r_j$ , resp.",
"at most $r_i$ , has nontrivial solutions.",
"Via Euclidean divisions $U_i = Q_i V_{i,n} + R_i$ , we obtain $G = \\sum _i (Q_i V_{i,n} + R_i) L_i= \\sum _i Q_i V_{n,i} L_n + \\sum _i R_i L_i= \\sum _i \\tilde{U}_i L_i$ where the $\\tilde{U}_i$ for $i \\le n - 1$ have order less than $r_n$ .",
"The $n-1$ first terms $\\tilde{U}_i L_i$ as well as $G$ itself have order less than $r_1 + r_n$ , hence the same must be true of $\\tilde{U}_n L_n$ .",
"Consider a Sylvester-like matrix $S\\in \\mathbb {K}[x]^{s \\times t}$ with rows $ \\mathcal {R}(L_1), \\mathcal {R}(M L_1), \\dots , \\mathcal {R}(M^{t-r_1-1} L_1),\\dots ,\\mathcal {R}(L_n), \\mathcal {R}(M L_n), \\dots , \\mathcal {R}(M^{t-r_n-1} L_n), $ where, for any operator $L = \\sum _k \\ell _k M^k$ , we denote $\\mathcal {R}(L) = (\\ell _{t-1}, \\dots , \\ell _{0})$ .",
"Call $C_0, C_1, \\dots , C_{t-1}$ the columns of $S$ , listed from right to left (so that $C_j$ contains the coefficients of $M^j$ in $M^k L_i$ ), and $C_{j,0}, C_{j,1}, \\dots , C_{j,s-1}$ the entries of $C_j$ .",
"Let $m$ denote the order of $G$ , and choose $J \\subseteq \\lbrace m + 1, \\dots , t - 1 \\rbrace $ of cardinality $|J| = \\operatorname{rk}S- 1$ in such a way that the columns $C_j$ with $j \\in J$ form a basis of the span of $C_{m+1}, \\dots , C_{t-1}$ , while the $C_j$ for $j \\in \\lbrace m \\rbrace \\cup J$ form a basis of the full column space of $S$ .",
"To see that such a $J$ exists, consider a row echelon form of $S$ : since $\\mathcal {R}(G)$ belongs to the left image of $S$ and $G$ has minimal order among the nonzero elements of the ideal $\\sum _i \\mathcal {M}(\\mathbb {K}) L_i$ , the rightmost pivot lies on column $m$ .",
"Further, let $I \\subseteq \\lbrace 0, \\dots , s-1\\rbrace $ be such that the submatrix $(C_{j,i})$ , $i \\in I$ , $j \\in J \\cup \\lbrace m \\rbrace $ of $S$ is nonsingular.",
"Call $D_m$ the corresponding minor, and more generally define $D_k$ as the determinant of the submatrix $(C_{j,i})$ , $i \\in I$ , $j \\in J$ , extended on the right by a copy of $C_k$ .",
"Expanding $D_k$ along the last column yields $D_k = \\sum _{i=0}^{s-1} u_i C_{k,i}$ , where the $u_i$ do not depend on $k$ .",
"For each $k > m$ , the determinant $D_k$ is zero, as $C_k$ is in the span of the $C_j$ for $j \\in J$ .",
"It follows that the vector $(0, \\dots , 0, D_m, \\dots , D_0)$ belongs to the left image of $S$ .",
"Thus, there is a gcrd of $L_1, \\dots , L_n$ with polynomial coefficients whose coefficients are minors of $S$ .",
"The entries of $\\mathcal {S}$ have degree bounded by $\\delta ^{\\prime } = \\max _{i=1}^n (b^{t-r_i-1} d_i)$ .",
"Therefore, the degree of $G$ is as most $t \\delta ^{\\prime } \\le 2 r_1 b^{r_1 - 1} \\delta \\le r_1 b^{r_1} \\delta $ .",
"Using fast polynomial linear algebra, it is plausible that one could actually compute $G$ based on this approach with a complexity of the type $\\operatorname{\\tilde{O}}(\\delta ^{\\prime } t^\\omega ) = \\operatorname{\\tilde{O}}(b^{r_1} \\delta )$ .",
"Now, the gcrd in the algorithm of [13] is that of a family of iterated sections of the input operator $L$ .",
"In terms of the order $r$ and degree $d$ of $L$ , this family can involve elements simultaneously of order $r-1$ and degree $d/2$ .",
"Thus, Grigor'ev's approach (at least in a straightforward way) would lead to a complexity bound similar to that of Proposition REF , but an exponentially worse bound on the degree of the output for large $r$ .",
"This result leaves open the question of devising algorithms for computing solutions of linear Mahler equations that run in polynomial time in $r$ and $d$ , for all possible combinations of these parameters, even when the trailing coefficient $\\ell _0$ of the equation is zero.",
"In particular, it would be interesting to see if the bounds on the size of an operator equivalent to $L$ implied by Algorithm REF would be enough to extend the algorithms of §– to the case where $\\ell _0$ is zero, without going through the explicit computation of such an operator.",
"We end the section by providing an extension of Algorithm REF , which computes a gcrd for a family of operators of arbitrary $M$ -valuations.",
"Theorem 4.11 Algorithm REF computes a gcrd of the input operators $L_1$ , ..., $L_s$ .",
"Observe that the minimal $M$ -valuation of operators in a family is the minimal $M$ -valuation of elements of the left ideal generated by the family, in particular, the $M$ -valuation of any gcrd of the family.",
"This justifies the general design of the algorithm, with the factorization of $M^w$ on the right at step REF .",
"By construction, the $L^{\\prime }_i$ 's thus obtained have orders at most $r-w$ and degrees at most $d$ , and at least one, say $L^{\\prime }_1$ , has $M$ -valuation zero.",
"Let $G^{\\prime }$ denote the monic gcrd of the $L^{\\prime }_i$ , which, as $L^{\\prime }_1$ , has $M$ -valuation zero.",
"By Lemma REF , $G^{\\prime }$ is a right-hand factor of all elements of the set $\\mathcal {L}$ computed at step REF .",
"By a proof similar to the one for Theorem REF , it remains so for all subsequent values of $\\mathcal {L}$ , so for the $\\tilde{L}$ of step REF as well.",
"As $\\tilde{L}$ is also obviously a right-hand factor of all previously computed operators, including the $L^{\\prime }_i$ 's, $\\tilde{L}$ is a gcrd of the latter.",
"This concludes the proof."
]
] | 1612.05518 |
[
[
"On the Non-existence of certain classes of perfect p-ary sequences and\n perfect almost p-ary sequences"
],
[
"Abstract We obtain new non-existence results of perfect p-ary sequences with period n (called type $[p, n]$).",
"The first case is a class with type [p\\equiv5\\pmod 8,p^aqn'].",
"The second case contains five types [p\\equiv3\\pmod 4,p^aq^ln'] for certain $p, q$ and $l$.",
"Moreover, we also have similar non-existence results for perfect almost p-ary sequences."
],
[
"Introduction",
"Let $n$ be a positive integer, $p$ a rational prime and $\\zeta _p$ a primitive $p$ -th root of unity (we can take $\\zeta _p$ to be $\\exp (\\frac{2\\pi i}{p})$ ).",
"Definition 1.1 A complex sequence $\\mathbf {a}=(a_0, a_1,\\dots ,a_{n-1},\\dots )$ with period $n$ is called a $p$ -ary sequence (resp.",
"an almost $p$ -ary sequence) if $a_j=\\zeta _p^{b_j}$ where $b_j\\in \\mathbb {Z}$ for all $i\\ge 0$ (resp.",
"$a_0=0$ and $a_j=\\zeta _p^{b_j}$ where $b_j\\in \\mathbb {Z}$ for all $1\\le i \\le n-1$ ).",
"A complex sequence $\\mathbf {a}=(a_0, a_1,\\dots ,a_{n-1},\\dots )$ with period $n$ is called perfect if $C_\\mathbf {a}(t)=0$ for all $1\\le t\\le n-1$ , where $C_\\mathbf {a}(t)=\\sum _{k=0}^{n-1}a_k\\bar{a}_{k+t}$ is the autocorrelation with a bar meaning the complex conjugation.",
"For simplicity, we denote a perfect $p$ -ary (resp.",
"an perfect almost $p$ -ary) sequence with period $n$ as a PPS (resp.",
"PAPS) with type $[p, n]$ .",
"A natural question is when PPSs (PAPSs) do exist.",
"This is equivalent to the existence of certain kinds of relative difference sets.",
"See [2], [9], [12] for details.",
"Their results imply that PPSs (PAPSs) can be constructed if the corresponding relative difference sets exist.",
"By using various techniques in combinatorial design theory, several classes of such sequences have been constructed (see [2], [9], [12], [10]).",
"On the other hand, there are some nonexistence results on such sequences (and related difference sets), see [2], [9], [14], [17].",
"Here we need the concept of “self-conjugate\".",
"See [21], [12].",
"Definition 1.2 Let $p$ be a prime integer, $m = p^a m^{\\prime }$ where $a\\ge 0$ and $(p, m^{\\prime }) = 1$ .",
"We call $p$ to be self-conjugated with respect to $m$ if there exists $s \\in \\mathbb {Z}$ such that $p^s \\equiv −1\\pmod {m^{\\prime }}$ .",
"Namely, if $-1\\in \\left<p\\right>\\subseteq (\\mathbb {Z}/m^{\\prime }\\mathbb {Z})^$ .",
"Now we give a list of typical non-existence results of PPSs (PAPSs) with reference at the beginning of each item: (Ma and Ng [14]) PPSs with type $[p, q^ln^{\\prime }]$ where $p\\ne q$ are two primes, $p\\ge 3,\\ (q,n^{\\prime })=1,\\ q$ is self-conjugate w.r.t.",
"$p$ and $l\\ge 1$ is odd.",
"(Liu and Feng [12]) PPSs with type $[p, p^aq^ln^{\\prime }]$ where $p\\equiv 3\\pmod {4}$ is a prime, $q$ is another prime with $(q-1, p) = 1,\\ \\genfrac(){}{}{q}{p}=1$ and $\\operatorname{ord}_p(q)$ being odd, $n^{\\prime }$ satisfies that $n^{\\prime }=1$ or $\\genfrac(){}{}{p^{\\prime }}{p}=-1$ for all prime divisor $p^{\\prime }$ of $n^{\\prime }$ , $a\\ge 1$ and $l$ is odd such that $ l < \\lambda /s$ where $s = (p-1)/\\operatorname{ord}_p(q)$ and $\\lambda $ is the smallest odd integer such that $x^2 + py^2 = 4q^\\lambda $ has solution $(x, y),\\ x, y \\in \\mathbb {Z}$ .",
"(Liu and Feng [12]) PAPSs with type $[p, q^ln^{\\prime }+1]$ where $p\\equiv 3\\pmod {4},\\ p\\mid q^ln^{\\prime }-1$ is a prime, $q,n^{\\prime },a$ and $l$ are the same as the above (REF ).",
"In this article we have two main results.",
"The first one shows the non-existence of PPSs with type $[p,p^aqn^{\\prime }]$ , where $p\\equiv 5\\pmod {8}$ is a prime, $q$ runs through a infinite set of primes and $n^{\\prime }$ is the same as (REF ) in the above list.",
"Theorem 1.3 Let $p\\equiv 5\\pmod {8}$ be a prime and $\\tilde{Q}_p=\\mathinner {\\lbrace \\,{q\\text{ is a prime } }\\mid { \\operatorname{ord}_p(q)=(p-1)/4}\\,\\rbrace }$ .",
"Then there exists a lower bound $p_0$ , and an infinite set $Q_p\\subseteq \\tilde{Q}_p$ for each $p$ , such that if $p>p_0$ , there is no PPSs with type $[p, n=p^aqn^{\\prime }]$ for all integers $a\\ge 1$ , $q\\in Q_p$ and $n^{\\prime }$ such that $n^{\\prime }=1$ or $\\genfrac(){}{}{p^{\\prime }}{p}=-1$ for all prime divisor $p^{\\prime }$ of $n^{\\prime }$ .",
"Remark 1.4 Since $p\\equiv 5\\pmod {8}$ , we have $\\operatorname{ord}_p(q)=(p-1)/4$ is odd for all $q\\in Q_p$ .",
"It follows that $q$ is not self-conjugate w.r.t $p$ , which says that our case is not contained in [14] ((REF ) in the above list).",
"Moreover, our case is also different from [12] ((REF ) in the above list) since $p\\lnot \\equiv 3\\pmod {4}$ .",
"Thus our result is new.",
"In the second main result we obtain the non-existence of PPSs with five types: Theorem 1.5 Let $p\\equiv 3$ be a prime, $q\\ne p$ another prime and $f=\\operatorname{ord}_p(q)$ .",
"Suppose that the triple $(p, f, l_0)$ equals to one of the following value: $(31 , 5 , 1),\\ (127, 9 , 1),\\ (127, 21, 3),\\ (139, 23, 1),\\ (151, 15, 3).$ Define $\\Xi _{31}(x) &= x^3 + x - 1,\\\\\\Xi _{127}(x) &= x^5 - x^4 - 2x^3 + x^2 + 3x - 1,\\\\\\Xi _{139}(x) &= x^3 - x^2 + x + 2,\\\\\\text{and}\\quad \\Xi _{151}(x) &= x^7 - x^6 + x^5 + 3x^3 - x^2 + 3x + 1.$ Suppose further that for each $p\\in \\mathinner {\\lbrace \\,{31,127,139,151}\\,\\rbrace }$ , the corresponding $q$ satisfies that $\\Xi _p(x)\\equiv 0\\pmod {q}$ is not solvable.",
"Then there is no PPPs with type $[p, n=p^aq^ln^{\\prime }]$ for all integers $a\\ge 1$ , $l$ odd, $1\\le l\\le l_0$ and $n^{\\prime }$ such that $n^{\\prime }=1$ or $\\genfrac(){}{}{p^{\\prime }}{p}=-1$ for all prime divisor $p^{\\prime }$ of $n^{\\prime }$ .",
"Remark 1.6 For the same reason, this case is also not contained in [14] ((REF ) in the above list), and the result [12] ((REF ) in the above list) can only deal with type $[p, p^aq^ln^{\\prime }]$ where $ l < \\lambda /s$ .",
"By direct calculation for $(p,f)$ in the cases listed in Theorem REF , the corresponding $\\lambda /s\\le l_0$ .",
"Thus the results in Theorem REF are also new.",
"For the proofs of the two theorems, we need some facts in algebraic number theory which are contained in Section .",
"With these preparations, we can prove Theorem REF and REF in Section and , respectively.",
"We also have corresponding non-existence results for PAPSs, which are similar to Theorem REF and REF .",
"See the last section."
],
[
"Basic Facts in Algebraic Number Theory",
"The methods for proving non-existence results of PPSs often involve algebraic number theory, mainly the basic arithmetic (ideals, units, class groups etc.)",
"of cyclotomic fields and their subfields.",
"The standard reference are [8] and [21].",
"In this section, we list some facts needed later, with proofs or references.",
"The reader who does not care the proofs may skip to the next section.",
"For any number field $F$ , denote by $\\mathfrak {o}_F$ the ring of integers of $F$ .",
"The latter ring is a Dedekind domain and we often consider the fractional ideals in it, which are $\\mathfrak {o}_F$ modules of the form $\\mathfrak {a}/\\alpha $ , where $\\mathfrak {a}\\subseteq \\mathfrak {o}_F$ is an integral ideal and $\\alpha \\in \\mathfrak {o}_F$ is a nonzero element.",
"Denote by $I_F$ the set of nonzero fractional ideals of $F$ , which, one can show, under multiplication, is a free abelian group generated by all prime ideals.",
"By a principal fractional ideal we mean a fractional ideal of the form $\\alpha \\mathfrak {o}_F$ where $\\alpha \\in F^$ .",
"Clearly, $P_F\\subseteq I_F$ as a subgroup, and the quotient $I_F/P_F$ , denoted by $Cl(F)$ , is called the class group of $F$ .",
"Class groups play an important role in classical algebraic number theory.",
"One of the nontrivial facts is that $Cl(F)$ is a finite abelian group for all $F$ , and by $h(F)$ we denote the cardinality of $Cl(F)$ , called the class number of $F$ .",
"We need the basic knowledge of the decompositions of prime ideals in extension fields, the decomposition groups and the decomposition fields.",
"We refer the reader to [8].",
"We also use properties of Artin maps and we need a corollary of class field theory, that is, there exists a finite unramified abelian extension $H_F/F$ (called the Hilbert class field) for every $F$ , such that the map $Cl(F)\\longrightarrow \\operatorname{Gal}(H_F/F)$ induced by Artin map is an isomorphism (see [8]).",
"In particular, a prime ideal $\\mathfrak {p}$ of $F$ is principal if and only if $\\mathfrak {p}$ splits completely in $H_F$ , and we have $h(F)=[H_F:F]$ .",
"For two subfields of the cyclotomic field $\\mathbb {Q}(\\zeta _{p^e})$ where $p^e$ is a prime power, we have the divisibility of class numbers.",
"Lemma 2.1 Let $L=\\mathbb {Q}(\\zeta _{p^e})$ and $F\\subseteq E\\subseteq L$ be two subfields of $L$ .",
"Then we have $h(F)\\mid h(E)$ .",
"Since $E/F$ is abelian and $p$ is totally ramified in $L/\\mathbb {Q}$ , the result follows from [21].",
"Lemma 2.2 Let $E/F$ be two number fields.",
"Then the canonical morphism $j_{E/F}: Cl(F)\\longrightarrow Cl(E)$ sending $\\mathfrak {a}$ to $\\mathfrak {a}\\mathfrak {o}_E$ is injective, provided that $\\gcd (h(F),[E:F])=1$ .",
"The argument is quite simple.",
"Let $\\mathfrak {a}$ be a fractional ideal of $F$ such that $\\mathfrak {a}\\mathfrak {o}_E$ is trivial in $Cl(E)$ .",
"Then $\\mathfrak {a}\\mathfrak {o}_E=\\alpha \\mathfrak {o}_E$ for some $\\alpha \\in E$ .",
"Taking norm to $F$ gives $\\mathfrak {a}^{[E:F]}=N_{E/F}(\\alpha )\\mathfrak {o}_F$ .",
"But $\\gcd (h(F),[E:F])=1$ , therefore raising to the power to $[E:F]$ is an automorphism on $Cl(F)$ .",
"Hence $\\mathfrak {a}$ is also trivial in $Cl(F)$ .",
"This prove the injectivity.",
"For some cases, we have the following more strong statements.",
"Proposition 2.3 Let $p\\equiv 3\\pmod {4},\\ p>3$ be a prime and $L=\\mathbb {Q}(\\zeta _{p^e}),\\ F:=\\mathbb {Q}(\\sqrt{-p})$ .",
"It is well-known that $F$ is a subfield of $L$ .",
"Let $E$ be any number field such that $F\\subseteq E\\subseteq L$ .",
"Then $j_{E/F}: Cl(F)\\longrightarrow Cl(E)$ is injective.",
"The statement of [19] says that if $M$ is any subfield of $\\mathbb {Q}(\\zeta _n)$ with the only roots of unity $\\pm 1$ , and $\\mathfrak {a}$ is an ideal of $M$ such that $\\mathfrak {a}\\bar{\\mathfrak {a}}$ is principal in $M$ and $\\mathfrak {a}$ is principal in $\\mathbb {Q}(\\zeta _n)$ , then $\\mathfrak {a}^4$ is principal in $M$ .",
"Now we apply this result with $M=F$ and $n=p$ .",
"Let $\\mathfrak {a}$ be any ideal of $F$ that is principal in $E$ .",
"Then $\\mathfrak {a}$ become principal in $L$ .",
"Also $\\mathfrak {a}\\bar{\\mathfrak {a}}$ is clearly principal in $F$ since $F$ is imaginary quadratic.",
"It follows that $\\mathfrak {a}^4$ is principal in $F$ .",
"On the other hand, by Gauss' genus theory (c.f.",
"[21]) or Lemma REF below, we know that $h(F)$ is odd.",
"Thus $\\mathfrak {a}$ is principal in $F$ .",
"The injectivity follows.",
"To show that the set $Q_p$ in Theorem REF is infinite in the subsequent section, we need a special case of Chebotarev's density theorem and compare class numbers.",
"We first introduce Definition 2.4 Let $K$ be any number field and $S$ be a set of prime ideals of $\\mathfrak {o}_K$ .",
"Denote all prime ideals of $\\mathfrak {o}_K$ by $\\mathcal {P}_K$ .",
"The Dirichlet density of $S$ is the limit (if exists) $\\delta =\\lim _{s\\rightarrow 1^+}\\frac{\\sum _{\\mathfrak {p}\\in S} \\frac{1}{N_{K/\\mathbb {Q}}(\\mathfrak {p})^s}}{\\sum _{\\mathfrak {p}\\in \\mathcal {P}_K} \\frac{1}{N_{K/\\mathbb {Q}}(\\mathfrak {p})^s}},$ denoted as $\\delta (S)=\\delta $ .",
"There may exists some other definitions but they are equivalent.",
"Now we have the statement: Proposition 2.5 Let $L/K$ be abelian extension of two number fields with Galois group $G$ and fix an element $\\sigma \\in G$ .",
"Let $S$ be the set of prime ideal $\\mathfrak {p}$ of $K$ whose Artin map $(\\mathfrak {p},L/K)$ is $\\sigma $ .",
"Then $S$ has Dirichlet density $\\delta (S)=1/\\#G$ .",
"See, for example [16].",
"Next we consider a wider class of number fields containing cyclotomic fields, namely: Definition 2.6 A CM-field $E$ is a totally imaginary quadratic extension of a totally real number field $E^+$ .",
"The field $E^+$ is the maximal real subfield of $E$ .",
"That a field is totally real (resp.",
"imaginary) means that all embeddings of the field into $\\mathbb {C}$ is real (resp.",
"imaginary).",
"As mentioned above, we want to compare certain class numbers.",
"For this purpose, we mainly use the following facts about CM-fields: Proposition 2.7 (c.f.",
"[21], Section 4, pp.",
"38-43) Let $E$ be CM and $E^+$ its maximal real subfield.",
"For convenience, let $h, U, W, R$ and $d$ be the class number, unit group, group of roots of unity, regulator and discriminant of $E$ respectively, and let $h^+, U^+, R^+$ and $d^+$ denote the corresponding objects for $E^+$ .",
"Then we have: (a) The class number $h^+$ divides $h$ , and the quotient $h^-$ is called the relative class number.",
"(b) The index $Q:=[U:WE^+]=1\\text{ or }2$ .",
"(c) The quotient $R/R^+=\\frac{1}{Q}2^r$ , where $r:=\\frac{1}{2}[E:\\mathbb {Q}]-1$ .",
"(d) (Brauer-Siegel theorem) Suppose $E$ runs through a sequence of number fields normal over $\\mathbb {Q}$ (not necessary CM) such that $\\frac{[E:\\mathbb {Q}]}{\\log |d_E|}\\rightarrow 0.$ Then $\\frac{\\log (h(E) R_E)}{\\log \\sqrt{|d_E|}}\\rightarrow 1.$ We also need a result for the parity of the class numbers of a special class of CM-fields.",
"Lemma 2.8 (See [5], Corollary 13.13) Let $E$ be CM which is Galois over $\\mathbb {Q}$ with $\\operatorname{Gal}(E/\\mathbb {Q})$ a cyclic group of order $2^k,\\ k\\ge 1$ .",
"Then $h(E)$ is odd if and only if exactly one finite rational prime ramifies in $E/\\mathbb {Q}$ .",
"Next we introduce Stickelberger ideals.",
"Suppose $p$ is a prime, $K=\\mathbb {Q}(\\zeta _p)$ and $G=\\operatorname{Gal}(K/\\mathbb {Q})\\cong (\\mathbb {Z}/p\\mathbb {Z})^$ .",
"Definition 2.9 The Stickelberger element $\\theta =\\theta _p\\in \\mathbb {Q}[G]$ is defined by $\\theta =\\sum _{a\\in (\\mathbb {Z}/p\\mathbb {Z})^}\\left\\lbrace \\frac{a}{p}\\right\\rbrace \\sigma _a^{-1}$ where $\\lbrace \\frac{a}{p}\\rbrace = \\frac{a}{p}- [\\frac{a}{p}]$ , and the Stickelberger ideal $S_p$ of $\\mathbb {Z}[G]$ is defined by $S_p=\\mathbb {Z}[G]\\theta \\cap \\mathbb {Z}[G].$ We mainly use these following properties of the Stickelberger ideal: Proposition 2.10 We have: (a) For $(c, p) = 1$ , the element $(c-\\sigma _c)\\theta $ are in $S_p$ .",
"(b) The Stickelberger ideal $S_p$ annihilates the ideal class group $Cl(M)$ , where $M$ is a subfield of $K$ such that $p$ is the minimal integer with the property that $M\\subseteq \\mathbb {Q}(\\zeta _p)$ .",
"See [21].",
"Notation.",
"Through this paper, we fix the following notation.",
"Let $p$ be an odd prime and denote $\\zeta _k$ a primitive $k$ -th root of unity.",
"Let $K=\\mathbb {Q}(\\zeta _p)$ .",
"In the remaining of this paper we mainly deal with $\\mathbb {Q}(\\zeta _p)$ and write $\\zeta = \\zeta _p$ for simplicity.",
"Let $G=\\operatorname{Gal}(K/\\mathbb {Q})$ .",
"It's well-known that $G\\cong (\\mathbb {Z}/N\\mathbb {Z})^$ , the isomorphism being $c\\mapsto (\\sigma _c: \\zeta \\mapsto \\zeta ^c)$ for $c\\in (\\mathbb {Z}/N\\mathbb {Z})^$ .",
"The starting point of our method is the following Proposition 2.11 If there exist PPS with type $[p, n]$ , then $p\\mid n$ and $\\alpha \\bar{\\alpha }=n$ for some $\\alpha \\in \\mathbb {Z}[\\zeta _p]$ .",
"The result is obtained by applying different sets.",
"See, for example, [12] and the remarks after it.",
"Thus for our purpose we need to investigate the equation $\\alpha \\bar{\\alpha }=n$ where $\\alpha \\in \\mathbb {Z}[\\zeta _p]$ .",
"So we mainly study the idealic behaviour of each $p$ dividing $n$ , in the cyclotomic field $K$ ."
],
[
"Non-existence result for PPSs with type $[p\\equiv 5\\pmod {8},p^aqn^{\\prime }]$",
"In this section, we will prove Theorem REF .",
"We start with the definition of $Q_p$ .",
"As the assumptions in the theorem, let $p\\equiv 5\\pmod {8}$ be a prime and $\\tilde{Q}_p=\\mathinner {\\lbrace \\,{q\\text{ is a prime } }\\mid { \\operatorname{ord}_p(q)=(p-1)/4}\\,\\rbrace }.$ Let $q\\in \\tilde{Q}_p$ so $\\operatorname{ord}_p(q)=(p-1)/4$ .",
"Let $K = \\mathbb {Q}(\\zeta _p)$ and $E$ be the unique subfield of $K$ having degree 4 over $\\mathbb {Q}$ .",
"Then the order of $q$ modulo $p$ tells us that $E$ is the decomposition group of $q$ in $K$ and depends only in $p$ .",
"Thus we write $E_p=E$ and it is well known that $K$ contains the unique real quadratic subfield $F_p=\\mathbb {Q}(\\sqrt{p})\\subset E_p$ Actually one can define $Q_p = \\left\\lbrace q \\in \\tilde{Q}_p \\Biggm | \\begin{split}&\\text{ there is a prime ideal $\\mathfrak {Q}$ in $E_p$ lying over $q$ such that }\\\\&\\text{ $\\mathfrak {Q}$ is not principal while $\\mathfrak {q}=\\mathfrak {P}\\cap \\mathfrak {o}_{F_p}$ is principal}\\end{split} \\right\\rbrace .$ To show that $Q_p$ is infinite, we only need to show that the Dirichlet density $\\delta (Q_p)>0$ , since any finite set has zero density by the definition.",
"Lemma 3.2 Let $L/M$ be cyclic extension of two number fields such that they are both Galois over $\\mathbb {Q}$ and there is some finite prime in $M$ totally ramified in $L$ .",
"Define $S(f, L/M)= \\left\\lbrace p \\text{ is a prime number } \\Biggm | \\begin{split}&\\text{ $p$ split completely in $M$ and there is a principal }\\\\&\\text{ prime ideal $\\mathfrak {p}$ in $M$ lying over $p$ and the order }\\\\&\\text{ of the Artin map $(\\mathfrak {p}, L/M)$ is $f$ }\\end{split} \\right\\rbrace .$ Then we have $\\delta (S(f, L/M)) = \\frac{\\varphi (f)}{[L:\\mathbb {Q}]\\ h(M)},$ with $\\varphi $ being the Euler's totient function.",
"Let $H_M$ be the Hilbert class field of $M$ .",
"Since there is a finite prime totally ramified in $L/M$ and $H_M/M$ is unramified, we have $H_M\\cap L=M$ .",
"Hence we have a natural isomorphism $\\operatorname{Gal}(LH_M/M)\\cong \\operatorname{Gal}(L/M)\\operatorname{Gal}(H_M/M)$ and that $LH_M/M$ is an abelian extension of degree $[L:~M]\\ h(M)$ .",
"Let $S$ denote the set of prime ideal $\\mathfrak {p}$ in $M$ such that $\\mathfrak {p}$ is principal and $(\\mathfrak {p}, L/M)$ has order $f$ .",
"Fix an element $\\sigma \\in \\operatorname{Gal}(L/M)$ having order $f$ .",
"Since $L/M$ is cyclic, we know that $\\sigma ^k,\\ k\\in (\\mathbb {Z}/f\\mathbb {Z})^$ are exactly all the element in $Gal(L/M)$ having order $f$ .",
"Moreover, we can interpret the constraint that $\\mathfrak {p}$ is principal as $(\\mathfrak {p}, H_M/M)=1$ .",
"Under the isomorphism (REF ), we know that $S=\\mathinner {\\lbrace \\,{ \\mathfrak {p}\\text{ in } M }\\mid { (\\mathfrak {p}, LH_M/M) = (\\sigma ^k, 1)\\in \\operatorname{Gal}(L/M)\\operatorname{Gal}(H_M/M),\\ k\\in (\\mathbb {Z}/f\\mathbb {Z})^}\\,\\rbrace }.$ A direct application of Proposition REF yields $\\delta (S) = \\frac{\\varphi (f)}{[L:M]\\ h(M)}.$ Let $S_1$ be the set of primes of $M$ having relative degree one over $\\mathbb {Q}$ .",
"An elementary argument (c.f.",
"[8] tells us that $\\delta (S\\cap S_1) = \\delta (S)$ .",
"Let $\\mathfrak {p}\\in S\\cap S_1$ and $p=\\mathfrak {p}\\cap \\mathbb {Z}$ .",
"Since $M/\\mathbb {Q}$ is Galois, $p$ splits completely in $M$ and every $\\mathfrak {p}^{\\prime }$ in $M$ lying over $p$ is also principal.",
"Moreover, the assumption that $L/\\mathbb {Q}$ is Galois ensures that all $(\\mathfrak {p}^{\\prime }, L/M)$ are conjugate and hence having the same order $f$ .",
"It follows that $\\delta (S(f, L/M)) = \\frac{1}{[M:\\mathbb {Q}]}\\ \\delta (S\\cap S_1) = \\frac{\\varphi (f)}{[L:\\mathbb {Q}]\\ h(M)}.$ The proof is complete.",
"The following lemma gives a lower bound for the density of $Q_p$ .",
"Lemma 3.4 Let $p\\equiv 5\\pmod {8}$ be a prime and $Q_p$ defined by (REF ).",
"Then we have $\\delta (Q_p) \\ge \\frac{\\varphi ((p-1)/4)}{p-1} \\left( \\frac{1}{h(F_p)}-\\frac{1}{h(E_p)} \\right),$ Clearly $K$ and $E_p$ are both Galois over $\\mathbb {Q}$ .",
"Let $q\\in Q_p$ and $\\mathfrak {Q}$ be any prime in $E_p$ lying over $q$ .",
"Since $E_p$ is the decomposition field of $q$ in $K$ , $q$ splits completely in $E_p$ .",
"Thus we have $(\\mathfrak {Q}, K/F_p)=(q, K/\\mathbb {Q})$ , which has order $(p-1)/4$ .",
"Applying Lemma REF we obtain $\\delta (S(\\frac{p-1}{4}, K/E_p)) = \\frac{\\varphi ((p-1)/4)}{(p-1)h(E_p)}.$ A similar analyze for $K/F_p$ yields $\\delta (S(\\frac{p-1}{4}, K/F_p)) = \\frac{\\varphi ((p-1)/4)}{(p-1)h(F_p)}.$ In view of $Q_p = S(\\frac{p-1}{4}, K/F_p) \\setminus S(\\frac{p-1}{4}, K/E_p)$ , we have $\\delta (Q_p) &= \\delta ( S(\\frac{p-1}{4}, K/F_p) \\setminus S(\\frac{p-1}{4}, K/E_p) )\\\\&\\ge \\delta ( S(\\frac{p-1}{4}, K/F_p) ) - \\delta ( S(\\frac{p-1}{4}, K/E_p) )\\\\&= \\frac{\\varphi ((p-1)/4)}{p-1} \\left( \\frac{1}{h(F_p)}-\\frac{1}{h(E_p)} \\right),$ where the second line is due to the fact that we can sum the densities of two disjoint sets, which is easily seen by the definition.",
"So we finish the proof.",
"Our next goal is to show that if $p>p_0$ for some $p_0$ , the density $\\delta (Q_p)$ is positive.",
"Recall that $E_p\\subseteq K=\\mathbb {Q}(\\zeta _p)$ contains $F_p=\\mathbb {Q}(\\sqrt{p})$ .",
"Since $\\operatorname{ord}_p(q)=(p-1)/4$ is odd, the complex conjugation does not fix $E_p$ .",
"It follows that $E_p$ is a totally imaginary cyclic extension of $\\mathbb {Q}$ , and hence a CM-field with $E_p^+=F_p$ being the maximal real subfield.",
"We write $h_p$ for $h(E_p)$ , and $h_p^+$ for $h(E_p^+)=h(F_p)$ .",
"Thus from Proposition REF (a) we know that $h_p=h_p^+h_p^-$ for a positive integer $h_p^-$ , which is the relative class number for $E_p$ .",
"We now consider the asymptotic behaviour of $h_p^-$ .",
"Lemma 3.5 With the previous notation we have $\\log h_p^- \\ge \\frac{1}{2}(\\log p)(1+o(1)) \\quad \\text{ as $p\\rightarrow \\infty $}.$ We follow the same method in [21].",
"But the case here is simpler.",
"Let $U_p, W_p, R_p$ and $d_p$ be the unit group, group of roots of unity, regulator and discriminant of $E_p$ respectively, and let $U_p^+, R_p^+$ and $d_p^+$ denote the corresponding objects for $E_p^+$ .",
"The ideal is to use Brauer-Siegel theorem (Proposition REF (d)) for $E_p/E_p^+$ .",
"To verify the assumption of the theorem, we need to estimate the discriminants of $E_p^+$ and $E_p$ .",
"Recall that $E_p^+=F_p=\\mathbb {Q}(\\sqrt{p})$ and clearly we know that $d_p^+=p$ .",
"Then the relative discriminant formula (c.f.",
"[11]) gives $|d_p|=N_{E_p/\\mathbb {Q}}(\\mathcal {D}(E_p/E_p^+)) |d_p^+|^{[E_p:E_p^+]},$ where $\\mathcal {D}(E_p/E_p^+)$ is the relative different, which is a integral ideal in $\\mathfrak {o}_{E_p}$ .",
"Thus we have $|d_p|\\ge |d_p^+|^{[E_p:E_p^+]} =p^2.$ Since the we have $[E_p:\\mathbb {Q}] = 2[E_p^+:\\mathbb {Q}]=4$ for all $p$ , we know that $\\frac{[E_p:\\mathbb {Q}]}{\\log |d_p|}\\rightarrow 0\\quad \\text{ and }\\quad \\frac{[E_p^+:\\mathbb {Q}]}{\\log |d_p^+|}\\rightarrow 0$ and Brauer-Siegel theorem applies.",
"It follows that $\\log (h_p R_p)&= \\frac{1}{2}\\log d_p + o(\\log d_p)\\\\\\text{and } \\log (h_p^+ R_p^+)&= \\frac{1}{2}\\log d_p^+ + o(\\log d_p^+)$ By Proposition REF (b) and (c) we have $\\log \\left( \\frac{R_p}{R_p^+} \\right) = O(1).$ Hence, noting that $\\log d_p^+\\le \\frac{1}{2} \\log d_p$ by (REF ), we have $\\log h_p^- &= \\log (h_p R_p) - \\log (h_p^+ R_p^+) - \\log \\left( \\frac{R_p}{R_p^+} \\right)\\\\&= \\frac{1}{2}\\log d_p - \\frac{1}{2}\\log d_p^+ + o(\\log d_p) + O(1)\\\\&\\ge \\frac{1}{2}\\log d_p - \\frac{1}{4}\\log d_p + o(\\log d_p)\\\\&= \\frac{1}{4}(\\log d_p)(1+o(1))\\\\&\\ge \\frac{1}{2}(\\log p)(1+o(1)).$ Proposition 3.7 Let notation be as before, $p\\equiv 5\\pmod {8}$ a prime and $Q_p$ defined by (REF ).",
"Then we have (a) the equation $\\beta \\bar{\\beta }=q,\\quad \\beta \\in \\mathfrak {o}_{E_p}$ has no solution for all $q\\in Q_p$ ; (b) if $h_p>h_p^+$ then the set $Q_p$ is infinite; (c) there exists a lower bound $p_0$ such that if $p>p_0$ , then the set $Q_p$ is infinite.",
"Let $p\\equiv 5\\pmod {8}$ and $q\\in Q_p$ .",
"Recall that $\\operatorname{ord}_p(q)=(p-1)/4$ is odd and $E_p$ is the decomposition field of $q$ in $K=\\mathbb {Q}(\\zeta _p)$ with $[E_p:\\mathbb {Q}]=4$ , so we have the prime decomposition $q\\mathfrak {o}_{E_p}=\\mathfrak {Q}_1\\mathfrak {Q}_2\\mathfrak {Q}_3\\mathfrak {Q}_4.$ It is also noted before that the complex conjugation is not in the decomposition group of $q$ .",
"Thus we may assume $\\mathfrak {Q}_3=\\bar{\\mathfrak {Q}}_1$ and $\\mathfrak {Q}_4=\\bar{\\mathfrak {Q}}_2$ .",
"Now we assume that the equation (REF ) has solution $\\beta \\in \\mathfrak {o}_{E_p}$ , so we have $\\beta \\bar{\\beta }\\mathfrak {o}_{E_p}=q\\mathfrak {o}_{E_p}=\\mathfrak {Q}_1\\mathfrak {Q}_2\\bar{\\mathfrak {Q}}_1\\bar{\\mathfrak {Q}}_2.$ It follows that the only possible decompositions of $\\beta $ are $\\beta \\mathfrak {o}_{E_p} = \\mathfrak {Q}_1\\mathfrak {Q}_2,\\ \\mathfrak {Q}_1\\bar{\\mathfrak {Q}}_2,\\ \\bar{\\mathfrak {Q}}_1\\mathfrak {Q}_2\\text{ or }\\bar{\\mathfrak {Q}}_1\\bar{\\mathfrak {Q}}_2.$ Write $\\operatorname{Gal}(E_p/\\mathbb {Q})=\\left<\\sigma \\right>$ with $\\sigma $ of order 4.",
"It follows that we can assume $\\mathfrak {Q}_1^{\\sigma ^t}=\\mathfrak {Q}_{t+1},\\quad t=0,1,\\dots ,3.$ Then (REF ) tells us that $1=\\mathfrak {Q}_1^{1+\\sigma },\\ \\mathfrak {Q}_1^{1+\\sigma ^3},\\ \\mathfrak {Q}_1^{\\sigma ^2+\\sigma }\\text{ or }\\mathfrak {Q}_1^{\\sigma ^2+\\sigma ^3}\\text{ in $Cl(E_p)$.",
"}$ Correspondingly, rising to the power to $1-\\sigma ,\\ 1-\\sigma ^3,\\ \\sigma ^2-\\sigma \\text{ or }\\sigma ^2-\\sigma ^3$ , we obtain the same equation $\\mathfrak {Q}_1^{1-\\sigma ^2}=1\\text{ in }Cl(E_p).$ On the other hand, by the definition (REF ), we know that there is a prime ideal $\\mathfrak {Q}$ in $E_p$ over $q$ such that $\\mathfrak {Q}$ is not principal while $\\mathfrak {q}=\\mathfrak {P}\\cap \\mathfrak {o}_{F_p}$ is principal.",
"With loss of generality, we may assume that $\\mathfrak {Q}_1=\\mathfrak {Q}$ .",
"Since $\\mathfrak {q}$ is principal, so is $\\mathfrak {q}\\mathfrak {o}_{E_p}=\\mathfrak {Q}_1\\bar{\\mathfrak {Q}}_1$ , which means that $\\mathfrak {Q}_1^{1+\\sigma ^2}=1\\text{ in }Cl(E_p).$ Combining with (REF ), we have $\\mathfrak {Q}_1^2=1$ in $Cl(E_p)$ .",
"However, since $p$ is the unique finite rational prime that ramifies in $E_p/\\mathbb {Q}$ , Lemma REF tells us that $h(E_p)$ is odd.",
"It follows that $\\mathfrak {Q}_1=1$ in $Cl(E_p)$ , which is an contradiction because $Q_1$ is not principal by the previous argument.",
"Thus the assumption we made before is false and we complete the proof for (a).",
"Next, let $p\\equiv 5\\pmod {8}$ and suppose that $h_p>h_p^+$ .",
"By Lemma REF , we know that $\\delta (Q_p)\\ge \\frac{\\varphi ((p-1)/4)}{p-1} \\left( \\frac{1}{h_p^+}-\\frac{1}{h_p} \\right) > 0.$ Thus $Q_p$ is a infinite set and (b) is correct.",
"For the last assertion (c), recall that $E_p/F_p$ is CM and we use Lemma REF to obtain $h_p^- \\rightarrow \\infty \\quad \\text{as $p\\rightarrow \\infty $}.$ It follows that there exists a lower bound $p_0$ such that if $p>p_0$ , then $h_p^->1$ , i.e.",
"$h_p>h_p^+$ .",
"It follow by (b) that $Q_p$ is infinite for all $p>p_0$ .",
"That's all the proof.",
"Although we have shown that $Q_p$ is infinite, one do not know whether a given $q$ is in $Q_p$ .",
"However, this can be done when we know the Hilbert class field $H_{F_p}$ and $H_{E_p}$ of $F_p$ and $E_p$ , respectively.",
"We now describe this method as follows.",
"Suppose in general, $L$ is a number field.",
"Let $\\Xi _L(x)\\in \\mathfrak {o}_L[x]$ be an irreducible polynomial having a root that generates the Hilbert class field $H_L$ over $L$ , and we call $\\Xi _L$ the Hilbert class polynomial.",
"If there is a subfield $M$ such that $L/M$ is cyclic with $h(M)=1$ , then there exist an auxiliary field $K=K_{L/M}$ such that $H_L=KM$ and $M=K\\cap M$ .",
"See [6].",
"It follows that we can choose $\\Xi _L(x)$ with coefficients in $\\mathfrak {o}_M$ .",
"Lemma 3.11 Let $L/M$ , $\\Xi _L(x)\\in \\mathfrak {o}_M[x]$ be as before, and $\\mathfrak {P}$ a prime ideal of $L$ having relative degree one over $M$ not dividing the discriminant of $\\Xi _L$ .",
"Let $\\mathfrak {p}=\\mathfrak {P}\\cap \\mathfrak {o}_M$ .",
"Then $\\mathfrak {P}$ is principal if and only if $\\Xi _L(x)=0$ has a solution over $\\mathfrak {o}_M/\\mathfrak {p}$ .",
"Since $H_L$ is the Hilbert class field of $L$ , so we know that $\\mathfrak {P}$ is principal if and only if $\\mathfrak {P}$ splits completely in $H_L$ .",
"Note that $\\mathfrak {P}$ does not divide the discriminant of $\\Xi _L$ .",
"Then a direct application of Kummer theorem (c.f.",
"[11]) tells us that $\\mathfrak {P}$ splits completely in $H_L$ if and only if $\\Xi _L(x)=0$ has a solution over $\\mathfrak {o}_L/\\mathfrak {P}$ , which is to say that $\\Xi _L(x)=0$ has a solution over $\\mathfrak {o}_E/\\mathfrak {p}$ , since $\\mathfrak {P}$ is a prime ideal of $L$ having relative degree one over $M$ .",
"In our case where $E_p/\\mathbb {Q}$ is cyclic and $F_p$ is real quadratic, we know that both $K_{E_p/\\mathbb {Q}}$ and $K_{F_p/\\mathbb {Q}}$ exist.",
"Hence we have $\\Xi _{E_p}$ and $\\Xi _{F_p}$ with integral coefficients.",
"Then we have Corollary 3.12 Fix a prime $p\\equiv 5\\pmod {8}$ and let the irreducible polynomials $\\Xi _{E_p}(x)$ and $\\Xi _{F_p}(x)$ in $\\mathbb {Z}[x]$ be as before.",
"Given $q\\in \\tilde{Q}_p$ not dividing the discriminants of the two polynomials, then we have $q\\in Q_p$ if and only if $\\Xi _{F_p}(x)\\equiv 0 \\pmod {q}$ is solvable while $\\Xi _{E_p}(x)\\equiv 0 \\pmod {q}$ is not.",
"Apply Lemma REF to $E_p/\\mathbb {Q}$ and $F_p/\\mathbb {Q}$ and then the result follows from the definition of $Q_p$ .",
"Now the problem left is to find the Hilbert class polynomials for $E_p$ and $F_p$ .",
"For the real quadratic field $F_p$ , the polynomial $\\Xi _{F_p}$ is quite easy to obtain (c.f.",
"Stark’s method described by Cohen [4], who also gives a list of these polynomials).",
"As for $E_p$ , which is a CM-field and is cyclic of degree 4 over $\\mathbb {Q}$ , we could use complex multiplication to calculate $\\Xi _{E_p}$ .",
"This is more complicated than the imaginary quadratic case where elliptic curves and the $j$ -invariant are enough.",
"In the case of degree 4 CM-fields, we work with curves of genus 2, three $j$ -invariants and three igusa class polynomials.",
"See Streng [20] or Enge et al.",
"[7] for complete description of the method.",
"There is also an implementation of the algorithm by Enge et al., CMH [1], which enables us to calculate the individual igusa class polynomials.",
"We can use igusa class polynomials instead of the Hilbert class polynomial, or calculate the Hilbert class polynomial by them.",
"This solves the whole problem.",
"We give an example to illustrate this method.",
"Example 3.13 Let $p=101$ , and $\\Xi (x)\\ =\\ x^5\\ -\\ 1237224274356339549352800\\ x^4\\ +\\ 57176933499148\\qquad &\\\\833882237435031573248869838360576\\ &x^3\\\\+\\ 2514056979190981026432576749022147825857609219676093\\qquad &\\\\86630877092098080768\\ &x^2\\\\-\\ 1023671146645480759972364788108250129958692705554245\\qquad &\\\\7706457967352624977868378319649505280\\ &x\\\\+\\ 1530499568113365603805244886351320629567046080073893\\qquad &\\\\31920814045884605474516662499805087162052695075848192.&$ If $q\\in \\tilde{Q}_p$ and $q \\ne 2542000616863$ , then $q\\in Q_p$ if and only if $\\Xi (x)\\equiv 0 \\pmod {q}$ is solvable.",
"For $p=101$ we have $F_p=\\mathbb {Q}(\\sqrt{101})$ and $E_p=\\mathbb {Q}(\\alpha )$ where $\\alpha $ is a root of $x^4+101x^2+101$ .",
"Using GP calculator (see [18]) we obtain that $h_p^+=h(F_p)=1$ and $h(E_p)=5$ .",
"Using CMH (see above), we obtain that the first igusa class polynomial of $E_p$ is $\\Xi (x)$ , which is also the Hilbert class polynomial of $E_p$ since it already has degree 5.",
"Also we know $q=2542000616863$ is the only prime in $\\tilde{Q}_p$ that divides the discriminant of $\\Xi (x)$ .",
"The assertion follows by Corollary REF .",
"Let us turn to the If there exist PPS with type $[p, n]$ , where $p\\equiv 5\\pmod {8}$ be a prime and $ n=p^aqn^{\\prime }$ , then by Proposition REF we know that $\\alpha \\bar{\\alpha }=n=p^aqn^{\\prime }\\text{ for some }\\alpha \\in \\mathfrak {o}_K=\\mathbb {Z}[\\zeta _p].$ Here $K=\\mathbb {Q}(\\zeta _p)$ .",
"[12] tells us that $\\alpha _1\\bar{\\alpha }_1=p^aq\\text{ for some }\\alpha _1\\in \\mathfrak {o}_K.$ Next by [12] we obtain that $\\alpha _2\\bar{\\alpha }_2=q\\text{ for some }\\alpha _2\\in \\mathfrak {o}_K.$ We may assume $p>5$ .",
"Then $\\operatorname{ord}_p(q)=(p-1)/4>1$ and so $(p,q-1)=1$ .",
"Recall that $E_p$ is the decomposition field of $q$ in $K$ , so we use [12] to obtain that $\\beta \\bar{\\beta }=q\\text{ for some }\\beta \\in \\mathfrak {o}_K\\text{ and }\\beta ^2\\in \\mathfrak {o}_{E_p}.$ But $[K:E_p]=\\operatorname{ord}_p(q)=(p-1)/4$ is odd, so in fact we have $\\beta \\in \\mathfrak {o}_{E_p}$ .",
"Thus the theorem follows from Proposition REF ."
],
[
"Non-existence result for PPSs with type $[p\\equiv 3\\pmod {4},p^aq^ln^{\\prime }]$ for certain {{formula:2b6bd4d0-de68-4bf7-a4ed-15331615eb6d}} and {{formula:bd2bc968-56f9-44c7-8455-fb7c08838474}}",
"We will prove Theorem REF in this section.",
"The main ideal is the application of Stickelberger relations, which was used by the first author and Jianing Li [13] for showing non-existence results for some bent functions.",
"First we fix some additional notation.",
"Suppose $n=p^aq^ln^{\\prime }$ and $p\\equiv 3 \\pmod {4}$ be as in the theorem.",
"Then we know that $f:=\\operatorname{ord}_p(q)$ is odd.",
"Thus $g:=\\frac{\\varphi (p)}{f}$ is even and we set $u=g/2$ .",
"Recall that $K = \\mathbb {Q}(\\zeta _p)$ and let $E$ be the unique subfield of $K$ having degree $g$ over $\\mathbb {Q}$ .",
"Then $E$ is the decomposition group of $q$ in $K$ and is CM with $E^+=E\\cap \\mathbb {R}$ its maximal real subfield (the argument is similar as before, see Section ).",
"It is well known that $K$ contains the unique imaginary quadratic subfield $F=\\mathbb {Q}(\\sqrt{-p})\\subset E$ .",
"Suppose the prime decomposition of $q$ in $E$ is $q\\mathfrak {o}_E=\\mathfrak {Q}_1\\mathfrak {Q}_2\\dots \\mathfrak {Q}_g.$ If there is a PPS with type $[p,n]$ , then as in the proof of Theorem REF , a similar argument using Proposition REF and [12] yields the equation $\\beta \\bar{\\beta }=q^l,\\ \\beta \\in \\mathfrak {o}_E.$ Since $f$ is odd, the complex conjugation is not in the decomposition group of $q$ .",
"Thus we may assume $\\mathfrak {Q}_{u+k}=\\bar{\\mathfrak {Q}}_k,\\ k=1,2,\\dots ,u$ .",
"Then we have $\\beta \\bar{\\beta }\\mathfrak {o}_E=\\prod _{k=1}^u\\mathfrak {Q}_k^l\\bar{\\mathfrak {Q}}_k^l.$ So $\\beta \\mathfrak {o}_E=\\prod _{k=1}^u\\mathfrak {Q}_k^{l_k}\\bar{\\mathfrak {Q}}_k^{\\bar{l}_k}$ where $l_k, \\bar{l}_k$ are nonnegative integer such that $l_k+\\bar{l}_k=l$ for all $k=1,2,\\dots ,u$ .",
"For convenience we write $x_k$ for $\\mathfrak {Q}_k$ in $Cl(E)$ and view $Cl(E)$ additively.",
"Hence (REF ) becomes $\\sum _{k=1}^u(l_k x_k + \\bar{l}_k \\bar{x}_k) = 0 \\\\\\text{where }l_k+\\bar{l}_k=l,\\ k=1,2,\\dots ,u.\\nonumber $ Thus we obtain the Proposition 4.4 With the above notation, if (REF ) has no nonnegative integral solution $(l_1,l_2,\\dots ,l_g)$ , where $l_k+\\bar{l}_k=l_0$ and $l_{u+k}:=\\bar{l}_k,\\ k=1,2,\\dots ,u$ , then there is no PPS with type $[p,n]$ .",
"To show (REF ) is not solvable in the above sense, we have to exploit the relations between $x_k$ 's in $Cl(E)$ .",
"By (REF ) we have $\\sum _{k=1}^g x_k=0.$ We want to find more relations.",
"Let $K^+=K\\cap \\mathbb {R}=\\mathbb {Q}(\\zeta _p+\\zeta _p^{-1})$ .",
"Then Miller's work on class number of $K^+$ gives Theorem 4.6 ([15], Theorem $1.1$ ) The class number of $\\mathbb {Q}(\\zeta _p+\\zeta _p^{-1})$ is 1 if $p\\le 151$ is a prime.",
"From now on we suppose $p\\le 151$ .",
"Clearly, $E^+\\subseteq K^+$ .",
"Then by Miller's result and Lemma REF , we have $h(E^+)=h(K^+)=1$ .",
"Now $q\\mathfrak {o}_{E^+}=\\mathfrak {q}_1\\mathfrak {q}_2\\dots \\mathfrak {q}_u$ where $\\mathfrak {q}_k\\mathfrak {o}_E=\\mathfrak {Q}_k\\bar{\\mathfrak {Q}}_k$ , and all $\\mathfrak {q}_k$ 's are principal since $h(E^+)=1$ .",
"This implies the relations $x_k+x_{u+k}=0,\\quad k=1,2,\\dots ,u.$ However, these relations above are not enough.",
"We need the Stickelberger ideal introduced in Section .",
"Let $\\mathfrak {Q}=\\mathfrak {Q}_1$ and correspondingly $x=x_1$ .",
"Let $c$ be an integer not divisible by $p$ .",
"Since it is well-known that $p$ is the minimal integer such that $F\\subseteq \\mathbb {Q}(\\zeta _p)$ , it follows that $p$ is also the minimal one such that $E\\subseteq \\mathbb {Q}(\\zeta _p)$ .",
"By Proposition REF , we have $(c-\\sigma _c)\\theta \\ \\mathfrak {Q}=1\\text{ in }Cl(E).$ Let $w$ be a primitive root mod $p$ .",
"Recall $G=\\operatorname{Gal}(K/\\mathbb {Q})$ .",
"Then the decomposition group of $q$ in $K$ is $\\left<q\\right>=\\left<w^g\\right>\\subseteq G=(\\mathbb {Z}/p\\mathbb {Z})^$ .",
"It follows that we can assume $\\sigma _w^{tg+s}(x)=x_{s+1},\\quad t\\in \\mathbb {Z},\\ s=0,1,\\dots ,g-1.$ Let $k_{c,a}=[\\frac{ca}{p}]$ for any integer $a$ .",
"We have $(c-\\sigma _c)\\theta &=(c-\\sigma _c)\\sum _{a\\in (\\mathbb {Z}/p\\mathbb {Z})^}\\left\\lbrace \\frac{a}{p}\\right\\rbrace \\sigma _a^{-1} \\\\&=\\sum _a \\left(c\\left\\lbrace \\frac{a}{p}\\right\\rbrace -\\left\\lbrace \\frac{ca}{p}\\right\\rbrace \\right)\\sigma _a^{-1}\\\\&=\\sum _{a=1}^{p-1}k_{c,a}\\sigma _a^{-1} \\quad \\text{ (the definition of $k_{c,a}$) }\\\\&=\\sum _{s=0}^{p-2}k_{c,w^{-s}}\\sigma _w^s \\quad \\text{ ($w^{-s}$ means $w^{-s} \\mod {p}$) }\\\\&=\\sum _{t=0}^{f-1}\\sum _{s=0}^{g-1}k_{c,w^{-(tg+s)}}\\sigma _w^{tg+s}$ Then by (REF ) we have $1&=\\mathfrak {Q}^{\\sum _{t=0}^{f-1}\\sum _{s=0}^{g-1}k_{c,w^{-(tg+s)}}\\sigma _w^{tg+s}},\\\\\\text{ i.e., }\\quad 0&=\\sum _{t=0}^{f-1}\\sum _{s=0}^{g-1}k_{c,w^{-(tg+s)}}\\sigma _w^{tg+s}(x)\\\\&=\\sum _{t=0}^{f-1}\\sum _{s=0}^{g-1}k_{c,w^{-(tg+s)}}x_{s+1} \\quad \\text{ (by (\\ref {eq_sigma_x}))}\\\\&=\\sum _{s=1}^g m_{c,s} \\sum _{t=0}^{f-1}k_{c,w^{-tg-s+1}} x_s.$ If we set $m_{c,s}= \\sum _{t=0}^{f-1}k_{c,w^{-tg-s+1}}$ we have $p-1$ linear equations $\\sum _{s=1}^g m_{c,s}x_s = 0,\\quad c=1,2,\\dots p-1.$ We now combine these $p-1$ equations, together with the equation (REF ) and the $u$ equations (REF ), to give a whole collection of equations $XM_{p,f}^T=0$ where $M_{p,f}$ is a $(p+u)g$ matrix with integer entries made of the coefficients of all the $p+u$ equations and $X=(x_1, x_2,\\dots , x_g)$ .",
"Note that $M_{p,f}$ depends only on $p$ and $f$ .",
"To simplify these relations of $x_1,x_2, \\dots ,x_g$ , we need to calculate the Hermite normal form of $M_{p,f}$ .",
"By the well-known result (c.f.",
"[3]) for the existence of the Hermite normal form, there exists a unique matrix $U_{p,f}\\in \\operatorname{GL}_{p+u}(\\mathbb {Z})$ , such that $H_{p,f}=M_{p,f}^TU_{p,f}$ is a Hermite normal form.",
"It follows from (REF ) that $XH_{p,f}=0.$ In fact, $H_{p,f}$ can be obtained by applying a finite sequence of elementary row operations over $\\mathbb {Z}$ from $M_{p,f}^T$ .",
"Now with the help of a computer and using a simple program or a computer algebra system, we can calculate the individual Hermite normal form $H_{p,f}$ for $(p, f)\\in \\mathinner {\\lbrace \\,{(31 , 5),\\ (127, 9),\\ (127, 21),\\ (139, 23),\\ (151, 15)}\\,\\rbrace }.$ Let us take $(p,f)=(31,5)$ and $(151,15)$ for example.",
"Thus we obtain the relation $(x_1, x_2,x_3)\\begin{pmatrix}18&8&15\\\\0&2&1\\\\0&0&1\\end{pmatrix}=0$ for $(p,f)=(31,5)$ and $X_{151}\\begin{pmatrix}3934&1304&3470&3544&1477\\\\0&2&0&0&1\\\\0&0&2&0&1\\\\0&0&0&2&1\\\\0&0&0&0&1\\end{pmatrix}=0$ for $(p,f)=(151,15)$ , where $X_{151}:=(x_1, x_2,\\dots ,x_5)$ and we omit $x_{u+1},\\dots ,x_g$ and other parts of $H_{p,f}$ since $x_{u+k}=-x_k$ .",
"Using these computational results, we can turn to the We have to verify the assumption in Proposition REF .",
"If $(p,f)=(31,5)$ the first column of the matrix in (REF ) tells us that $18x_1=0$ in $Cl(E)$ .",
"Recall that $K=\\mathbb {Q}(\\zeta _p), h_p=h(K)$ and $h_p^+=h(\\mathbb {Q}(\\zeta _p+\\zeta _p^{-1}))=1$ .",
"We can write $h_p = h_p^+ h_p^-$ (see Proposition REF (a)).",
"By [21] we know $h_{31}^-$ is odd.",
"Since $h(E)\\mid h(K)=h_p=h_p^-$ , we know $h(E)$ is also odd.",
"It follows that $9x_1=0$ and $\\operatorname{ord}(x_1)=1,3$ or 9 in $Cl(E)$ .",
"We claim that $\\operatorname{ord}(x_1)=9$ .",
"Recall $F=\\mathbb {Q}(\\sqrt{-p})=\\mathbb {Q}(\\sqrt{-31})\\subseteq E$ and let $\\mathfrak {q}_F=\\mathfrak {Q}_1\\cap \\mathfrak {o}_F$ .",
"By the table in [4] we know that $\\Xi _{31}(x)$ is the Hilbert class polynomial of $F$ .",
"Thus $h(F)=\\deg (\\Xi _{31}(x))=3$ and the same argument as in the proof of Lemma REF tells us that $\\mathfrak {q}_F$ is not principal if and only if $\\Xi _{31}(x)\\equiv 0\\pmod {q}$ is not solvable.",
"By the assumption in Theorem REF we know this is the case and then $\\mathfrak {q}_F$ has order 3 in $Cl(F)$ .",
"If $\\operatorname{ord}(x_1)=1$ , i.e.",
"$\\mathfrak {Q}_1=1$ in $Cl(E)$ , then taking norm gives $\\mathfrak {q}_F=1$ in $Cl(F)$ , which is a contradiction.",
"If $\\operatorname{ord}(x_1)=3$ , then $\\left<x_1\\right>\\cong \\mathbb {Z}/3\\mathbb {Z}$ and we may assume $x_1=1\\mod {3}$ .",
"The second column of the matrix reads $8x_1+2x_2=0$ .",
"Since 2 can be canceled from every equation, we have $x_2=-4x_1$ .",
"Hence $x_2=-x_1=-1\\mod {3}$ and similarly $x_3=1\\mod {3}$ .",
"Thus $x_k=\\pm 1\\mod {3}\\in \\mathbb {Z}/3\\mathbb {Z}$ for all $k=1,2,\\dots ,6$ .",
"But we know three of all six $x_k$ 's (i.e.",
"$\\mathfrak {Q}_k$ 's) lie over $\\mathfrak {q}_F$ .",
"Suppose that $\\mathfrak {q}_F\\mathfrak {o}_E = \\mathfrak {Q}_{k_1} \\mathfrak {Q}_{k_2} \\mathfrak {Q}_{k_3}$ .",
"If all these three $x_{k_1},x_{k_2},x_{k_3}$ are the same, say $1\\mod {3}$ , then $\\mathfrak {q}\\mathfrak {o}_E=1$ in $Cl(E)$ .",
"Since $Cl(F)\\longrightarrow Cl(E)$ is injective (Proposition REF ), we have $\\mathfrak {q}_F=1$ in $Cl(F)$ , a contradiction.",
"Otherwise we may assume $x_{k_1}=-x_{k_2}=1$ and then $x_{k_1}+x_{k_2} = 0$ .",
"Taking norm gives $\\mathfrak {q}_F^2=1$ , which is also false.",
"Thus we have $\\operatorname{ord}(x_1)=9$ and using the matrix again we obtain $(x_1,x_2,\\dots ,x_6)=(1,-4,-2,-1,4,2)$ are all in $\\left<x_1\\right>\\cong \\mathbb {Z}/9\\mathbb {Z}$ .",
"We now apply Proposition REF .",
"Let $l=1,3\\dots $ and solve the equation (REF ) modulo 9.",
"A simple calculation tells us that $l_0=1$ is the maximal nonnegative odd number such that (REF ) is not solvable in $Cl(E)$ .",
"Hence we obtain by Proposition REF the non-existence of GBFs with type $[31, 31^aq^ln^{\\prime }]$ .",
"The argument for $(p,f)=(151,15)$ is similar.",
"Using the matrix in (REF ) we know that $27281x_1=0$ .",
"The same method yields the fact that $h(E)$ is also odd.",
"Thus we find that $\\operatorname{ord}(x_1)=7,281$ or 1967.",
"In this case $F=\\mathbb {Q}(\\sqrt{-157})$ .",
"Knowing that $h(F)=7$ and $\\mathfrak {q}_F$ has order 7 in $Cl(F)$ since $\\Xi _{151}(x)\\equiv 0\\pmod {q}$ is not solvable, the candidate order 1 and 7 can be removed by the previous method.",
"If we have $281x_1=0$ , taking norm gives $\\mathfrak {q}_F^{281}=1$ , which contradicts to $\\operatorname{ord}(\\mathfrak {q}_F)=7$ .",
"Thus $\\operatorname{ord}(x_1)=1967$ and we obtain $x_1,\\dots ,x_{10}\\in \\left<x_1\\right>\\cong \\mathbb {Z}/1967\\mathbb {Z}$ and $(x_1,x_2,\\dots ,x_5)=( 1, -652, 232, 195, 715 )\\\\x_{5+k}=-x_k,\\ k=1,2,\\dots ,5.$ Let $l=1,3\\dots $ and solve the equation (REF ) modulo 1967.",
"We find that $l_0=5$ is the maximal nonnegative odd number such that (REF ) is not solvable in $Cl(E)$ .",
"Again Proposition REF implies the non-existence of GBFs with type $[151, 151^aq^ln^{\\prime }]$ .",
"For other $(p,f)\\in \\mathinner {\\lbrace \\,{(127, 9),\\ (127, 21),\\ (139, 23)}\\,\\rbrace }$ , the proofs are similar."
],
[
"Corresponding non-existence results for PAPSs",
"In this section, we give briefly tow non-existence results for PAPSs, which are similar to Theorem REF and REF , respectively.",
"Their proofs are also similar.",
"Proposition 5.1 (See [12] Theorem 1.4 (2)) If there exist PAPS with type $[p, n+1]$ , then $p\\mid n-1$ and $\\alpha \\bar{\\alpha }=n$ for some $\\alpha \\in \\mathbb {Z}[\\zeta _p]$ .",
"Theorem 5.2 Let $p\\equiv 5\\pmod {8}$ be a prime and $\\tilde{Q}_p=\\mathinner {\\lbrace \\,{q\\text{ is a prime } }\\mid { \\operatorname{ord}_p(q)=(p-1)/4}\\,\\rbrace }$ .",
"Then there exists a lower bound $p_0$ , and an infinite set $Q_p\\subseteq \\tilde{Q}_p$ for each $p$ , such that if $p>p_0$ , there is no PAPSs with type $[p, qn^{\\prime }+1]$ for all integers $q\\in Q_p$ , $n^{\\prime }$ such that $n^{\\prime }=1$ or $\\genfrac(){}{}{p^{\\prime }}{p}=-1$ for all prime divisor $p^{\\prime }$ of $n^{\\prime }$ and $p\\mid qn^{\\prime }-1$ .",
"If there exist PAPS with type $[p, qn^{\\prime }+1]$ , where $p\\equiv 5\\pmod {8}$ be a prime, then by Proposition REF we know that $p\\mid qn^{\\prime }-1$ and $\\alpha \\bar{\\alpha }=qn^{\\prime }\\text{ for some }\\alpha \\in \\mathfrak {o}_K=\\mathbb {Z}[\\zeta _p].$ Here $K=\\mathbb {Q}(\\zeta _p)$ .",
"By [12] we obtain that $\\alpha _2\\bar{\\alpha }_2=q\\text{ for some }\\alpha _2\\in \\mathfrak {o}_K.$ Then the remaining argument is totally the same as the proof of Theorem REF .",
"See Section .",
"Theorem 5.3 Let $p\\equiv 3$ be a prime, $q\\ne p$ another prime and $f=\\operatorname{ord}_p(q)$ .",
"Suppose that the triple $(p, f, l_0)$ equals to one of the following value: $(31 , 5 , 1),\\ (127, 9 , 1),\\ (127, 21, 3),\\ (139, 23, 1),\\ (151, 15, 3).$ Define $\\Xi _{31}(x) &= x^3 + x - 1,\\\\\\Xi _{127}(x) &= x^5 - x^4 - 2x^3 + x^2 + 3x - 1,\\\\\\Xi _{139}(x) &= x^3 - x^2 + x + 2,\\\\\\text{and}\\quad \\Xi _{151}(x) &= x^7 - x^6 + x^5 + 3x^3 - x^2 + 3x + 1.$ Suppose further that for each $p\\in \\mathinner {\\lbrace \\,{31,127,139,151}\\,\\rbrace }$ , the corresponding $q$ satisfies that $\\Xi _p(x)\\equiv 0\\pmod {q}$ is not solvable.",
"Then there is no PPPs with type $[p, q^ln^{\\prime }+1]$ for all integers $l$ odd, $1\\le l\\le l_0$ , $n^{\\prime }$ such that $n^{\\prime }=1$ or $\\genfrac(){}{}{p^{\\prime }}{p}=-1$ for all prime divisor $p^{\\prime }$ of $n^{\\prime }$ and $p\\mid q^ln^{\\prime }-1$ .",
"The argument is totally the same as the proof of Theorem REF (in Section ), except that we use Proposition REF instead of Proposition REF ."
],
[
"Acknowledgment",
"The author would like to thank Jianing Li for many helpful discussions and comments."
]
] | 1612.05682 |
[
[
"Weak Hopf algebras and the distribution of involutions in symmetric\n groups"
],
[
"Abstract By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup."
],
[
"Introduction",
"Let ${{\\mathfrak {S}}_n}$ be the symmetric group of $n$ -letters.",
"Then we have the following classical identity in combinatorial representation theory: $\\bigl |\\lbrace a \\in {{\\mathfrak {S}}}_n \\,|\\, a^2 = 1 \\rbrace \\bigr |=\\bigl |{\\mathrm {STab}}(n)\\bigr |,$ where ${\\mathrm {STab}}(n)$ denotes the set of all standard tableaux of size $n$ .",
"Besides a proof based on RSK correspondence, there is a proof of this identity, which is based on Frobenius-Schur indicators.",
"Let $G$ be an arbitrary finite group.",
"Then the $r$ -th root number function $R_{G}^r (a) := \\bigl |\\lbrace c \\in G\\,|\\, c^r = a \\rbrace \\bigl |$ is given by $R_G^r=\\sum _{\\chi \\in \\mathrm {Irr}\\, G} \\mathrm {FS}_r (\\chi )\\, \\chi ,$ where $\\mathrm {Irr}\\, G$ denotes the set of (complex) irreducible characters of $G$ and $\\mathrm {FS}_r (\\chi )$ denotes the $r$ -th Frobenius-Schur indicator of $\\chi $ .",
"Hence (REF ) follows from $\\mathrm {FS}_2 (\\chi ) = 1$ $(\\chi \\in \\mathrm {Irr}\\, {{\\mathfrak {S}}_n})$ and $\\sum _{\\chi \\in \\mathrm {Irr}\\, {{\\mathfrak {S}}_n}} \\chi (1) = \\bigl |{\\mathrm {STab}}(n)\\bigr |$ .",
"Let $H$ be a subgroup of $G$ and let $b$ be an element of $G$ .",
"In this paper, we consider the following coset-wise root number function: $R_{G,bH}^r (a):= \\bigl |\\bigl \\lbrace c\\in b H\\,|\\, c^r = a \\bigr \\rbrace \\bigl |.$ The support $K$ of the restricted function $R_{G,bH}^r\\bigl |_H$ becomes a subgroup of $H$ and the restriction of $R_{G,bH}^r$ on $K$ has the expansion $R_{G,bH}^r \\Bigl |_K=\\sum _{\\chi \\in \\mathrm {Irr} K} \\mathrm {FS}_r (L_\\chi ) \\chi ,$ where $\\mathrm {FS}_r (L_\\chi )$ denotes the $r$ -th Frobenius-Schur indicator of certain simple module $L_\\chi $ of a weak Hopf algebra (WHA) ${\\cal F} (G,X)$ attached to $G$ and $X := G/H$ .",
"When $G = {{\\mathfrak {S}}_n}$ and $H = {{\\mathfrak {S}}}_{n-1}$ , we give an explicit formulas of $\\mathrm {FS}_r (L_\\chi )$ and $R_{G,bH}^r$ for every $r>0$ .",
"When $G = {{\\mathfrak {S}}_n}$ and $H$ is a Young subgroup ${{\\mathfrak {S}}}_{\\alpha }$ , we determine the value $\\mathrm {FS}_2 (L_\\chi )$ and give an explicit formula for the number of involutions in $b {{\\mathfrak {S}}}_\\alpha $ .",
"As a special case, we obtain $\\bigl |\\bigl \\lbrace a \\in b {{\\mathfrak {S}}}_m \\,|\\, a^2 = 1 \\bigr \\rbrace \\bigl |={\\left\\lbrace \\begin{array}{ll}\\,\\bigl |{\\mathrm {STab}}(m - k)\\bigl |\\, \\quad ({{\\mathfrak {S}}}_m b {{\\mathfrak {S}}}_m = {{\\mathfrak {S}}}_m b^{-1}{{\\mathfrak {S}}}_m)\\\\\\, 0 \\qquad \\qquad \\qquad \\quad ({{\\mathfrak {S}}}_m b {{\\mathfrak {S}}}_m \\ne {{\\mathfrak {S}}}_m b^{-1}{{\\mathfrak {S}}}_m),\\end{array}\\right.",
"}$ for each $0<m<n$ and $b \\in {{\\mathfrak {S}}_n}$ , where $k := \\bigl |\\lbrace 1,2,\\ldots , m\\rbrace \\setminus b\\lbrace 1,2,\\ldots , m\\rbrace \\bigl |$ .",
"Also, we obtain $\\bigl |\\bigl \\lbrace a \\in b ({{\\mathfrak {S}}}_m \\times {{\\mathfrak {S}}}_{m^\\prime })\\,|\\, a^2 = 1 \\bigr \\rbrace \\bigl |=k !\\, \\bigl |{\\mathrm {STab}}(m - k)\\bigl |\\, \\bigl |{\\mathrm {STab}}(m^\\prime - k)\\bigl |,$ where $m^\\prime := n-m$ .",
"By counting $R^2_{{\\mathfrak {S}}_n}(1)$ using (REF ), we obtain $\\bigl |{\\mathrm {STab}}(n)\\bigl |=\\sum _{0\\le k\\le \\mathrm {min}\\lbrace m, m^{\\prime } \\rbrace }\\frac{m!\\, m^{\\prime } !",
"}{k !\\, (m-k)!\\, (m^{\\prime }-k)!",
"}\\bigl |{\\mathrm {STab}}(m - k)\\bigl |\\, \\bigl |{\\mathrm {STab}}(m^{\\prime } - k)\\bigl |.$ In [15], Ng and Schauenburg defined Frobenius-Schur indicators as invariants of (objects of) pivotal fusion categories (See also [6]).",
"Also, Schauenburg [17], [18], [19] gave several results for Frobenius-Schur indicators of group-theoretical fusion categories.",
"Since the representation category of ${\\cal F} (G,X)$ is group-theoretical by Andruskiewitsch-Natale [1] and Mombelli-Natale [13], our general results for Frobenius-Schur indicators of ${\\cal F} (G,X)$ -modules overlap with Schauenburg's results significantly.",
"Nevertheless, we give WHA counterparts of his results, since WHA approach seems to be more elementary than his category-theoretic approach.",
"In fact, our approach clarifies the importance of the G-sets $X$ and $X \\times X$ , which did not play important roles in his paper.",
"The outline of the paper is as follows.",
"In Section 2 we give the definition of the algebra ${\\cal F} (G,X)$ .",
"In Section 3, we define and study Frobenius-Schur indicators of ${\\cal F} (G,X)$ -modules.",
"In Section 4 we give a relation between second indicators of ${\\cal F} (G,X)$ -modules and Kawanaka-Matsuyama indicators [10] of $K̏$ -modules.",
"In Section 5 and Section 6, we compute second indicators of ${\\cal F} ({{\\mathfrak {S}}_n},{{\\mathfrak {S}}}_\\alpha )$ -modules and indicators of ${\\cal F} ({{\\mathfrak {S}}_n},{{\\mathfrak {S}}}_{n-1})$ -modules, respectively.",
"Also, we give the corresponding results for $R^r_{G,bH}$ in these sections.",
"In Section 7 we give a correspondence between invariant bilinear forms on ${\\cal F} (G,X)$ -modules and invariant bilinear pairings of some $K̏$ -modules.",
"In Section 8 we verify that our definition of Frobenius-Schur indicators of ${\\cal F} (G,X)$ -modules coincides with that of Ng-Schauenburg [15].",
"The author thank K. Shimizu for telling him about the Frobenius-Schur indicators."
],
[
"Preliminaries",
"Throughout this paper, all modules are assumed to be finite dimensional over the complex number field $.Let $ G$ be a finite group and let $ X$ be a finite left $ G$-set.For $ x X$, we denote by $ Gx$ the {\\it stabilizer} of $ G$ at $ x$,that is, $ Gx = { a G | a x = x}$.Let $ X G$ be the $ -linear span of the symbols $e_x\\, a\\,\\, (a \\in G,\\, x \\in X)$ .",
"Then $X\\!",
"\\rtimes G$ becomes an algebra via $(e_x \\, a)(e_y \\, b)=\\delta _{x,ay}\\, e_x \\, ab.$ By identifying $a \\in G$ with $\\sum _{x\\in X} e_x\\, a$ , $G̏$ becomes a subalgebra of $X\\!",
"\\rtimes G$ .",
"The elements $e_x := e_x\\, 1_G\\,\\, (x \\in X)$ are mutually orthogonal idempotents and give a partition of unity of $X\\!",
"\\rtimes G$ .",
"Let $M$ be a left $X\\!",
"\\rtimes G$ -module and let $$ be an orbit of the $G$ -set $X$ .",
"We say that $M$ is of type $$ if $M = \\bigoplus _{x \\in } e_x M$ .",
"We note that each $X\\!",
"\\rtimes G$ -module has a unique decomposition $M = \\sum _{\\in G\\backslash X} M_$ such that $M_$ is of type $$ .",
"It seems that the following is a folklore among some communities of Hopf algebraists.",
"Proposition 2.1 (cf.",
"[12] page 241, [11] Lemma 3.2, Theorem 3.3) (1) Let $= G x$ be an orbit of $X$ and let $V$ be a left $G̏_{x}$ -module.",
"Then ${\\cal I}_{x} (V) := G̏ \\otimes _{G̏_{x}} V$ becomes a $X\\!",
"\\rtimes G$ -modules via $a (b\\otimes v) = ab \\otimes v,\\quad e_y (b\\otimes v) = \\delta _{y,bx} b\\otimes v\\quad (a,b \\in G, y\\in X, v \\in V).$ (2) The correspondence ${\\cal I}_{x}$ gives an equivalence between the category of $G̏_{x}$ -modules and the category of $X\\!",
"\\rtimes G$ -modules of type $$ .",
"Let ${\\cal F} = {\\cal F} (G,X)$ be the $-linear span of the symbols$ exy a (a G, x, y X)$.Then $ F$ becomes an algebra via$$(e^x_y \\, a)(e^z_w \\, b)=\\delta _{x,az} \\delta _{y,aw} e^x_y \\, ab.$$Let$ : FFF$and $ : F be linear maps given by $\\Delta (e^x_y \\, a)=\\sum _{z \\in X} e^x_z \\, a \\otimes e^z_y \\, a,\\qquad \\varepsilon (e^x_y \\, a) = \\delta _{xy}.$ Then $\\cal F$ becomes a $X$ -face algebra with antipode $S:\\!",
"{\\cal F} \\rightarrow {\\cal F}; e^x_y a \\mapsto a^{-1} e^y_x$ (cf.",
"[7]).",
"Hence $\\cal F$ is a weak Hopf algebra (cf.",
"[2]).",
"We call ${\\cal F} (G,X)$ the group-like face algebra of $(G,X)$ .",
"Let $= G(x,y)$ be an orbital of $X$ , that is, $\\in G \\backslash (X\\times X)$ .",
"Since ${\\cal F} (G,X)\\cong {X\\times X}\\!",
"\\rtimes G$ as algebras, we have an equivalence ${\\cal I}_{xy}$ between the category of $G̏_{xy}$ -modules and the category of ${\\cal F} (G,X)$ -modules of type $$ , where $G_{xy}$ stands for the two-point stabilizer $G_x \\cap G_y$ .",
"In particular, if $\\lbrace V(\\lambda ) \\rbrace $ is a set of representatives for the isomorphism classes of simple $G̏_{xy}$ -modules, then $\\lbrace {\\cal I}_{xy} (V(\\lambda ))\\rbrace $ is a set of representatives for the isomorphism classes of simple ${\\cal F} (G,X)$ -modules of type $\\Omega $ ."
],
[
"Frobenius-Schur indicators",
"We define elements ${\\textstyle \\int }^{[r]}\\,\\,(r\\ge 1)$ of ${\\cal F} (G,X)$ by ${\\textstyle \\int }^{[1]} = \\int := \\frac{1}{|G|}\\sum _{a\\in G} \\sum _{x \\in X} e^x_x\\, a$ and ${\\textstyle \\int }^{[r]} := (m^{(r)}\\circ \\Delta ^{(r)}) ({\\textstyle \\int })\\,\\, (r \\ge 2) $ respectively, where $m^{(r)}\\!",
":{\\cal F} (G,X)^{\\otimes r} \\rightarrow {\\cal F} (G,X)$ and $\\Delta ^{(r)}\\!",
":{\\cal F} (G,X)\\rightarrow {\\cal F} (G,X)^{\\otimes r}$ denote the iterations of the product and the coproduct of ${\\cal F} (G,X)$ respectively, that is, $m^{(3)} (\\alpha ,\\beta ,\\gamma ) =\\alpha \\beta \\gamma $ and $\\Delta ^{(3)} (\\alpha ) = (\\Delta \\otimes {\\mathrm {id}})(\\Delta (\\alpha ))$ , for example.",
"Then, ${\\textstyle \\int }$ is an idempotent two-sided integral of ${\\cal F} (G,X)$ (cf.",
"[2]), that is, ${\\textstyle \\int }^2 = {\\textstyle \\int }$ and $\\alpha {\\textstyle \\int }= \\varepsilon ^L ( \\alpha ) {\\textstyle \\int },\\quad {\\textstyle \\int }\\alpha = {\\textstyle \\int }\\varepsilon ^R (\\alpha )\\quad (\\alpha \\in {\\cal F} (G,X)),$ where, $\\varepsilon ^L ( \\alpha ) = \\sum _{x,y,z\\in X} \\varepsilon ( e^x_z \\alpha ) e^z_y,\\quad \\varepsilon ^R ( \\alpha ) = \\sum _{x,y,z\\in X} e^x_z \\varepsilon (\\alpha e^z_y).$ Let $M$ be a finite-dimensional ${\\cal F} (G,X)$ -module.",
"We define the $r$ -th Frobenius-Schur indicator of $M$ by $\\mathrm {FS}_r (M) := {\\mathrm {Tr}}_M ({\\textstyle \\int }^{[r]})$ .",
"Lemma 3.1 (1) Explicitly, the elements ${\\textstyle \\int }^{[r]}$ are given by ${\\textstyle \\int }^{[r]}=\\frac{1}{|G|}\\sum _{a\\in G}\\sum _{x \\in X} \\delta _{x,a^r x}e^x_{a^{-1} x} a^r=\\frac{1}{|G|}\\sum _{a\\in G}\\sum _{x \\in X} \\delta _{x,a^r x} a^r e^x_{a^{-1} x}.$ (2) The element ${\\textstyle \\int }^{[r]}$ is central.",
"Proof.",
"Part (1) follows from the following computations: $& (m^{(r)}\\circ \\Delta ^{(r)}) (e^x_x\\, a)\\\\= &\\sum _{y_1 \\in X}\\cdots \\sum _{y_{r-1} \\in X}\\left( e^x_{y_1} a \\right) \\left( e^{y_1}_{y_2} a \\right) \\cdots \\left( e^{y_{r-2}}_{y_{r-1}} a \\right) \\left( e^{y_{r-1}}_x a \\right)\\\\= &\\sum _{y_1 \\in X}\\cdots \\sum _{y_{r-1} \\in X}\\left( e^x_{y_1} a \\right) \\left( e^{y_1}_{y_2} a \\right) \\cdots \\left( e^{y_{r-3}}_{y_{r-2}} a \\right)\\delta _{y_{r-2},a y_{r-1}} \\delta _{y_{r-1}, ax}\\left( e^{y_{r-2}}_{y_{r-1}} a^2 \\right)\\\\= &\\sum _{y_1 \\in X}\\cdots \\sum _{y_{r-3} \\in X}\\left( e^x_{y_1} a \\right) \\left( e^{y_1}_{y_2} a \\right) \\cdots \\left( e^{y_{r-3}}_{a^2 x} a \\right)\\left( e^{a^2 x}_{ax} a^2 \\right)\\\\= &\\cdots \\\\= &\\left( e^x_{a^{r-1} x} a \\right)\\left( e^{a^{r-1} x}_{a^{r-2} x} a^{r-1} \\right)=\\delta _{x,a^r x}\\,e^x_{a^{-1} x}\\, a^r=\\delta _{x,a^r x}\\, a^r\\, e^x_{a^{-1} x}.$ Let $c$ be an element of $G$ .",
"Replacing $a$ and $x$ in (REF ) by $cbc^{-1}$ and $cy$ respectively, we obtain ${\\textstyle \\int }^{[r]} c= &\\left(\\frac{1}{|G|}\\sum _{b\\in G}\\sum _{y \\in X} \\delta _{c y, c b^r y }\\,e^{cy}_{c b^{-1} y}\\, c b^r c^{-1}\\right) c \\\\= &\\frac{1}{|G|}\\sum _{b\\in G}\\sum _{y \\in X} \\delta _{y, b^r y }\\,c\\, e^{y}_{b^{-1} y}\\, b^r=c {\\textstyle \\int }^{[r]}.$ For $y, z \\in X$ , we have $e^y_z {\\textstyle \\int }^{[r]}=\\frac{1}{|G|} \\sum _{a\\in G} \\delta _{y,a^r y} \\delta _{z,a^{-1}y} e^y_{a^{-1} y} a^r=\\frac{1}{|G|} \\sum _{a\\in G} \\delta _{y,a^r y} \\delta _{z,a^{-1}y} a^r e^y_{a^{-1} y}={\\textstyle \\int }^{[r]} e^y_z.$ Since $g$ 's and $e^y_z$ 's generate the algebra ${\\cal F} (G,X)$ , this proves Part (2).",
"$\\hfill \\square $ Theorem 3.2 (cf.",
"Schauenburg [17], Theorem 4.1) For each $x, y \\in X$ and $G̏_{xy}$ -module $V$ , we have $\\mathrm {FS}_r ({\\cal I}_{xy} (V))=\\frac{1}{| G_{xy} |} \\sum _{a \\in G[x,y;\\,r]} {\\mathrm {Tr}}_V (a^{-r}),$ where $G[x,y;r] := \\lbrace a \\in G\\,|\\, a x = y,\\, a^r x= x \\rbrace $ .",
"Proof.",
"We first note that the right-hand side of (REF ) is well-defined, since $a^{-r} \\in K:= G_{xy}$ for each $a \\in G[x,y;\\,r]$ .",
"Also, we note that we may assume that $V$ is a simple $K̏$ -module, since both the right-hand side and the left-hand side of (REF ) are additive with respect to $V$ .",
"Then, by Proposition REF (2), ${\\cal I}_{xy} (V)$ is a simple ${\\cal F} (G,X)$ -module.",
"Hence by Schur's lemma, the action of the central element ${\\textstyle \\int }^{[r]}$ on ${\\cal I}_{xy} (V)$ is given by some scalar.",
"Therefore, we have ${\\mathrm {Tr}}_{{\\cal I}_{xy} (V)} ({\\textstyle \\int }^{[r]})=\\frac{\\dim {\\cal I}_{xy} (V)}{\\dim ( K̏ \\otimes _{K̏} V)}{\\mathrm {Tr}}_{K̏ \\otimes _{K̏} V} ({\\textstyle \\int }^{[r]})\\\\=\\frac{|G|}{|K|}{\\mathrm {Tr}}_{K̏ \\otimes _{K̏} V} ({\\textstyle \\int }^{[r]}).$ By Lemma REF (1), we have ${\\textstyle \\int }^{[r]} (1_G \\otimes v)=\\frac{1}{|G|}\\sum _{a,z} \\delta _{z,a^r z}\\delta _{z, x}\\delta _{a^{-1}z,y} a^r\\otimes v \\\\=\\frac{1}{|G|}\\sum _{c \\in G[x,y;\\,r]} 1_G \\otimes c^{-r} v.$ Therefore, ${\\mathrm {Tr}}_{K̏ \\otimes _{K̏} V} ({\\textstyle \\int }^{[r]})=\\frac{1}{|G|}\\sum _{c \\in G[x,y;\\,r]} {\\mathrm {Tr}}_V (c^{-r}).$ This proves (REF ).",
"$\\hfill \\square $ Let $H$ be a subgroup of $G$ .",
"We define the $r$ -th root number function $R^r_{G}$ and the $r$ -th coset-wise root number function $R^r_{G,\\,bH}$ by $R^r_{G} (a) = \\bigl |\\lbrace c \\in G \\,|\\, c^r = a \\rbrace \\bigl |,\\\\R^r_{G,\\,bH} (a) = \\bigl |\\lbrace c \\in b H \\,|\\, c^r = a \\rbrace \\bigl |,$ respectively.",
"We note that $R^r_G$ is a class function and that $R^r_{G,\\,hbH} (a) = R^r_{G,\\,bH} (h^{-1} a h)$ for each $a,b \\in G$ and $h \\in H$ .",
"In particular, we have $R^r_{G,\\,hbH} (1) = R^r_{G,\\,bH} (1).$ By (REF ), the assignment $HbH \\mapsto R^r_{G,HbH} (1):= R^r_{G,bH} (1)$ gives a well-defined function on $H\\backslash G/H$ .",
"Proposition 3.3 The root number function satisfy $R^r_G (1)=\\sum _{HbH \\in H\\backslash G/H} \\frac{\\bigl |H \\bigl |}{\\bigl |H \\cap bHb^{-1} \\bigl |}\\,R^r_{G,\\,HbH} (1).$ Proof.",
"By definition, we have $R^r_G (1)& =\\sum _{bH \\in G/H} R^r_{G,bH} (1)\\\\& =\\sum _{H b_1 H \\in H\\backslash G/H} c_{H b_1 H}R^r_{G, H b_1 H} (1),$ where $c_{H b_1 H} := \\bigl |\\lbrace bH \\in G/H\\,|\\, HbH = H b_1 H \\rbrace \\bigl |$ .",
"Since $c_{H b_1 H}$ is equals to the size of the $H$ -orbit through $y := b_1 H \\in X:=G/H$ , it equals $|H|/|H_{y}|=|H|/|H \\cap b_1 H b_1^{-1}|$ .",
"This proves (REF ).",
"$\\hfill \\square $ Theorem 3.4 (cf.",
"Schauenburg [17], Lemma 4.5) For each $ a \\in H$ and $y = b H \\in X := G/H$ , we have $\\Bigl | \\bigl \\lbrace c \\in b H \\,|\\, c^r = a \\bigr \\rbrace \\Bigr |\\nonumber \\\\={\\left\\lbrace \\begin{array}{ll}\\sum _{\\lambda }\\, \\mathrm {FS}_r ({\\cal I}_{xy} (V(\\lambda )))\\, \\chi _\\lambda (a) & (ay = y)\\\\0 & (ay \\ne y),\\end{array}\\right.",
"}$ where ${x} = H \\in X$ , $\\lbrace { V(\\lambda )} \\rbrace $ is as in Section 2 and $\\chi _{\\lambda } = {\\mathrm {Tr}}_{V(\\lambda )}$ denotes the character of $V(\\lambda )$ .",
"Proof.",
"To begin with, we show that the left-hand side of (REF ) is non-zero only if $ay =y$ .",
"Suppose that $c^r = a$ for some $c \\in bH$ .",
"Since $c \\in bH$ , we have $cx = b x =y$ .",
"Hence $ay = c^r y = c^{r+1} x = c a x = cx = y.$ Let $K$ be the two-point stabilizer $G_{xy}$ .",
"By (REF ), $R^r_{G,bH}\\bigl |_K$ is a class function on $K$ .",
"Hence $R^r_{G,bH}\\bigl |_K=\\sum _\\lambda (R^r_{G,bH}\\,|\\,\\chi _\\lambda )_K\\, \\chi _\\lambda ,$ where $(\\,|\\,)_K$ denotes the usual inner product of the space of class functions on $K$ , that is, $(f|g)_K := |K|^{-1} \\sum _{a\\in K} f(a)\\overline{g(a)}$ .",
"Therefore, it suffices to show that $(R^r_{G,bH}\\,|\\,\\chi _\\lambda )_K = \\mathrm {FS}_r ({\\cal I}_{xy} (V)(\\lambda )$ .",
"By definition, we have $(R^r_{G,bH}\\,|\\,\\chi _\\lambda )_K=\\frac{1}{|K|} \\sum _{a \\in K}\\bigl |\\lbrace c \\in bH\\,|\\, c^r = a\\rbrace \\bigl |\\,\\chi _\\lambda (a^{-1})\\nonumber \\\\=\\frac{1}{|K|} \\sum _{c \\in bH; c^r \\in K}\\chi _\\lambda (c^{-r}).$ Since $\\lbrace c \\in bH\\,|\\, c^r \\in K \\rbrace =\\lbrace c \\in G\\,|\\, cx =y, c^r y = y \\rbrace =G[x,y;r],$ the right-hand side of (REF ) coincides with that of (REF ) for $V = V(\\lambda )$ .",
"$\\hfill \\square $"
],
[
"Twisted Frobenius-Schur indicators",
"Let $K$ be a finite group.",
"Let $\\phi $ be an automorphism of $K$ and let $k_0$ be an element of $K$ .",
"We say that $(\\phi , k_0)$ is an outer involution of $K$ if $\\phi ^2 (k) = k_0^{-1} k k_0\\quad (k\\in K)$ and $\\phi (k_0) = k_0$ .",
"We note that if $K \\le G$ and $t \\in G$ satisfies $t^{-1} K t = K$ and $t^2 \\in K$ , then $((-)^t, t^2)$ is an outer involution of $K$ , where $(-)^t\\!",
":K\\rightarrow K; k \\rightarrow t^{-1} k t$ .",
"Conversely, for an outer involution $(\\phi ,k_0)$ , there exists a group $G \\ge K$ and $t \\in G \\setminus K$ such that $(\\phi , k_0) = ((-)^t, t^2)$ .",
"Explicitly, $G$ is given by $G = K \\coprod tK$ , which is equipped with product $(tk)(t k^\\prime ) = k_0 (\\phi (k)k^\\prime )$ , $(tk)k^\\prime = t(kk^\\prime )$ , $k(tk^\\prime ) = t (\\phi (k)k^\\prime )$ $(k,k^\\prime \\in K)$ , where $t K =\\lbrace tk\\,|\\, k \\in K\\rbrace $ is a copy of the set $K$ .",
"Let $V$ be a finite-dimensional $K̏$ -module.",
"We define $(\\phi ,k_0)$ -twisted second Frobenius-Schur indicator of $V$ by $\\mathrm {FS}_2 (V,\\phi ,k_0) = \\frac{1}{|K|}\\,\\sum _{k \\in K}\\,{\\mathrm {Tr}}_V (k_0 \\phi (k) k).$ When $(\\phi , k_0) = ((-)^t, t^2)$ , we write $\\mathrm {FS}_2 (V,t) = \\mathrm {FS}_2 (V,\\phi ,k_0)$ .",
"It agrees with Kawanaka-Matsuyama's indicator (cf.",
"[10]), that is, $\\mathrm {FS}_2 (V,t) = \\frac{1}{|K|}\\,\\sum _{k \\in K}\\,{\\mathrm {Tr}}_V ((tk)^2).$ We say that an orbital $$ is symmetric (or self-paired) if $^{\\sf T}= $ , where $^{\\sf T}:= \\lbrace (y,x)\\,|\\, (x,y) \\in \\rbrace $ .",
"Proposition 4.1 (1) An orbital $=G (x,y)$ is symmetric if and only if there exists an element $t\\in G$ such that $t x = y$ , $ty = x$ .",
"In this case, $K:= G_{xy}$ satisfies $t^{-1} K t = K$ and $t^2 \\in K$ .",
"(2) Let $H$ be a subgroup of $G$ and let $b$ be an element of $G$ .",
"Let $x_0$ be the element $H$ of $X = G/H$ and let $$ be $G (x_0,bx_0)$ .",
"Then $$ is symmetric if and only if $HbH = Hb^{-1}H$ .",
"Proof.",
"Part (1) is obvious.",
"Since there exists a bijection $H\\backslash G /H \\cong G\\backslash (X\\times X)$ ; $HbH \\mapsto G(x_0,bx_0)$ (cf.",
"[4] p240), $G(x_0,bx_0)$ is symmetric if and only if $HbH = Hb^{-1} H$ .",
"$\\hfill \\square $ Theorem 4.2 (cf.",
"Schauenburg [19], Proposition 3.2) Let $V$ be a $G̏_{xy}$ -module.",
"(1) If $= G(x, y)$ is not symmetric, then $\\mathrm {FS}_2 ({\\cal I}_{x,y} (V)) =0$ .",
"(2) Suppose that $$ is symmetric and that $t \\in G$ satisfies $t(x,y) = (y,x)$ .",
"Then, $\\mathrm {FS}_2 ({\\cal I}_{x,y} (V))=\\mathrm {FS}_2 (V,t).$ Proof.",
"Since $G[x,y;2] = \\lbrace a \\in G\\,|\\, ax = y, ay = x\\rbrace $ is empty if $^{\\sf T}\\ne $ , Part (1) follows from (REF ).",
"If $t \\in G$ satisfies $t(x,y) = (y,x)$ , then we have $G[x,y;2] = G_{xy}\\, t^{-1}$ .",
"Hence Part (2) also follows from (REF ).",
"$\\hfill \\square $"
],
[
"Symmetric groups I",
"For each subset $S$ of $[n] := \\lbrace 1,2,\\ldots , n\\rbrace $ , we define a subgroup ${{\\mathfrak {S}}}(S)\\cong {{\\mathfrak {S}}}_{|S|}$ of $G:={{\\mathfrak {S}}_n}$ by ${{\\mathfrak {S}}}(S):= \\lbrace a\\in {{\\mathfrak {S}}_n}\\,|\\, a i = i\\,\\, (i \\in [n]\\setminus S)\\rbrace $ .",
"For a set $S\\subset {Z}$ and an integer $\\epsilon $ , we set $\\epsilon + S = \\lbrace \\epsilon + s\\,|\\, s\\in S\\rbrace $ .",
"Let $\\alpha = (\\alpha _1,\\alpha _2,\\ldots ,\\alpha _\\ell )$ be a sequence of positive integers such that $\\alpha _1 + \\cdots + \\alpha _\\ell = n$ .",
"Let ${{\\mathfrak {S}}}_\\alpha = {{\\mathfrak {S}}}_{\\alpha _1}\\times \\cdots \\times {{\\mathfrak {S}}}_{\\alpha _\\ell }$ be the corresponding Young subgroup of ${{\\mathfrak {S}}_n}$ .",
"Here, we identify ${{\\mathfrak {S}}}_\\alpha $ with ${{\\mathfrak {S}}}(A_1) \\cdots {{\\mathfrak {S}}}(A_\\ell )$ as usual, where $A_1 = [\\alpha _1],\\, A_2 = \\alpha _1+[\\alpha _2],\\, \\ldots ,\\,A_\\ell = \\alpha _1+\\cdots \\alpha _{\\ell -1} + [\\alpha _\\ell ].$ Next, define a set $X= \\binom{[n]}{\\alpha }$ by $X:=\\bigl \\lbrace B = (B_1,\\ldots ,B_\\ell ) \\in (2^{[n]})^\\ell \\,\\bigm |\\,[n] = \\coprod _i B_i, \\quad \\bigl |B_i \\bigl |= \\alpha _i\\quad (1\\le i \\le \\ell )\\bigr \\rbrace .$ Then $X$ becomes a transitive $G$ -set via $a (B_1,\\ldots ,B_\\ell ) := (a(B_1),\\ldots ,a(B_\\ell ))$ .",
"Since the stabilizer $G_A$ of $G$ at $A := (A_1,\\ldots ,A_\\ell )$ is ${{\\mathfrak {S}}}_\\alpha $ , ${{\\mathfrak {S}}_n}/{{\\mathfrak {S}}}_\\alpha \\cong X;$ $b{{\\mathfrak {S}}}_\\alpha \\mapsto b A$ as $G$ -sets.",
"It is known that $G (B,C) \\mapsto [\\bigl |B_i\\cap C_j\\bigl |]_{ij}$ gives a bijection from $G \\backslash (X \\times X)$ onto $M_\\alpha :=\\bigl \\lbrace = [\\gamma _{ij}]_{ij}\\in \\mathrm {Mat}(\\ell ,{Z}_{\\ge 0})\\, \\bigm |\\,\\sum _i \\gamma _{ij} = \\alpha _j = \\sum _i \\gamma _{ji}\\quad (1 \\le j \\le \\ell )\\bigr \\rbrace .$ See e.g.",
"[9].",
"Note that $G (B, C)$ is a symmetric orbital if and only if $[\\bigl |B_i\\cap C_j\\bigl |]_{ij}$ is a symmetric matrix.",
"Let $B = (B_1,\\ldots ,B_\\ell ) = b A$ be an element of $X$ .",
"For $1 \\le i, j \\le \\ell $ , we set $B_{ij} := A_i \\cap B_j$ and $\\gamma _{ij} := \\bigl |B_{ij} \\bigr |$ .",
"Also, we set $A_{ij} = \\epsilon _{ij} + [\\gamma _{ij}]$ , where $\\epsilon _{11} =0$ , $\\epsilon _{12} =\\gamma _{11}$ , ..., $\\epsilon _{1\\ell } =\\gamma _{11}+\\cdots + \\gamma _{1,\\ell -1}$ , $\\epsilon _{21} = \\gamma _{11}+\\cdots + \\gamma _{1\\ell } = \\alpha _1$ , $\\epsilon _{22} = \\alpha _1 + \\gamma _{21}$ , ..., $\\epsilon _{ij} = \\alpha _{1}+\\cdots + \\alpha _{i-1}+\\gamma _{i1}+\\cdots + \\gamma _{i,j-1} $ , ... .",
"By definition, we have $A_i = A_{i1}\\coprod \\cdots \\coprod A_{i\\ell }$ for each $1 \\le i\\le \\ell $ .",
"For each $i,j$ , we fix a bijection $u_{ij}\\!",
": A_{ij} \\cong B_{ij}$ and define $u \\in {{\\mathfrak {S}}_n}$ by $u\\bigl |_{A_{ij}} = u_{ij}$ .",
"Let $\\gamma = (\\gamma _{1}, \\gamma _{2},\\ldots , \\gamma _{\\ell ^2})$ be the sequence $(\\gamma _{11}, \\gamma _{12},\\ldots , \\gamma _{1\\ell }, \\gamma _{21},\\ldots ,\\gamma _{2\\ell },\\ldots ,\\gamma _{\\ell 1},\\ldots ,\\gamma _{\\ell \\ell })$ and let $K_0$ be the subgroup ${{\\mathfrak {S}}}_\\gamma = {{\\mathfrak {S}}}_{\\gamma _1}\\times \\cdots \\times {{\\mathfrak {S}}}_{\\gamma _{\\ell ^2}}$ of ${{\\mathfrak {S}}_n}$ , where ${{\\mathfrak {S}}}_0 = \\lbrace 1\\rbrace $ .",
"Note that we have $a(\\epsilon _{ij} + s) = \\epsilon _{ij} + a_{ij}s$ for $a = (a_{11}, \\ldots , a_{1\\ell },\\ldots , a_{\\ell 1},\\ldots ,a_{\\ell \\ell })\\in {{\\mathfrak {S}}}_\\gamma $ and $s \\in [\\gamma _{ij}]$ .",
"Let $K$ the two-point stabilizer $G_{AB}$ .",
"Lemma 5.1 (1) $K = \\prod _{ij} {{\\mathfrak {S}}}(B_{ij})$ .",
"(2) The correspondence $k \\mapsto u^{-1} k u$ gives a group isomorphism $\\psi \\!",
":K \\cong K_0$ .",
"Proof.",
"(1) For each $a \\in {{\\mathfrak {S}}_n}$ , $a \\in K$ if and only if $a A_i = A_i$ and $a B_j = B_j$ for all $i,j$ , if and only if $a B_{ij} = B_{ij}$ for all $i,j$ .",
"Hence, we have Part (1).",
"(2) Since $K_0 = \\prod _{ij}{{\\mathfrak {S}}}(A_{ij})$ and $\\bigl |{{\\mathfrak {S}}}(A_{ij})\\bigl |= \\gamma _{ij}!",
"= \\bigl |{{\\mathfrak {S}}}(B_{ij})\\bigl |$ , it suffices to show that $u k u^{-1} \\in {{\\mathfrak {S}}}(B_{ij})$ for each $k \\in {{\\mathfrak {S}}}(A_{ij})$ .",
"Let $s$ be an element of $[n] \\setminus B_{ij}$ .",
"Since $u^{-1} s \\in [n]\\setminus A_{ij}$ , we have $k u^{-1} s = u^{-1} s$ .",
"This proves the assertion.",
"$\\hfill \\square $ Now suppose that $G(A,B)$ is a symmetric orbital.",
"We define $t_0, t \\in {{\\mathfrak {S}}_n}$ by $t_0 (\\epsilon _{ij} + s) = \\epsilon _{ji} + s$ $(s \\in [\\gamma _{ij}])$ and $t = u t_0 u^{-1}$ .",
"Then we have $t_0^2 = 1_G$ , $t_0 \\bigl |_{A_{ii}} = {\\mathrm {id}}$ and $t_0 (A_{ij}) = A_{ji}$ .",
"Moreover, $t_0 a t_0 = a^{\\sf T}$ for $a = (a_{11}, \\ldots , a_{1\\ell },\\ldots , a_{\\ell 1},\\ldots ,a_{\\ell \\ell })\\in K_0$ , where $a^{\\sf T}:= (a_{11}, a_{21},\\ldots , a_{\\ell 1}, ,\\ldots ,a_{1\\ell },\\ldots ,a_{\\ell \\ell }).$ Since $t(B_{ij}) = B_{ji}$ , we have $t A = B$ , $t B = A$ .",
"Moreover, we have $t \\psi ^{-1} (a) t = \\psi ^{-1} (a^{\\sf T})$ for each $a \\in K_0$ .",
"For $m \\ge 0$ , let $ {\\mathcal {P}}(m)$ be the set of partitions of $m$ and let $\\lbrace V(\\lambda )\\,|\\, \\lambda \\in {\\mathcal {P}}(m)\\rbrace $ be a complete representatives of simple $_m$ -modules such that $\\dim V(\\lambda ) = \\bigl |{\\mathrm {STab}}(\\lambda )\\bigl |$ , where ${\\mathrm {STab}}(\\lambda )$ denotes the set of standard tableaux of shape $\\lambda $ .",
"We denote the character of $V(\\lambda )$ by $\\chi _\\lambda $ .",
"Note that $ {\\mathcal {P}}(0)$ is a single element set $\\lbrace ()\\rbrace $ and that $V(())$ is a one-dimensional module of ${{\\mathfrak {S}}}_0 = \\lbrace 1 \\rbrace $ .",
"Let $ {\\mathcal {P}}()$ be the set of matrices $= [\\lambda _{ij}]_{1\\le i,j \\le \\ell }$ of partitions such that $\\lambda _{ij} \\in {\\mathcal {P}}(\\gamma _{ij})$ .",
"For each $=[\\lambda _{ij}] \\in {\\mathcal {P}}()$ , define a simple $K̏_0$ -module $V()$ by the following outer tensor product: $V() = V(\\lambda _{11})\\boxtimes V(\\lambda _{12})\\boxtimes \\cdots \\boxtimes V(\\lambda _{1\\ell })\\boxtimes \\cdots \\boxtimes V(\\lambda _{\\ell 1})\\boxtimes \\cdots \\boxtimes V(\\lambda _{\\ell \\ell }).$ Then $\\lbrace V()\\,|\\, \\in {\\mathcal {P}}()\\rbrace $ gives a complete representative of simple $K̏_0$ -modules.",
"Hence $\\lbrace V()^{\\psi }\\,|\\, \\in {\\mathcal {P}}()\\rbrace $ and $\\lbrace {\\cal I}_{A,B}(V()^{\\psi })\\,|\\, \\in {\\mathcal {P}}()\\rbrace $ give complete representatives of simple $K̏$ -modules and simple ${\\cal F} (G,X)$ -modules of type $G(A,B)$ , respectively.",
"Here, the action of $K$ on $V()^{\\psi }:=V()$ is given by $(k,v) \\mapsto \\psi (k) v$ ($k \\in K$ , $v \\in V()^{\\psi }$ ).",
"Theorem 5.2 For each $\\in {\\mathcal {P}}()$ , the second Frobenius-Schur indicator of the ${\\cal F} (G,X)$ -module ${\\cal I}_{A,B}(V()^{\\psi })$ is given by $\\mathrm {FS}_2 ({\\cal I}_{A,B} (V()^\\psi ))={\\left\\lbrace \\begin{array}{ll}1\\, & (^{\\sf T}= \\,\\, \\mathrm {and}\\,\\, ^{\\sf T}=)\\\\0 & (^{\\sf T}\\ne \\,\\,\\,\\, \\mathrm {or}\\,\\,\\,\\, ^{\\sf T}\\ne ).\\end{array}\\right.",
"}$ Proof.",
"By Theorem REF (1), we may assume $^{\\sf T}= $ .",
"Hence $\\mathrm {FS}_2 ({\\cal I}_{A,B} (V(\\Lambda )^\\psi ))= \\mathrm {FS}_2 (V(\\Lambda )^\\psi ,t)= \\frac{1}{\\bigl |K \\bigl |} \\sum _{k\\in K}{\\mathrm {Tr}}_{V()} (\\psi ((tk)^2))\\\\$ by Theorem REF (2).",
"Since $\\psi ((tk)^2) = t_0 a t_0 a = a^{\\!",
"{\\sf T}} a$ for $k = \\psi ^{-1} (a)$ and $a \\in K_0$ , the right-hand side equals $&\\frac{1}{\\bigl |K_0 \\bigl |} \\sum _{(a_{11},\\ldots ,a_{\\ell \\ell })\\in K_0}{\\mathrm {Tr}}_{V()} (a_{11} a_{11},a_{21}a_{12},\\ldots ,a_{\\ell 1}a_{1 \\ell },\\ldots a_{1 \\ell }a_{\\ell 1},\\ldots , a_{\\ell \\ell }a_{\\ell \\ell })\\\\= &\\frac{1}{\\prod _{ij}\\gamma _{ij}!",
"}\\sum _{a_{11}\\in {{\\mathfrak {S}}}_{\\gamma _{11}}}\\sum _{a_{12}\\in {{\\mathfrak {S}}}_{\\gamma _{12}}}\\cdots \\sum _{a_{\\ell \\ell }\\in {{\\mathfrak {S}}}_{\\gamma _{\\ell \\ell }}}\\prod _{ij}\\chi _{\\lambda _{ij}} (a_{ji} a_{ij}).$ Since $\\frac{1}{\\gamma _{ii}!}",
"\\sum _{a_{ii}} \\chi _{\\lambda _{ii}} (a_{ii}^2) = \\mathrm {FS}_2 (V(\\lambda _{ii})) = 1$ , the right-hand side equals $& \\prod _{i < j}\\frac{1}{(\\gamma _{ij}!",
")^2}\\sum _{a,a^{\\prime }\\in {{\\mathfrak {S}}}_{\\gamma _{ij}}}\\chi _{\\lambda _{ij}} (a^{\\prime } a)\\chi _{\\lambda _{ji}} (a a^{\\prime })\\\\= &\\prod _{i < j}\\frac{1}{\\gamma _{ij}!",
"}\\sum _{a\\in {{\\mathfrak {S}}}_{\\gamma _{ij}}}\\chi _{\\lambda _{ij}} (a )\\overline{\\chi _{\\lambda _{ji}} (a)}\\\\= &\\prod _{i < j}\\delta _{\\lambda _{ij},\\lambda _{ji}}=\\delta _{,^{\\sf T}},$ where the first equality follows from the fact that $\\chi _{\\lambda _{ji}}$ is a real-valued class function.",
"The second equality follows from the orthogonality relation of irreducible characters.",
"$\\hfill \\square $ Theorem 5.3 Let $\\alpha =(\\alpha _1,\\ldots ,\\alpha _\\ell )$ be a sequence of positive integers such that $\\sum _i \\alpha _i = n$ and let ${{\\mathfrak {S}}}_\\alpha $ be the corresponding Young subgroup of ${{\\mathfrak {S}}_n}$ .",
"Then, for each $b \\in {{\\mathfrak {S}}_n}$ , we have $\\bigl | \\bigl \\lbrace a \\in b\\, {{\\mathfrak {S}}}_{\\alpha }\\,|\\, a^2 = 1 \\bigr \\rbrace \\bigr |={\\left\\lbrace \\begin{array}{ll}\\left(\\prod _{i < j} \\gamma _{ij} !\\right)\\,\\prod _{i} \\bigl | \\mathrm {STab} (\\gamma _{ii}) \\bigr |& (^{\\sf T}= )\\\\0 & (^{\\sf T}\\ne ),\\end{array}\\right.",
"}$ where $= [\\gamma _{ij}]$ , $\\gamma _{ij} = \\bigl |A_i \\cap b A_j \\bigl |$ and $A_i$ is as in (REF ).",
"Proof.",
"By Theorem REF and Theorem REF , the left-hand side of (REF ) equals $&\\sum _{= [\\lambda _{ij}]\\in {\\mathcal {P}}()}\\delta _{, ^{\\sf T}}\\, \\delta _{, ^{\\sf T}} \\dim V(\\lambda _{11}) \\dim V(\\lambda _{12})\\cdots \\dim V(\\lambda _{\\ell \\ell })\\\\= & \\,\\,\\delta _{, ^{\\sf T}} \\Bigl ( \\prod _{i}\\sum _{\\lambda _{ii} \\in {\\mathcal {P}}(\\gamma _{ii}) } \\bigl |{\\mathrm {STab}}(\\lambda _{ii}) \\bigl |\\Bigr )\\,\\,\\Bigl ( \\prod _{i < j}\\sum _{\\lambda _{ij} \\in {\\mathcal {P}}(\\gamma _{ij}) }\\bigl |{\\mathrm {STab}}(\\lambda _{ij}) \\bigl |^2 \\Bigr )\\\\= & \\,\\,\\delta _{, ^{\\sf T}} \\Bigl (\\prod _{i}\\, \\bigl |{\\mathrm {STab}}(\\gamma _{ii}) \\bigl |\\Bigr )\\,\\,\\Bigl ( \\prod _{i < j}\\gamma _{ij} !",
"\\Bigr )$ as desired.",
"$\\hfill \\square $ Example.",
"(1) Suppose $\\alpha = (m,m^\\prime )$ , where $m^\\prime = n-m$ .",
"Since $= [\\gamma _{ij}] \\in M_\\alpha $ satisfies $\\gamma _{11} + \\gamma _{12} = m = \\gamma _{11} + \\gamma _{21}$ , it is a symmetric matrix of the form $\\begin{bmatrix}m-k &k\\\\k & m^\\prime - k\\end{bmatrix}.$ Hence each orbital of $X \\cong {{\\mathfrak {S}}_n}/ ({{\\mathfrak {S}}}_m\\times {{\\mathfrak {S}}}_{m^\\prime })$ is symmetric.",
"When $$ corresponds to $G(A,bA)$ , $k$ equals $\\bigl |A_1 \\cap bA_2\\bigl |=\\bigl |[m]\\setminus b[m] \\bigl |$ .",
"Hence the number of involutions in $b({{\\mathfrak {S}}}_m\\times {{\\mathfrak {S}}}_{m^\\prime })$ is given by (REF ).",
"Since $\\bigl |H \\cap b H b^{-1}\\bigl |= \\bigl |G_{A,bA} \\bigl |= (m-k)!",
"(k!",
")^2 (m^\\prime -k)!$ , (REF ) follows from (REF ).",
"(2) Suppose $\\alpha = (m,1^{n-m})$ , so that ${{\\mathfrak {S}}}_\\alpha \\cong {{\\mathfrak {S}}}_m$ and $A = ([m],\\lbrace m+1 \\rbrace ,\\ldots ,\\lbrace n \\rbrace )$ .",
"Then, $\\gamma _{ij} = 0,1$ unless $(i,j) = (1,1)$ .",
"Assume $$ is symmetric.",
"Since $\\bigl | {\\mathcal {P}}(0) \\bigl |= \\bigl | {\\mathcal {P}}(1) \\bigl |= 1$ , every matrix $\\in {\\mathcal {P}}()$ is necessarily symmetric.",
"Hence $\\mathrm {FS}_2 ({\\cal I}_{A,B} (V)) = 1$ for each simple $G_{AB}$ -module $V$ .",
"We note that this result gives a characterization of “null indicator double coset\" of Schauenburg [19] Theorem 4.2.",
"Also, we note that this gives a generalization of computations of Frobenius-Schur indicators of Hopf algebra representations given by Kashina, G. Mason, S. Montgomery [11], Jedwab, S. Montgomery [8] and Timmer [20] (cf.",
"[19]).",
"Since $\\gamma _{11} = m - \\bigl |[m]\\setminus b[m]\\bigl |$ , we obtain (REF )."
],
[
"Symmetric groups II",
"Let $G = {{\\mathfrak {S}}_n}$ be the symmetric group of $n$ -letters.",
"As usual, we identify the two-point stabilizer $G_{n,n-1}$ with ${{\\mathfrak {S}}}_{n-2}$ .",
"Theorem 6.1 For each $_{n-2}$ -module $V$ , $\\mathrm {FS}_r ({\\cal I}_{n,n-1}(V))=\\sum _{2 \\le s \\le n;\\,\\,s|r}\\mathrm {FS}_r ( V|_{\\mathfrak {S}_{n-s}}),$ where $V|_{\\mathfrak {S}_{n-s}}$ denotes the restriction of $V$ to ${{\\mathfrak {S}}}_{n-s}$ .",
"Here, for convenience, we set $\\mathrm {FS}_r ( V|_{\\mathfrak {S}_{0}}) = \\dim V$ .",
"Proof.",
"Let $a$ be an element of $G[x,y;r]$ , where $G[x,y;r]$ is as in Theorem REF , $i_1 = x: = n$ and $i_2 = y := n-1$ .",
"Let $s \\ge 2$ be the smallest integer such that $a^s x = x$ .",
"It is easy to see that $s$ divides $r$ and that $s$ agrees with the size of the orbit $\\langle a \\rangle x$ .",
"Hence we have the following decomposition: $G[x,y;r]=\\coprod _{2 \\le s \\le n;\\,\\,s|r}G_s [x,y;r], \\\\G_s [x,y;r] := \\bigl \\lbrace a \\in G[x,y;r]\\,\\,\\bigl |\\,\\,\\bigl |\\langle a \\rangle x\\bigl |= s \\bigr \\rbrace .$ Suppose that $a$ belongs to $G_s [x,y;r]$ .",
"We define integers $i_3,i_4,\\ldots ,i_{s} \\in [n-2]$ by $i_{3} = a^{2} x,\\ldots ,i_s = a^{s-1} x$ .",
"Then, we have $h:=a(i_1,i_2,\\ldots ,i_{s})^{-1} \\in {{\\mathfrak {S}}}([n]\\setminus \\lbrace i_1,\\ldots ,i_s\\rbrace )$ , that is, $h$ fixes each element of $\\lbrace i_1,\\ldots ,i_s \\rbrace $ .",
"Conversely, if $i_3,\\ldots ,i_{s}$ are distinct elements of $[n-2]$ and $h$ is an element of ${{\\mathfrak {S}}}([n]\\setminus \\lbrace i_1,\\ldots ,i_s\\rbrace )$ , $a = h (i_1,\\ldots ,i_{s})$ gives an element of $G_s [x,y;r]$ .",
"Therefore, we have $G_s [x,y;r]=\\coprod _{i_3,\\ldots ,i_{s}}\\bigl \\lbrace h (i_1,\\ldots ,i_{s}) \\,\\,\\bigl |\\,\\,h \\in {{\\mathfrak {S}}}([n]\\setminus \\lbrace i_1,\\ldots ,i_s\\rbrace ) \\bigr \\rbrace ,$ where $i_3,\\ldots ,i_{s}$ run over distinct elements of $[n-2]$ .",
"By (REF ), this implies $& \\mathrm {FS}_r ({\\cal I}_{n,n-1}(V))\\\\=\\,\\, &\\frac{1}{(n-2)!",
"}\\sum _{2 \\le s \\le n;\\,\\,s|r}\\,\\sum _{i_3,\\ldots ,i_{s}} \\sum _{h}{\\mathrm {Tr}}_V ((h (i_1,\\ldots ,i_{s}))^{-r}).$ where $h$ runs over ${{\\mathfrak {S}}}([n]\\setminus \\lbrace i_1,\\ldots ,i_s\\rbrace )$ .",
"Since $h$ commutes with the permutation $(i_1,\\ldots ,i_{s}) \\in {{\\mathfrak {S}}}(\\lbrace i_1,\\ldots ,i_s \\rbrace )$ , the right-hand side of the above equality is $\\frac{1}{(n-2)!",
"}\\sum _{2 \\le s \\le n;\\,\\,s|r}\\,\\sum _{i_3,\\ldots ,i_{s}} \\sum _{h}{\\mathrm {Tr}}_V (h^{-r})\\\\= \\frac{1}{(n-2)!",
"}\\sum _{2 \\le s \\le n;\\,\\,s|r}\\frac{(n-2)!}{(n-s)!}",
"\\sum _{h\\in {{\\mathfrak {S}}}_{n-s}}{\\mathrm {Tr}}_V (h^{-r})\\\\= \\sum _{2 \\le s \\le n;\\,\\,s|r}\\mathrm {FS}_r ( V|_{\\mathfrak {S}_{n-s}}).$ $\\hfill \\square $ Theorem 6.2 (1) We have $R^r_{{{\\mathfrak {S}}_n},(n-1,n){{\\mathfrak {S}}}_{n-1}}|_{\\mathfrak {S}_{n-2}}=\\sum _{2 \\le s \\le n;\\,\\,s|r}{\\mathrm {Ind}}_{\\mathfrak {S}_{n-s}}^{\\mathfrak {S}_{n-2}}(R^r_{\\mathfrak {S}_{n-s}}),$ where ${\\mathrm {Ind}}_{\\mathfrak {S}_{n-s}}^{\\mathfrak {S}_{n-2}}(R^r_{\\mathfrak {S}_{n-s}})$ denotes the induced class function of $R^r_{\\mathfrak {S}_{n-s}}$ on ${{\\mathfrak {S}}}_{n-2}$ .",
"(2) The class function $R^r_{{{\\mathfrak {S}}_n},(n-1,n){{\\mathfrak {S}}}_{n-1}}|_{\\mathfrak {S}_{n-2}}$ is a character of a certain representation of $\\mathfrak {S}_{n-2}$ .",
"Proof.",
"Let $H$ be a finite group.",
"By (REF ) and the orthogonal relation of the irreducible characters, we have $\\mathrm {FS}_r (W) = (R^r_H | \\chi _W)_H$ for each simple $H̏$ -module $W$ .",
"Since $\\mathrm {FS}_r$ is additive, this relation also holds for every finite-dimensional $H̏$ -module $W$ .",
"Suppose $H$ is a subgroup of a finite group $G$ .",
"By Frobenius reciprocity, we have $\\mathrm {FS}_r(V \\bigl |_H) = ({\\mathrm {Ind}}^G_H (R^r_H)\\,|\\,\\chi _V)_G$ for each finite-dimensional $G̏$ -module $V$ .",
"Applying this equality to $G = {{\\mathfrak {S}}}_{n-2}$ and $H = {{\\mathfrak {S}}}_{n-s}$ together with (REF ), we find that $\\mathrm {FS}_r ({\\cal I}_{n,n-1}(V))=\\sum _{2 \\le s \\le n;\\,\\,s|r}({\\mathrm {Ind}}_{\\mathfrak {S}_{n-s}}^{\\mathfrak {S}_{n-2}}(R^r_{\\mathfrak {S}_{n-s}})\\,|\\,\\chi _V)_{{{\\mathfrak {S}}}_{n-2}}.$ Hence by Theorem REF , we get $R^r_{{{\\mathfrak {S}}_n},(n-1,n){{\\mathfrak {S}}}_{n-1}}|_{\\mathfrak {S}_{n-2}}& =\\sum _{\\mu \\in {\\mathcal {P}}_{n-2}}\\sum _{2 \\le s \\le n;\\,\\,s|r}({\\mathrm {Ind}}_{\\mathfrak {S}_{n-s}}^{\\mathfrak {S}_{n-2}}(R^r_{\\mathfrak {S}_{n-s}})\\,|\\,\\chi _\\mu )_{{{\\mathfrak {S}}}_{n-2}}\\, \\chi _\\mu \\\\& =\\sum _{2 \\le s \\le n;\\,\\,s|r}{\\mathrm {Ind}}_{\\mathfrak {S}_{n-s}}^{\\mathfrak {S}_{n-2}}(R^r_{\\mathfrak {S}_{n-s}}),$ where the last equality follows from the fact that $\\lbrace \\chi _\\mu \\,|\\,\\mu \\in {\\mathcal {P}}_{n-2}\\rbrace $ is an orthonormal basis of the space of class functions of ${{\\mathfrak {S}}}_{n-2}$ .",
"Thus we get Part (1).",
"Part (2) follows immediately from Part (1) and [16].",
"$\\hfill \\square $ Corollary 6.3 (1) For each $b \\in {{\\mathfrak {S}}_n}\\setminus {{\\mathfrak {S}}}_{n-1}$ , $\\bigl |\\bigl \\lbrace a \\in b {{\\mathfrak {S}}}_{n-1}\\,|\\, a^r = 1 \\bigr \\rbrace \\bigl |=\\sum _{2\\le s \\le n;\\, s | r}\\frac{(n-2)!}{(n-s)!",
"}\\,\\bigl |\\bigl \\lbrace a \\in {{\\mathfrak {S}}}_{n-s}\\,|\\, a^r = 1 \\bigr \\rbrace \\bigl |.$ (2) (cf.",
"[3]) The root number $R^r_{{{\\mathfrak {S}}_n}} (1)$ satisfies the recurrence relation $R^r_{{{\\mathfrak {S}}_n}} (1) =\\sum _{1\\le s \\le n;\\, s | r}\\frac{(n-1)!}{(n-s)!",
"}\\,R^r_{{{\\mathfrak {S}}}_{n-s}} (1).$ Proof.",
"Since the induced class function ${\\mathrm {Ind}}^G_H (f)$ satisfies ${\\mathrm {Ind}}^G_H (f)(1) = \\frac{|G|}{|H|} f(1)$ , (REF ) immediately follows from (REF ) when $b = (n-1,n)$ .",
"On the other hand, since ${{\\mathfrak {S}}}_{n-1} \\backslash {{\\mathfrak {S}}_n}/ {{\\mathfrak {S}}}_{n-1}= \\lbrace {{\\mathfrak {S}}}_{n-1}, {{\\mathfrak {S}}}_{n-1}(n-1,n){{\\mathfrak {S}}}_{n-1}\\rbrace $ , we have $R^r_{G, b{{\\mathfrak {S}}}_{n-1}} (1) = R^r_{G,(n-1,n) {{\\mathfrak {S}}}_{n-1}} (1)$ by (REF ).",
"This proves Part (1).",
"Part (2) follows from Part (1) and (REF ).",
"$\\hfill \\square $"
],
[
"A correspondence between bilinear pairings",
"As well as the group case, the second Frobenius-Schur indicator of an ${\\cal F} (G,X)$ -module $M$ has a close connection to invariant bilinear forms on $M$ .",
"In this section, we show it by giving a correspondence between invariant bilinear forms on ${\\cal F} (G,X)$ -modules and certain bilinear pairings on $G̏_{xy}$ -modules.",
"Let $G$ be a finite group and let $K$ be its subgroup.",
"Let $t$ be an element of $G$ such that $t^{-1} K t = K$ and $t^2 \\in K$ .",
"For a $K̏$ -module $V$ , we denote by ${}^t V =\\lbrace {}^t v\\,|\\,v\\in V\\rbrace $ a copy of $V$ with $K̏$ -action given by $k\\, {}^t v := {}^t (t^{-1} k t v)$ .",
"Let $B : V \\times {}^t V \\rightarrow be a bilinear pairing.",
"We say that $ B$ is $ K$-{\\it invariant} if$ B(k v,t w) = B(v, k-1 t w)$ for each $ k K$ and $ v,w V$.We denote by $ B(V,t)$ the set of $ K$-invariant bilinear pairings$ B : V t V .",
"For $B \\in {\\cal B}(V,t)$ , we set $B^{\\sf T}(v, {}^t w) := B(t^2 w, {}^t v)$ .",
"Since $B^{\\sf T}(kv, {}^t w) = B(t^2 w, t k t^{-1}\\, {}^t v)= B(t k^{-1} t w, {}^t v)= B^{\\sf T}(v, k^{-1}\\, {}^t w),\\\\(B^{\\sf T})^{\\sf T}(v, {}^t w) = B(t^2v, t^2\\, {}^t w) = B (v,{}^t w),$ we have $B^{\\sf T}\\in {\\cal B}(V,t)$ and $(B^{\\sf T})^{\\sf T}= B$ .",
"Similarly to [10], we have the following result.",
"Proposition 7.1 Let $V$ be a simple $K̏$ -module.",
"Then $\\mathrm {FS}_2 (V,t) \\in \\lbrace 0, \\pm 1 \\rbrace $ and $\\dim {\\cal B} (V,t) \\le 1$ .",
"Moreover, we have $\\mathrm {FS}_2 (V,t) ={\\left\\lbrace \\begin{array}{ll}1 & \\dim {\\cal B} (V,t)_+ = 1\\\\-1 & \\dim {\\cal B} (V,t)_- = 1\\\\0 & \\dim {\\cal B} (V,t) = 0,\\\\\\end{array}\\right.",
"}$ where ${\\cal B} (V,t)_\\pm := \\lbrace B \\in {\\cal B} (V,t)\\,|\\, B^{\\sf T}= \\pm B \\rbrace $ .",
"Let $M$ be a ${\\cal F} (G,X)$ -module and let $C\\!",
": M \\times M \\rightarrow be abilinear form on $ M$.",
"We say that $ C$ is $ F (G,X)$-{\\it invariant} if it is $ G$-invariant and satisfies$ C( exy , ) = C(, eyx )$ for each $ x, y X$ and $ , M$.We denote by $ B(M)$ the set of $ F (G,X)$-invariant bilinear forms.For $ C B(M)$, we define $ CTB(M)$ by$ CT(, ) = C(,)$.$ Theorem 7.2 Let $\\Omega = G(x,y)$ be a symmetric orbital and let $t$ be an element of $G$ such that $t(x,y) = (y,x)$ .",
"(1) For each $G̏_{xy}$ -module $V$ , we have a bijective correspondence ${\\mathrm {Res}}\\!",
": {\\cal B} ({\\cal I}_{x,y} (V)) \\cong {\\cal B}(V,t)$ given by ${\\mathrm {Res}}(C)(v,{}^t w) = C(1\\otimes v, t\\otimes w)$ .",
"The inverse ${\\mathrm {Ind}}$ of ${\\mathrm {Res}}$ is given by ${\\mathrm {Ind}}(B) (a\\otimes v, b \\otimes w)=\\sum _{k \\in K} \\delta _{akt, b}\\, B(v, k\\, {}^t w)$ , where $K = G_{xy}$ .",
"(2) For each $C \\in {\\cal B} ({\\cal I}_{x,y} (V))$ , we have ${\\mathrm {Res}}(C^{\\sf T}) = {\\mathrm {Res}}(C)^{\\sf T}$ .",
"(3) A pairing $B \\in {\\cal B} (V,t)$ is non-degenerate if and only if ${\\mathrm {Ind}}(B)$ is non-degenerate.",
"Proof.",
"It is straightforward to verify that ${\\mathrm {Res}}$ and ${\\mathrm {Ind}}$ give well-defined maps between ${\\cal B} ({\\cal I}_{x,y} (V))$ and ${\\cal B}(V,t)$ .",
"Also, it is easy to verify that ${\\mathrm {Res}}\\circ {\\mathrm {Ind}}= {\\mathrm {id}}$ .",
"Hence, to show Part (1), it suffices to prove that ${\\mathrm {Ind}}({\\mathrm {Res}}(C)) (a\\otimes v, b \\otimes w) = C(a\\otimes v, b \\otimes w)$ for each $a, b \\in G$ and $v, w \\in V$ .",
"By ${\\cal F} (G,X)$ -invariance, the left and right-hand sides of (REF ) is zero unless $a(x,y) = b(y,x)$ .",
"Suppose $a(x,y) = b(y,x)$ .",
"Since $k:=a^{-1} b t^{-1} \\in K$ , the left-hand side of (REF ) is ${\\mathrm {Res}}(C)(v,k\\,{}^t w)=C (1 \\otimes v, t \\otimes t^{-1} k t w)=C (1 \\otimes v, k t \\otimes w)=C(a\\otimes v, b \\otimes w).$ This proves Part (1).",
"Part (2) follows from $t^2 \\in K$ .",
"Suppose that $C = {\\mathrm {Ind}}(B) \\in {\\cal B} ({\\cal I}_{x,y} (V))$ is non-degenerate and that $v \\in V$ satisfies $B(v,{}^t w) = 0$ for every $w \\in V$ .",
"To prove the non-degeneracy of $B$ , it suffices to show that $C(1 \\otimes v, b \\otimes w) = 0$ for each $b \\in G$ and $w \\in V$ .",
"By ${\\cal F} (G,X)$ -invariance, we may assume $b (x, y) = (y,x)$ , or $b = t k$ for some $k \\in K$ .",
"Then, we obtain $C(1 \\otimes v, b \\otimes w) = B (v,{}^t (kw)) = 0$ and prove the non-degeneracy of $B$ .",
"Conversely, suppose that $B$ is non-degenerate and that $m \\in e^z_w {\\cal I}_{x,y} (V)$ satisfies $C(m, n) = 0$ for every $n \\in e^w_z {\\cal I}_{x,y} (V)$ .",
"Let $a$ be an arbitrary element of $G$ such that $a(x,y) = (z,w)$ .",
"Then, we have that $m = a \\otimes v$ and $n = at \\otimes w$ for some $v, w \\in V$ , and that $B(v,{}^t w) = C(m, n) = 0$ .",
"Hence, the non-degeneracy of $B$ implies $m = 0$ .",
"Since $C$ is ${\\cal F} (G,X)$ -invariant, this proves the non-degeneracy of $C$ .",
"$\\hfill \\square $ Corollary 7.3 Let $M$ be a simple ${\\cal F} (G,X)$ -module.",
"Then $\\mathrm {FS}_2 (M) \\in \\lbrace 0, \\pm 1 \\rbrace $ and $\\dim {\\cal B} (M) \\le 1$ .",
"Moreover, we have $\\mathrm {FS}_2 (M) ={\\left\\lbrace \\begin{array}{ll}1 & \\dim {\\cal B} (M)_+ = 1\\\\-1 & \\dim {\\cal B} (M)_- = 1\\\\0 & \\dim {\\cal B} (M) = 0,\\\\\\end{array}\\right.",
"}$ where ${\\cal B} (M)_\\pm := \\lbrace C \\in {\\cal B} (M)\\,|\\, C^{\\sf T}= \\pm C \\rbrace $ .",
"Proof.",
"Suppose $M$ is of type $\\Omega $ .",
"When $\\Omega $ is symmetric, the assertion follows immediately from Theorem REF (2), Proposition REF and Theorem REF .",
"When $\\Omega $ is not symmetric, ${\\cal B} (M) = 0$ by the definition of ${\\cal F} (G,X)$ -invariance.",
"Hence the assertion follows from Theorem REF (1).",
"$\\hfill \\square $ Proposition 7.4 Let $x, y $ and $b$ be as in Theorem REF and let $$ be $G(x,y)$ .",
"Then the following conditions are equivalent: (1) $^{\\sf T}= $ .",
"(2) $R^2_{G,bH}\\ne 0$ .",
"(3) $\\mathrm {FS}_2 (M) \\ne 0$ for some ${\\cal F} (G,X)$ -module $M$ of type $\\Omega $ .",
"Proof.",
"The equivalence of (2) and (3) follows from Theorem REF and the linear independence of the characters.",
"Since the unit $G̏_{xy}$ -module $ satisfies$ B (t)+ = 1$,the equivalence of (1) and (3) follows from Theorem \\ref {BCorresp} (1),Corollary \\ref {FS2M} and Theorem \\ref {FS2I} (1).$$$"
],
[
"Frobenius-Schur indicators of Ng and Schauenburg",
"In [15], Ng and Schauenburg have defined higher Frobenius-Schur indicators $\\nu _r (M)$ for each pivotal tensor category ${\\cal C}$ and its object $M$ .",
"In this section, we verify that $\\mathrm {FS}_r$ coincides with $\\nu _r$ when ${\\cal C}$ is the category ${}_{{\\cal F}} {\\bf mod}$ of finite-dimensional left ${\\cal F}$ -modules, where ${\\cal F}={\\cal F} (G,X)$ for some $(G,X)$ .",
"We refer to [5] for terminology for tensor categories.",
"To begin with, we give an explicit description of operations on ${}_{{\\cal F}} {\\bf mod}$ .",
"For each $M, N \\in \\mathrm {ob} {}_{{\\cal F}} {\\bf mod}$ , let $M {\\overline{\\otimes }}N$ be a subspace $M \\otimes N$ defined by $M {\\overline{\\otimes }}N :=\\Delta (1) (M\\otimes N) = \\bigoplus _{z\\in X} e_z M \\otimes e^z N$ , where $e_y = \\sum _x e^x_y$ and $e^x = \\sum _y e^x_y$ .",
"Then $M {\\overline{\\otimes }}N$ becomes an ${\\cal F}$ -module via $e^x_y\\,a \\sum _z e_z m \\otimes e^z n= \\sum _z e_z e^x a m \\otimes e^z e_y a n\\,\\,\\, (a \\in G,\\, x, y,z \\in X,\\,m \\in M, n \\in N).$ The linear span ${1}:= X̏$ of $X$ becomes an ${\\cal F}$ -module via $e^x_y\\, a \\otimes z \\mapsto \\delta _{x,az}\\delta _{y,az}az$ .",
"Moreover it becomes a unit object with respect to ${\\overline{\\otimes }}$ via $M \\cong {1}{\\overline{\\otimes }}M& =\\bigoplus _x x̏ \\otimes e^x M;\\,\\,m \\mapsto \\sum _x x \\otimes e^x m,\\nonumber \\\\M \\cong M{\\overline{\\otimes }}{1}& =\\bigoplus _x e_x M \\otimes x̏ ;\\,\\, m \\mapsto \\sum _x e_x m \\otimes x.$ The linear dual $M^*$ of $M$ has an ${\\cal F}$ -module structure, which is determined by $\\langle e^x_y\\, a\\, m^\\prime ,\\, m \\rangle =\\langle m^\\prime ,\\, a^{-1}\\, e^y_x m \\rangle \\quad (a \\in G,\\, x, y \\in X,\\,m^\\prime \\in M^*,\\, m \\in M).$ The module $M^*$ becomes a left dual object of $M$ via $&ev \\!",
":M^* {\\overline{\\otimes }}M \\rightarrow {1};\\,\\,\\sum _x e_x m^\\prime \\otimes e^x m\\mapsto \\sum _y \\langle e^y m^\\prime , m\\rangle y,\\nonumber \\\\&coev \\!",
": {1}\\rightarrow M {\\overline{\\otimes }}M^* ;\\,\\,x\\mapsto \\sum _i e^x m_i \\otimes m^i,$ where $\\lbrace m_i \\rbrace $ denotes a basis of $M$ and $\\lbrace m^i \\rbrace $ denotes its dual basis.",
"The canonical linear isomorphism $j_M \\!",
":M \\cong M^{**}$ becomes an isomorphism of ${\\cal F}$ -modules.",
"Hence ${\\cal C}={}_{{\\cal F}} {\\bf mod}$ becomes a pivotal tensor category.",
"For each $M, N \\in {\\rm ob}\\, {\\cal C}$ , we define linear maps $A_{M,N} \\!",
": {\\cal C}({1}, M {\\overline{\\otimes }}N) \\rightarrow {\\cal C}(M^*, N)$ , $T_{M,N} \\!",
": {\\cal C}(M^*,N) \\rightarrow {\\cal C}(N^*,M)$ , $E_{M,N} \\!",
": {\\cal C}({1}, M {\\overline{\\otimes }}N) \\rightarrow {\\cal C}({1}, N {\\overline{\\otimes }}M)$ by ${\\begin{matrix}A_{M,N}(f)\\!",
": M^* \\cong M^* {\\overline{\\otimes }}{1}&\\xrightarrow{}&M^* {\\overline{\\otimes }}M {\\overline{\\otimes }}N&\\xrightarrow{}& {1}{\\overline{\\otimes }}N \\cong N.\\end{matrix}}\\\\T_{M,N} (g) = j_M^{-1} \\circ g^*,\\quad E_{M,N} (f) = (A_{N,M}^{-1} \\circ T_{M,N} \\circ A_{M,N}) (f),$ respectively, where $f \\in {\\cal C}({1}, M {\\overline{\\otimes }}N)$ and $g \\in {\\cal C}(M^*,N)$ .",
"Then, the $r$ -th indicator $\\nu _r (M)$ of $M \\in \\mathrm {ob} {}_{{\\cal F}} {\\bf mod}$ is defined by $\\nu _r (M):= {\\mathrm {Tr}}(E_{M,M^{{\\overline{\\otimes }}r-1}})$ .",
"Let $M$ be a finite-dimensional vector space.",
"We say that $M$ is an ${\\cal F}$ -space if it is equipped with an associative action ${\\cal F}\\otimes M \\rightarrow M$ , that is, the corresponding linear map $\\pi _M\\!",
": {\\cal F}\\rightarrow \\mathrm {End} (M)$ satisfies $\\pi _M (\\alpha \\beta ) = \\pi _M (\\alpha ) \\pi _M (\\beta )$ $(\\alpha , \\beta \\in {\\cal F})$ .",
"Let $N$ be another ${\\cal F}$ -space.",
"Then $M \\otimes N$ becomes an ${\\cal F}$ -space via $\\pi _{M\\otimes N} (\\alpha ) = (\\pi _M \\otimes \\pi _N)(\\Delta (\\alpha ))$ $(\\alpha \\in {\\cal F})$ .",
"For each ${\\cal F}$ -space $M$ , we set $\\overline{M} := \\pi _M (1) M$ and $M^{\\cal F}:= \\pi _M ({\\textstyle \\int }) M$ .",
"Then $\\overline{M}$ becomes an ${\\cal F}$ -module.",
"Lemma 8.1 Let $M$ and $N$ be ${\\cal F}$ -spaces.",
"(1) We have $\\overline{\\overline{M} \\otimes N}=\\overline{M \\otimes N}=\\overline{M \\otimes \\overline{N}}$ .",
"(2) Let $\\varepsilon ^L$ be as in (REF ).",
"then, we have $M^{\\cal F}= \\lbrace m \\in M\\,|\\,\\alpha m = \\varepsilon ^L ( \\alpha ) m\\quad ( \\alpha \\in {\\cal F}) \\rbrace .$ (3) The twist map $\\mathrm {tw}_{M,N}\\!",
": M\\otimes N \\rightarrow N\\otimes M;$ $m\\otimes n \\mapsto n\\otimes m$ satisfies $\\mathrm {tw}_{M,N} \\circ \\pi _{M \\otimes N} ({\\textstyle \\int })=\\pi _{N \\otimes M} ({\\textstyle \\int }) \\circ \\mathrm {tw}_{M,N}$ .",
"In particular, it gives a linear isomorphism $(M \\otimes N)^{\\cal F}\\cong (N \\otimes M)^{\\cal F}$ .",
"Proof.",
"Part (1) is obvious.",
"Let $N$ be the right-hand side of (REF ).",
"By (REF ), we have $M^{\\cal F}\\subseteq N$ .",
"On the other hand, since $\\varepsilon ^L ({\\textstyle \\int }) = 1$ , we have $n = \\pi _M({\\textstyle \\int }) n \\in M^{\\cal F}$ for each $n \\in N$ .",
"Part (3) follows from $(\\mathrm {tw}_{M,N} \\circ \\pi _{M \\otimes N} ({\\textstyle \\int }))(m \\otimes n)=\\frac{1}{|G|} \\sum _{x,y,a} e^y_x a n \\otimes e^x_y a m=(\\pi _{N \\otimes M} ({\\textstyle \\int }) \\circ \\mathrm {tw}_{M,N})(m \\otimes n).$ $\\hfill \\square $ For each ${\\cal F}$ -space $M$ , there exists a linear isomorphism $\\iota _{M} \\!",
": M^{\\cal F}\\cong {\\cal C}({1}, \\overline{M})$ such that $\\iota _{M} (m) (x) =e^x m$ for each $m \\in M^{\\cal F}$ and $x \\in X$ .",
"The inverse of $\\iota _{M}$ is given by $\\iota _{M}^{-1} (f) = \\sum _{x \\in X} f(x)$ .",
"Lemma 8.2 For each ${\\cal F}$ -spaces $M$ and $N$ , the diagram ${\\begin{matrix}(M \\otimes N)^{\\cal F}&\\xrightarrow{}&(N \\otimes M)^{\\cal F}\\\\{\\scriptstyle \\iota _{M \\otimes N} }\\downarrow \\mathbox{mphantom}{\\scriptstyle \\iota _{M \\otimes N} }&& \\mathbox{mphantom}{\\scriptstyle \\iota _{N \\otimes M} }\\downarrow {\\scriptstyle \\iota _{N \\otimes M} }&&\\\\{\\cal C}({1}, \\overline{M \\otimes N}) &\\xrightarrow{}&{\\cal C}({1}, \\overline{N \\otimes M})\\end{matrix}}$ is commutative.",
"Proof.",
"Let $\\sum _i m_i \\otimes n_i$ be an element of $(M \\otimes N)^{\\cal F}$ .",
"Set $g_1 = (A_{M,N} \\circ \\iota _{M \\otimes N}) ( \\sum _i m_i \\otimes n_i )$ and $g_2 = (A_{N,M} \\circ \\iota _{N \\otimes M}) ( \\sum _i n_i \\otimes m_i )$ .",
"It is straightforward to verify that $g_1 (m^\\prime ) = \\sum _i \\langle m^\\prime ,m_i \\rangle n_i$ for each $m^\\prime \\in {\\overline{M}}^{\\,*}$ .",
"Hence $\\langle m^\\prime , T_{M,N} (g_1) (n^\\prime )\\rangle =\\sum _i \\langle n^\\prime , n_i \\rangle \\langle m^\\prime , m_i \\rangle =\\langle m^\\prime , g_2 (n^\\prime )\\rangle $ for each $m^\\prime \\in \\overline{M}^{\\, *}$ and $n^\\prime \\in \\overline{N}^{\\,*}$ .",
"This proves the assertion.",
"$\\hfill \\square $ Proposition 8.3 For each $M \\in \\mathrm {ob} {}_{{\\cal F}} {\\bf mod}$ , we have $\\mathrm {FS}_r (M) = \\nu _r (M)$ .",
"Proof.",
"Apllying Lemma REF to $N = M^{\\otimes r-1}$ and using Lemma REF (1), we obtain $\\nu _r (M) = {\\mathrm {Tr}}_{(M^{\\otimes r})^{\\cal F}} (\\mathrm {tw}_{M,M^{\\otimes r-1}})$ .",
"Since ${\\textstyle \\int }$ is an idempotent, this equals to ${\\mathrm {Tr}}\\left(\\pi _{M^{\\otimes r}} ({\\textstyle \\int }) \\circ \\mathrm {tw}_{M,M^{\\otimes r-1}} \\right)={\\mathrm {Tr}}\\left(\\pi _{M}^{\\,\\,\\,\\,\\,\\,\\otimes r} (\\Delta ^{(r)} ({\\textstyle \\int }) ) \\circ \\mathrm {tw}_{M,M^{\\otimes r-1}} \\right) $ by Lemma REF (3).",
"Hence, the assertion follows from the formula ${\\mathrm {Tr}}((f_1 \\otimes \\cdots \\otimes f_r)\\circ \\mathrm {tw}_{M,M^{\\otimes r-1}})= {\\mathrm {Tr}}(f_1 \\circ \\cdots \\circ f_r)$ , which holds for each $f_1,\\ldots , f_r \\in \\mathrm {End} (M)$ .",
"$\\hfill \\square $"
]
] | 1612.05731 |
[
[
"Inferring Gravitational Potentials from Mass Densities in Cluster-sized\n Halos"
],
[
"Abstract We use N-body simulations to quantify how the escape velocity in cluster-sized halos maps to the gravitational potential in a LambdaCDM universe.",
"Using spherical density-potential pairs and the Poisson equation, we find that the matter density inferred gravitational potential profile predicts the escape velocity profile to within a few percent accuracy for group and cluster-sized halos (10^13 < M_200 < 10^15 M_sun, with respect to the critical density).",
"The accuracy holds from just outside the core to beyond the virial radius.",
"We show the importance of explicitly incorporating a cosmological constant when inferring the potential from the Poisson equation.",
"We consider three density models and find that the Einasto and Gamma profiles provide a better joint estimate of the density and potential profiles than the Navarro, Frenk and White profile, which fails to accurately represent the escape velocity.",
"For individual halos, the 1 sigma scatter between the measured escape velocity and the density-inferred potential profile is small (<5%).",
"Finally, while the sub-halos show 15% biases in their representation of the particle velocity dispersion profile, the sub-halo escape velocity profile matches the dark matter escape velocity profile to high accuracy with no evidence for velocity bias outside 0.4r_200."
],
[
"Introduction",
"Cosmological N-body simulations are a theoretical tool to understand how gravity in a dynamical space-time governs the formation of massive objects.",
"Cluster-sized halos are recently-formed (if not still forming) objects, with sizes that reach beyond the scale of the effects from baryonic physics.",
"In the cores of clusters where the baryonic cooling time is short, localized disturbances are not yet well understood, but researchers model them using hydrodynamics and astrophysical feedback mechanisms .g., Martizzi12, Martizzi14, Pike14.",
"From these simulations we have learned that within cluster cores, baryonic physics can affect the local density and the gravitational potential and thus affect the dynamics of the tracers [14].",
"Outside cluster cores, it is only gravity and the expanding space-time which govern the potential and the dynamics.",
"Clusters grow through infall and accretion [32], [17].",
"Particles and smaller sub-halos are held, captured, or released over time as the systems grow in mass and the gravitational potential deepens.",
"Under Newtonian dynamics, the escape velocity is related to the gravitational potential of the system, $v_{esc}^2(r) = -2\\phi (r) .$ The extrema of the tracer velocities in the radius/velocity phase space define a surface which we call the escape velocity edge.",
"In other words, particles, sub-halos, semi-analytic galaxies, etc, all exist in a well-defined region of the radius/velocity ($r-v$ ) phase-space bounded by a sharp escape velocity edge.",
"By determining this escape velocity surface, one is directly measuring the projected potential profile.",
"We can then use the Poisson equation to infer the mass density profile from the potential via $\\nabla ^2\\phi (r) = 4\\pi G \\rho (r),$ where $G$ is the gravitational constant, $\\phi $ is the gravitational potential, and $\\rho $ is the matter density.",
"In practice, the escape-edge is used to estimate $\\phi $ and infer cluster masses [23], [24], [10], [12].",
"[11] (hereafter GM) find that when calibrated through an argument based on virial equilibrium, the escape velocity technique allows one to infer unbiased cluster-sized halo masses with low scatter ($\\sim 10\\%$ ) in three dimensional simulated data.",
"The GM result suggests that the actual gravitational potential is precisely traced by the escape edge.",
"Our primary goal for this paper is to test this hypothesis.",
"However, GM also showed that when using the [19] (hereafter NFW) mass profile to predict the potential profile via the Poisson equation, the masses are biased low by $\\sim 10\\%$ .",
"[25] show that the NFW potential over-predicts the numerically evaluated potential by $>$ 10%.",
"This is consistent with the mass underestimation found by GM (see their equation 6) and relates to how the caustic technique is applied within the NFW formalism.",
"[25] propose that the mass outside the cluster exerts a pull which would lower the numerical value of potential and explain the difference, but this is not a satisfactory explanation because the presence of mass would only increase the fractional difference between the numerical and the NFW-inferred potential profile.",
"Another aim of this paper is to reconcile this reported discrepancy between the expected and actual accuracy of the escape-velocity technique as a representation of the gravitational potential in cluster-sized halos.",
"When using the Poisson equation to infer the potential one needs to be concerned with the accuracy of the mass density profile.",
"Because the potential is determined from an integration over the density to well beyond the virial radius, we require that the spherically averaged cluster density profile be reasonably accurate over a wide range of scales.",
"However, even if the density profile is not entirely accurate, a steep drop-off in the density means that there is little mass in the outskirts to contribute to the deepening of the potential.",
"For instance, while many authors have shown that the NFW is a good measure of the density profile outside the core to the virial radius [3], [30], [2], it has a shallower outer profile than the Einasto profile [8].",
"The Einasto profile is also a better model of the density profile [18].",
"This motivates us to consider multiple density models when applying the Poisson equation to infer the potential.",
"In this paper, we focus on understanding the precision and accuracy of the gravitational potential as measured by the escape velocity surface for individual cluster-sized halos in N-body simulations.",
"We utilize observables that can in principal be measured, such as the density profile (e.g., through gravitational lensing) and the phase-space edge (e.g., through spectroscopic surveys).",
"We make extensive use of parametric models of the density and potential via the Poisson equation (Equation REF ).",
"We focus only on 3D information in this paper, leaving the challenges of projected measurements and experiment-specific configurations to other efforts [13], [12].",
"Figure: Left: The spherically averaged density profile of a halo from the Millennium Simulation (M 200 =6.3×10 14 _{200} = 6.3\\times 10^{14}M ⊙ _{\\odot } and r 200 =1.34_{200} = 1.34 Mpc at 200×\\times the critical density).",
"The three lines are fits to the density profile (squares) over the range 0≤r 200 ≤10 \\le r_{200} \\le 1 using Equations ,, and .",
"Middle and Right: The radius-velocity phase-space of the particles.",
"These are the radial components of the particle velocities and include the Hubble flow.",
"The lines in the middle and right panels are the predicted escape velocity profile from the Poisson equation and the fits to the density profiles (Equations ,, and ).",
"In the middle panel, the Newtonian potential is integrated to infinity.",
"In the right panel the Λ\\Lambda CDM potential is integrated to the radius r eq _{eq} where the gravitational force from the halo balances the expansion of the space in a Λ\\Lambda CDM universe.",
"No dynamical information from the particles is used in the prediction of the escape-edge in the middle and right panels."
],
[
"Theory",
"Consider a mass distribution described by a spherical profile such that the mass density $\\rho $ and the potential $\\phi $ radial profiles are related by the Poisson equation (REF ): $\\phi (r) =-{\\rm 4\\pi G}\\Big {[} \\frac{1}{r}\\int _{0}^{r}\\rho (r^{\\prime })r^{\\prime 2}dr^{\\prime } + \\int _{r}^{\\infty }\\rho (r^{\\prime })r^{\\prime }dr^{\\prime }\\Big {]}.$ Equation REF allows one to analytically calculate the potential profile $\\phi $ for spherical density models in a static universe and for isolated systems."
],
[
"Analytical Density-Potential Pairs",
"There exist analytic formulae which have been shown to fit the density profiles of halos in N-body simulations.",
"We consider the following three: the NFW profile, the Gamma profile [4], and the Einasto profile [8], [21].",
"Using equation REF , we have: (r) = 0(r/r0)(1+r/r0)2 (r) = -4G 0 (r0)2 (1+r/r0)r/r0 (r) = (3-n) M4r0rn1(r+r0)4-n (r) = GMr0-12-n[1-(rr+r0)2-n], n 2 = GMr0lnrr+r0, n=2 (r) = 0 exp [-(rr0)1/n] (r) = -GMr [ 1 - (3n,rr0(1/n))(3n) + rr0(2n,rr0(1/n))(3n)] where $\\rho _0$ or $\\rm {M}$ is the normalization, $r_0$ is the scale radius, and $n$ is the index.",
"Equations REF , REF and REF are examples of density - potential pairs which share the same values for the shape parameters in the radial profiles of both the density and the potential.",
"In other words, given a fit to the spherical density profile, one can infer the shape of the gravitational potential through these equations.",
"In Figure REF we show an example halo with M$_{200} = 6.3\\times 10^{14}$ M$_{\\odot }$ and r$_{200} = 1.34$ Mpc from the Millennium Simulation [27].",
"We use radii and masses with respect to 200$\\times $ the critical density throughout.",
"The left panel shows the spherically averaged density profile and the three model fits from Equations REF , REF , and REF .",
"The models are fit over the range $0.0 \\le r/r_{200} \\le 1$ .",
"While the models are nearly identical within $r_{200}$ , they differ significantly in the outskirts.",
"We note that this is a single halo and is meant to illustrate the model differences.",
"A statistical analysis is conducted in Section REF .",
"The middle panel shows the radius/velocity phase space of the particles within this halo.",
"We use the radial components of the velocities of each particle and include the Hubble flow in the velocities.",
"Notice that the particle edge contains a fair amount of localized structure due to infall.",
"This cluster is dynamically active.",
"The lines in the middle panel of Figure REF show the predicted escape velocity edge for the three models using the Poisson equation and the fits to the density profiles and using Equations REF , REF and REF .",
"We consider each of the three models separately and infer model parameters by minimizing the $\\chi ^2$ difference to the density profiles.",
"Notice that the naive use of the Poisson equation and equation REF over-predicts the escape edge."
],
[
"Integration Limit on $\\phi $ in a {{formula:2afb1d9a-580a-40c2-9bd7-a47e8f2ee002}} CDM universe",
"In the middle panel of Figure REF , the density-inferred escape velocities use the simple Newtonian formalism (equation REF ) and an integration radius that requires escape to infinity.",
"Equation REF ignores the added potential term from the cosmological constant, $\\Lambda $ .",
"As shown in [1], there exists a radius r$_{eq}$ where the radial inward pull from gravity balances the radial outward pull of the expanding universe.",
"This radius can be derived in a simple way using only the radial components of the tracer velocities: $r_{eq}^3 = -GM/qH^2$ , where $H$ is the Hubble parameter and $q$ is the deceleration parameter, $\\Omega _m$ /2 - $\\Omega _{\\Lambda }$ .",
"Behroozi et al.",
"also consider non-radial motion, but in this work, we focus only on the radial component for both the theory and the measured velocities.",
"We now revisit the limits of the integral on Equation REF and require tracers to escape only to $r_{eq}$ .",
"We also include the $\\Lambda $ -term and apply the same integration limit to the effective potential (Equation REF ).",
"Following Behroozi et al.",
"(2013), the radial component of the escape velocity in an accelerating universe should be: $\\Phi = \\frac{v^2_{esc}}{2} = -(\\phi (r) - \\phi (r_{eq})) - \\frac{qH^2}{2} (r^2 - r_{eq}^2).",
"$ A similar derivation is provided by [1] for a point sourceNote the sign difference compared to Behroozi et al.",
"(2013), where we use the classical definition such that q is a deceleration.",
"In other words, the radial component of the escape velocity is zero (relative to the cluster) at $r_{eq}$ , where a tracer is then picked up by the expanding universe.",
"Equation REF means that the escape speed from a galaxy cluster in a $\\Lambda $ CDM universe is less than the Newtonian escape speed (equation REF .)",
"We show the revised $\\Lambda $ CDM-specific escape velocity profile using Equation REF in the right panel of Figure REF .",
"Notice that both the shape and the amplitude of the predicted escape edge match the phase-space edge using the tracer particles.",
"We note that Equation REF utilizes the definition of $r_{eq}$ derived for a point mass, such that we require the entire cluster mass to be contained within $r_{eq}$ .",
"Figure: The radius-velocity phase spaces of a low mass (top- 2.4×10 14 2.4\\times 10^{14}M ⊙ _{\\odot }) and high mass (bottom- 1.0×10 15 1.0\\times 10^{15}M ⊙ _{\\odot }) cluster in the Millennium simulation.",
"The dots are particle radial positions and radial velocities.",
"The orange circles are sub-halo radial positions and radial velocities.",
"The lines are the measured escape edges for the particles (blue) or the sub-halos (orange).",
"Notice the increasing statistical bias in the sub-halo edges compared to the dark matter edges towards the core where the sampling is low.",
"The dotted vertical bar is the location of r 200 r_{200}."
],
[
"Analysis",
"In the previous section, we showed qualitatively that we can predict the potential profile, and thus the phase-space escape velocity edge profile, from the density profile alone.",
"We showed a few clusters to highlight the the theoretical expectations.",
"However, in order to quantify how well the measured escape velocity profiles match the profiles predicted from the density-inferred potential, we need to measure the phase-space escape edges.",
"We also need to conduct the analysis over a larger sample of clusters in the simulations."
],
[
"Measuring the Phase-space Edges",
"We follow [5], where the edges are defined by the minimum of the two maxima in the positive and negative velocity sectors of the radially binned phase-spaces.",
"Beyond the core (0.2h$^{-1}$ Mpc) we enforce each min/max edge to be equal to or lower than the previous edge, such that the edge profiles are monotonically decreasing.",
"This is slightly different than [5], who use an additional free parameter to limit the radial up and down variations in the edge profile in order to mitigate the effects from local structure in the phase-spaces (see Section ).",
"In Figure REF we show the measured edge for two halos of different mass.",
"The velocities in the phase-spaces are the radial components of the tracers and include the Hubble expansion with respect to the cluster.",
"We also measure the edges using the sub-halos identified within the main halos (orange circles and lines).",
"We discuss the sub-halos in Section REF ."
],
[
"Predicting the Escape Edges",
"Our next goal is to quantify how well the predictions of the $\\Lambda $ CDM potential, using Equations REF , REF , and REF and exemplified in Figure REF , compare to the measured escape velocity edge, exemplified in Figure REF .",
"We use the 100 halos from the Millennium Simulation and their particle data as described in [12], which have a uniform mass sampling from $1\\times 10^{14} \\le M_{\\odot } \\le 2\\times 10^{15}$ .",
"We work in physical units of km/s, i.e., $\\sqrt{-2\\phi }$ .",
"For the remaining analyses we follow the same procedures.",
"First, we fit the spherical radial density profiles to each of the three models and for each halo separately.",
"We then take the best-fit density model parameters to make a prediction of the escape velocity edge.",
"We then compare the predicted escape velocity profile to the measured phase-space edge.",
"We measure the accuracy using a radial average of the fractional differences between the predicted and the measured escape surfaces.",
"We do the same for the scatter, which is determined using all 100 halos.",
"In Figure REF , we show fractional differences between the model and the data.",
"We calculate errors on the median (solid line) using bootstrap re-sampling with replacement.",
"We also show the cluster-to-cluster scatter as the light and dark gray bands (67% and 90% respectively).",
"We find that all of the profiles perform well when measuring the density utilizing all particles within the range $0.3 \\le r/r_{200} \\le 1$ .",
"Beyond $r_{200}$ it is clear that the Gamma and Einasto density profiles fall off much more quickly compared to the NFW.",
"We also find that the Einasto and Gamma potential profiles perform better than the NFW when using the density profile to predict the escape edge.",
"The NFW predicts an escape edge that is biased high at a level of 10-15% out to a few times $r_{200}$ .",
"This is due to the fact that the density profile is over-estimated out to 4$\\times $ $r_{200}$ .",
"On the other hand, the Gamma and Einasto density profiles are more accurate than the NFW beyond $r_{200}$ and drop off quickly to produce potential profiles that are nearly unbiased ($\\sim $ 3% or less) out to 3$\\times $ $r_{200}$ .",
"Regardless of the model, the cluster-cluster variation (or scatter) between the predicted and measured escape edges is $< 5\\%$ percent ($< r_{200}$ ) for most clusters.",
"Figure REF also shows what happens when we move the center of the halo.",
"The solid line uses halo centers defined by the mean position and velocity of all particles within $r_{200}$ .",
"The dotted line uses the position of the central halo defined by SUBFIND [26].",
"The dashed line uses the mean position of all particles within 0.5$r_{200}$ .",
"The Einasto profile is most sensitive to the positional choice of the main halo, whereas the Gamma and NFW profiles show the least variation (the differences are hardly noticeable in Figure REF ).",
"We conclude that the joint accuracy and precision of the density and the potential depends on the choice of the parametrized density model used.",
"Both the Gamma and the Einasto profiles produce nearly unbiased density-potential pairs when compared to observables, while the Einasto profiles are most sensitive to how the halo centers are defined.",
"As noted in the Introduction, [25] attribute the lower numerically integrated potential (equation REF ) compared to the NFW potential (equation REF ) to mass outside the cluster that is not accounted for by the NFW profile shape.",
"We show that the opposite is true.",
"The NFW model density profile over predicts the true density profile outside $r_{200}$ and thus over predicts the true gravitational potential via the Poisson equation by 10-15%.",
"Figure: The fractional difference between theory and simulation observables for the NFW potential-density Poisson pair in cluster-sided halos.",
"Left shows the fractional difference between the particle density profiles and the model profile fits.",
"The median of the 100 halos is the solid line and the error bars are determined from boot-strap re-sampling of the median.",
"The dark grey band encompasses 90% of the individual halo profiles and the light grey band 67%.",
"Right shows the fractional difference between the measured v esc v_{esc} edges and the inferred Λ\\Lambda CDM potential (-2Φ\\sqrt{-2\\Phi }) based on the best model fits to the density profiles.",
"The individual profiles are determined relative to the average particle velocities and positions (solid) and only small differences appear when we use the sub-halo positions or re-define the particle mean velocity and position using only particles within 0.5 r 200 _{200} (dashed, dotted).",
"Note that the NFW model density profile over-predicts the measured density from 1≤r 200 ≤4×1 \\le r_{200} \\le 4\\times r 200 _{200}.",
"As a consequence, an over-abundance of mass is integrated into the NFW potential profile, thus inflating the expected potential compared to the measured escape velocity profile.",
"The Gamma and Einasto fits to the density profiles provide a more accurate representation of their respective potential profiles."
],
[
"Mass and Redshift Dependence",
"Next, we examine how the edge varies as a function of halo mass and redshift.",
"In this case, we use a new sub-set of the Millennium simulation with the 100 most massive halos halos smaller than $1\\times 10^{14}$ M$_{\\odot }$ .",
"The minimum mass of this new subset is $\\sim 1\\times 10^{13}$ M$_{\\odot }$ , i.e.",
"a factor of 10 smaller than the previous sample.",
"We measure the spherically averaged density profiles and infer the escape edges via the potential from Equation REF .",
"The edges are measured using the same algorithm as applied to the more massive subset studied previously.",
"We find no statistical or systematic difference between the high mass and low mass halo datasets.",
"The density profile predicts the escape edge via the Poisson equation to the same level of accuracy and precision for over two orders of magnitude in cluster halo mass.",
"We study the low mass clusters at four different simulation snapshot outputs, corresponding to $z=0, 0.25, 0.5$ , and $0.75$ .",
"We keep the 100 most massive halos in each snapshot as provided by the “millimil” subset of the Millennium data.",
"At $z > 0.75$ , the deceleration parameter goes from negative to positive.",
"As the cluster density profiles evolve with redshift, we fit the profiles separately for each halo at each redshift.",
"Instead of physical coordinates, we use radial coordinates with respect to the $r_{200}$ of each cluster for the profiles, due to the fact that the cluster sizes also evolve with redshift.",
"In Figure REF we show that within $r_{200}$ , the edge can be accurately predicted from the density and Poisson equation to z=0.75.",
"However outside the virial radius, the edge is increasingly under-predicted compared to the model, as a function of increasing redshift.",
"We find the same result when using the mean background density as opposed to the critical background density when defining the cluster masses and radii.",
"We can explain this by the growing influence of the halo gravitational potential well on the dynamics of the infall regions around the clusters.",
"Over time, the measured escape edge in the outskirts of galaxy clusters grows in amplitude to represent the predicted escape velocity defined by the potential.",
"This dynamical evolution of the escape edge in the cluster infall regions is important for studies which use the escape velocity technique to measure mass profiles to well beyond the virial radius [22], [24].",
"Figure: The variation in the fractional difference between the measured v esc v_{esc} edge and the predictions from the density profile from redshift 0 to 0.75.",
"The light (dark) grey bands represent the 67% (90%) scatter of the individual halos."
],
[
"Particles vs. Subhalos",
"Having defined the baseline accuracy and precision of the escape velocity technique for cluster-sized halos using the particles, we ask whether other tracers of cluster potential can be used.",
"We use resolved sub-halos defined for the Millennium simulation by SUBFIND [26].",
"As an example, the sub-halos for two clusters are shown as the orange circles in Figure REF .",
"There are two important issues with the sub-halos that are evident in this figure.",
"First, the sub-halos decrease in density towards the core while the particles increase in density.",
"Second, the sub-halos do not track the phase-space near $\\Delta v = 0$ within $r_{200}$ .",
"Both of these effects are the result of sub-halo destruction through gravitational interactions with the density field: galaxies would not be destroyed so easily.",
"However, by using only the sub-halos which have survived mergers as a tracer of the particle phase-space, one is weighting the velocity distribution in an unfair way with respect to both the dark matter particles as well as any realistic galaxy populations.",
"[31] review the current consensus on velocity bias in simulated halos.",
"As measured by the velocity dispersion, Wu et al.",
"find that sub-halos typically show 10-15% positive biases .",
"This is a manifestation of how the radius/velocity phase-space is sampled by the resolved sub-halos.",
"One can draw from the phase-space in any number of ways, any of which may show positive or negative biases compared to the full representation of the phase-space by the particles.",
"In the case of sub-halos, they can easily be destroyed through interactions causing a paucity of tracers with low velocities.",
"The end result is a sub-halo velocity dispersion that is biased with respect to the particles.",
"[12] showed that the virial masses and the caustic masses of halos in the Millennium simulation are biased high when using only the sub-halos (by $\\sim $ 35% and 30% respectively for well-sampled phase-spaces).",
"These biases are always a result of the velocity dispersion.",
"The virial mass is biased simply because it is directly related to the velocity dispersion [9].",
"The caustic mass is biased because the standard “caustic” technique calibrates the escape edge to the velocity dispersion [6], [11].",
"In this work, we do not calibrate the escape surface according to virial equilibrium, but we measure it directly.",
"Therefore, we can test whether the sub-halos are in fact biased tracers of the escape-edge by comparing to the particle edges.",
"First, we need to separate systematic velocity biases (i.e., along the vertical axis of the phase-space diagrams) from statistical sampling biases (i.e., along the horizontal axis).",
"While most of our halos have thousands of particles in each radial bin of the phase-space, there are only tens of sub-halos in any bin.",
"This can cause a sampling bias as a function of radius due to the small number of objects per bin.",
"This bias is purely statistical and is visible in Figure REF .",
"We can determine the level of this bias by sub-sampling from the particles to match the number of sub-halos.",
"We use 100 uniformly random sub-samples of the particles per bin per cluster.",
"We then calculate the difference between the sub-sampled edge and the full particle edge.",
"Not surprisingly, we find a statistical sampling bias that gets worse as we move into the core of the clusters and the sub-halo density relative to the particle density decreases.",
"We calculate the radial difference between the full and sub-sampled edges with respect to the particle edge as determined beyond $r_{200} = 3h^{-1}$ Mpc, well beyond the radius where sub-halo interactions are common.",
"We then apply this statistical sampling correction to the measured sub-halo escape edges.",
"We note that the sampling bias results in an escape edge that is biased low and in the opposite direction of the halo bias reported in [14], [12] and [31].",
"In Figure REF , we show the sampling corrected sub-halo velocity dispersion and edge bias determined as the fractional difference from the particles.",
"To ensure a fair comparison to the velocity dispersion, we apply the same sampling correction procedure as we did for the escape edges.",
"Notice that the velocity dispersion profile shows positive biases $\\sim $ 15%, identical to what is presented in [12].",
"However, the escape-edge based on the sub-halos is unbiased beyond $\\sim $ 0.4 h$^{-1}$ Mpc to at least $\\sim $ 2$\\times r_{200}$ .",
"Figure: The fractional sub-halo velocity bias profile with respect to the particles for the velocity dispersion and the edge.",
"In both cases, we remove the statistical bias with results from the low sampling of the sub-halo population.",
"While the sub-halos have a biased velocity dispersion with respect to the dark matter particles, the escape edge is well-constrained by the sub-halos.",
"The vertical line is the average r 200 r_{200} for the sample.",
"The gray band represents the ±3%\\pm {3}\\% scatter on how well the density-inferred potential predicts the measured escape edge from the particles.The fact that the sub-halo edge is unbiased is an important result.",
"The edge is a well-defined and sharp feature of the phase-spaces of halos in simulations and so long as the sampling is high enough, the edge will be detected regardless of how the sampling is done.",
"Gravity insists that there can be no population of tracers which exist above the escape edge.",
"We note that there can be sub-halos which momentarily live above the edge while they are escaping (see the top panel of Figure REF ), but these are rare and fleeting and do not systematically bias the edges for all halos over all radii."
],
[
"Non-Radial Escape",
"Throughout this work we focus on radial escape.",
"However, it is known that escape along tangential orbits requires more kinetic energy than radial escape [1].",
"Therefore, we investigate the escape edge as measured through the velocity vectors along the $\\theta $ and $\\phi $ directions.",
"We treat the analysis of the non-radial motion identically to the radial motion and identify the edge as described in Section REF .",
"The only difference is that the phase-spaces utilize particle velocities tangential to the sphere.",
"We take $v_{esc}$ (non-radial) as ($v_{esc}(\\theta )$ +$v_{esc}(\\phi )$ )/2.",
"In Figure REF we show the fraction of the radial versus the non-radial components of the velocity in the escape edge.",
"As expected, we see that tangential components grow with respect to the radial component with increasing radius.",
"Within the virial radius, the fractional difference is small (a few percent) growing to a $>$ 10% at a few virial radii.",
"Figure REF is also a representation of the velocity anisotropy of the particles which comprise the edge.",
"Notice that the edge is nearly isotropic, such that radial and non-radial components of the edge velocities are nearly equal."
],
[
"Summary",
"We quantified the density-inferred gravitational potential compared to the escape velocity surface for individual cluster-sized halos in N-body simulations.",
"Throughout, we utilized observables, such as the density profile (e.g., through gravitational lensing) and the phase-space edge (e.g., through spectroscopic surveys).",
"We then applied the Poisson equation on potential-density pairs to predict the potential and thus the escape velocity profile of cluster-sized halos.",
"Our main conclusions are: The upper limit on the integral over the density in the Poisson equation needs to be physically meaningful and must incorporate the added potential from the cosmological constant.",
"Specifically, we find that particles and sub-halos are escaping to the radius at which the gravitational force on a tracer is balanced by the pull of the expanding universe.",
"The Einasto and Gamma density profiles can predict the escape edge of the radius-velocity phase-space to within 3% accuracy and 5% precision from outside the core to $\\sim 3$ virial radii for low and high mass clusters.",
"Within the virial radius, this precision and accuracy holds to $z = 0.75$ .",
"The NFW profile over-predicts the halo density profile beyond $r_{200}$ and thus the potential profile at all radii by 10-15%.",
"In other words, the NFW model is not a true potential-density pair in the context of the Poisson equation.",
"The sub-halo velocity dispersion profile is biased high compared to the dark matter particles by as much as 15%.",
"However, the sub-halo escape velocities trace the dark matter escape edge to high accuracy outside the core.",
"We conclude that the Poisson equation for clusters in a $\\Lambda $ CDM universe results in a well-defined phase-space edge for the particles and sub-halos.",
"The density profile alone can be used to predict the dynamically-inferred potential of groups and clusters.",
"These results are informative and encouraging for mass estimation techniques based on the dynamical potential of clusters.",
"In this work we utilize the 3-dimensional positions and velocities of the tracers to match the radial escape velocity to its prediction.",
"However, the real universe is subject to projection effects and line-of-sight observables, both of which smear the edge.",
"Fortunately, it has been shown that stacked phase-spaces can recover this sharp phase-space caustic even for poorly sampled individual phase-spaces [13].",
"By using stacked phase-spaces and stacked weak-lensing mass profiles, one can use current data to explore the very precise agreement expected for density-inferred escape edges.",
"The methods discussed here have recently been applied to make predictions on how well phase-space edges can constrain modifications to gravity in the local Universe [28].",
"We also report an important implication regarding the wide range of halo density models discussed in the literature.",
"Previous research has focused on the inner core regions of clusters when identifying differences between universal density profiles .g., Merritt06,Diemer15.",
"Our work does not focus on which of the NFW, Gamma, or Einasto-shaped profiles are the best when measuring the density.",
"Instead, we focus on the joint recovery of the density and potential profiles.",
"The density profile beyond $r_{200}$ plays an important role in the accuracy of the predicted the phase-space edge.",
"It is possible that there is an even more accurate functional form which describes the Poisson-pair profiles of cluster-sized halos.",
"Figure: The fractional difference between the radial and non-radial components of the velocities that comprise the escape edge.",
"Tangential motion increases the escape edge compared to radial motion.",
"The light (dark) grey bands represent the 67% (90%) scatter of the individual halos."
],
[
"Acknowledgements",
"The authors want to thank Jessica Kellar and August Evrard for their helpful comments and discussion.",
"We also thank the anonymous referee who made useful suggestions to improve the manuscript.",
"This material is based upon work supported by the National Science Foundation under Grant No.",
"1311820.",
"The Millennium Simulation databases used in this paper and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory (GAVO).",
"The authors want to especially thank Gerard Lemson for his assistance and access to the particle data."
]
] | 1612.05565 |
[
[
"An Alternative Softmax Operator for Reinforcement Learning"
],
[
"Abstract A softmax operator applied to a set of values acts somewhat like the maximization function and somewhat like an average.",
"In sequential decision making, softmax is often used in settings where it is necessary to maximize utility but also to hedge against problems that arise from putting all of one's weight behind a single maximum utility decision.",
"The Boltzmann softmax operator is the most commonly used softmax operator in this setting, but we show that this operator is prone to misbehavior.",
"In this work, we study a differentiable softmax operator that, among other properties, is a non-expansion ensuring a convergent behavior in learning and planning.",
"We introduce a variant of SARSA algorithm that, by utilizing the new operator, computes a Boltzmann policy with a state-dependent temperature parameter.",
"We show that the algorithm is convergent and that it performs favorably in practice."
],
[
"Introduction",
"There is a fundamental tension in decision making between choosing the action that has highest expected utility and avoiding “starving” the other actions.",
"The issue arises in the context of the exploration–exploitation dilemma [34], non-stationary decision problems [31], and when interpreting observed decisions [3].",
"In reinforcement learning, an approach to addressing the tension is the use of softmax operators for value-function optimization, and softmax policies for action selection.",
"Examples include value-based methods such as SARSA [27] or expected SARSA [32], [36], and policy-search methods such as REINFORCE [37].",
"An ideal softmax operator is a parameterized set of operators that: has parameter settings that allow it to approximate maximization arbitrarily accurately to perform reward-seeking behavior; is a non-expansion for all parameter settings ensuring convergence to a unique fixed point; is differentiable to make it possible to improve via gradient-based optimization; and avoids the starvation of non-maximizing actions.",
"Let $\\textbf {X}= x_1,\\ldots ,x_n$ be a vector of values.",
"We define the following operators: $\\max (\\textbf {X}) = \\max _{i \\in \\lbrace 1,\\ldots ,n\\rbrace } x_i\\ ,$ $\\mbox{\\rm mean}(\\textbf {X}) = \\frac{1}{n}\\; \\sum _{i=1}^nx_i\\ ,$ $\\mbox{\\rm eps}_\\epsilon (\\textbf {X}) = \\epsilon \\; \\mbox{\\rm mean}(\\textbf {X}) + (1-\\epsilon ) \\max (\\textbf {X})\\ ,$ $\\mbox{\\rm boltz}_\\beta (\\textbf {X}) = \\frac{\\sum _{i=1}^nx_i\\ e^{\\beta x_i}}{\\sum _{i=1}^ne^{\\beta x_i}}\\ .$ The first operator, $\\max (\\textbf {X})$ , is known to be a non-expansion [18].",
"However, it is non-differentiable (Property REF ), and ignores non-maximizing selections (Property REF ).",
"The next operator, $\\mbox{\\rm mean}(\\textbf {X})$ , computes the average of its inputs.",
"It is differentiable and, like any operator that takes a fixed convex combination of its inputs, is a non-expansion.",
"However, it does not allow for maximization (Property REF ).",
"The third operator $\\mbox{\\rm eps}_\\epsilon (\\textbf {X})$ , commonly referred to as epsilon greedy [32], interpolates between $\\max $ and $\\mbox{\\rm mean}$ .",
"The operator is a non-expansion, because it is a convex combination of two non-expansion operators.",
"But it is non-differentiable (Property REF ).",
"The Boltzmann operator $\\mbox{\\rm boltz}_\\beta (\\textbf {X})$ is differentiable.",
"It also approximates $\\max $ as $\\beta \\rightarrow \\infty $ , and $\\mbox{\\rm mean}$ as $\\beta \\rightarrow 0$ .",
"However, it is not a non-expansion (Property REF ), and therefore, prone to misbehavior as will be shown in the next section.",
"In the following section, we provide a simple example illustrating why the non-expansion property is important, especially in the context of planning and on-policy learning.",
"We then present a new softmax operator that is similar to the Boltzmann operator yet is a non-expansion.",
"We prove several critical properties of this new operator, introduce a new softmax policy, and present empirical results."
],
[
"Boltzmann Misbehaves",
"We first show that $\\mbox{\\rm boltz}_\\beta $ can lead to problematic behavior.",
"To this end, we ran SARSA with Boltzmann softmax policy (Algorithm ) on the MDP shown in Figure REF .",
"The edges are labeled with a transition probability (unsigned) and a reward number (signed).",
"Also, state $s_2$ is a terminal state, so we only consider two action values, namely $\\hat{Q}(s_1,a)$ and $\\hat{Q}(s_2,b)$ .",
"Recall that the Boltzmann softmax policy assigns the following probability to each action: $\\pi (a|s)=\\frac{e^{\\beta \\hat{Q}(s,a)}}{\\sum _a e^{\\beta \\hat{Q}(s,a)}}\\ .$ Figure: A simple MDP with two states, two actions, andγ=0.98\\gamma ~=~0.98\\ .",
"The use of a Boltzmann softmax policy is not sound in this simple domain.Input: initial $\\hat{Q}(s,a)\\ \\forall s\\in \\mathcal {S}\\ \\forall a \\in \\mathcal {A}$ , $\\alpha $ , and $\\beta $ each episode Initialize $s$ $a \\sim $ Boltzmann with parameter $\\beta $ Take action $a$ , observe $r,s^\\prime $ $a^{^{\\prime }} \\sim $ Boltzmann with parameter $\\beta $ $\\hat{Q}(s,a) \\leftarrow \\hat{Q}(s,a) + \\alpha \\Big [r+\\gamma \\hat{Q}(s^{\\prime },a^{\\prime })-\\hat{Q}(s,a)\\Big ]$ $s \\leftarrow s^{^{\\prime }}, a \\leftarrow a^{^{\\prime }}$ $s$ is terminal SARSA with Boltzmann softmax policy In Figure REF , we plot state–action value estimates at the end of each episode of a single run (smoothed by averaging over ten consecutive points).",
"We set $\\alpha =.1$ and $\\beta =16.55$ .",
"The value estimates are unstable.",
"Figure: Values estimated by SARSA with Boltzmann softmax.",
"The algorithm never achieves stable values.SARSA is known to converge in the tabular setting using $\\epsilon $ -greedy exploration [18], under decreasing exploration [29], and to a region in the function-approximation setting [14].",
"There are also variants of the SARSA update rule that converge more generally [22], [2], [36].",
"However, this example is the first, to our knowledge, to show that SARSA fails to converge in the tabular setting with Boltzmann policy.",
"The next section provides background for our analysis of the example."
],
[
"Background",
"A Markov decision process [24], or MDP, is specified by the tuple $\\langle \\mathcal {S,A,R,P,\\gamma }\\rangle $ , where $\\mathcal {S}$ is the set of states and $\\mathcal {A}$ is the set of actions.",
"The functions $\\mathcal {R:S\\times A\\rightarrow }\\ \\mathbb {R}$ and $\\mathcal {P:S\\times \\ A \\times S\\rightarrow }\\ [0,1]$ denote the reward and transition dynamics of the MDP.",
"Finally, $\\gamma \\in [0,1)$ , the discount rate, determines the relative importance of immediate reward as opposed to the rewards received in the future.",
"A typical approach to finding a good policy is to estimate how good it is to be in a particular state—the state value function.",
"The value of a particular state $s$ given a policy $\\pi $ and initial action $a$ is written $Q_{\\pi }(s,a)$ .",
"We define the optimal value of a state–action pair $Q^{\\star }(s,a)= \\max _{\\pi } Q_{\\pi }(s,a) .$ It is possible to define $Q^{\\star }(s,a)$ recursively and as a function of the optimal value of the other state–action pairs: $Q^{\\star }(s,a)=\\mathcal {R}(s,a)+\\sum _{s^{\\prime }\\in \\mathcal {S}}\\gamma \\ \\mathcal {P}(s,a,s^{\\prime }) \\max _{a^{\\prime }}Q^{\\star }(s^{\\prime },a^{\\prime })\\ .$ Bellman equations, such as the above, are at the core of many reinforcement-learning algorithms such as Value Iteration [5].",
"The algorithm computes the value of the best policy in an iterative fashion: $\\hat{Q}(s,a) \\leftarrow \\mathcal {R}(s,a)+ \\gamma \\sum _{s^{\\prime }\\in \\mathcal {S}} \\mathcal {P}(s,a,s^{\\prime }) \\max _{a^{\\prime }}\\hat{Q}(s^{\\prime },a^{\\prime }) .$ Regardless of its initial value, $\\hat{Q}$ will converge to $Q^*$ .",
"[18] generalized this algorithm by replacing the $\\max $ operator by any arbitrary operator $\\bigotimes $ , resulting in the generalized value iteration (GVI) algorithm with the following update rule: $\\hat{Q}(s,a) \\leftarrow \\mathcal {R}(s,a)+\\gamma \\sum _{s^{\\prime }\\in \\mathcal {S}}\\ \\gamma \\mathcal {P}(s,a,s^{\\prime }) \\bigotimes _{a^{\\prime }} \\hat{Q}(s^{\\prime },a^{\\prime }) .",
"$ GVI algorithm Input: initial $\\hat{Q}(s,a)\\ \\forall s\\in \\mathcal {S}\\ \\forall a \\in \\mathcal {A}$ and $\\delta \\in \\mathcal {R^{+}}$ $\\textrm {diff} \\leftarrow 0$ each $s\\in \\mathcal {S}$ each $a\\in \\mathcal {A}$ $Q_{copy}\\leftarrow \\hat{Q}(s,a)$ $\\hat{Q}(s,a) \\leftarrow \\sum _{s^{\\prime }\\in \\mathcal {S}} \\mathcal {R}(s,a,s^{\\prime })$ $\\hspace{28.45274pt}+\\ \\gamma \\mathcal {P}(s,a,s^{\\prime }) \\bigotimes \\hat{Q}(s^{\\prime },.",
")$ $\\textrm {diff} \\leftarrow \\max \\big \\lbrace \\textrm {diff},|Q_{copy}-\\hat{Q}(s,a)|\\big \\rbrace $ $\\textrm {diff}<\\delta $ Crucially, convergence of GVI to a unique fixed point follows if operator $\\bigotimes $ is a non-expansion with respect to the infinity norm: $\\Big |\\bigotimes _a \\hat{Q}(s,a) - \\bigotimes _a \\hat{Q}^{\\prime }(s,a)\\Big |\\le \\max _a \\Big |\\hat{Q}(s,a) - \\hat{Q}^{\\prime }(s,a)\\Big | ,$ for any $\\hat{Q}$ , $\\hat{Q}^{\\prime }$ and $s$ .",
"Figure: max\\max is a non-expansion under the infinity norm.As mentioned earlier, the $\\max $ operator is known to be a non-expansion, as illustrated in Figure REF .",
"$\\mbox{\\rm mean}$ and $\\mbox{\\rm eps}_\\epsilon $ operators are also non-expansions.",
"Therefore, each of these operators can play the role of $\\bigotimes $ in GVI, resulting in convergence to the corresponding unique fixed point.",
"However, the Boltzmann softmax operator, $\\mbox{\\rm boltz}_\\beta $ , is not a non-expansion [19].",
"Note that we can relate GVI to SARSA by observing that SARSA's update is a stochastic implementation of GVI's update.",
"Under a Boltzmann softmax policy $\\pi $ , the target of the (expected) SARSA update is the following: ${\\operatornamewithlimits{\\mathbb {E}}_\\pi \\big [r+\\gamma \\hat{Q}(s^{\\prime },a^{\\prime })\\big |s,a\\big ]=}\\\\&&\\mathcal {R}(s,a)+\\gamma \\sum _{s^{\\prime } \\in \\mathcal {S}}\\mathcal {P}(s,a,s^{\\prime })\\underbrace{\\sum _{a^{\\prime }\\in \\mathcal {A}}\\pi (a^{\\prime }|s^{\\prime })\\hat{Q}(s^{\\prime },a^{\\prime })}_{\\mbox{\\rm boltz}_{\\beta }\\big (\\hat{Q}(s^{\\prime },\\cdot )\\big )}.$ This matches the GVI update (REF ) when $\\bigotimes = \\mbox{\\rm boltz}_\\beta $ ."
],
[
"Boltzmann Has Multiple Fixed Points",
"Although it has been known for a long time that the Boltzmann operator is not a non-expansion [19], we are not aware of a published example of an MDP for which two distinct fixed points exist.",
"The MDP presented in Figure REF is the first example where, as shown in Figure REF , GVI under $\\mbox{\\rm boltz}_\\beta $ has two distinct fixed points.",
"We also show, in Figure REF , a vector field visualizing GVI updates under $\\mbox{\\rm boltz}_{\\beta =16.55}$ .",
"The updates can move the current estimates farther from the fixed points.",
"The behavior of SARSA (Figure REF ) results from the algorithm stochastically bouncing back and forth between the two fixed points.",
"When the learning algorithm performs a sequence of noisy updates, it moves from a fixed point to the other.",
"As we will show later, planning will also progress extremely slowly near the fixed points.",
"The lack of the non-expansion property leads to multiple fixed points and ultimately a misbehavior in learning and planning.",
"Figure: Fixed points of GVI under boltz β \\mbox{\\rm boltz}_\\beta for varying β\\beta .",
"Two distinct fixed points (red and blue) co-exist for a range of β\\beta .Figure: A vector field showing GVI updates under boltz β=16.55 \\mbox{\\rm boltz}_{\\beta =16.55}.",
"Fixed points are marked in black.",
"For some points, such as the large blue point, updates can move the current estimates farther from the fixed points.",
"Also, for points that lie in between the two fixed-points, progress is extremely slow."
],
[
"Mellowmax and its Properties",
"We advocate for an alternative softmax operator defined as follows: $\\mbox{\\rm mm}_\\omega (\\textbf {X}) =\\frac{ \\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i})}{\\omega }\\ ,$ which can be viewed as a particular instantiation of the quasi-arithmetic mean [4].",
"It can also be derived from information theoretical principles as a way of regularizing policies with a cost function defined by KL divergence [35], [26], [12].",
"Note that the operator has previously been utilized in other areas, such as power engineering [28].",
"We show that $\\mbox{\\rm mm}_\\omega $ , which we refer to as mellowmax, has the desired properties and that it compares quite favorably to $\\mbox{\\rm boltz}_\\beta $ in practice."
],
[
"Mellowmax is a Non-Expansion",
"We prove that $\\mbox{\\rm mm}_\\omega $ is a non-expansion (Property REF ), and therefore, GVI and SARSA under $\\mbox{\\rm mm}_\\omega $ are guaranteed to converge to a unique fixed point.",
"Let $\\textbf {X}= x_1,\\ldots ,x_n$ and $\\textbf {Y}= y_1,\\ldots ,y_n$ be two vectors of values.",
"Let $\\Delta _i=x_i - y_i$ for $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ be the difference of the $i$ th components of the two vectors.",
"Also, let $i^*$ be the index with the maximum component-wise difference, $i^*=\\operatornamewithlimits{argmax}_i \\Delta _i$ .",
"For simplicity, we assume that $i^*$ is unique and $\\omega >0$ .",
"Also, without loss of generality, we assume that $x_{i^*} - y_{i^*} \\ge 0$ .",
"It follows that: ${\\big |\\mbox{\\rm mm}_\\omega (\\textbf {X})-\\mbox{\\rm mm}_\\omega (\\textbf {Y})\\big |}\\\\&=& \\big |\\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i})/\\omega \\ -\\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega y_i})/\\omega \\ \\big |\\\\&=& \\big |\\log \\frac{\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i}}{\\frac{1}{n} \\sum _{i=1}^ne^{\\omega y_i}}/\\omega \\ \\big |\\\\&=& \\big |\\log \\frac{\\sum _{i=1}^ne^{\\omega \\big (y_i+\\Delta _i\\big )}}{ \\sum _{i=1}^ne^{\\omega y_i}}/\\omega \\ \\big |\\\\&\\le &\\big | \\log \\frac{\\sum _{i=1}^ne^{\\omega \\big (y_i+\\Delta _{i^*}\\big )}}{ \\sum _{i=1}^ne^{\\omega y_i}}/\\omega \\ \\big | \\\\$ $&=& \\big |\\log \\frac{e^{\\omega \\Delta _{i^*}}\\sum _{i=1}^ne^{\\omega y_i}}{ \\sum _{i=1}^ne^{\\omega y_i}}/\\omega \\ \\big |\\\\&=&\\big |\\log (e^{\\omega \\Delta _{i^*}})/\\omega \\big |= \\big |\\Delta _{i^*}\\big |= \\max _i \\big |x_i - y_i\\big |\\ ,$ allowing us to conclude that mellowmax is a non-expansion under the infinity norm."
],
[
"Maximization",
"Mellowmax includes parameter settings that allow for maximization (Property REF ) as well as for minimization.",
"In particular, as $\\omega $ goes to infinity, $\\mbox{\\rm mm}_\\omega $ acts like $\\max $ .",
"Let $m = \\max (\\textbf {X})$ and let $W = |\\lbrace x_i = m | i \\in \\lbrace 1, \\ldots ,n\\rbrace \\rbrace |$ .",
"Note that $W \\ge 1$ is the number of maximum values (“winners”) in $\\textbf {X}$ .",
"Then: $\\lim _{\\omega \\rightarrow \\infty } \\mbox{\\rm mm}_\\omega (\\textbf {X})&=& \\lim _{\\omega \\rightarrow \\infty } \\frac{\\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i})}{\\omega } \\\\&=& \\lim _{\\omega \\rightarrow \\infty } \\frac{\\log (\\frac{1}{n} e^{\\omega m} \\sum _{i=1}^ne^{\\omega (x_i-m)})}{\\omega }\\\\&=& \\lim _{\\omega \\rightarrow \\infty } \\frac{\\log (\\frac{1}{n} e^{\\omega m} W)}{\\omega } \\\\&=& \\lim _{\\omega \\rightarrow \\infty } \\frac{\\log (e^{\\omega m}) -\\log (n) + \\log (W)}{\\omega } \\\\&=& m + \\lim _{\\omega \\rightarrow \\infty } \\frac{-\\log (n) + \\log (W)}{\\omega } \\\\&=& m= \\max (\\textbf {X})\\ .$ That is, the operator acts more and more like pure maximization as the value of $\\omega $ is increased.",
"Conversely, as $\\omega $ goes to $-\\infty $ , the operator approaches the minimum."
],
[
"Derivatives",
"We can take the derivative of mellowmax with respect to each one of the arguments $x_i$ and for any non-zero $\\omega $ : $\\frac{\\partial \\mbox{\\rm mm}_\\omega (\\textbf {X})}{\\partial x_i}=\\frac{e^{\\omega x_i}}{\\sum _{i=1}^ne^{\\omega x_i}} \\ge 0\\ .$ Note that the operator is non-decreasing in each component of $\\textbf {X}$ .",
"Moreover, we can take the derivative of mellowmax with respect to $\\omega $ .",
"We define $n_\\omega (\\textbf {X})=\\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i})$ and $d_\\omega (\\textbf {X})=\\omega $ .",
"Then: $\\frac{\\partial n_\\omega (\\textbf {X})}{\\partial \\omega }=\\frac{\\sum _{i=1}^nx_i e^{\\omega x_i}}{\\sum _{i=1}^ne^{\\omega x_i}}\\quad \\textrm {and} \\quad \\frac{\\partial d_\\omega (\\textbf {X})}{\\partial \\omega }=1\\ ,$ and so: $\\frac{\\partial \\mbox{\\rm mm}_\\omega (\\textbf {X})}{\\partial \\omega }=\\frac{\\frac{\\partial n_\\omega (\\textbf {X})}{\\partial \\omega }d_\\omega (\\textbf {X}) - n_\\omega (\\textbf {X}) \\frac{\\partial d_\\omega (\\textbf {X})}{\\partial \\omega }}{d_\\omega (\\textbf {X})^2} \\ ,$ ensuring differentiablity of the operator (Property REF )."
],
[
"Averaging",
"Because of the division by $\\omega $ in the definition of $\\mbox{\\rm mm}_\\omega $ , the parameter $\\omega $ cannot be set to zero.",
"However, we can examine the behavior of $\\mbox{\\rm mm}_\\omega $ as $\\omega $ approaches zero and show that the operator computes an average in the limit.",
"Since both the numerator and denominator go to zero as $\\omega $ goes to zero, we will use L'Hôpital's rule and the derivative given in the previous section to derive the value in the limit: $\\lim _{\\omega \\rightarrow 0} \\mbox{\\rm mm}_\\omega (\\textbf {X})&=& \\lim _{\\omega \\rightarrow 0} \\frac{ \\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i})}{\\omega } \\\\&\\stackrel{\\text{L'Hôpital}}{=}& \\lim _{\\omega \\rightarrow 0} \\frac{\\frac{1}{n} \\sum _{i=1}^nx_i e^{\\omega x_i}}{\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i}}\\\\&=& \\frac{1}{n} \\sum _{i=1}^nx_i= \\mbox{\\rm mean}(\\textbf {X})\\ .$ That is, as $\\omega $ gets closer to zero, $\\mbox{\\rm mm}_\\omega (\\textbf {X})$ approaches the mean of the values in $\\textbf {X}$ ."
],
[
"Maximum Entropy Mellowmax Policy",
"As described, $\\mbox{\\rm mm}_\\omega $ computes a value for a list of numbers somewhere between its minimum and maximum.",
"However, it is often useful to actually provide a probability distribution over the actions such that (1) a non-zero probability mass is assigned to each action, and (2) the resulting expected value equals the computed value.",
"Such a probability distribution can then be used for action selection in algorithms such as SARSA.",
"In this section, we address the problem of identifying such a probability distribution as a maximum entropy problem—over all distributions that satisfy the properties above, pick the one that maximizes information entropy [10], [23].",
"We formally define the maximum entropy mellowmax policy of a state $s$ as: $&&\\pi _{\\rm mm}(s)=\\operatornamewithlimits{argmin}_\\pi \\sum _{a\\in \\mathcal {A}}\\pi (a|s)\\log \\big (\\pi (a|s)\\big ) \\\\&&\\textrm {subject to}\\ \\Big \\lbrace \\begin{array}{l}\\sum _{a\\in \\mathcal {A}}\\pi (a|s)\\hat{Q}(s,a)=\\mbox{\\rm mm}_\\omega (\\hat{Q}(s,.))",
"\\\\\\pi (a|s) \\ge 0 \\\\\\sum _{a\\in \\mathcal {A}}\\pi (a|s)=1 \\ .",
"\\end{array}\\nonumber $ Note that this optimization problem is convex and can be solved reliably using any numerical convex optimization library.",
"One way of finding the solution, which leads to an interesting policy form, is to use the method of Lagrange multipliers.",
"Here, the Lagrangian is: $&&L(\\pi ,\\lambda _1,\\lambda _2)=\\sum _{a\\in \\mathcal {A}}\\pi (a|s)\\log \\big (\\pi (a|s)\\big )\\\\ &&-\\lambda _1\\big (\\sum _{a\\in \\mathcal {A}}\\pi (a|s)-1\\big )\\\\&&- \\lambda _2\\Big (\\sum _{a\\in \\mathcal {A}}\\pi (a|s)\\hat{Q}(s,a)-\\mbox{\\rm mm}_\\omega \\big (\\hat{Q}(s,.",
")\\big )\\Big )\\ .\\\\$ Taking the partial derivative of the Lagrangian with respect to each $\\pi (a|s)$ and setting them to zero, we obtain: $\\frac{\\partial L}{\\partial \\pi (a|s)}=\\log \\big (\\pi (a|s)\\big )+1-\\lambda _1-\\lambda _2\\hat{Q}(s,a)=0\\quad \\forall \\ a \\in \\mathcal {A}\\ .$ These $|\\mathcal {A}|$ equations, together with the two linear constraints in (REF ), form $|\\mathcal {A}|+2$ equations to constrain the $|\\mathcal {A}|+2$ variables $\\pi (a|s)\\ \\forall a \\in \\mathcal {A}$ and the two Lagrangian multipliers $\\lambda _1$ and $\\lambda _2$ .",
"Solving this system of equations, the probability of taking an action under the maximum entropy mellowmax policy has the form: $\\pi _{mm}(a|s)=\\frac{e^{\\beta \\hat{Q}(s,a)}}{\\sum _{a\\in \\mathcal {A}}e^{\\beta \\hat{Q}(s,a)}}\\quad \\forall a \\in \\mathcal {A} \\ ,$ where $\\beta $ is a value for which: $\\sum _{a \\in \\mathcal {A}} e^{\\beta \\big (\\hat{Q}(s,a)-\\mbox{\\rm mm}_\\omega \\hat{Q}(s,.",
")\\big )}\\big (\\hat{Q}(s,a)-\\mbox{\\rm mm}_\\omega \\hat{Q}(s,.",
")\\big )=0\\ .$ The argument for the existence of a unique root is simple.",
"As $\\beta \\rightarrow \\infty $ the term corresponding to the best action dominates, and so, the function is positive.",
"Conversely, as $\\beta \\rightarrow -\\infty $ the term corresponding to the action with lowest utility dominates, and so the function is negative.",
"Finally, by taking the derivative, it is clear that the function is monotonically increasing, allowing us to conclude that there exists only a single root.",
"Therefore, we can find $\\beta $ easily using any root-finding algorithm.",
"In particular, we use Brent's method [7] available in the Numpy library of Python.",
"This policy has the same form as Boltzmann softmax, but with a parameter $\\beta $ whose value depends indirectly on $\\omega $ .",
"This mathematical form arose not from the structure of $\\mbox{\\rm mm}_\\omega $ , but from maximizing the entropy.",
"One way to view the use of the mellowmax operator, then, is as a form of Boltzmann policy with a temperature parameter chosen adaptively in each state to ensure that the non-expansion property holds.",
"Finally, note that the SARSA update under the maximum entropy mellowmax policy could be thought of as a stochastic implementation of the GVI update under the $\\mbox{\\rm mm}_\\omega $ operator: ${\\operatornamewithlimits{\\mathbb {E}}_{\\pi _{mm}}\\big [r+\\gamma \\hat{Q}(s^{\\prime },a^{\\prime })\\big |s,a\\big ]=}\\\\&&\\sum _{s^{\\prime } \\in \\mathcal {S}}\\mathcal {R}(s,a,s^{\\prime })+\\gamma \\mathcal {P}(s,a,s^{\\prime })\\underbrace{\\sum _{a^{\\prime }\\in \\mathcal {A}}\\pi _{mm}(a^{\\prime }|s^{\\prime })\\hat{Q}(s^{\\prime },a^{\\prime })\\big ]}_{\\mbox{\\rm mm}_{\\omega }\\big (\\hat{Q}(s^{\\prime },.",
")\\big )}$ due to the first constraint of the convex optimization problem (REF ).",
"Because mellowmax is a non-expansion, SARSA with the maximum entropy mellowmax policy is guaranteed to converge to a unique fixed point.",
"Note also that, similar to other variants of SARSA, the algorithm simply bootstraps using the value of the next state while implementing the new policy."
],
[
"Experiments on MDPs",
"We observed that in practice computing mellowmax can yield overflow if the exponentiated values are large.",
"In this case, we can safely shift the values by a constant before exponentiating them due to the following equality: $\\frac{ \\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega x_i})}{\\omega }=c+\\frac{ \\log (\\frac{1}{n} \\sum _{i=1}^ne^{\\omega (x_i-c)})}{\\omega }\\ .$ A value of $c=\\max _i x_i$ usually avoids overflow.",
"We repeat the experiment from Figure REF for mellowmax with $\\omega =16.55$ to get a vector field.",
"The result, presented in Figure REF , show a rapid and steady convergence towards the unique fixed point.",
"As a result, GVI under $\\mbox{\\rm mm}_\\omega $ can terminate significantly faster than GVI under $\\mbox{\\rm boltz}_\\beta $ , as illustrated in Figure REF .",
"Figure: GVI updates under mm ω=16.55 \\mbox{\\rm mm}_{\\omega =16.55}.",
"The fixed point is unique, and all updates move quickly toward the fixed point.Figure: Number of iterations before termination of GVI on the example MDP.",
"GVI under mm ω \\mbox{\\rm mm}_\\omega outperforms the alternatives.We present three additional experiments.",
"The first experiment investigates the behavior of GVI with the softmax operators on randomly generated MDPs.",
"The second experiment evaluates the softmax policies when used in SARSA with a tabular representation.",
"The last experiment is a policy gradient experiment where a deep neural network, with a softmax output layer, is used to directly represent the policy."
],
[
"Random MDPs",
"The example in Figure REF was created carefully by hand.",
"It is interesting to know whether such examples are likely to be encountered naturally.",
"To this end, we constructed 200 MDPs as follows: We sampled $|\\mathcal {S}|$ from $\\lbrace 2, 3, ... ,10\\rbrace $ and $|\\mathcal {A}|$ from $\\lbrace 2, 3, 4, 5\\rbrace $ uniformly at random.",
"We initialized the transition probabilities by sampling uniformly from $[0,.01]$ .",
"We then added to each entry, with probability 0.5, Gaussian noise with mean 1 and variance 0.1.",
"We next added, with probability 0.1, Gaussian noise with mean 100 and variance 1.",
"Finally, we normalized the raw values to ensure that we get a transition matrix.",
"We did a similar process for rewards, with the difference that we divided each entry by the maximum entry and multiplied by 0.5 to ensure that $R_{\\max }=0.5\\ $ .",
"We measured the failure rate of GVI under $\\mbox{\\rm boltz}_\\beta $ and $\\mbox{\\rm mm}_\\omega $ by stopping GVI when it did not terminate in 1000 iterations.",
"We also computed the average number of iterations needed before termination.",
"A summary of results is presented in the table below.",
"Mellowmax outperforms Boltzmann based on the three measures provided below.",
"Table: NO_CAPTION"
],
[
"Multi-passenger Taxi Domain",
"We evaluated SARSA on the multi-passenger taxi domain introduced by [11].",
"(See Figure REF .)",
"Figure: Multi-passenger taxi domain.",
"The discount rate γ\\gamma is 0.99.",
"Reward is +1+1 for delivering one passenger, +3+3 for two passengers, and +15+15 for three passengers.",
"Reward is zero for all the other transitions.",
"Here FF, SS, and DD denote passengers, start state, and destination respectively.One challenging aspect of this domain is that it admits many locally optimal policies.",
"Exploration needs to be set carefully to avoid either over-exploring or under-exploring the state space.",
"Note also that Boltzmann softmax performs remarkably well on this domain, outperforming sophisticated Bayesian reinforcement-learning algorithms [11].",
"Figure: Comparison on the multi-passenger taxi domain.",
"Results are shown for different values of ϵ\\epsilon , β\\beta , and ω\\omega .",
"For each setting, the learning rate is optimized.",
"Results are averaged over 25 independent runs, each consisting of 300000 time steps.As shown in Figure REF , SARSA with the epsilon-greedy policy performs poorly.",
"In fact, in our experiment, the algorithm rarely was able to deliver all the passengers.",
"However, SARSA with Boltzmann softmax and SARSA with the maximum entropy mellowmax policy achieved significantly higher average reward.",
"Maximum entropy mellowmax policy is no worse than Boltzmann softmax, here, suggesting that the greater stability does not come at the expense of less effective exploration."
],
[
"Lunar Lander Domain",
"In this section, we evaluate the use of the maximum entropy mellowmax policy in the context of a policy-gradient algorithm.",
"Specifically, we represent a policy by a neural network (discussed below) that maps from states to probabilities over actions.",
"A common choice for the activation function of the last layer is the Boltzmann softmax policy.",
"In contrast, we can use maximum entropy mellowmax policy, presented in Section , by treating the inputs of the activation function as $\\hat{Q}$ values.",
"We used the lunar lander domain, from OpenAI Gym [8] as our benchmark.",
"A screenshot of the domain is presented in Figure REF .",
"This domain has a continuous state space with 8 dimensions, namely x-y coordinates, x-y velocities, angle and angular velocities, and leg-touchdown sensors.",
"There are 4 discrete actions to control 3 engines.",
"The reward is +100 for a safe landing in the designated area, and $-100$ for a crash.",
"There is a small shaping reward for approaching the landing area.",
"Using the engines results in a negative reward.",
"An episode finishes when the spacecraft crashes or lands.",
"Solving the domain is defined as maintaining mean episode return higher than 200 in 100 consecutive episodes.",
"The policy in our experiment is represented by a neural network with a hidden layer comprised of 16 units with RELU activation functions, followed by a second layer with 16 units and softmax activation functions.",
"We used REINFORCE to train the network.",
"A batch episode size of 10 was used, as we had stability issues with smaller episode batch sizes.",
"We used the Adam algorithm [16] with $\\alpha =0.005$ and the other parameters as suggested by the paper.",
"We used Keras [9] and Theano [33] to implement the neural network architecture.",
"Figure: A screenshot of the lunar lander domain.For each softmax policy, we present in Figure REF the learning curves for different values of their free parameter.",
"We further plot average return over all 40000 episodes.",
"Mellowmax outperforms Boltzmann at its peak.",
"Figure: Comparison of Boltzmann (top) and maximum entropy mellowmax (middle) in Lunar Lander.",
"Mean return over all episodes (bottom).",
"Results are 400-run averages."
],
[
"Related Work",
"Softmax operators play an important role in sequential decision-making algorithms.",
"In model-free reinforcement learning, they can help strike a balance between exploration (mean) and exploitation (max).",
"Decision rules based on epsilon-greedy and Boltzmann softmax, while very simple, often perform surprisingly well in practice, even outperforming more advanced exploration techniques [17] that require significant approximation for complex domains.",
"When learning “on policy”, exploration steps can [27] and perhaps should [15] become part of the value-estimation process itself.",
"On-policy algorithms like SARSA can be made to converge to optimal behavior in the limit when the exploration rate and the update operator is gradually moved toward $\\max $ [29].",
"Our use of softmax in learning updates reflects this point of view and shows that the value-sensitive behavior of Boltzmann exploration can be maintained even as updates are made stable.",
"Analyses of the behavior of human subjects in choice experiments very frequently use softmax.",
"Sometimes referred to in the literature as logit choice [30], it forms an important part of the most accurate predictor of human decisions in normal-form games [38], quantal level-$k$ reasoning (QLk).",
"Softmax-based fixed points play a crucial role in this work.",
"As such, mellowmax could potentially make a good replacement.",
"Algorithms for inverse reinforcement learning (IRL), the problem of inferring reward functions from observed behavior [21], frequently use a Boltzmann operator to avoid assigning zero probability to non-optimal actions and hence assessing an observed sequence as impossible.",
"Such methods include Bayesian IRL [25], natural gradient IRL [20], and maximum likelihood IRL [1].",
"Given the recursive nature of value defined in these problems, mellowmax could be a more stable and efficient choice.",
"In linearly solvable MDPs [35], an operator similar to mellowmax emerges when using an alternative characterization for cost of action selection in MDPs.",
"Inspired by this work [12] introduced an off-policy G-learning algorithm that uses the operator to perform value-function updates.",
"Instead of performing off-policy updates, we introduced a convergent variant of SARSA with Boltzmann policy and a state-dependent temperature parameter.",
"This is in contrast to [12] where an epsilon greedy behavior policy is used."
],
[
"Conclusion and Future Work",
"We proposed the mellowmax operator as an alternative to the Boltzmann softmax operator.",
"We showed that mellowmax has several desirable properties and that it works favorably in practice.",
"Arguably, mellowmax could be used in place of Boltzmann throughout reinforcement-learning research.",
"A future direction is to analyze the fixed point of planning, reinforcement-learning, and game-playing algorithms when using the mellowmax operators.",
"In particular, an interesting analysis could be one that bounds the sub-optimality of the fixed points found by GVI.",
"An important future work is to expand the scope of our theoretical understanding to the more general function approximation setting, in which the state space or the action space is large and abstraction techniques are used.",
"Note that the importance of non-expansion in the function approximation case is well-established.",
"[13] Finally, due to the convexity of mellowmax [6], it is compelling to use it in a gradient-based algorithm in the context of sequential decision making.",
"IRL is a natural candidate given the popularity of softmax in this setting."
],
[
"Acknowledgments",
"The authors gratefully acknowledge the assistance of George D. Konidaris, as well as anonymous ICML reviewers for their outstanding feedback."
]
] | 1612.05628 |
[
[
"A Type II Supernova Hubble diagram from the CSP-I, SDSS-II, and SNLS\n surveys"
],
[
"Abstract The coming era of large photometric wide-field surveys will increase the detection rate of supernovae by orders of magnitude.",
"Such numbers will restrict spectroscopic follow-up in the vast majority of cases, and hence new methods based solely on photometric data must be developed.",
"Here, we construct a complete Hubble diagram of Type II supernovae combining data from three different samples: the Carnegie Supernova Project-I, the Sloan Digital Sky Survey-II SN, and the Supernova Legacy Survey.",
"Applying the Photometric Colour Method (PCM) to 73 Type II supernovae (SNe~II) with a redshift range of 0.01--0.5 and with no spectral information, we derive an intrinsic dispersion of 0.35 mag.",
"A comparison with the Standard Candle Method (SCM) using 61 SNe~II is also performed and an intrinsic dispersion in the Hubble diagram of 0.27 mag is derived, i.e., 13\\% in distance uncertainties.",
"Due to the lack of good statistics at higher redshifts for both methods, only weak constraints on the cosmological parameters are obtained.",
"However, assuming a flat Universe and using the PCM, we derive a Universe's matter density: $\\Omega_{m}$=0.32$^{+0.30}_{-0.21}$ providing a new independent evidence for dark energy at the level of two sigma."
],
[
"Introduction",
"One of the most important investigation in astronomy is to understand the formation and the composition of our Universe.",
"To achieve this goal is very challenging but can be done by measuring distances using astrophysical sources for which the absolute magnitude is known (aka standard candles), and using the Hubble diagram as a classical cosmological test.",
"For more than two decades, Type Ia supernovae (hereafter SNe Ia; , , and references therein) have been used as standard candles in cosmology (e.g.",
", , , ), and led to the revolutionary discovery of the accelerated expansion of the Universe driven by an unknown force attributed to dark energy , , .",
"SNe Ia cosmology today has reached a mature state in which the systematic errors dominate the overall error budget of the cosmological parameters (e.g.",
", , ) and further improvement to constrain the nature of the dark energy requires developing as many independent methods as possible.",
"One of the most interesting independent techniques to derive accurate distances and measure cosmological parameters is the use of Type II supernova (hereafter SNe II)Throughout the rest of the text we refer to SNe II as the two historical groups, SNe IIP and SNe IIL, since recent studies showed that SNe II family forms a continuous class , , .",
"Note that and , have argued for two separate populations..",
"Even if both SNe Ia and SNe II cosmology use in general the same surveys and share some systematic uncertainties like the photometric calibration, other systematic errors are different such as the redshift evolution uncertainties.",
"Furthermore, SNe II are the result of the same physical mechanism, and their progenitors are better understood than those of SNe Ia .",
"To date several methods have been developed to standardise SNe II, such as: the “Expanding Photosphere Method” (EPM) developed by the “Spectral-fitting Expanding Atmosphere Method” (SEAM, and updated in ) the “Standard Candle Method” (SCM) introduced by the “Photospheric Magnitude Method” (PMM) which is a generalisation of the SCM over various epochs and the most recent technique the “Photometric Colour Method” (PCM; ).",
"In this paper, we focus our effort on two different methods: the SCM which is the most common method used to derive SNe II distances and thus makes easier the comparison with other works, and the PCM being the only purely photometric method in the literature, i.e., which does not require observed spectra.",
"The EPM and the SEAM methods are not discussed in this paper because they require corrections factors computed from model atmospheres (, ).",
"The SCM is a powerful method based on both photometric and spectroscopic input parameters which enables a decrease of the scatter in the Hubble diagram from $\\sim $ 1 mag to levels of 0.3 mag and to derive distances with a precision of $\\sim $ 14%.",
"This method is mainly built on the correlation between the SN II luminosity and the photospheric expansion velocity 50 days post-explosion.",
"More luminous SNe II have the hydrogen recombination front at a larger radius and thus, the velocity of the photosphere will be greater in a homologous expansion .",
"Many other works have used an updated version of the SCM where a colour correction is added in order to take into account the host-galaxy extinction.",
"All these studies , , , , , have confirmed the use of SNe II as distance indicators finding similar dispersion in the Hubble diagram (0.25-0.30 mag).",
"Recently, suggested a new method using corrected magnitudes derived only from photometry.",
"In this method, instead of using the photospheric expansion velocity, the standardisation is done using the second, shallower slope in the light curve after maximum, $s_{2}$ , which corresponds to the plateau for the SNe IIP .",
"found that more luminous SNe II have higher $s_{2}$ (steeper decline, $>$ 1.15 mag per 100 days) confirming previous studies finding that traditional SNe IIL are more luminous than SNe IIP , .",
"Using this correlation and adding a colour term, succeeded to reduce the scatter in the low-redshift SNe II Hubble diagram ($z=$ 0.01-0.04) to a level of $\\sim $ 0.4 mag ($\\pm $ 0.05 mag), which corresponds to a precision of 18% in distances.",
"A better comprehension of our Universe requires the observation of more distant SNe II.",
"Differences between the expansion histories are extremely small and distinguishing between them will require measurements extending far back in time.",
"The main purpose of the current work is to build a Hubble diagram using the SCM and the PCM as achieved in but adding higher redshift SN samples such as the Sloan Digital Sky Survey II Supernova Survey (SDSS-II SN; , ), and the Supernova Legacy Survey (SNLS; , ).",
"The paper is organised as follows.",
"In Section 2 we give a description of the data set and Section 3 describes our procedure to perform K-corrections and S-correction, together with line of sight extinction corrections from our own Milky Way.",
"Section 4 presents the Hubble diagram obtained using the PCM while in Section 5 we use the SCM.",
"In Section 6 we discuss our results and conclude with a summary in Section 7."
],
[
"Data Sample",
"In this paper, we use data from three different projects: the Carnegie Supernova Project-Ihttp://CSP-I.obs.carnegiescience.edu/ (CSP-I; ), the SDSS-II SN Surveyhttp://classic.sdss.org/supernova/aboutsupernova.html , and the Supernova Legacy Surveyhttp://cfht.hawaii.edu/SNLS/ , .",
"These three surveys all used very similar Sloan optical filters permitting a minimisation of the systematic errors.",
"Our sample is listed in Table REF ."
],
[
"Carnegie Supernova Project-I",
"The CSP-I had guaranteed access to $\\sim $ 300 nights per year between 2004-2009 on the Swope 1-m and the du Pont 2.5-m telescopes at the Las Campanas Observatory (LCO), both equipped with high-performance CCD and IR cameras and CCD spectrographs.",
"This observation time allowed the CSP-I to obtain optical-band light-curves 67 SNe II (with $z \\le 0.04$ ) with good temporal coverage and more than 500 visual-wavelength spectra for these same objects.",
"The optical photometry ($u$ , $g$ , $r$ , $i$ ) was obtained after data processing via standard reduction techniques.",
"The final magnitudes were derived relative to local sequence stars and calibrated from observations of standard stars () and are expressed in the natural photometric system of the Swope+CSP-I bands.",
"The spectra were also reduced and calibrated in a standard manner using IRAF.IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.",
"A full description can be found in , , , and .",
"From the CSP-I sample, we remove six outliers.",
"Three were described in but SN 2005hd has no clear explosion date defined, SN 2008bp is identified as an outlier by , and SN 2009au was classified at the beginning as a SNe IIn showing strong interaction.",
"Thus, the total sample used is composed of 61 SNe II."
],
[
"Sloan Digital Sky Survey-II SN Survey",
"The SDSS-II SN Survey was operated during 3-years, from September 2005 to November 2007.",
"Using the 2.5-m telescope at Apache Point Observatory in New Mexico , repeatedly imaged the same region of the sky around the Southern equatorial stripe 82 .",
"This survey observed about 80 spectroscopically confirmed core-collapse SNe but the main driver of this project was the study of SNe Ia, involving the acquisition of only one or two spectra per SNe II.",
"The images were obtained using the wide-field SDSS-II CCD camera , and the photometry was computed using the five $ugriz$ filters defined in .",
"More information about the data reduction can be found in , , and .",
"A spectroscopic follow up program was performed and uncertainties derived on the redshift measurement are about 0.0005 when the redshift is measured using the host-galaxy spectra, and about 0.005 when the SN spectral features are used.",
"The total SDSS-II SN sample is composed of 16 spectroscopically confirmed SNe II of which 15 SNe II are from , and we add one SN II (SN 2007ny) removed by in his SCM sample due to the absence of explosion date estimation.",
"We derive an explosion date using the rise model.",
"From the SDSS-II sample, we exclude SN 2007nv due to its large $i$ -band uncertainties .",
"Note that the majority of spectra were obtained soon after explosion, ans therefore they only exhibit clearly $H_{\\alpha }$ $\\lambda 6563$ and $H_{\\beta }$ $\\lambda 4861$ lines but very weak Fe2 $\\lambda 5018$ or Fe2 $\\lambda 5169$ lines which are often used for the SCM."
],
[
"Supernova Legacy Survey",
"In order to obtain a more complete Hubble diagram, we also use higher redshift SNe II from the SNLS.",
"The SNLS was designed to discover SNe and to obtain a photometric follow-up using the MegaCam imager on the 3.6-m Canada-France-Hawaii Telescope.",
"The observation strategy consisted of obtaining images of the same field every 4 nights during 5 years (between 2003 and 2008), thus, in total more than 470 nights were allocated to this project.",
"Even though the sample was designed for SNe Ia cosmology and was very successful , , , , the observation of many SNe II with 0.1$\\le $ z $\\le $ 0.5, with good explosion date constraints, and good photometric coverage allowed the use of this sample to previously construct a SN II Hubble diagram , to constrain SN II rise-times , and to derive a precise measurement of the core-collapse SN rate .",
"Photometry was obtained in four pass-bands ($g, r, i, z$ ) similar to those used by the SDSS-II and CSP-I .",
"After each run, the images are pre-processed using the Elixir pipeline and then, sky background subtraction, astrometry, and photometric correction have been performed using two different and independent pipelines.",
"The description of all the data reduction steps can be found in , , , , , and .",
"Due to the redshift of the SNe, and their faintness, spectroscopy was obtained using different large telescopes.",
"All spectra were reduced in a standard way as described in , , , , and .",
"We select only SNe II from the full photometric sample (more than 6000 objects), as achieved by particulary SNe II with spectroscopic redshift from the SN or the host, a spectroscopic classification or a good photometric classificatio (based on the Gonzalez method described in ), and a well-defined explosion date.",
"The total SNLS sample is composed of 28 SNe II, 4 of them were used in to derive the first SNe II high redshift Hubble diagram.",
"For this sample, 16 SNe II have a spectrum and could potentially be used for the SCM.",
"SN 07D2an was identified as SN 1987A-like event by and is removed from the SNLS sample.",
"As for the SDSS-II sample, the majority of the spectra do not show clear Fe2 $\\lambda 5018$ or Fe2 $\\lambda 5169$ absorption lines which prevented us to measure the photospheric expansion velocities using these lines.",
"Fortunately, many SNe II also exhibit a strong $H_{\\beta }$ absorption line ($\\lambda 4861$ ) which is useful for the SCM.",
"Table: Supernovae sampleTable: Conclusions"
]
] | 1612.05636 |
[
[
"Floating zone growth of {\\alpha}-Na$_{0.90}$MnO$_2$ single crystals"
],
[
"Abstract Single crystal growth of {\\alpha}-Na$_x$MnO$_2$ (x = 0.90) is reported via the floating zone technique.",
"The conditions required for stable growth and intergrowth-free crystals are described along with the results of trials under alternate growth atmospheres.",
"Chemical and structural characterizations of the resulting {\\alpha}-Na$_{0.90}$MnO$_2$ crystals are performed using ICP-AES, NMR, XANES, XPS, and neutron diffraction measurements.",
"As a layered transition metal oxide with large ionic mobility and strong correlation effects, {\\alpha}-Na$_x$MnO$_2$ is of interest to many communities, and the implications of large volume, high purity, single crystal growth are discussed."
],
[
"Introduction",
"Two-dimensional layered transition metal oxides of the form ABO$_2$ (A=alkali metal, B=transition metal) have drawn the attention of scientists from a variety of backgrounds due to their wide array of novel electronic and functional properties.",
"For instance, in the realm of novel cathode materials, $\\alpha $ -Na$_x$ MnO$_2$ ($\\alpha $ -NMO) with the monoclinic NaNiO$_2$ structure type is widely studied as a potential Na-based cathode platform due to its superior cycling performance and operating potential [1], [2].",
"At the same time, $\\alpha $ -NMO also holds interest for researchers in the area of fundamental condensed matter physics due to its underlying anisotropic triangular lattice of Mn$^{3+}$ moments and its rich electronic phase diagram accessible via deintercalation [3].",
"While small volume crystals of the $\\alpha $ -NMO system have been produced via hydrothermal and sealed crucible techniques [4], [5], high purity floating zone (FZ) growth of large volume crystals has remained elusive.",
"This is primarily due to the challenge of dual Na and Mn volatility during growth as well as two competing polymorphic forms for NMO, namely ${\\alpha }$ -NMO and ${\\beta }$ -NaMnO$_2$ (${\\beta }$ -NMO), with very close energetics [6], [7], [8], [9].",
"The successful synthesis of large volume FZ grown crystals of the form ABO$_2$ , such as Na$_x$ CoO$_2$ [10], [11], [12], [13], have historically provided access to deeper experimental insights.",
"The current absence of high purity, large volume FZ crystals of $\\alpha $ -NMO presents an impasse to the community's understanding of this system's rich phase behavior, where recent studies have been limited to polycrystalline specimens [3], [7], [14], [15], [16], [17].",
"$\\alpha $ -NMO is composed of alternating layers of two-dimensional manganese oxide and sodium sheets as shown in Fig.",
"1.",
"It crystallizes in the monoclinic $C2/m$ spacegroup with an $O3$ layering sequence where Na ions occupy octahedrally coordinated sites between the MnO$_6$ layers.",
"MnO$_6$ octahedra within these layers form an edge sharing triangular lattice where the Mn$^{3+}$ cations undergo a large, cooperative Jahn-Teller distortion [14].",
"This results in an anisotropic triangular lattice of manganese cations in the high-spin $S=2$ state and a ($d^4$ , $t_{2g}^{3}e_{g}^{1}$ ) electronic configuration.",
"In contrast, the $\\beta $ polymorph is made up of alternating zigzag-like layers of Na and MO$_2$ sheets and has orthorhombic symmetry ($Pnmn$ space group).",
"The removal of Na from the $\\alpha $ polymorph provides a means of hole doping, which introduces Mn$^{4+}$ cations and locally relaxes the Jahn-Teller distortion in the manganese oxide planes.",
"This can lead to a rich interplay between Na ion/vacancy ordering, charge ordering, magnetic correlations, and Jahn-Teller lattice distortions as the Na-site occupancy is tuned [3].",
"While studies of this interplay in $\\alpha $ -NMO are just beginning, detailed investigations of single crystal specimens are notably lacking.",
"Successful FZ crystal growth was previously harnessed to explore and gain considerable physical insight into the structurally related Na$_x$ CoO$_2$ system [18], suggesting a similar approach for the crystal growth of $\\alpha $ -NMO.",
"Here we report the FZ growth of single crystals of $\\alpha $ -NMO with $x = 0.90$ .",
"While $\\alpha $ -NMO is prone to disorder from stacking faults (SF) and intergrowths of the competing polymorph ${\\beta }$ -NMO—both of which originate from the tendency of the structure to twin [6], [7], [8]—by tailoring the crystal growth speed we were able to mitigate this intergrowth contamination.",
"Specifically, $^{23}$ Na nuclear magnetic resonance (NMR) measurements characterizing the degree of structural faulting demonstrate that the growth rate correlates to the relative phase fractions of $\\alpha $ - and $\\beta $ - polymorphs and the number of stacking faults within the resulting crystals.",
"This fact, along with neutron powder and single crystal diffraction data, show that the optimized, large volume $\\alpha $ -NMO crystals are free of both local and long-range ${\\beta }$ -phase intergrowths with a good quality mosaic.",
"Combined x-ray absorption near edge spectroscopy (XANES), x-ray photoelectron spectroscopy (XPS), and inductively coupled plasma atomic emission spectroscopy (ICP-AES) data determine the stoichiometry of crystals grown under optimal conditions to be Na$_{0.90}$ MnO$_{2}$ .",
"Our work opens the $\\alpha $ -NMO system to new avenues of investigation via single crystal studies harnessing a variety of experimental techniques, such as neutron scattering where large volume single crystals are required.",
"Starting powders were prepared from Na$_2$ CO$_3$ and MnCO$_3$ powders (Alfa Aesar, Puratronic® 99.997$\\%$ and 99.985$\\%$ , respectively).",
"The powders were mixed with a 1:1 molar ratio, plus $10\\%$ weight excess of Na$_2$ CO$_3$ to account for sodium loss during synthesis.",
"The mixed powder was sintered in an alumina crucible at 350 $^{\\circ }$ C for 15 hours, reground and then sintered at 750 $^{\\circ }$ C for 15 hours.",
"The powder was then reground, formed into a rod with a diameter of 5 mm, and pressed at 50,000 psi in an isostatic press.",
"The pressed rod was then sintered in a vertical furnace at 1000 $^{\\circ }$ C for 15 hours and then quenched in air.",
"At this point in the synthesis process, the polycrystalline rod is comprised of a majority of $\\beta $ -NMO.",
"We note here that quenching was used as a preventative measure to avoid decomposition of the rod into mixed phases, and the effect of alternatively slow cooling the sintered feed rod is not explored here.",
"The polycrystalline sintered $\\beta $ -rod was then cut and used as both the feed rod and as a polycrystalline seed for floating zone growth in a four mirror optical floating zone furnace with 500 W halogen lamps (Crystal Systems Corp. Model FZ-T-10000-H-VI-VPO-I-HR-PC).",
"A 4:1 ratio of Ar:O$_2$ was used to pressurize the chamber to 0.15 MPa in order to help mitigate Na volatility, and gases were flowed through the growth chamber at rates of 80 SCCM and 20 SCCM for Ar and O$_2$ , respectively.",
"Once grown and cooled, crystals were immediately transferred to an Ar-filled glovebox for storage and further analysis."
],
[
"Inductively coupled plasma atomic emission spectroscopy (ICP-AES)",
"To determine sodium and manganese concentrations in $\\alpha $ -NMO crystals and polycrystalline samples, ICP-AES measurements were performed in a Thermo iCap 6300.",
"Samples for analysis were prepared by first massing the starting materials on a 0.01 mg resolution balance and then dissolving the crystals in concentrated trace metals grade hydrochloric acid (High-Purity Standards).",
"Heat was applied via a hot water bath over a hotplate to the samples in a closed container containing HCl until no particulates could be seen and the solution became clear.",
"This reduced the room temperature dissolution time in HCl from 4-10 days to only 1-3 hours.",
"Upon cooling, the dissolved sodium manganese oxide was diluted with deionized water to obtain a $5\\%$ HCl matrix.",
"Instrument calibrations for Na and Mn were done using blank, low, and high PPM solutions within a $5\\%$ HCl matrix, which were prepared using standard analysis grade solutions of Na (1000 $\\mu $ g/mL in $1\\%$ HCl) and Mn (1000 $\\mu $ g/mL in $2\\%$ HCl) from High-Purity Standards."
],
[
"$^{23}$ Na solid-state NMR (ssNMR)",
"$^{23}$ Na ssNMR spectra were acquired at room temperature on a Bruker Advance III 200 wide-bore spectrometer (4.7 T external magnetic field) at a Larmor frequency of -53.0 MHz.",
"All NMR experiments were performed under 60 kHz magic angle spinning (MAS) using a 1.3 mm double-resonance HX probe and a recycle delay of 30 ms. $^{23}$ Na NMR data were acquired on finely ground samples of single crystal NMO.",
"$^{23}$ Na NMR chemical shifts were referenced against solid $^{23}$ NaCl at 7.21 ppm.",
"$^{23}$ Na spin echo NMR spectra were acquired using a 90$^{\\circ }$ radiofrequency (RF) pulse of 1.03 $\\mu $ s and a 180$^{\\circ }$ RF pulse of 2.06 $\\mu $ s at 25.04 W. Transverse (T$_2$$^{\\prime }$ ) relaxation times were obtained from an exponential fit of the decay of the signal intensity obtained as the echo delay was increased in an NMR spin echo pulse sequence."
],
[
"X-Ray Absorption Near Edge Spectroscopy (XANES)",
"XANES data were taken at beamline 20-BM-B at the Advanced Photon Source at Argonne National Laboratory with an incident energy tuned to the Mn K-edge.",
"Single crystals of NMO were finely ground and a thin, uniform layer of powder was sealed between pieces of kapton tape under an inert environment.",
"The standards used, LiMn$_2$ O$_4$ and Mn$_2$ O$_3$ , were prepared in a similar manner.",
"Data were deglitched, calibrated, and normalized using the software Athena [19].",
"Mn foil was used as a reference, and a simultaneous spectrum of the foil was collected in transmission mode during each run of the sample and standards.",
"Calibrations to each data set were made by matching the absorption edge of the Mn foil to 6539 eV and then shifting the data set by that amount [20]."
],
[
"X-ray photoelectron spectroscopy (XPS)",
"Data were taken using a Kratos Axis Ultra X-ray Photoelectron Spectroscopy system with a pass energy of 40 eV and step size of 0.1 eV.",
"Data were analyzed using the splitting of the Mn 3$s$ peak, which is a result of the exchange coupling between 3$s$ holes and 3$d$ electrons.",
"The NMO spectrum was corrected using a Shirley background and peaks were fit to a Gaussian-Lorentzian line shape."
],
[
"Neutron diffraction measurements",
"Neutron powder diffraction data were collected using the BT-1 neutron powder diffractometer at the NIST Center for Neutron Research (NCNR).",
"A Cu(311) monochromator with a 90$^{\\circ }$ take-off angle, ${\\lambda }=1.5397(2)$ Å, and in-pile collimation of 60$^{\\prime }$ were used.",
"Data were collected over the 2${\\theta }$ range of 3-168$^{\\circ }$ with a step size of 0.05$^{\\circ }$ .",
"About 3 g of crystal from a single growth run was ground and sealed in a vanadium container of length 50 mm and diameter 9.2 mm inside a dry He-filled glovebox.",
"A fit to the data was calculated using the Le Bail refinement [21] option in FullProf [22].",
"The triple-axis instrument BT-7 [23] at NCNR was used to demonstrate the mosaic of the typical crystals using a vertically focused PG(002) monochromator and an incident energy of 14.7 meV.",
"A single crystal of ${\\sim }$ 0.5 g was aligned in the HK0 plane using open–25$^{\\prime }$ –25$^{\\prime }$ –120$^{\\prime }$ collimators placed before the monochromator, before the sample, after the sample, and before the detector, respectively.",
"Uncertainties where indicated represent one standard deviation."
],
[
"Results and Discussion",
"A number of varying growth speeds and translation rates were attempted with key results summarized in Table 1.",
"The optimal growth conditions for phase pure ${\\alpha }$ -NMO were found to be a 20 mm/hr mirror translation rate, 2 mm/hr feed rod translation rate, 30 rpm seed rod rotation, and 20 rpm feed rod rotation.",
"Under these conditions, attempts to seed from a previously grown ${\\alpha }$ -NMO crystal were unsuccessful, likely due to substantial decomposition (i.e.",
"Na loss) of the seed crystal during the initial heating process.",
"However, seeding from a polycrystalline rod was able to repeatedly nucleate a single grain crystal after ${\\approx }$ 4 cm of growth.",
"Facets form readily after the start of growth leading to the formation of a single domain within 4 cm, negating the need for a seed crystal.",
"Specifically, a stable molten zone which leads to ${\\alpha }$ -phase growth with minimal stacking faults (the determination of which is discussed later) was achieved by starting mirror translation at 50 mm/hr from the initial polycrystalline seed and then stepping it down gradually toward 20 mm/hr, where steady state growth was performed.",
"We note that our attempts at seeding growth at this eventual lower growth rate failed to maintain a stable molten zone.",
"This is illustrated in Fig.",
"2, where the boule's cross section becomes more elliptical in shape with flat facets forming perpendicular to the direction of seed translation at the point where a stable molten zone was achieved at 20 mm/hr.",
"These perpendicular facets are oriented along the (-101) lattice plane, and the crystal growth direction is along the short $b$ -axis.",
"Substantial evaporation of both Na and Mn occurred during FZ growth, and depositions composed of a mixture of Na and Mn oxides built up on the inner quartz walls of the growth chamber.",
"We found it necessary to increase the power of the lamps slightly over the course of growth ($\\approx $ 1-2%) to compensate for the decreasing transparency of the tube.",
"ICP-AES analysis of samples grown at a rate of 20 mm/hr indicated a Na:Mn ratio of 0.90:1.",
"We note here that the absolute values of the measured Na and Mn content in our samples also matched the reported ratios ($i.e.$ the measured Mn content was stoichiometric within experimental error), and that various sections of the rod were tested to check for consistency.",
"The portions of the crystals grown under the stepped down growth rates showed uniform Na content across each crystal.",
"The results are summarized as Na:Mn ratios in Table 1 for a series of representative samples as well as the starting polycrystalline feed rod.",
"As a further step, the relative fraction of Mn$^{3+}$ versus Mn$^{4+}$ was probed via XANES measurements, which when combined with ICP-AES results are capable of resolving substantial oxygen non-stoichiometry.",
"XANES data (Fig.",
"3(a)) on a 20 mm/hr grown crystal show the white-line peak position close to that of the Mn$_2$ O$_3$ standard with an oxidation state of Mn$^{3+}$ .",
"There is, however, a resolvable shift of the NMO spectrum toward the LiMn$_2$ O$_4$ standard with an average valence of Mn$^{3.5+}$ , consistent with the known Na deficiency of the sample.",
"The pre-edge region of the XANES spectra is associated with transitions from the 1$s$ states to the split $t_{2g}$ and $e_{g}$ $d$ -orbitals, resulting in varying peak shapes for Mn$^{4+}$ and Mn$^{3+}$ cations in varying local environments [24].",
"A double peak structure in this energy range is conventionally indicative of Mn$^{4+}$ , and a single broad peak is associated with Jahn-Teller distorted MnO$_{6}$ octahedra [25], [26].",
"The double peak is resolvable in the Mn$^{3.5+}$ LiMn$_2$ O$_4$ standard as shown in the inset of Fig.",
"3(a), but is not in the ${\\alpha }$ -NMO sample, again, indicating the majority of manganese in the sample is Mn$^{3+}$ .",
"We therefore performed XPS measurements in order to gain a more quantitative understanding of the manganese valence state.",
"There exists a linear relationship between the manganese oxidation state and the exchange splitting of the Mn 3$s$ peak, ${\\Delta }E_{3s}$ where $V_{Mn}=7.875-0.893{\\Delta }E_{3s}$ [27], [28].",
"Using this relation to evaluate the Mn valence for a typical ${\\alpha }$ -Na$_x$ MnO$_{2{\\pm }{\\delta }}$ crystal grown at 20 mm/hr, the data shown Fig.",
"3(b) reveal ${\\Delta }E_{3s}=5.38$ eV, which corresponds to an average Mn valence of $+3.07$ $\\pm $ $0.04$ .",
"The combined XPS and ICP-AES analysis of optimal ${\\alpha }$ -phase crystals determines the oxygen to be stoichiometric within error.",
"The lattice structure of ${\\alpha }$ -NMO crystals was verified by cutting a crystal from the end of the growth boule, crushing the crystal into powder, and then performing neutron powder diffraction.",
"Neutron powder data collected at 300 K are shown in Fig.",
"4(a) and can be fully indexed to the reported ${\\alpha }$ -NMO space group, $C2/m$ , with Le Bail refined lattice parameters $a=5.6672$ Å $\\pm $ $0.0003$ Å, $b=2.8606$ Å $\\pm $ $0.0001$ Å, $c=5.8007$ Å $\\pm $ $0.0003$ Å, and ${\\beta }=113.143^{\\circ }$ $\\pm $ $0.003^{\\circ }$ .",
"Separate single crystal neutron diffraction measurements on a crystal observed only a single grain with an observed full-with-at-half-maximum (FWHM) of $0.41^{\\circ }$ $\\pm $ $0.01^{\\circ }$ as plotted in Fig.",
"4(b), which after correction for the instrumental resolution indicates a mosaic spread of $0.35^{\\circ }$ .",
"Together these measurements establish the $long$ -$range$ ordered lattice structure of crystals grown under optimal conditions to phase pure ${\\alpha }$ -NMO single crystals; however, they are not directly sensitive to $local$ intergrowths of ${\\beta }$ -NMO which may arise as a series of stacking faults within the $O3$ layered structure.",
"To investigate the presence of local intergrowths of ${\\beta }$ -NMO and stacking faults within the lattice of ${\\alpha }$ -NMO crystals, $^{23}$ Na solid-state NMR (ssNMR) data were collected with results plotted in Fig.",
"5.",
"If a number of $^{23}$ Na resonant frequencies are resolved in the NMR spectra collected on crushed ${\\alpha }$ -NMO crystals, it suggests the presence of multiple chemical environments reflective of the formation of stacking faults (twin planes) between nanodomains of the ${\\alpha }$ and ${\\beta }$ polymorphs of NMO.",
"While structural intergrowths and the formation of stacking faults between the $\\alpha $ - and $\\beta $ -polymorphs of NMO have been reported previously [6], [7], [8], quantifying their relative abundance across a macroscopic sample presents a challenge.",
"Recently, a $^{23}$ Na NMR study of ${\\beta }$ -NMO identified three resonances with isotropic shifts of ca.",
"750, 530 and 320 ppm and assigned them to Na nuclei in ${\\alpha }$ -NMO domains, in ${\\beta }$ -NMO domains, and Na atoms in the direct vicinity of localized stacking faults, respectively [8].",
"This assignment was confirmed by recent first principles calculations of Na NMR parameters in various NMO structures containing twin planes between ${\\alpha }$ - and ${\\beta }$ -type structural domains [9].",
"In the present work, we use these assignments to quantify the proportion of Na nuclei in these three different regions within our FZ grown NMO crystals.",
"Relative fractions of Na site occupations were determined by integration of spin echo spectra shown in Fig.",
"5, and contributions from individual Na sites were scaled by a transverse relaxation factor accounting for the loss of NMR signal intensity over the signal acquisition time.",
"Fig.",
"5(b) was collected on a crystal grown at 20 mm/hr and indicates the dominance of a single Na crystallographic environment where ca.",
"$96\\%$ of Na in the sample resides in an ${\\alpha }$ -NMO environment with a small percentage (ca.",
"$4\\%$ ) of Na near stacking faults.",
"A nearly negligible fraction ($<1\\%$ ) of Na in ${\\beta }$ -like environments indicates local ${\\beta }$ -NMO regions.",
"This demonstrates that the lattice structure of crystals grown at the lowest rate of 20 mm/hr is largely free of faulting and that the local structure is consistent with the long-range ${\\alpha }$ -NMO crystal structure determined via neutron diffraction.",
"The fraction of Na$^{+}$ ions in an ${\\alpha }$ -like environment can be further broken down into Na$^{+}$ ions close to a Mn$^{4+}$ ion (ca.",
"6% of all Na) and into Na$^{+}$ ions surrounded by Mn$^{3+}$ ions only (ca.",
"90% of all Na).",
"The former environment is indicated by a Na resonance at 950 ppm (see small peak on the left of the alpha peak in Fig.",
"5(b)) which determines the proportion of Mn$^{4+}$ ions/Na vacancies to be in relatively good agreement with the total Na content obtained with XPS/ICP-AES.",
"At this time, control over the sodium content is limited to $x$ = 0.90 for quality, phase-pure samples.",
"Having established that phase pure ${\\alpha }$ -NMO single crystals can be grown via FZ, one further question explored was the degree through which the NMO polymorphs can be selected via the crystal growth rate.",
"While the ${\\beta }$ -phase of NMO is nominally the higher temperature structure [1], [2], [29], ${\\beta }$ -NMO is known to persist at ambient conditions through quenching the system into a metastable state [1], [7], [8].",
"As a result, an increased crystal pull rate can potentially be harnessed to increase the relative phase fraction of ${\\beta }$ -NMO within Na$_x$ MnO$_2$ crystals.",
"To investigate this, crystals were grown under identical conditions as the optimal ${\\alpha }$ -Na$_{0.90}$ MnO$_{2}$ crystals discussed previously with the exception of an increase in the sustained mirror translation rate to 50 mm/hr.",
"The $^{23}$ Na ssNMR spectrum collected on crystals grown under this increased rate is plotted in Fig.",
"5(a) and is dominated by the characteristic signal from Na ions in ${\\beta }$ -NMO type local environments; specifically, the majority phase fraction of the more rapidly grown sample is $66\\%$ ${\\beta }$ -NMO, which far exceeds the relative fractions of $15\\%$ ${\\alpha }$ -NMO and $19\\%$ of Na near locally faulted regions.",
"This demonstrates that the dominant growth mode has switched to the metastable ${\\beta }$ -NMO polymorph.",
"We note that this local phase mixture between ${\\alpha }$ -NMO, ${\\beta }$ -NMO, and faulted regions is consistent with the composition of powders whose long-range lattice structure is ${\\beta }$ -NMO—a lattice known to be highly defect prone [8], [9] and intermixed with regions of the competing ${\\alpha }$ -phase.",
"Due to the defect prone lattice of ${\\beta }$ -NMO, it is currently unclear whether even higher growth rates ($>$ 50 mm/hr) would result in more locally phase pure ${\\beta }$ -NMO crystals."
],
[
"Conclusions",
"Floating zone crystal growth in an optical image furnace was utilized to produce large volume, single crystals of ${\\alpha }$ -phase Na$_x$ MnO$_2$ with minimal stacking faults.",
"ICP-AES, XANES, and XPS measurements determined that crystals grown via the parameters reported here possess a $10\\%$ Na deficiency and a final stoichiometry of Na$_{0.90}$ MnO$_{2}$ .",
"Further characterization of crystals grown at slower growth rates via combined neutron diffraction and ssNMR studies determined both the long-range and local structure of these crystals to be single-phase ${\\alpha }$ -NMO.",
"By varying the crystal growth rate ($i.e.$ the mirror translation rate), the mixture of polymorphs present in Na$_{0.90}$ MnO$_{2}$ crystals can be selected/tuned—a finding of potential interest for the creation of tailored cathode materials with a tunable intermixture of ${\\alpha }$ - and ${\\beta }$ - phases.",
"Furthermore, the large volume growth of high purity ${\\alpha }$ -NMO crystals opens the compound to detailed exploration via a new array of probes such as single crystal neutron scattering and single crystal muon spin relaxation."
],
[
"Acknowledgements",
"SDW gratefully acknowledges support from the Hellman Foundation, and SDW and RD acknowledge support from ARO Award W911NF-16-1-0361.",
"VDN is supported by the University of California President's Postdoctoral Fellowship and the University of California, Santa Barbara California NanoSystems Institute Elings Prize Fellowship.",
"This work was partially supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH11231, under the Batteries for Advanced Transportation Technologies (BATT) Program subcontract No.",
"7057154 (RJC and CPG).",
"CPG and RJC thank the EU ERC for an Advanced Fellowship for CPG.",
"The MRL Shared Experimental Facilities are supported by the MRSEC Program of the NSF under Award No.",
"DMR 1121053; a member of the NSF-funded Materials Research Facilities Network (www.mrfn.org).",
"This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No.",
"DE-AC02-06CH11357.",
"Sector 20 operations are supported by the US Department of Energy and the Canadian Light Source.",
"The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology.",
"Table: Summary of growth trials using varying crystal pull rates and growth environments.",
"Compositional analyses of crystals grown under each condition as well as the polycrystalline feed material are also summarized.Figure: The structure of monoclinic α\\alpha -NaMnO 2 _2 is illustrated via projections of the (a) abab-plane and the (b) acac-plane, with the unit cell outlined in black for both figures.",
"Panel (a) shows just a single plane of the manganese cations and panel (b) shows the MnO 6 _6 polyhedra as shaded purple regions, oxygen atoms as orange spheres, and Na atoms as green spheres.Figure: As grown α\\alpha -Na 0.90 _{0.90}MnO 2 _2 crystal.",
"Stable crystal growth begins after ≈{\\approx }4 cm of translation, where there is a visible change in shape.",
"The inset shows a close-up of the flat (-101) facet that was formed during stable growth.Figure: (a) XANES data show the spectra collected for a 20 mm/hr grown α{\\alpha }-Na 0.90 _{0.90}MnO 2 _2 sample and standards at the Mn K-edge, and the inset details the pre-edge region of the spectra.",
"(b) XPS data collected on a α{\\alpha }-Na 0.90 _{0.90}MnO 2 _2 sample showing the multiplet splitting of the Mn 3ss peak, which exhibits a ΔE=5.38\\Delta E=5.38 eV, corresponding to an average valence of Mn 3.07+ ^{3.07+}.Figure: a) Neutron powder diffraction data and corresponding Le Bail refinement for a crushed single crystal of α\\alpha -Na 0.90 _{0.90}MnO 2 _2 (b) Bulk averaged mosaic of a crystal shown through the rocking curve collected at the (200) nuclear Bragg peak of a typical crystal of α\\alpha -Na 0.90 _{0.90}MnO 2 _2.",
"Solid line shows a Gaussian fit to the peak of the form I∝exp(-1 2(x-x 0 w) 2 I\\propto exp(-\\frac{1}{2}{(\\frac{x-x_0}{w})}^2) and where the FWHM=2w2ln(2)FWHM=2w\\sqrt{2ln(2)} defines the mosaic, which after taking into account the instrumental resolution defines the intrinsic mosaic spread of the crystal.Figure: 23 ^{23}Na ssNMR spectra obtained at room temperature at 4.7 T on two samples of single crystal NMO grown at different rates.",
"The peaks corresponding to Na nuclei in α\\alpha -NaMnO 2 _2 domains (Na α\\alpha ), in β\\beta -NaMnO 2 _2 domains (Na β\\beta ), and in the vicinity of a stacking fault (Na SF), are shown on the figure.",
"Spinning sidebands due to fast sample rotation are indicated by (*)."
]
] | 1612.05618 |
[
[
"A spectroscopic study of the open cluster NGC 6250"
],
[
"Abstract We present the chemical abundance analysis of 19 upper main-sequence stars of the young open cluster NGC 6250 (log t ~ 7.42 yr).",
"This work is part of a project aimed at setting observational constraints on the theory of atomic diffusion in stellar photospheres, by means of a systematic study of the abundances of the chemical elements of early F-, A- and late B-type stars of well-determined age.",
"Our data set consists of low-, medium- and high-resolution spectra obtained with the Fibre Large Array Multi Element Spectrograph (FLAMES) instrument of the ESO Very Large Telescope (VLT).",
"To perform our analysis, we have developed a new suite of software tools for the chemical abundance analysis of stellar photospheres in local thermodynamical equilibrium.",
"Together with the chemical composition of the stellar photospheres, we have provided new estimates of the cluster mean radial velocity, proper motion, refined the cluster membership, and we have given the stellar parameters including masses and fractional age.",
"We find no evidence of statistically significant correlation between any of the parameters, including abundance and cluster age, except perhaps for an increase in Ba abundance with cluster age.",
"We have proven that our new software tool may be successfully used for the chemical abundance analysis of large data sets of stellar spectra."
],
[
"Introduction",
"The spectra of early F-, A- and late B-type stars frequently show a wealth of signatures of various physical phenomena of comparable magnitude, such as, for instance, pulsation, the presence of a magnetic field and a non-homogeneous distribution of the chemical elements .",
"The latter is an effect of the diffusion of the chemical elements, a mechanism that is particularly important to study because it affects the apparent chemical composition of stars.",
"In principle, effects of the diffusion that operate at a large time-scale (i.e., comparable to the stellar lifetime) may even mimic those due to the Galactic chemical evolution.",
"Therefore, it is important to understand whether the relative chemical composition of the photosphere appear systematically different from that of younger stars.",
"In order to obtain information about time-dependent processes acting in stellar photospheres, we have chosen to study stars that are member of open clusters of various ages.",
"This is because the age of an open cluster may be determined with a much better accuracy than that of individual stars in the field, in particular when the star is in the first-half of its main-sequence lifetime .",
"A second advantage is that open cluster stars are presumably formed with the same chemical composition, so that any difference in the observed chemical composition between cluster members may be directly linked to one of the stellar properties (e.g., effective temperature and/or rotation).",
"We have considered the low- and mid-resolution spectra of 32 stars observed with the FLAMES instrument of the ESO VLT in the field of view of the open cluster NGC 6250, and we performed a detailed chemical abundance analysis of the 19 member stars.",
"Our observations are part of a data set containing the spectra of approximately 1000 stars observed as part of a larger effort to explore how various physical effects change as a function of stellar age, in particular to set observational constraints to the theory of atomic diffusion in stellar photospheres , both in the cases of magnetic and non-magnetic atmospheres.",
"The overall project includes data for potential members of various open clusters, covering ages from $\\log t= 6.8$ to 8.9 and distance moduli from 6.4 to 11.8.",
"The full list of observed open clusters is given by .",
"The analysis of three of these clusters has been performed by (NGC6405); , , and (Praesape cluster and NGC 5460).",
"found NGC 6405 to have an age of $\\log t \\sim 7.88$ , a distance of 400 pc $\\pm $ 50 pc and an [Fe/H] metallicity of 0.07 $\\pm $ 0.03.",
"The Praesape cluster has an age of $\\log t \\sim 8.85 \\pm 0.15$ and it is at a distance of 180 pc $\\pm $ 10 pc .",
"found NGC 5460 to have an age of $\\log t \\sim 8.2 \\pm 0.1$ , a distance of 720 pc $\\pm $ 50 pc and a near solar metallicity.",
"With an age of $\\log t \\sim 7.42$ yr and a distance of 865 pc , NGC 6250 is both the youngest and most distant cluster analysed as part of this project so far.",
"Further studies completed by different groups, with data that can be used as part of this study, include those by and , (Coma Berenices, $\\log t$ = 8.65; the Pleiades, $\\log t$ = 8.13; and Hyades, $\\log t$ = 8.9); and (NGC 6475, $\\log t$ = 8.48); and (IC 2391, $\\log t$ = 7.66).",
"The study by searched for trends between chemical abundance and stellar parameters of chemically peculiar Ap stars, to determine whether chemical peculiarities change as a star evolves.",
"This data will allow us to compare the behaviour of chemically peculiar magnetic stars with our sample of chemically normal stars.",
"To analyse the remaining clusters for this project in a more efficient manner, and in particular to deal with the especially interesting case of magnetic stars, we have developed sparti (SpectroPolarimetric Analysis by Radiative Transfer Inversion), a software tool based on the radiative transfer code cossam , .",
"sparti will be presented in a forthcoming paper (Martin et al., in preparation).",
"In this work, we introduce its simplest version, sparti_simple, specifically designed to deal with the non-magnetic case.",
"sparti_simple is based around cossam_simple, which in turn is a modified version of the code cossam for the spectral synthesis of magnetic atmospheres.",
"This approach has the advantage that both magnetic and non-magnetic stars may be analysed in a homogeneous way.",
"Eventually, the comparison of the chemical composition of magnetic and non-magnetic stars belonging to the same cluster will allow us a more accurate analysis of the effects of magnetic fields on the diffusion of the chemical elements in a stellar photosphere.",
"Our new software suite is fully parallelized, which reduces the CPU time required to analyse each star.",
"In this paper, we first describe cossam_simple and sparti_simple (Sections and ), then we present the observations (Section ), we establish cluster membership (Section ) and we determine the fundamental parameters of the cluster members (Section ).",
"We then present new spectroscopic observations of the cluster NGC 6250 (Section ).",
"Finally we present and discuss our results (Section ).",
"Our conclusions are summarized in Section ."
],
[
"The Cossam code",
"cossam, the `Codice per la sintesi spettrale nelle atmosfere magnetiche' is an object-oriented and fully parallelized polarized spectral line synthesis code, under GNU copyleft since the year 2000.",
"It allows the calculation of detailed Stokes IQUV spectra in the Sun and in rotating and/or pulsating stars with dipolar and quadrupolar magnetic geometries.",
"Software archaeology reveals that cossam harks back to the algol 60 code analyse 65 by and to the fortran code adrs3 by .",
"cossam is the first code of its kind that takes advantage of the sophisticated concurrent constructs of the ada programming language that make it singularly easy to parallelize the line synthesis algorithms without having recourse to message passing interfaces.",
"`Tasks', each of which has its own thread of control and each of which performs a sequence of actions – such as opacity sampling and solving the polarized radiative transfer equation over a given spectral interval – can execute concurrently within the same program on a large number of processor cores.",
"Protected objects, which do not have a thread of control of their own, are accessed in mutual exclusion (i.e.",
"only one process can update a variable at a time) and provide efficient synchronisation with very little overhead."
],
[
"Physics and numerics",
"cossam assumes a plane-parallel atmosphere and local thermodynamic equilibrium (LTE).",
"It is convenient to use the VALD data base extracting atomic transition data including radiation damping, Stark broadening and van der Waals broadening constants.",
"The atomic partition functions are calculated with the help of the appropriate routines in atlas12 .",
"Landé factors and $J$ -values for the lower and the upper energy levels provided by VALD make it possible to determine the Zeeman splitting and the individual component strengths of each line; in the case the Landé factors are missing, a classical Zeeman triplet is assumed.",
"For the continuous opacity $\\kappa _{\\rm c}$ at a given wavelength, cossam employs atlas12 routines rewritten in ada by .",
"The total line opacities required in the formal solver are determined by full opacity sampling of the $\\sigma _{-}$ , $\\sigma _{+}$ and ${\\mathbf {\\pi }}$ components separately.",
"The opacity profiles – Voigt and Faraday functions – of metallic lines are based on the rational expression found in .",
"The approximation to the hydrogen line opacity profiles given in tlusty has proved highly satisfactory and easy to implement.",
"The higher Balmer series members are treated according to ; this recipe is based on the occupation probability formalism , , .",
"By default, cossam employs the Zeeman Feautrier method , reformulated by [1] in order to treat blends in a static atmosphere.",
"Alternatively, the user can choose the somewhat faster but less accurate DELO method .",
"Since most of the CPU time is spent on opacity sampling, the overall cost of Zeeman Feautrier is only slightly higher compared to DELO.",
"In the local (`solar') case, the emerging Stokes spectrum is calculated for one given point on the solar surface – specified by the position $\\mu = \\cos \\theta =(1 - r^2)^{1/2} $ – and the attached magnetic vector.",
"In the `stellar' (disc-integrated) case, cossam has to integrate the emerging spectrum over the whole visible hemisphere, i.e.",
"over $0 < \\mu \\le 1$ , taking into account rotation, (non-)radial pulsation and a global dipolar or quadrupolar magnetic field structure.",
"Different spatial grids are provided to best cater for the different spectral line synthesis problems usually encountered.",
"We may distinguish corotating from observer-centred grids.",
"The former are extensively used in Doppler mapping (see e.g.",
"); the entire stellar surface is split into elements of approximately equal size.",
"Chemical and/or magnetic spots can easily be modelled with the help of these corotating spatial grids.",
"Observer-centred grids usually are of a fixed type where neither the magnetic field geometry nor rotation and/or pulsation determine the distribution of the quadrature points.",
"cossam also provides a third type of grid, namely an adaptive grid as discussed in and .",
"A special algorithm provides optimum 2D-integration by ensuring that the change in the monochromatic opacity matrix between two adjacent quadrature points does not exceed a certain percentage.",
"The point distribution can become very non-uniform, depending on the direction of the magnetic field vector, its azimuth, the Doppler shifts due to rotation and/or pulsation, and on the amount of limb darkening.",
"At the same level of accuracy of the resulting Stokes profiles, it is thus possible to greatly reduce the number of quadrature points compared to fixed grids."
],
[
"In principle, the original version of cossam may naturally deal with the non-magnetic case, just by setting the magnetic field strength to zero.",
"Practically, cossam would still perform a number of time-consuming numerical computations ending into flat-zero Stokes $QUV$ profiles.",
"Therefore, the original code was modified to take advantage of the various symmetries and simplifications of the non-magnetic case, and to allow extremely fast, but nevertheless highly accurate integration of intensity profiles even in rapidly rotating stars.",
"cossam_simple calculates local spectra at high wavelength resolution at various positions $\\mu = \\cos \\theta $ ; the stellar spectrum is then derived by integration over the appropriately shifted local spectra.",
"Instead of the hundreds or even thousands of local spectra to be calculated for the general-purpose 2D-grid, a few dozen local spectra prove sufficient.",
"After calculating a synthetic spectrum, we convolve it with a Gaussian matching the instrument resolution, the wavelength sampling of the resulting spectrum is then matched to the observed wavelength grid to allow for comparison."
],
[
"Inversion Method",
"sparti_simple is an inversion code that uses cossam_simple to calculate synthetic stellar spectra, and the Levenberg–Marquardt algorithm (LMA) to find the best-fitting parameters to observed stellar spectra.",
"Its free parameters are the chemical abundances of an arbitrary number of chemical elements (including independent abundances for each ionization stage), assuming a fixed model for the stellar atmosphere (hence fixed values of effective stellar temperature $T_{\\rm eff}$ and gravity $\\log g$ )."
],
[
"Levenberg–Marquardt algorithm",
"The Levenberg–Marquardt algorithm , is a least-squares technique that combines the Gauss–Newton and gradient-descent methods.",
"It allows one to determine the minimum of a multivariate function minimizing the expression $\\chi ^2 = \\sum \\limits _{i=1}^{n} \\frac{\\left[F_{\\rm mod}\\left(\\lambda _i,{\\mathbf {x}}\\right)-F_{\\rm obs}\\left(\\lambda _i\\right)\\right]^2}{\\sigma _i^2},$ where $F_{\\rm obs}$ is the observed spectrum, $\\sigma $ is the error associated with each spectral bin $i$ , $F_{\\rm mod}$ is the synthetic spectrum, convolved with the instrument response, ${\\mathbf {x}}$ is the array of free parameters assumed in the spectral synthesis, and $n$ is the number of spectral points.",
"sparti_simple initially calculates a synthetic spectrum with the chemical abundances set to solar values from [2].",
"Initial estimate values of the projected equatorial velocity $v\\sin i$ , the radial velocity $v_{\\rm rad}$ and the microturbulence $v_{\\rm mic}$ are also required.",
"After the initial spectrum is generated, the LMA is run and convergence to the best solution is usually reached within four to eight iterations.",
"The algorithm is stopped when the following conditions are met $\\begin{aligned}\\sum \\limits _{i=1}^n [\\nabla \\chi ^2]^2 &< \\sqrt{\\epsilon }\\sum \\limits _{i=1}^n [1+f(i)]\\\\\\sum \\limits _{i=1}^n \\left[x(i) - x_0(i)\\right]^2 &<\\sqrt{\\epsilon }\\sum \\limits _{i=1}^n \\left[1 + x(i)^2\\right]\\\\\\sum \\limits _{i=1}^n \\left[F(i) - F_0(i)\\right]^2 &<\\epsilon \\sum \\limits _{i=1}^n \\left[1 + F(i)^2\\right],\\end{aligned}$ where $x_0$ is the previous parameter set, $F_0$ is the model spectrum calculated with $x_0$ and $\\epsilon = 10^{-4}$ .",
"The maximum number of iterations is set to 50, if this number is reached, we reassess the starting parameters and rerun sparti_simple.",
"To quickly check which elements may be identified in the observed spectrum, after the best-fit is found, we re-calculate a number of synthetic spectra, each of which obtained after setting to zero the abundance of a single element.",
"We compare each of these new synthetic spectra with the observed one and we check if the reduced $\\chi ^2$ has varied by more than the signal-to-noise (S/N) threshold.",
"If there is no change we consider that the element cannot be measured in the observed spectrum.",
"If $\\chi ^2$ has varied, we check the spectrum to determine whether the element has visible spectral lines."
],
[
"Abundance uncertainties",
"We estimate the uncertainties of the best-fitting parameters, including $v_{\\rm mic}$ and $v\\sin i$ by taking the square root of the diagonal values of the covariance matrix ${\\rm cov}({\\mathbf {\\widehat{x}}}) = s^2{\\mathbf {\\Delta F \\Delta F}}^T,$ where $\\mathbf {\\widehat{x}}$ is the vector of best-fit and $s = \\sqrt{\\frac{\\sum \\limits _{i=0}^n\\left[F_{\\rm mod}\\left(\\lambda _i\\right)-F_{\\rm obs}\\left(\\lambda _i\\right)\\right]^2}{m-n}},$ where $n$ is the number of wavelength points in the spectrum and $m$ the number of best-fitting parameters.",
"The error so estimated is only a lower limit since sparti_simple assumes fixed values of $T_{\\rm eff}$ and $\\log g$ .",
"Therefore, the covariance matrix does not contain information on the effects of the uncertainties of $T_{\\rm eff}$ or $\\log g$ .",
"To take these uncertainties into account, we run the inversion another four times, setting $T_{\\rm eff}= T^0_{\\rm eff} \\pm \\Delta T_{\\rm eff}$ and $\\log g= \\log g^0\\pm \\Delta \\log g$ , where $T_{\\rm eff}^0$ and $\\log g^0$ are our best estimates for $T_{\\rm eff}$ and $\\log g$ , respectively, and $\\Delta T_{\\rm eff}$ and $\\Delta \\log g$ are their errors.",
"As abundance uncertainty we finally adopt $\\sigma = \\sqrt{\\sigma _{\\rm cov}^2 + \\sigma _{T_{\\rm eff}}^2 +\\sigma _{\\log g}^2}.$ where $\\sigma _{\\rm cov}$ is the error values calculated from equation (REF ), $\\sigma _{T_{\\rm eff}}$ is half of the difference between the best-fitting values for the abundances obtained assuming $T_{\\rm eff}= T_{\\rm eff}^0 + \\Delta T_{\\rm eff}$ and $T_{\\rm eff}= T_{\\rm eff}^0 - \\Delta T_{\\rm eff}$ , and $\\sigma _{\\log g}$ half of the difference between the best-fitting values for the abundances obtained assuming $\\log g= \\log g^0 + \\Delta \\log g$ and $\\log g= \\log g^0 - \\Delta \\log g$ ."
],
[
"Test cases: the Sun, HD 32115 and 21 Peg",
"In order to check the full consistency between the results obtained with sparti_simple and those obtained in the previous spectral analysis of open cluster members, we have performed a test spectral analysis of the Sun, HD 32115 and the star 21 Peg.",
"We chose these stars because they represent three different regimes: in the solar spectrum, the Balmer lines are only sensitive to temperature; in the spectrum of HD 32115, the Balmer lines are sensitive to temperature and surface gravity; and in the spectrum of 21 Peg the Balmer lines are more sensitive to surface gravity.",
"We calculate the fundamental parameters for each star shown in Tables REF –REF .",
"Good agreement is seen between the parameters we calculated and the previously published results.",
"The higher value of $v \\sin i$ that we measure for the Sun is due to the fact that we do not account for macroturbulence broadening, while the technique used to calculate the value of $v \\sin i$ given in does.",
"For HD 32115, we derived the $T_{\\rm eff}$ and $\\log g$ from the Balmer lines, since showed the ionization balance of Fe is not sufficient in this star to determine $\\log g$ accurately.",
"The $v_{\\rm mac}$ of the Sun ranges from 1 to 4 kms$^{-1}$ ; however, we convolved the solar spectrum to a resolution of 25900 to simulate the analysis of our GIRAFFE spectra.",
"At this resolution, the effect of $v_{\\rm mac}$ is not visible in our spectra.",
"The analysis of the chemical abundances of the Sun shows good agreement with the results of [2] within the error bars shown in Table REF .",
"To perform the comparison between the chemical abundances determined by and those determined by sparti_simple for 21 Peg, we choose the same lines, atomic parameters and atlas12 model atmosphere as and removed any lines that show NLTE effects and/or have evidence of a hyperfine structure.",
"The results of our test are shown in Fig.",
"REF (which shows a comparison between our newly calculated H $\\beta $ profile and the observed spectrum) and in Table REF .",
"The numerical results of Table REF demonstrate that our results agree within the errors of those calculated by .",
"Figure: The fit of the H β\\beta line in the spectrum of 21 Peg.",
"The solid red line is the observed spectrum.",
"The dashed black is the synthetic spectrum calculated using T eff =10400T_{\\rm eff}=10400 K and logg=3.55\\log g = 3.55.",
"The two dashed blue lines are ±\\pm 200KTable: The fundamental parameters associated with the Sun.",
"Calculatedboth by and by the method usedin this work.Table: The fundamental parameters associated with HD32115.",
"Calculatedboth by and by the method usedin this work.Table: The fundamental parameters associated with the 21 Peg.",
"Calculatedboth by and by the method usedin this work.Table: A comparison between the chemical abundances of the Sun calculated by and those calculated by sparti_simple.Table: A comparison between the chemical abundances of 21 Peg calculated by and those calculated by sparti_simple, along with the solar abundances from .Table: Instrument settings information with the useful spectral linesgiven for stellar T eff T_{\\rm eff} between ∼6000\\sim 6000 Å and ∼25000\\sim 25000\\,ÅTable: List of programme stars.",
"The proper motion(μ\\mu ) in right ascension (RA) and declination (DEC), for each star, is taken fromthe UCAC2 catalogue .",
"The radial velocities (v r v_{\\rm r}) are calculated using the methodin Section and for the member stars using sparti_simple.Δ\\Delta is the number of standard deviations the proper motions and radialvelocity are away from the cluster mean.The S/N (column 8) corresponds to theS/N per spectral bin of either the GIRAFFE settings or the UVES 520 nmsetting.",
"The BVBV photometry is taken from the APASS catalogue and theJKJK photometry is taken from the UCAC2 catalogue .",
"The Memb.",
"column gives the results of our membership analysis, where `y' means the star is a member, `n' means the star is not a member.",
"The final column gives the membership probabilities given by ."
],
[
"Target",
"The cluster NGC 6250 is located in the constellation Ara in the Southern hemisphere.",
"Using proper motions and photometry from PPMXL and 2MASS $JHK$ photometry data, have estimated the age of the cluster to be $\\log t \\sim 7.42$ yr and the distance from the Sun to the cluster as 865 pc.",
"calculated the cluster proper motion as $0.7\\,\\pm \\,0.4$ mas yr$^{-1}$ in right ascension (RA) and $-4.1\\,\\pm \\,0.4$ mas yr$^{-1}$ in declination (DEC) with a cluster radial velocity of $-8.0\\,\\pm \\,0.8$ km s$^{-1}$ .",
"Previously, estimated $\\log t \\sim 7.146$ yr for the age and 1025 pc for the distance.",
"estimated $d=950$ pc.",
"NGC 6250 is located in a dust-rich region of space, with $E$ ($B$ – $V$ ) = 0.385 and $E$ ($J$ – $H$ ) = 0.123 .",
"To our best knowledge, the cluster has not previously been studied in detail spectroscopically, except for the purpose of classification spectra and radial velocity measurements."
],
[
"Instrument",
"The observations of NGC 6250 were obtained in service mode on 2007 May 27 and 30 using FLAMES, the multi-object spectrograph attached to the Unit 2 Kueyen of the ESO/VLT.",
"The FLAMES instrument is able to access targets over a field of view of 25 arcmin in diameter.",
"Its 138 fibres feed two spectrographs, GIRAFFE and the Ultraviolet and Visual Echelle Spectrograph (UVES).",
"This makes it possible to observe up to 138 stars, 130 using GIRAFFE linked to FLAMES with MEDUSA fibres, and 8 using UVES linked to FLAMES with UVES fibres.",
"FLAMES-GIRAFFE can obtain low- or medium-resolution spectra ($R$ = 7500–30 000), within the spectral range 3700–9000 Å. Low-resolution spectra may be obtained within wavelength intervals 500–1200 Å wide; medium-resolution spectra are obtained in wavelength intervals 170–500 Å wide.",
"FLAMES-UVES can obtain high-resolution spectra ($R$ = 47 000), with central wavelengths of 5200, 5800 or 8600 Å each covering a wavelength range $\\sim 2000$ Å wide."
],
[
"Instrument Settings",
"For the observations, we chose the instrument set-up that allowed us: (1) to observe two hydrogen lines, which are essential to the determination of $T_{\\rm eff}$ and $\\log $ $g$ and (2) to maximize the number of metal lines and consequently the number of chemical elements available for spectral analysis.",
"The GIRAFFE settings were chosen such that they are as close to the guiding wavelength of 520 nm as possible, in an effort to minimize light losses due to atmosphere differential refraction.",
"The higher spectral resolution of UVES means that, to achieve a high enough S/N ratio for each spectra, UVES requires an exposure time typically three or four times that of GIRAFFE.",
"As a result during UVES observations we are able to obtain observations with three GIRAFFE settings.",
"The settings we used are shown in Table REF .",
"We observed with the HR9B and HR11 settings both observing nights and one L- setting each night.",
"To prevent saturation of the observations each UVES observation was divided into four sub-exposures and each GIRAFFE setting was divided into two sub-exposures."
],
[
"Data reduction",
"Our data were obtained in service mode, and the package released included the products reduced by ESO with the instrument dedicated pipelines.",
"In this work, we used the low- and mid resolution GIRAFFE data as reduced by ESO and re-reduced UVES spectra.",
"For the normalization we followed a standard procedure.",
"First, we fit the observed spectrum with a function $G(\\lambda )$ , clipping in an iterative way all points 3 $\\sigma $ above and 1 $\\sigma $ below the spectrum.",
"Then, we compare the normalized spectrum with a synthetic spectrum calculated adopting a similar stellar model, in order to test the quality of our normalization.",
"Experience has shown for the low-resolution settings LR3 and LR6, a third-order polynomial gives the best normalization.",
"For UVES and the high-resolution settings HR9B and HR11, a cubic spline gives the best normalization.",
"In the case of UVES the Balmer lines spread across two different orders, the merging of the orders was therefore done before the normalization."
],
[
"Cluster membership",
"Previous studies by , and give cluster membership information for the stars in the vicinity of NGC 6250. estimate cluster membership using the statistical method of .",
"The probabilities are shown in Table REF .",
"Note that some of the results of show inconsistencies: three stars (UCAC 12065057, UCAC 12284608 and UCAC 12284626) have been assigned 100 % likelihood of being members despite their proper motion values being far from the cluster mean .",
"Furthermore, UCAC 12065064 has proper motion values very close to the cluster mean values and its photometry fits well with the theoretical isochrone, it only has a 3 % likelihood of being a member.",
"The determination of cluster membership from previous literature was used as a guess for an initial target selection.",
"Our new spectroscopic data allow a refined membership study, which is critical for our analysis."
],
[
"Proper motion and radial velocity membership",
"The proper motions and radial velocities of the observed stars are shown in Table REF and are plotted in Fig.",
"REF .",
"In our membership analysis we have followed two methods.",
"First, we have identified those stars that are within 1 $\\sigma $ of the mean proper motion and radial velocity values for our sample.",
"This way we have identified 19 stars to be members of the cluster.",
"We calculate the cluster mean proper motions as, 0.1 $\\pm $ 2.9 mas yr$^{-1}$ in RA, $-$ 6.1 $\\pm $ 4.4 mas yr$^{-1}$ in DEC and radial velocity of $-$ 10 $\\pm $ 11 km s$^{-1}$ .",
"As a cross-check, we also use the partitional clustering technique, $K$ -means clustering .",
"It is a technique to find common data points based on the analysis of the variables that define the data.",
"In general, the method is to set a predicted number of data clusters and give an initial guess for the centres of these clusters.",
"The algorithm then assigns points to the closest cluster centre and recalculates until the cluster centre values do not change.",
"Our problem is simplified by the fact that we can define one cluster centre close to the literature value of cluster proper motion and radial velocity.",
"In practice, we have set the number of clusters to 5: one initially centred in the literature values $ (\\mu _0, v_{r_0}) $ and the other four initially centred respectively at $(\\mu _0 +1.5\\,\\sigma _\\mu , v_{r_0})$ , $(\\mu _0 - 1.5\\,\\sigma _\\mu , v_{r_0})$ , $(\\mu _0, v_{r_0} + 1.5\\,\\sigma _{v_{r}})$ and $(\\mu _0, v_{r_0} +1.5\\,\\sigma _{v_r})$ , where $\\sigma _{v_{r}}$ and $\\sigma _\\mu $ are the standard deviation in our sample.",
"We have performed this computation using the CLUSTER function in idl.",
"Fig.",
"REF shows the results of our cluster analysis.",
"We found 15 members and we calculated the cluster mean proper motions as 0.4 $\\pm $ 3.0 mas yr$^{-1}$ in RA, $-$ 4.80 $\\pm $ 3.2 mas yr$^{-1}$ in DEC and radial velocity of $-$ 10 $\\pm $ 6 km s$^{-1}$ .",
"Both methods give the same results apart from four stars.",
"The discrepancy between the two methods is likely because the spread of radial velocity is large and asymmetric, and the $K$ -means clustering is able to deal with this more effectively."
],
[
"Photometry",
"The magnitude and colour of our sample of stars are a good indicator of cluster membership.",
"Using the $B$ and $V$ magnitudes from and the $J$ and $K$ magnitudes from , we have produced two colour magnitude diagrams, displayed in Fig.",
"REF .",
"Each diagram shows the photometry and a theoretical isochrone calculated with CMD 2.7 , , , for an age of $\\log t = 7.42$ .",
"The photometry has been corrected for the extinction and distance to the cluster .",
"In general, we see very good agreement between the member stars determined using the kinematics approach and those that fit the isochrones.",
"Notable exceptions are UCAC 12065058, UCAC 12065075 and UCAC 12284628, which do not agree with the kinematics of the cluster mean, but agree very well with the photometry.",
"This maybe as a result of a re-ejection event and so we consider these stars as members."
],
[
"Radial velocity and rotational velocity",
"We have calculated initial values of $v_{\\rm rad}$ and $v\\sin i$ from the FLAMES spectra obtained in the range 5150–5350 Å with the highest resolution setting HR9B.",
"We used a least-squares deconvolution of the observed spectrum with lines selected from the VALD list to calculate an average line profile in velocity space.",
"To do the line selection, we must have an estimate of the temperature, so we use a combination of Balmer lines and photometry to determine an estimate of the temperature.",
"Figure: The least-square deconvolution of the spectrum ofUCAC 12284746 (T eff =7200T_{\\rm eff} = 7200 K) (solid black line), plotted againstthe Gaussian fit (red dashed line).",
"The radial velocity is givenby the position of the centre of the Gaussian, and it isindicated by a solid vertical line.Figure: The fast Fourier transform of the LSD profileof Fig.",
"(black solid line), plotted with theFTT of a model LSD profile withT eff =7200T_{\\rm eff}= 7200 K and vsini=19.5v\\sin i=19.5 km s -1 ^{-1} (red dashed lines).The radial velocity of each star has been estimated using a Gaussian fit, as shown in Fig.",
"REF .",
"In order to measure $v\\sin i$ the LSD profile is then shifted to the rest frame, and the fast Fourier transform (FFT) is calculated.",
"An example of the FFT is shown in Fig.",
"REF .",
"Following , the first minimum of the FFT corresponds to the stellar $v\\sin i$ value.",
"showed that it is possible to use the LSD profile, in place of the more noisy profiles of single lines, to derive the $v\\sin i$ value using the FFT method described by .",
"Because we expect low $v_{\\rm mac}$ values , at the resolution of FLAMES $v_{\\rm mac}$ cannot be distinguished from $v \\sin i$ and we therefore ignore it."
],
[
"Fundamental parameters from Balmer lines",
"We deduced the fundamental parameters $T_{\\rm eff}$ and $\\log g$ by fitting synthetic and observed Balmer lines H $\\beta $ and H $\\gamma $ for UVES spectra and H $\\alpha $ and H $\\beta $ for GIRAFFE spectra.",
"Model atmospheres were computed with atlas9 assuming plane parallel geometry, local thermodynamical equilibrium and opacity distribution function (ODF) for solar abundances .",
"The synthetic spectra were computed with cossam_simple.",
"Examples of the fit between our model and the observed Balmer lines are shown in Fig.",
"REF .",
"We used hydrogen line as both temperature and gravity indicator because for $T_{\\rm eff}8000$ K they are more sensitive to temperature and for higher temperature they are more sensitive to $\\log g$ variation but temperature effects can still be visible in the part of the wing close to the line core, according to .",
"To check our values of $\\log g$ for each star we determined the $\\log g$ which provided the best ionization balance between Fe i and Fe ii lines.",
"To test this method, we calculate the abundance of Fe i and Fe ii for the Sun using $T_{\\rm eff}$ = 5800 K and varying the $\\log g$ between 3.8 and 4.5.",
"The value of $\\log g$ where the abundances of Fe i and Fe ii are equal is 4.49 compared with 4.44 found by .",
"We performed a similar analysis for 21 Peg, varying the $\\log g$ between 3.5 and 4.2 and using $T_{\\rm eff}$ = 10400 K. As a result we calculate a value for $\\log g$ of 3.5 compared with 3.55 found by .",
"The results of this analysis are shown in Figs REF and Figs REF .",
"For each of the NGC 6250 stars, the $\\log g$ values found using both methods agree within the uncertainties.",
"As a result of the low S/N, we were unable to measure any abundances for TYC 8327-565-1 and UCAC 12284638; however, we were able to estimate $T_{\\rm eff}$ and $\\log g$ that are given in Table REF .",
"Figure: The difference between the abundance of Fe i and Fe iiplotted as a function of surface gravity for the Sun determined using sparti_simple by varying the logg\\log g of the model atmosphere at T eff T_{\\rm eff} = 5800K.Figure: The difference between the abundance of Fe i and Fe iiplotted as a function of surface gravity for the star 21Peg determined using sparti_simple by varying the logg\\log g of the model atmosphere at T eff T_{\\rm eff} = 10400K.Figure: A sample of the observed H β\\beta lines (black solid lines) fittedwith the model spectra (red dashed line).",
"From top to bottomthe stars are:UCAC 12284594 (T eff =6200T_{\\rm eff} = 6200 K),UCAC 12065075 (T eff =6300T_{\\rm eff} = 6300 K),UCAC 12065064 (T eff =7600T_{\\rm eff} = 7600 K) andUCAC 12284536 (T eff =9800T_{\\rm eff} = 9800 K).",
"Each profile is calculatedwith vsiniv\\sin i as shown in Table .Table: Atmospheric parameters for the sample of stars from NGC 6250.Table: The abundance of elements for the analysed stars of NGC6250 (ordered by decreasing T eff T_{\\rm eff}), given in log(NN/H)where H = 12.00.",
"In parenthesis, the first number is the error calculated using equation () and thesecond in the error calculating using equation ().",
"Both errors are in units of 0.01dex.",
"The last rowof each set gives the solar abundances from ."
],
[
"Results and Discussion",
"The results of the abundance analysis are given in Table REF .",
"Since this is a young cluster, there is the potential for some of the stars to still have discs.",
"If discs were present, we would expect to see the presence of emission lines, particularly in the core of H $\\alpha $ and H $\\beta $ .",
"We do not see any evidence of emission lines in any of the stars."
],
[
"UCAC 12284546",
"UCAC 12284546 shows an overabundance of C, Ca, Cr, Fe and Ni and an underabundance of Mg.",
"However, this abundance pattern does not match any standard chemically peculiar star in this temperature range.",
"To better understand this star, it would be necessary to collect and analyse a higher resolution spectrum with higher S/N.",
"As a result of the abundance anomalies we observe in this star, we do not consider this star in the global analysis of the results."
],
[
"Stellar Metallicity",
"For the evolutionary tracks and isochrones we adopted the metallicity calculated as $Z_{\\rm cluster} = 10^{[{\\rm Fe/H}]_{\\rm stars} - [{\\rm Fe/H}]_{\\rm \\odot }}Z_{\\rm \\odot },$ where $Z_{\\rm cluster}$ and $[{\\rm Fe/H}]_{\\rm stars}$ are respectively the clusters metallicity and average Fe abundance.",
"This formulation does not follow the definition of $Z$ , which is $Z=\\sum _{i=1}^{n}m_iX_i,$ where $n$ is the number of elements, $m_i$ is the atomic mass of each element and $X_i$ the abundance of each element.",
"In equation (REF ), $Z$ is driven mostly by the abundance of C and O, which are the most abundant elements following H and He, but in stellar evolutionary calculations the relevant factor is the Fe opacity.",
"This is why for the cluster metallicity we adopt the expression given by equation (REF ).",
"When using equation (REF ) to infer the metallicity, it is important to use as Z$_{\\rm \\odot }$ the value adopted by the considered stellar evolution tracks.",
"In this work we use the stellar evolutionary tracks by , which adopt Z$_{\\rm \\odot } = 0.0152$ .",
"Using the average Fe abundance obtained from the non-chemically peculiar stars, we obtain $Z_{\\rm cluster} = 0.018\\pm 0.005$ , which is consistent with the solar value within the uncertainty."
],
[
"Spectroscopic H-R diagram",
"We plot a spectroscopic H-R diagram (Fig.",
"REF ) using the $T_{\\rm eff}$ and $\\log g$ values calculated for each star.",
"We calculate the flux weighted luminosity, $\\log \\mathcal {L}/\\mathcal {L}_\\odot $ , following with $\\log \\mathcal {L}/\\mathcal {L}{\\rm \\odot } = \\log \\left(\\frac{T_{\\rm eff}^4}{g}\\right)-\\log \\left(\\frac{T_{\\rm eff\\odot }^4}{g_\\odot }\\right)$ where $T_{\\rm eff}$ and $\\log g$ are taken from Table REF , $T_{\\rm eff\\odot }$ is the solar effective temperature and $g_\\odot $ is the solar surface gravity.",
"The isochrones are from .",
"Based on the H-R diagram, we are not able to constrain the age of this cluster; however, the age of $\\log t$ = 7.42 given by fits well our data.",
"As a result, we use this age in the remainder of the paper.",
"We also give the flux-weighted luminosity, masses and fractional age of each of the stars in Table REF calculated by fitting evolutionary tracks to each star.",
"Table: logℒ/ℒ ⊙ \\log \\mathcal {L}/\\mathcal {L}_\\odot , logT eff \\log T_{\\rm eff}, M/M ⊙ M/M_\\odot and fractional age (τ\\tau ) with associatederror bars for the stars of the NGC 6250 open cluster.Figure: An H-R diagram of NGC6250, the stars plotted (black plus-signs)with theoretical isochrones at logt\\log t = 7.40 (dashed black line) andlogt\\log t = 7.45 (solid black line).",
"Both isochrones have solar metallicity.Figure: The mean abundances of each element relative to solarfor F- (red circles),A-(green squares) and B- (blue diamonds) type stars.",
"The error barsare calculated by taking the standard deviations of the calculated meanabundances.Figure: The abundances of C, Na, Mg, and Si relative to the solarabundance against T eff T_{\\rm eff}.",
"For F- (red circles),A-(green squares) and B- (blue diamonds) type stars.Figure: Same as Fig.",
", but for Ca, Sc, Ti andCr.Figure: Same as Fig.",
", but for Mn, Fe, Ni andBa.Figure: The abundances of C, Na, Mg, Si, Ca and Sc relative to the solarabundance against vsiniv \\sin i.",
"For F- (red circles),A-(green squares) and B- (blue diamonds) type stars.Figure: Same as Fig.",
", but for Ti, Cr, Mn, Fe, Ni andBa."
],
[
"Analysis of chemical abundances",
"Fig.",
"REF shows the mean abundance of each element obtained for the F-, A- and B-type stars.",
"The error bars are calculated as the standard deviation about the mean abundance.",
"We consider only the measurements from Table REF with maximum errors smaller than 0.5.",
"To determine whether there is any correlation with the stellar fundamental parameters, we have compared each set of element abundances with $T_{\\rm eff}$ , $v \\sin i$ , $M/$ M$_\\odot $ and fractional main-sequence age.",
"In Figs REF –REF , we show abundance as a function $T_{\\rm eff}$ and in Figs REF and REF , we show abundance as a function of $v \\sin i$ .",
"After comparison between abundance and each of the fundamental parameters, we see no statistically significant patterns.",
"This is consistent with the findings of .",
"In addition, we have compared our results with the previous studies of the open clusters NGC6405, NGC 5460 and Praesape performed by , and , , , respectively.",
"This allows us to determine whether there is any evidence for correlation between cluster age and abundance.",
"We compare our results with only these clusters since they have all been analysed within this project, and the analysis has been either fully carried out (Praesape and NGC 5460) or supervised by one of us (LF) (NGC 6405 and NGC 6250), to minimize the possibility of systematic differences between the results.",
"To compare the results from each cluster analysis, we have offset the abundance values of the individual chemical elements according to the cluster metallicities as estimated from Fe abundances of the cluster F and later type stars, which should be less affected by diffusion than earlier type stars.",
"Figure: A comparison between the mean C, O, Mg, Si, Ca and Scabundances foundfor each of the previous studies and those found in this paper.Mean abundances forF- (red circles) and A-type (green squares) stars areplotted against cluster age (logt=7.42\\log t=7.42 for NGC 6250; logt=7.88\\log t=7.88 for NGC 6405;logt=8.20\\log t=8.20 for NGC5460; and logt=8.85\\log t=8.85 for Praesepe).",
"The error is given by the standarddeviation of all the measured abundances.Figure: Same as Fig.",
"but for Ti, Cr,Mn, Fe, Ni and Ba.In NGC 6250, we found that O, Na, Sc, Ti, Cr, Mn, Ni, Zn and Y all have solar abundances within the uncertainties, while S and V are overabundant.",
"These results are consistent with the findings of , and for the Praesepe cluster, NGC 5460 and NGC 6405, respectively (see Figs REF and REF ).",
"Similarly to what was found by , and in the Praesepe cluster, NGC 5460 and NGC 6405, we have found an overabundance of C in the F- and A-type stars of NGC 6250.",
"However, we do not see any trend with age (see Fig.",
"REF ).",
"For all of the F-type stars, we find a solar abundance of Mg, for the A-type and B-type stars there is an underabundance of Mg; however, there is a large spread in the results and all but two stars have approximately solar abundance, which matches with the results of the previous studies.",
"In agreement with previous studies, we found that in A-type stars Si is overabundant; however, at odds with previous studies, we found that in F-type stars Si is underabundant.",
"Fig.",
"REF indicates the presence of a possible correlation between $T_{\\rm eff}$ and the Si abundance, though a further analysis reveals that this apparent correlation is not statistically significant.",
"For all of the F-type stars, we find a solar abundance of Ca that is consistent with the previous results.",
"However, for the A-type stars, we find an overabundance, which is contrary to the findings of the previous studies; the origin of this is unclear.",
"We measure the abundance of Fe in all of our stars to be approximately solar.",
"For both Mn and Fe, found an increase in abundance with $T_{\\rm eff}$ , which we do not; this therefore may be the result of an age effect.",
"The narrow $T_{\\rm eff}$ range of the stars analysed by for the Praesepe cluster means we are unable to provide any definite conclusions until the remaining clusters are analysed.",
"We measure an almost solar abundance for Ba, albeit with relatively large uncertainties.",
"This is in contrast with the findings of , and who all report overabundances.",
"To understand each of the results together, we plot the mean abundance of Ba measured for each cluster in Fig.",
"REF .",
"We did not consider the stars HD 122983 and HD 123182 from NGC 5460 because of their apparent chemical peculiarities .",
"From Fig.",
"REF , we obtain a hint of a positive correlation of Ba abundance with age; however, the abundance uncertainties are too large to draw any concrete conclusion.",
"By analysing further clusters we will be able to determine whether this effect is the result of diffusion or the different chemistry of the star-forming region for each cluster.",
"We measure Nd to be overabundant in four stars; however, the data from previous papers are too sparse to provide any conclusion.",
"Finally, we have compared our results with the study of chemically peculiar magnetic Ap stars by .",
"This allows us to examine the differences and similarities between abundance trends of chemically normal and chemically peculiar stars.",
"found statistically significant trends between He, Ti, Cr, Fe, Pr and Nd and stellar age.",
"They also found a strong trend between the abundances of Cr and Fe, and $T_{\\rm eff}$ .",
"For Cr, an underabundance was observed for stars with $T_{\\rm eff} \\lesssim 7000\\,$ K, for stars with $T_{\\rm eff} \\gtrsim 7000\\,$ K the abundance of Cr sharply rises and peaks at $T_{\\rm eff} \\sim 10000\\,$ K before falling back to approximately solar.",
"For Fe, an underabundance was observed for stars with $T_{\\rm eff} \\lesssim 8000\\,$ K and an overabundance for the remaining stars.",
"These results are in stark contrast with what we observed for NGC 6250.",
"This suggests that the abundance of chemical elements in the photosphere of chemically normal F-, A- and B-type stars remains relatively constant during their main-sequence lifetime except when influenced by a magnetic field."
],
[
"Conclusions",
"We have presented the new code for spectral analysis, sparti_simple.",
"Based on cossam_simple, a modified version of the radiative transfer code cossam, sparti_simple employs the inversion algorithm LMA and allows one to recover the abundance of the chemical elements of non-magnetic stellar atmospheres.",
"To test our new code, we have performed the abundance analysis of the Sun, HD 32115 and 21 Peg and compared our results with those previously published in the thorough works by , [2], and , finding excellent agreement.",
"We have applied our new code for a spectroscopic study of the open cluster NGC 6250, which was observed with the FLAMES instrument of the ESO VLT.",
"From the observed sample of stars, we have performed cluster membership analysis based on a $K$ -means clustering procedure and analysis of the photometry.",
"As a result of our analysis, we selected 19 stars from our sample as members of the cluster.",
"We have computed the cluster mean proper motions of 0.4 $\\pm $ 3.0 mas yr$^{-1}$ in RA, $-$ 4.80 $\\pm $ 3.2 mas yr$^{-1}$ in DEC and radial velocity of $-$ 10 $\\pm $ 6 km s$^{-1}$ .",
"These values agree within the errors with the values calculated by .",
"The age and distance given by agree well with our photometric analysis of the cluster.",
"Finally, we have examined the chemical abundance measurements for each star and searched for any trend between abundance and the stellar fundamental parameters and between the abundance measured in this study and the abundance measured in the previous studies of older clusters by , , , and .",
"Our results for the abundance of O, Na, Sc, Ti, Cr, Mn, Ni, Zn and Y are solar within the uncertainties, while S and V are overabundant.",
"These results are consistent with previous studies.",
"We do not find evidence of the correlation between either the Fe or Mn abundance and $T_{\\rm eff}$ found by ; however, this may be evidence of an age effect and we need to study more clusters before being able to determine this.",
"We find hints of an increase in mean Ba abundance with cluster age but more clusters should be analysed to confirm this trend.",
"Comparing our results with those from , who searched for trends between chemical abundances and stellar parameters of chemically peculiar magnetic Ap stars, suggests that the abundance of chemical elements in the photosphere of chemically normal F-, A- and B- type stars remains relatively constant during their main-sequence lifetime except when influenced by a magnetic field."
],
[
"Acknowledgements",
"This paper is based on observations made with ESO Telescopes at the Paranal Observatory under programme ID 079.D-0178.",
"We thank Claudia Paladini for the re-reduction of the UVES spectra.",
"AM acknowledges the support of a Science and Technology Facilities Council (STFC) PhD studentship.",
"Thanks go to AdaCore for providing the GNAT GPL Edition of its Ada compiler.",
"This publication makes use of data products from the AAVSO Photometric All Sky Survey (APASS).",
"Funded by the Robert Martin Ayers Sciences Fund and the National Science Foundation.",
"We thank the referee Charles Proffitt for providing constructive comments that led to a significant improvement of the manuscript."
]
] | 1612.05739 |
[
[
"Resolved Stellar Populations as Tracers of Outskirts"
],
[
"Abstract Galaxy haloes contain fundamental clues about the galaxy formation and evolution process: hierarchical cosmological models predict haloes to be ubiquitous, and to be (at least in part) the product of past merger and/or accretion events.",
"The advent of wide-field surveys in the last two decades has revolutionized our view of our own Galaxy and its closest \"sister\", Andromeda, revealing copious tidal streams from past and ongoing accretion episodes, as well as doubling the number of their known faint satellites.",
"The focus shall now be shifted to galaxy haloes beyond the Local Group: resolving individual stars over significant areas of galaxy haloes will enable estimates of their ages, metallicities and gradients.",
"The valuable information collected for galaxies with a range of masses, morphologies and within diverse environments will ultimately test and quantitatively inform theoretical models of galaxy formation, and shed light onto the many challenges faced by simulations on galactic scales."
],
[
"The Importance of Haloes",
"Our understanding of galaxy formation and evolution has dramatically evolved in the past fifty years.",
"The first and simplest idea for the formation scenario of our own Milky Way (MW) Galaxy was put forward by [50], who proposed the bulk of a stellar halo to be formed in a rapid collapse of gas in the protogalaxy.",
"This scenario, often referred to as “monolithic” collapse, is a dissipative process and takes place on dynamical timescales of the order of $\\sim 10^8$ yr.",
"This process gives birth to a metal-poor stellar component in the halo outer regions, while the inner regions ends up being more metal-rich due to the reprocessing of the gas as it collapses deeper into the protogalaxy potential well.",
"This idea was later challenged by an alternative explanation, based on the observation that globular clusters (GCs) at different Galactocentric distances have a wide range of metallicities.",
"In this scenario, the halo is formed on longer timescales ($\\sim 10^9$ yr) and, instead of being a self-contained system, it comes together as the product of several protogalactic fragments ([163]).",
"These fragments can be pre-enriched before they are accreted.",
"While both scenarios are capable of explaining many observed quantities of the Galactic halo, they cannot individually give a comprehensive picture ([133], [22]), which has led to the development of hybrid “two-phase” models.",
"In the latter, the inner Galaxy regions are formed in a first phase as a result of a monolithic-like process, while the outer halo regions are built up over the Galaxy's lifetime through dissipationless accretion events ([60]).",
"In the past couple of decades, the most widely accepted paradigm of the hierarchical Lambda-Cold Dark Matter ($\\rm \\Lambda $ CDM) structure formation model has prevailed, favouring the predominance of merger and accretion events in the build-up of galactic haloes ([192], [18], [167], [92]).",
"These models predict the ubiquitous presence of haloes, which are characterized by old and metal-poor populations and often shows signs of recent interactions, in contrast with the smooth haloes predicted by dissipative models ([18], [1], [57]).",
"The interaction events provide a mine of information on the assembly of haloes: dynamical timescales become relatively long (up to several Gyr) in the outer regions of a galaxy, and thus accretion/merger events that occurred a long time ago are often still visible as coherent structures like disrupting galaxies or streams, which readily testify the past assembly history of their host.",
"The assembly itself depends on a variety of factors, such as number, mass, stellar content and structural properties of the accreted satellites, as well as orbital properties, timing and energy of the accretion event.",
"Even when the progenitor is completely dissolved in the host's halo (which is particularly true in the inner halo regions where dynamical timescales are relatively short), its stripped stellar content still retains a characteristic coherence in velocity space as well as in metallicity content, thus giving important clues about the progenitor's properties.",
"Observing the stellar “fossils” that populate galaxy haloes thus offers a unique opportunity to reconstruct the modes, timing, and statistics of the halo formation process.",
"Besides being taletellers of their host system's merger history, the shape and size of haloes also hold vital clues to the process of galaxy formation.",
"In particular, they can teach us about the primordial power spectrum of density fluctuations at the smallest scales; about the reionization process, that shall lead to faint and concentrated haloes for an early suppression of star formation in low-mass dark matter (DM) subhaloes; or about the triaxiality of DM haloes, which are predicted to be more flattened for dissipationless formation scenarios ([1]).",
"Despite only accounting for a mere $\\sim 1\\%$ of a galaxy's total mass (e.g., [122]), extended haloes are clearly extremely valuable to test and refine theoretical predictions on the halo assembly process.",
"Due to their extreme faintness, however, haloes have not been as fully exploited as they should have been as key tests of galaxy formation models: they are not easily detected above the sky level, i.e., surface brightness values of $\\mu _V\\sim 25$ mag arcsec$^{-2}$ , posing a serious observing challenge to their investigation.",
"Cosmological simulations predict the majority of past and ongoing accretion events to have surface brightness values well below this value (e.g., [18]).",
"According to some models, reaching a surface brightness of $\\mu _V\\sim 29$ mag arcsec$^{-2}$ should allow the detection of at least one stream per observed galaxy ([92], [28]).",
"How is it then possible to extract the information locked in the faint outskirts of galaxies?",
"The best method to study faint haloes and their substructure in nearby galaxies is to resolve individual stars.",
"Even when sparse and faint, resolved stars can be individually counted, and a stellar number density can easily be converted into a surface brightness value.",
"When the Galactic extinction presents a high degree of spatial inhomogeneity (possibly mimicking faint irregular substructures), and the sky level is higher than the integrated light signal coming from extremely faint sources, resolved populations provide a very powerful means to trace them.",
"This method is not free from complications: there will always be contamination coming both from foreground Galactic stars as well as from background unresolved galaxies.",
"This can be accounted for statistically, by observing “field” regions away from the main target and quantifying the contaminants, while a direct confirmation of a star's membership requires spectroscopy.",
"At the same time, resolving individual stars poses constraints on the inherent nature and on the distance of the putative targets: for systems where the stellar density is so high that stars fall on top of each other on the sky, the “crowding” prevents the resolution of individual objects.",
"This can of course occur also in the case of a relatively sparse galaxy which has a large line-of-sight distance, so that the stars are packed in a small region of the sky.",
"Distance is also the principal enemy of depth: the larger the distance, the brighter the detection limit, i.e., the absolute magnitude/surface brightness that we can reach for a fixed apparent magnitude.",
"Nonetheless, resolved stellar populations are able to deliver powerful information for galaxies located within $\\sim 10$ Mpc, i.e., within the so-called Local Volume.",
"The discovery of the Sagittarius dwarf galaxy by [84] from the identification of a comoving group of stars opened the door to the era of halo studies and their substructure: a galaxy resembling the properties of classical dwarf spheroidals was clearly in the process of being disrupted by its giant host, our own MW.",
"This evidence was the first to support theoretical predictions for the hierarchical assembly models and the existence of observable accretion events.",
"Soon thereafter, stellar density maps allowed the discovery of a prominent low surface brightness stream around the MW's closest giant spiral Andromeda (M31), the so-called Giant Stellar Stream ([81]).",
"This feature, invisible to the naked eye, is a clear example of the elusive nature of haloes and their substructure: the surface brightness of the Giant Stellar Stream is $\\mu _V\\sim 30$ mag arcsec$^{-2}$ , which is prohibitive for integrated light images.",
"As challenging as it is, the mere detection of haloes and their substructures is not enough to provide quantitative constraints on models of galaxy evolution.",
"From the stars' photometry and thus position in the colour-magnitude diagram (CMD), i.e., the observational counterpart of the Hertzsprung-Russel diagram, it is possible to characterize the properties of the considered stellar system.",
"First and foremost, in contrast to integrated light, accurate distance measurements can be obtained from CMD features that act as standard candles, e.g., the luminosity of the tip of the red giant branch (TRGB) or of the horizontal branch (HB).",
"Another key advantage of resolved populations is the possibility to constrain ages and metallicities more tightly than with integrated light alone.",
"The CMD is used to quantify the star formation rate as a function of lookback time, and thus derive the star formation history (SFH) of a composite stellar population (e.g., [62], and references therein).",
"Spectroscopy of individual stars is the ultimate method to constrain their metallicity content and kinematical properties, such as radial velocity and proper motion, which allows for the full six-dimensional phase space to be investigated.",
"The latter cannot, for the moment, be achieved beyond the LG limits, and still only occasionally for M31.",
"Besides giving precious insights into galaxy haloes and their accretion histories, resolved stellar populations can help us characterizing the “surviving” low-mass galaxies that have not been accreted to date and reside in the outskirts of giant hosts."
],
[
"The Low-mass End of the Galaxy Luminosity Function",
"The low-mass end of the galaxy luminosity function (LF) is of no less interest than haloes themselves.",
"Besides the MW and M31, the Local Group (LG) contains tens of smaller galaxies which can be studied in detail due to their proximity (see [178] for a review).",
"While the $\\rm \\Lambda $ CDM cosmological model has provided a convincing match to the large-scale structures observed in the high-redshift Universe, it falls short at the smallest, galactic scales, indicating an incomplete understanding of the physics involved in the evolution of galaxies: for example, the “missing-satellite problem” has been highlighted for the first time by [121] and [99].",
"Briefly, the number of DM subhaloes predicted in simulations exceeds the observed number of MW satellites by almost two orders of magnitude.",
"The shape of the DM profile in the innermost regions of dwarf galaxies is also a matter of debate (the “cusp-core” problem; [186]).",
"In addition, the more massive among the MW satellites are less dense than what is expected from simulations, which is puzzling because they should be affected by fewer observational biases than their smaller, sparser siblings (the “too-big-to-fail” problem; [15]).",
"In addition, the fact that many of the MW and M31 satellites are distributed along planes does not have a straightforward explanation in $\\rm \\Lambda $ CDM models (e.g., [136]).",
"From the theoretical point of view, the inclusion of baryonic physics in DM-only simulations is key to reconcile predictions with observations of the smallest galaxies.",
"In particular, effects such as supernova feedback, stellar winds, cosmic reionisation, and tidal/ram pressure stripping all concur to reduce star formation efficiency in the least massive DM haloes.",
"Tremendous progress is being made on this front, taking into account realistic physics as well as increasing the resolution of simulations (e.g., [168], [16], [160], [190]).",
"At the same time, new observational discoveries keep offering intriguing challenges at the smallest galactic scales, as further described in Sect.",
"REF and REF .",
"The galaxies closest to us give us the most detailed information because of the large number of stars that can be resolved.",
"Here, I will summarize what we have learnt in the past two decades about our own Galaxy (even though an extensive picture of the MW outskirts goes beyond the scope of this contribution and can be found in Figueras, this volume), about its closest spiral neighbour M31 and about their lower-mass satellites."
],
[
"Milky Way",
"The MW is traditionally divided into discrete components, i.e., the bulge, the disks (thin and thick) and the halo.",
"The spheroidal portion of the MW is given by the central bulge, which consists mainly of metal-rich populations, and an extended diffuse component, which has a lower mean metallicity.",
"Overall, stars and GCs in the halo have ages $\\sim 11-13$ Gyr ([20]).",
"The halo can be further deconstructed into an inner halo and an outer halo, even though the distinction could partly arise from observational biases ([162]).",
"The inner and outer haloes also seem to have different chemical composition ([Fe/H]$\\sim -1.6$ and [Fe/H]$\\sim -2.2$ , respectively; [151], [20]).",
"According to simulations, the two halo components should also have formed on different timescales: the inner halo ($<20$ kpc) is partly constituted by early-formed in-situ stars, partly due to a violent relaxation process, and partly assembled from early, massive merging events that provide metal-rich populations ([1], [58], [175], [141], [29]); the outer halo is assembled more recently, with its mass beyond $\\sim 30$ kpc being mainly accreted in the past $\\sim 8$ Gyr ([18], [28]).",
"These predictions are, however, still not sufficient at a quantitative level, and unconstrained as to the exact ratio of accreted stars versus in-situ populations.",
"At the same time, observations of the MW halo with better statistics and precision are needed to inform them.",
"Our position within the MW puts us at a clear disadvantage for global studies of its outskirts: the distant and sparse halo stars are observed from within the substantial disk component, which produces contamination both in terms of extinction and numerous disk stars along the line of sight, which completely “obscure” the sky at low Galactic latitudes.",
"Nonetheless, thanks to the advent of wide-field imagers, the past two decades have revolutionized the large scale view of our Galaxy.",
"Several stellar tracers can be used to dig into the MW halo at different distance ranges: old main sequence turnoff (MSTO) stars are identified mostly out to $\\sim 20$ kpc, brighter RGB stars out to $\\sim 40-50$ kpc, while RR Lyrae and blue horizontal branch (BHB) stars can be detected out to 100 kpc.",
"Spatial clustering of these stellar components indicate non-mixed substructure, which is often confirmed to be kinematically coherent.",
"After the cornerstone discovery of the disrupting Sagittarius dwarf, it became clear that substructure is not only present in the MW halo, but it also might constitute a big portion of it.",
"To put it in S. Majewski's words ([105]), “There is good reason to believe that within a decade we will have a firm handle on the contribution of satellite mergers in the formation of the halo, as we move observationally from serendipitous discoveries of circumstantial evidence to more systematic surveys for the fossils left behind by the accretion process” .",
"In the following decade, several stream-like features have indeed emerged from a variety of multi-band photometric and spectroscopic surveys indeed, and the Sloan Digital Sky Survey (SDSS) proved to be an especially prolific mine for such discoveries around the northern Galactic cap.",
"The Sagittarius stream has been traced further, including in the Galactic anti-centre direction (e.g., [114], [106]), and independent substructures have been uncovered ([89], [195], [132], [71], [93]), most notably the Monoceros ring, the Virgo overdensity, the Orphan stream and the Hercules-Aquila cloud.",
"Some of these have later been confirmed to be coherent with radial velocities ([46]).",
"Note that most of these substructures are discovered at Galactocentric distances $>15$ kpc, while the inner halo is smooth due to its shorter dynamical timescales.",
"During the past decade, one of the most stunning vizualisations of the ongoing accretion events in the MW halo was provided by the Field of Streams ([10]), reproduced in Fig.",
"REF .",
"The stunning stellar density map is derived from SDSS data of stars around the old MSTO at the distance of Sagittarius, with a range of magnitudes to account for a range in distances.",
"This map not only shows the Sagittarius stream and its distance gradient, but also a plethora of less massive streams, as well as an abundance of previously unknown dwarf satellites (see Sect.",
"REF ).",
"The Field of Streams has been now complemented with results from the latest state-of-the-art surveys, most notably the all-sky Panoramic Survey Telescope and Rapid Response System (PanSTARRS), which covers an area significantly larger than that of SDSS.",
"In Fig.",
"REF the first stellar density maps from PanSTARRS are shown, obtained in a similar way as the Field of Streams ([12]).",
"The map highlights the fact that the deeper and wider we look at the Galaxy halo, the more substructures can be uncovered and used to constrain its past accretion history and the underlying DM halo properties.",
"From this kind of maps, for example, the halo stellar mass that lies in substructures can be estimated, amounting to $\\sim 2-3 \\times 10^8\\,M_\\odot $ (see [9]).",
"Using SDSS, [8] also highlight the predominant role of accretion in the formation of the MW's halo based on MSTO star counts, adding to up to $\\sim 40\\%$ of the total halo stellar mass (note that, however, different tracers could indicate much smaller values; e.g., [39]).",
"Many of the known halo streams arise from tidally disrupting GCs, of which Palomar 5 is one of the most obvious examples ([134]).",
"This demonstrates the possible role of GCs, besides dwarf satellites, in building up the halo stellar population, and additionally implies that some of the halo GCs may be stripped remnants of nucleated accreted satellites (see [60], and references therein).",
"In order to discern between a dwarf or a cluster origin of halo stars, we need to perform chemical “tagging”, i.e., obtain spectroscopic abundances for tens of elements for these stars (e.g., [110]): stars born within the same molecular cloud will retain the same chemical composition and allow us to trace the properties of their birthplace.",
"A number of ambitious ongoing and upcoming spectroscopic surveys (SEGUE, APOGEE, Gaia-ESO, GALAH, WEAVE, 4MOST) is paving the path for this promising research line, even though theoretical models still struggle to provide robust predictions for the fraction of GC stars lost to the MW halo (e.g., [161], and references therein)."
],
[
"The Smooth Halo Component",
"Once the substructure in the halo is detected, it is important that it is “cut out” in order to gain insights into the smooth, in-situ stellar component (note that, however, the latter will inevitably suffer from residual contamination from accreted material that is now fully dissolved).",
"The stellar profile of the Galactic halo is, in fact, not smooth at all: several studies have found a break at a radius $\\sim 25$ kpc, with a marked steepening beyond this value ([187], [164]), in qualitative agreement with halo formation models.",
"Some of the explanations put forward suggest that a density break in the halo stellar profile is the likely consequence of a massive accretion event, corresponding to the apocentre of the involved stars ([40]).",
"The kinematics of halo stars, of GCs and of satellite galaxies, as well as the spatial distribution of streams and tidal features in satellites, can be further used as mass tracers for the DM halo.",
"The total MW mass is to date still poorly constrained, given the difficulty of evaluating it with a broad range of different tracers.",
"The general consensus is for a virial mass value of $\\sim 1.3\\pm 0.3 \\times 10^{12}\\,M_\\odot $ , even though values discrepant up to a factor of two have recently been suggested (see [14] for a compilation of estimates).",
"Besides providing estimates for the total MW mass, studies of SDSS kinematical data, of the Sagittarius stream and of GCs tidal streams have provided discording conclusions on the shape of the MW DM halo: nearly spherical from the modelling of streams or strongly oblate from SDSS kinematics at Galactocentric distances $<20$ kpc, while nearly spherical and oblate based on stream geometry or prolate from kinematical arguments for distances as large as $\\sim 100$ kpc (see [14] for details).",
"These constraints need a substantial improvement in the future to be able to inform cosmological models: the latter predict spherical/oblate shapes once baryons are included in DM-only flattened haloes (see [144])."
],
[
"Dwarf Satellites",
"As mentioned above, the SDSS has revolutionized our notions of dwarf satellites of the MW.",
"Bright enough to be easily recognized on photographic plates, a dozen “classical” MW dwarf satellites has been known for many decades before the advent of wide-field surveys ([114], [68]).",
"Starting with the SDSS, an entirely new class of objects has started to emerge with properties intermediate between the classical dwarfs and GCs (see [193], and references therein).",
"The so-called ultra-faint satellites have magnitudes higher than $M_V\\sim -8$ and surface brightness values so low that the only way to find them is to look for spatial overdensities of resolved main sequence/BHB stars.",
"Their discovery ten years ago doubled the number of known MW satellites and revealed the most DM-dominated galaxies in the Universe, with mass-to-light ratios of up to several times $10^3\\,M_\\odot /L_\\odot $ ([165]).",
"More recently, the interest in the low end of the galaxy LF has been revitalized once again with deep, wide-field surveys performed with CTIO/DECam, VST/Omegacam, and PanSTARRS: these have led to the discovery of more than 20 southern dwarfs in less than two years ([7], [101], [97], [44], [179], and references therein).",
"Some of these discoveries represent extremes in the properties of MW satellites, with surface brightness values as low as $\\sim 30$ mag arcsec$^{-2}$ , total luminosities of only a few hundred $L_\\odot $ and surprisingly low stellar density regimes.",
"One of the perhaps most intriguing properties of the newly discovered dwarfs is that many of them appear to be clustered around the Large Magellanic Cloud (LMC): this might be the smoking gun for the possible infall of a group of dwarfs onto the MW, which is predicted by simulations ([41], [156]).",
"Low-mass galaxies are expected to have satellites on their own and to provide a large fraction of a giant galaxy's dwarf companions (e.g., [189]).",
"The properties of the possible LMC satellites will give us a glimpse onto the conditions of galaxy formation and evolution in an environment much different from the LG as we know it today.",
"These faintest galaxies, or their accreted and fully dispersed counterparts, are also excellent testbeds to look for the very most metal-poor stars and to investigate the star formation process in the early stages of the Universe (e.g., [59]).",
"The study of the lowest mass galaxies holds the promise to challenge our knowledge of galaxy physics even further and pushes us to explore unexpected and exciting new limits."
],
[
"M31 (Andromeda)",
"Our nearest giant neighbour has received growing attention in the past decade.",
"Having a remarkable resemblance with the MW and a comparable mass (e.g., [184]), it is a natural ground of comparison for the study of spiral haloes.",
"In terms of a global perspective, the M31 halo is arguably known better than that of the MW: our external point of view allows us to have a panoramic picture of the galaxy and its surrounding regions.",
"The other side of the medal is that, at a distance of $\\sim 780$ kpc, we can only resolve the brightest evolved stars in M31, and we are mostly limited to a two-dimensional view of its populations.",
"Its proximity also implies a large angular size on the sky, underlining the need for wide field-of-view imagers to cover its entire area.",
"At the distance of M31, ground-based observations are able to resolve at best the uppermost $\\sim 3-4$ magnitudes below the TRGB, which is found at a magnitude $i\\sim 21$ .",
"The RGB is an excellent tracer for old ($>1$ Gyr) populations, but suffers from a degeneracy in age and metallicity: younger, metal-rich stars overlap in magnitude and colour with older, metal-poor stars ([100]).",
"Despite this, the RGB colour is often used as a photometric indicator for metallicity, once a fixed old age is assumed ([183], [30]).",
"This assumption is justified as long as a prominent young and intermediate-age population seems to be absent (i.e., as judged from the lack of luminous main sequence and asymptotic giant branch, AGB, stars), and it shows very good agreement with spectroscopic metallicity values where both methods have been applied.",
"The very first resolved studies of M31's halo introduced the puzzling evidence that the M31 halo stellar populations along the minor axis have a higher metallicity than that of the MW at similar galactocentric distances (e.g., [131]).",
"This was further confirmed by several studies targeting projected distances from 5 to 30 kpc and returning an average value of [Fe/H]$\\sim -0.8$ : in particular, [47] study a halo region at a galactocentric distance of $\\sim 20$ kpc and underline the difference between the properties of M31 and of the MW, suggesting that our own Galaxy might not represent the prototype of a typical spiral.",
"In fact, it has later been suggested that the MW is instead fairly atypical based on its luminosity, structural parameters and the metallicity of its halo stars when compared to spirals of similar mass ([73]).",
"This result was interpreted as the consequence of an abnormally quiet accretion history for the MW, which apparently lacked a major merger in its recent past.",
"The wide-area studies of M31's outskirts were pioneered $\\sim 15$ years ago with an Isaac Newton Telescope survey mapping $\\sim 40$ deg$^2$ around M31, reaching significantly beyond its disk out to galactocentric distances of $\\sim 55$ kpc ([81], [54]).",
"As mentioned before, the southern Giant Stream was first uncovered with this survey, and the halo and its substructures could be studied with a dramatically increased detail.",
"A metal-poor halo component ([Fe/H]$\\sim -1.5$ ) was finally uncovered for regions beyond 30 kpc and out to 160 kpc ([87], [94], [21]), similar to what had been observed for the MW both in terms of metallicity and for its stellar density profile.",
"These studies do not detect a significant gradient in metallicity across the covered radial range.",
"Nonetheless, the properties of the inner halo remained a matter of debate: while [21] found a metal-poor halo population within 30 kpc above the disc, [94] analysed a kinematically selected sample of stars within 20 kpc along the minor axis and derived a significantly higher value of [Fe/H]$\\sim -0.5$ .",
"At the same time, [17] used deep, pencil beam Hubble Space Telescope (HST) pointings in M31's inner halo to conclude that a significant fraction of its stellar populations have an intermediate age with an overall high metallicity.",
"These results were later interpreted by [82] in light of their wider-field dataset: the samples from [94] and [17] are simply part of regions dominated by an extended disc component and with a high contamination from various accretion events, respectively.",
"This underlines, once again, the importance of wide-field observations to reach a global understanding of halo properties.",
"Figure: Stellar density maps of metal-poor RGB populations at the distanceof M31, as derived from the PAndAS survey.",
"The large circles lie atprojected radii of 150 kpc and 50 kpc from M31 and M33, respectively.Upper panel: The Andromeda satellites are visible as clear overdensitiesand are marked with circles.",
"The vast majority of them wereuncovered by the PAndAS survey.",
"Reproduced by permission of the AASfrom , their Fig. 1.",
"Lower panel: The mainsubstructures around M31 are highlighted, showcasing a broad rangeof morphologies and likely progenitor type.",
"Tidal debris is alsopresent in the vicinities of the low-masssatellites M33 and NGC 147, indicating an ongoing interaction with M31.",
"Reproducedby permission of the AAS from , their Fig.",
"1Figure: Stellar density map of M31 (akin to Fig.",
"), thistime subdivided into photometric metallicity bins (as indicated ineach subpanel).",
"The upper panels show high metallicity cuts, wherethe Giant Stream and Stream C are the most prominent features; notethat the shape of the Giant Stream changes as a function ofmetallicity.",
"The lower panels show lower metallicity cuts: thelower left panel is dominated by substructure at large radii, whilethe most metal-poor panel (lower right) is smoother and believed tomostly contain in-situ populations.",
"Reproduced by permission of theAAS from , their Fig.",
"9.The M31 INT survey was further extended out to 150 kpc (200 kpc in the direction of the low-mass spiral M33) with the Canada-France-Hawaii Telescope/Megacam and dubbed Pan-Andromeda Archaeological Survey (PAndAS; [82], [116]).",
"This survey contiguously covered an impressive 380 deg$^2$ around M31, reaching 4 mag below the TRGB.",
"The PAndAS RGB stellar density map (see Fig.",
"REF ) is a striking example of an active accretion history, with a copious amount of tidal substructure at both small and large galactocentric radii.",
"PAndAS also constituted a mine for the discovery of a number of very faint satellites and GCs (see below; [150], [80], [112]).",
"Fig.",
"REF further shows the RGB stellar map broken into bins of photometric metallicity.",
"The parallel Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo (SPLASH) survey ([72], [94]) provides a comparison dataset with both photometric and spectroscopic information, the latter obtained with Keck/DEIMOS.",
"The SPLASH pointings are significantly smaller than the PAndAS ones but strategically cover M31 halo regions out to $\\sim 225$ kpc.",
"Deeper, pencil-beam photometric follow-up studies have further made use of the HST to target some of the substructures uncovered in M31's outskirts, resolving stars down to the oldest MSTO (e.g., [17], [11]).",
"These observations reveal a high complexity in the stellar populations in M31, hinting at a high degree of mixing in its outskirts.",
"Overall, M31 has evidently had a much richer recent accretion history than the MW (see also [53])."
],
[
"Streams and Substructures",
"As seen from the maps in Figs.",
"REF and REF , while the inner halo has a flattened shape and contains prominent, relatively metal-rich substructures (e.g., the Giant Stream), the outer halo ($>50$ kpc) hosts significantly less extended, narrow, metal-poor tidal debris.",
"The features in the innermost regions of M31 can be connected to its disk populations (e.g., the north-east structure or the G1 clump): kinematic studies show that a rotational component is present in fields as far out as 70 kpc, and they retain a fairly high metallicity ([43]).",
"This reinforces the possible interpretation as a vast structure, which can be explained as disk stars torn off or dynamically heated due to satellite accretion events.",
"Deep HST pointings of these features indeed reveal relatively young populations, likely produced from pre-enriched gas in a continuous fashion, comparable to the outer disk ([55], [17], [11]).",
"The most prominent feature in M31's outer halo, the Giant Stream, was initially thought to originate from the disruption of either M32 or NGC 205, the two dwarf ellipticals located at only $\\sim 25-40$ kpc from M31's centre ([81], [54]).",
"While both these dwarfs shows signs of tidal distortion, it was soon clear that none of them could produce the vast structure extending $\\sim 100$ kpc into M31's halo.",
"Great effort has been spent into mapping this substructure both photometrically and spectroscopically, in order to trace its orbit and define its nature: a gradient in its line-of-sight distance was first highlighted by [115], who found the outer stream regions to be located behind M31, the innermost regions at about the distance of M31, and an additional stream component on the opposite (northern) side of M31 to be actually in front of M31.",
"The stream presents a metallicity gradient, with the core regions being more metal-rich and the envelope more metal-poor (see also Fig.",
"REF ), as well as a very narrow velocity dispersion, with the addition of a puzzling second kinematic component ([65]); possible interpretations for the latter may be a wrap or bifurcation in the stream, as well as a component from M31's populations.",
"A number of increasingly sophisticated theoretical studies have tried to reproduce the appearance of the Giant Stream and picture its progenitor, which is undetected to date.",
"The general consensus seems to be that a relatively massive ($\\sim 10^9\\,M_\\odot $ ) satellite, possibly with a rotating disk, impacted M31 from behind with a pericentric passage around $1-2$ Gyr ago (most recently, [52], [154]).",
"In particular, simulations can reproduce the current extension and shape of the stream and predict the progenitor to be located to the north-east of M31, just beyond its disk ([52]).",
"This study also concludes that some of the substructures linked to M31's inner regions are likely to have arisen from the same accretion event, i.e., the north-east structure and the G1 clump (Fig.",
"REF ): these shelf features would trace the second and third passage around M31, which is also supported by their radial velocities.",
"CMDs of the Giant Stream populations are in agreement with these predictions: its stellar populations have mixed properties, consistent with both disk and stream-like halo features ([55], [149]).",
"Detailed reconstruction of its SFH indicate that most star formation occurred at early ages, and was possibly quenched at the time of infall in M31's potential (around 6 Gyr ago) ([11]).",
"Again, these studies deduce a likely origin of these populations as a dwarf elliptical or a spiral bulge.",
"Besides the Giant Stream, the only other tidal feature with a relatively high metallicity is Stream C (see Fig.",
"REF and REF ), which appears in the metal-poor RGB maps as well.",
"The origin of this feature is obscure, even though it is tempting to speculate that it could be part of the Giant Stream event.",
"The lower left panel of Fig.",
"REF , showing metal-poor populations, encompasses all of the narrow streams and arcs beyond 100 kpc, which extend for up to several tens of kpc in length.",
"All these substructures are extremely faint ($\\mu _V\\sim 31.5$ mag arcsec$^{-2}$ ), and their origin is mostly unknown because of the difficulty in following up such faint and sparse populations.",
"As part of the HST imaging of these features, [11] find that their populations are mainly formed at early ages and undergo a more rapid chemical evolution with respect to the disk populations.",
"Despite the metal-poor nature of these features, the hypothesis of a single accretion event producing most of the tidal features observed in the outer halo is not that unlikely, given the metallicity gradient present in the Giant Stream itself.",
"An efficient alternative to investigate the nature of these streams is to study the halo GC population: the wide-field surveys of M31 have allowed to uncover a rich population of GCs beyond a radius of $\\sim 25$ kpc (e.g., [80], and references therein), significantly more numerous than that of the MW halo.",
"[104] first highlighted a high spatial correlation between the streams in M31's halo and the GC population, which would be extremely unlikely in a uniform distribution.",
"Following the hypothesis that the disrupting satellites might be providing a high fraction of M31's halo GCs, [184] obtained spectroscopic follow-up: they were able to confirm that streams and GCs often have correlated velocities and remarkably cold kinematics.",
"This exciting result gives hope for studies of more distant galaxies, where halo populations cannot be resolved and GCs could be readily used to trace possible substructure."
],
[
"Smooth Halo",
"One of the first spatially extended datasets to investigate the halo of M31 in detail is described in [170]: their Subaru/SuprimeCam photometry along the minor axis in both directions are deeper, even though less extended, than PAndAS.",
"The stellar density profile derived in this study extends out to 100 kpc and shows a consistent power law for both directions.",
"The authors also suggest that, given the inhomogeneities in the stellar populations, the M31 halo is likely not fully mixed.",
"In the most metal-poor (lower right) panel of Fig.",
"REF , the substructures in the outer halo fade away, displaying a smoother component that can be identified with the in-situ M31 halo.",
"Once the substructures are decoupled based on the lack of obvious spatial correlation and with an additional photometric metallicity cut, [86] derive a stellar density profile out to 150 kpc.",
"Again, the profile follows a power-law, which turns out to be steeper when increasingly more metal-rich populations are considered.",
"[86] also conclude that only $5\\%$ of M31's total halo luminosity lies in its smooth halo, and the halo mass is as high as $\\sim 10^{10}\\,M_\\odot $ , significantly larger than what estimated for the MW.",
"The SPLASH survey extends further out than PAndAS, and benefits from kinematical information that is crucial to decontaminate the studied stellar samples from foreground stars and decreases the scatter in the radial profiles.",
"Based on this dataset, [66] find that the halo profile does not reveal any break out to 175 kpc.",
"This is somewhat surprising given the prediction from simulations that accreted M31-sized stellar haloes should exhibit a break beyond a radius of $\\sim 100$ kpc ([18], [28]).",
"Beyond a radius of 90 kpc, significant field-to-field variations are identified in their data, which suggests that the outer halo regions are mainly comprised of stars from accreted satellites, in agreement with previous studies.",
"At the outermost radii probed by SPLASH ($\\sim 230$ kpc), there is a tentative detection of M31 stars, but this is hard to confirm given the high contamination fraction.",
"Finally, the [66] stellar halo profile suggests a prolate DM distribution, which is also consistent with being spherical, in agreement with [86].",
"Both [86] and [67] investigate the existence of a metallicity gradient in the smooth halo of M31: they found a steady decrease in metallicity of about 1 dex from the very inner regions out to 100 kpc.",
"This might indicate the past accretion of (at least) one relatively massive satellite.",
"At the same time, a large field-to-field metallicity variation could mean that the outer halo has been mainly built up by the accretion of several smaller progenitors."
],
[
"Andromeda Satellites",
"Similarly to the boom of satellite discoveries around the MW, the vast majority of dwarfs in M31's extended halo has been uncovered by the SDSS, PAndAS, and PanSTARRS surveys in the past decade (see [112], and references therein).",
"The M31 satellites follow the same relations between luminosity, radius and metallicity defined by MW satellites, with the exception of systems that are likely undergoing tidal disruption ([26]).",
"Once more, the characterization of the lowest-mass galaxies raises new, unexpected questions: from the analysis of accurate distances and kinematics, [85] conclude that half of the M31 satellites lie in a vast ($\\sim 200$ kpc) and thin ($\\sim 12$ kpc) corotating plane, and share the same dynamical orbital properties.",
"The extreme thinness of the plane is very hard to reconcile with $\\rm \\Lambda $ CDM predictions, where such structures should not survive for a Hubble time.",
"While several theoretical interpretations have been offered (e.g., [56]), none is conclusive, and this reinforces the allure of mystery surrounding low-mass satellites.",
"Besides the detailed studies of the two LG spirals, increasing attention is being paid to lower-mass galaxies and their outskirts.",
"Given the self-similar nature of DM, low-mass galaxies should naively be expected to possess haloes and satellites of their own; however, our difficulty in constraining star formation efficiency and physical processes affecting galaxy evolution at these scales blurs these expectations.",
"In the last couple of years, the increasing resolution of cosmological simulations has allowed to make quantitative predictions about the halo and substructures in sub-MW-mass galaxies, and about the number of satellites around them ([191], [42]).",
"Observations are thus much needed to test these predictions.",
"Since the late 90s, numerous studies of star-forming dwarfs within or just beyond the LG have claimed the detection of an RGB component extending beyond the blue, young stars (see [168], and references therein), hinting at a generic mode of galaxy formation independent on galaxy size.",
"Such envelopes, however, were not characterized in detail, and in fact could not be identified uniquely as the product of hierarchical merging without, e.g., accurate age and metallicity estimates.",
"The presence of extended haloes in the most luminous satellites of the MW and M31, i.e., the irregular LMC and the low-mass spiral M33, respectively, has not been confirmed to date despite the availability of exquisite datasets.",
"[61] demonstrate how, out to a galactocentric distance of 7 kpc, the stellar density profile of the LMC disk does not show a clear break, in contrast to previous tentative claims.",
"Clearly, the question is complicated by the fact that the LMC is undergoing tidal disruption, and stripped stellar material could easily be misinterpreted as a halo component.",
"Nonetheless, [117] suggest to have found a sparse LMC halo population from a wide-field dataset around the nearby dwarf galaxy Carina, at galactocentric distances as large as 20 deg.",
"The question might be settled in the near future with the help of wide-field surveys such as the Survey of MAgellanic Stellar History ([111]).",
"With regard to possible low-mass satellites, there is now tantalizing indication that the LMC might have fallen onto the MW with its own satellite system, as mentioned in Sect.",
"REF .",
"As part of the PAndAS survey, deep imaging of M33 has revealed prominent substructure in its outer disk reminiscent of a tidal disturbance, and a faint, diffuse substructure possibly identified as a halo component ([25]).",
"This result was, however, carefully reconsidered by [118], who claim that a definitive sign of a halo structure cannot be confirmed, and if present it must have a surface brightness below $\\mu _V\\sim 35$ mag arcsec$^{-2}$ .",
"Besides the investigation of haloes and satellites, deep and wide-field views of low-mass galaxies are crucial to, e.g., assess the presence of tidal disturbances, which in turn are key to estimate mass values and constrain DM profiles (e.g., [157]).",
"As demonstrated by [32], a striking similarity in the global properties (luminosity, average metallicity, size) of two low-mass galaxies, such as the M31 satellites NCG 185 and NGC 147, can be quite misleading: once deep imaging was obtained around these galaxies (within PAndAS), NCG 147 revealed extended, symmetric tidal tails, returning a much larger extent and luminosity for this dwarf than what was previously thought.",
"This dataset further showed a flat metallicity gradient for NGC 147, in contrast with the marked gradient found in NGC 185.",
"All these pieces of evidence point at an ongoing interaction of NGC 147 with M31.",
"Large-scale studies of LG dwarfs also provide useful insights into their evolutionary history: by studying CMDs reaching below the MSTO, [77] trace significant age gradients that advocate an outside-in mode of star formation for dwarf galaxies.",
"Clearly, systematic deep searches are needed to detect and characterize the outskirts of low-mass satellites.",
"With this goal in mind, wide-field surveys of nearby ($<3$ Mpc) dwarfs have started to be pursued.",
"The first of these efforts targets NGC 3109, a sub-LMC-mass dwarf located just beyond the boundaries of the LG: several candidate satellites of NGC 3109 are identified from a CTIO/DECam survey targeting regions out to its virial radius ([159]).",
"One of them, confirmed to be at the distance of NGC 3109, is relatively bright ($M_V\\sim -10$ ), and is already in excess of the predicted number by [42] for this system.",
"Other ongoing surveys are similarly looking for halo substructures and satellites in several relatively isolated dwarfs, e.g., the SOlitary LOcal dwarfs survey ([78]) and the Magellanic Analog Dwarf Companions And Stellar Halos survey ([19]), by using wide-field imagers on large telescopes such as CFHT/MegaCam, Magellan/Megacam, CTIO/DECam and Subaru/HyperSuprimeCam.",
"These datasets will constitute a mine of information to constrain the role of baryonic processes at the smallest galactic scales.",
"The ground-breaking photometric and kinematic surveys carried out in the past two decades have significantly advanced our knowledge of haloes and their substructures within LG galaxies.",
"Nonetheless, the MW and M31 may not be representative of generic MW-sized haloes, given the stochasticity of the hierarchical assembly process: several marked differences in the stellar populations of their haloes underline the need for observations of a statistically significant sample of galaxy haloes with different morphologies, with surveys targeting large portions of their haloes.",
"Cosmological simulations of MW-mass analogues show a wide variation in the properties of their haloes.",
"As already mentioned, the relative contribution of in-situ star formation and disrupted satellites remains unclear: depending on the models (e.g., full hydrodynamical simulations, $N$ -body models with particle tagging), they can vary from a negligible number of accretion events for a MW-sized halo, to making up for most of a stellar halo content (e.g., [103], [174]).",
"Even within the same set of simulations, the number, mass ratio and morphology of accretion and merger events span a wide range of possible values ([18], [92], [63]).",
"The chemical content of extended haloes can provide useful insights into their assembly history: mergers or accretion events of similar-mass satellites will generally tend to produce mild to flat gradients; in-situ populations will feature increasingly metal-poor populations as a function of increasing galactocentric radius, similarly to the accretion of one or two massive companions (e.g., [28], [58]).",
"More extended merger histories are also expected to return younger and relatively metal-rich populations with respect to those coming from a shorter assembly, and to produce more massive stellar haloes, with the final result that the mean halo metallicities of MW-mass spirals can range by up to 1 dex (e.g., [148]).",
"Comprehensive observational constraints are key to guide future simulations of galaxy haloes: the past decade has seen a dramatic increase in the observational census of resolved galaxy haloes beyond the LG, thanks to deep imaging obtained with space facilities, as well as to the advent of wide-field imagers on large ground-based telescopes.",
"While the increasing target distance means that it is easier to survey larger portions of their haloes, the drawback is that the depth of the images decreases dramatically, and thus we are only able to detect the brightest surface brightness features in the haloes, i.e., the uppermost $\\sim 2-3$ mag below the TRGB in terms of resolved stars (see Fig.",
"6 in [143] for a schematic visualization of the different stellar evolutionary phases recognizable in such shallow CMDs).",
"A number of studies has surveyed relatively nearby and more distant galaxy haloes in integrated light despite the serious challenges posed by sky subtraction at such faint magnitudes, masking of bright stars, flat-fielding and scattered light effects, point spread function modelling, and/or spatially variable Galactic extinction.",
"A few early studies have been able to uncover a halo component and tidal debris in the target galaxies (e.g., [109], [123], [153]), without, however, settling the questions about their existence, nature or ubiquity.",
"Different approaches have been adopted to detect haloes and their substructures, i.e., targeting either individual galaxies (e.g., [197], [142], [90], [91], [113], [2], [3]) or stacking the images of thousands of objects (e.g., [198], [182], [169]).",
"A precise quantification of the occurrence of faint substructure in the outskirts of nearby galaxies seems as uncertain as it can be, ranging from a few percent to $\\sim 70\\%$ (see, e.g., [3], and references therein).",
"This is perhaps unsurprising given the heterogeneity of methods used, target galaxy samples, and surface brightness limits in such studies.",
"Besides the identification of such features, the characterization of unresolved halo stellar populations constitutes an even harder challenge: integrated colours and spectra can at most reach a few effective radii, thus missing the outer haloes.",
"Even for the available datasets, the degeneracies between age, metallicity and extinction are generally challenging to break (e.g., [37]); in addition, tidal features can rarely tell us about the mass ratio of a merger event or its orbit (with the exception of tails).",
"Here, we do not intend to discuss the detection of haloes and the variety of fractions and morphologies for tidal features observed in integrated light studies; Knapen & Trujillo (this volume) treat this topic in detail, while this contribution focusses on resolved populations.",
"Obtaining resolved photometry beyond the LG is a daunting task as well, due to the very faint luminosities involved—the brightest RGB stars for galaxies at $\\sim 4-10$ Mpc have magnitudes of $I\\sim 24-28.5$ , and thus this approach is so far really limited to the Local Volume.",
"Early attempts to perform photometry of individual stars in the outskirts of nearby galaxies have been made using large photographic plates and the first CCDs (e.g., [79], [36], [64]).",
"The brightest populations (i.e., the youngest) could often be reconciled with being members of the parent galaxy, but the critical information on the faint, old stars was still out of reach.",
"With the advent of wide-format CCDs in the mid 90s, photometry finally became robust enough to open up new perspectives on the resolved stellar content of our closest neighbours.",
"The first studies of this kind date back to twenty years ago and mainly focus on the inner regions of the target galaxies, most commonly their disks or inner haloes, with the goal of studying their recent star formation and of deriving TRGB distances (see, e.g., [166] for CenA, [155] for M81 and M82).",
"[51], in particular, resolved individual stars in the halo of the S0 galaxy NGC 3115 with HST.",
"By analysing the uppermost 1.5 mag of the RGB at a galactocentric distance of 30 kpc, they derived a distance of $\\sim 11$ Mpc, and additionally discovered for the first time a bimodality in the photometric metallicity distribution function of this early-type galaxy.",
"[172] studied for the first time the resolved content of the nearest ($\\sim 3.5$ Mpc) S0 galaxy NGC 404 with combined ground-based and HST imaging.",
"Their furthermost HST pointings ($\\sim 20$ kpc in projection) contain RGB stars that are clearly older than the main disk population, with similar colour (metallicity).",
"The authors conclude that the disk of NGC 404 extends out to this galactocentric distance, but they do not mention a halo component.",
"Beyond these early studies of individual galaxies, the need for systematic investigations of resolved stellar haloes was soon recognized.",
"Next we describe the design and results of some systematic surveys targeting samples of galaxies in the Local Volume."
],
[
"Systematic Studies",
"A decade ago, [126], [127], [128] started an effort to systematically observe the haloes of eight nearby ($<7$ Mpc) spiral galaxies with the resolution of HST.",
"In particular, they utilized WFPC2 to target fields off of the galaxies' disks (2 to 13 kpc in projection along the minor axis) with the goal of investigating their stellar populations, and obtaining accurate distance estimates as well as photometric metallicity distribution functions, to gain insights into the halo formation process.",
"[128] find the haloes to predominantly contain old populations, with no younger components and little to no intermediate-age populations.",
"Interestingly, [127] find a correlation between luminosity and metallicity for the target galaxies, where the metallicity is derived from the mean colour of the resolved RGB.",
"Both the spiral galaxies from their sample (NGC 253, NGC 4244, NGC 4945, NGC 4258, NGC 55, NGC 247, NGC 300, and NGC 3031 or M81) and the two ellipticals (NGC 3115 and NGC 5128 or Centaurus A, included in their comparison from previous literature data) fall on the same relation, indicating that haloes might have a common origin regardless of the galaxy morphological type.",
"Interestingly enough, the MW halo turns out to be substantially more metal-poor than those of the other galaxies of comparable luminosity, based on kinematically selected pressure-supported halo stars within $\\sim 10$ kpc above the disk (see also Sect.",
"REF ).",
"This relation is consistent with a scenario where halo field stars form in the potential well of the parent galaxy in a gradual way from pre-enriched gas.",
"Moreover, the relatively high metallicities of the target haloes seem to suggest that they likely originate from the disruption of intermediate-mass galaxies, rather than smaller metal-poor dwarf galaxies ([128]).",
"Interestingly, the dataset presented and studied in [126], [127], [128] is further analyzed by [124] to find that each spiral of the sample presents a bimodal metallicity distribution.",
"In particular, both a metal-poor and a metal-rich component are present in the outskirts of the target galaxies, and both components correlate with the host's luminosity.",
"This is taken as a hint that these populations are born in subgalactic fragments that were already embedded in the dark haloes of the host galaxy; the metal-poor component additionally has a broader dispersion than that of the metal-rich population.",
"These properties show similarities with GC subpopulations in the haloes of early-type galaxies (e.g., [140]).",
"[124] argues that the metal-poor component may arise from the accretion of low-mass satellites, while the metal-rich one could be linked to the formation of the bulge or the disk.",
"The shortcoming of this ambitious study is, however, twofold: first, the limited field of view (FoV) of HST hampers global conclusions on the galaxies' haloes, and the stellar populations at even larger radii may have different properties than those in the observed fields; second, perhaps most importantly, it is not obvious what structure of the galaxy is really targeted, i.e., the halo, the outer bulge or disk, or a mixture of these.",
"Along the same lines of these studies, [143] present an even more ambitious HST survey of 14 nearby disk galaxies within 17 Mpc, with a range of luminosities, inclinations and morphological types.",
"The Galaxy Halos, Outer disks, Substructure, Thick disks, and Star clusters (GHOSTS) survey aims at investigating radial light profiles, axis ratios, metallicity distribution functions (MDFs), SFHs, possible tidal streams and GC populations, all to be considered as a function of galaxy type and position within the galaxies.",
"The 76 ACS pointings of the survey are located along both major and minor axes for most of the targets, and reach $\\sim 2-3$ mag below the TRGB, down to surface brightness values of $V\\sim 30$ mag arcsec$^{-2}$ .",
"This dataset thus represents a very valuable resource for testing hierarchical halo formation models.",
"[120] investigate six of the galaxies in this sample (NGC 253, NGC 891, M81, NGC 4565, NGC 4945, and NGC 7814) and conclude that all of them contain a halo component out to 50 kpc, and two of them out to 70 kpc along their minor axis.",
"The colour (i.e., photometric metallicity) distribution of RGB stars in the target haloes is analysed and reveals a non-homogeneity which likely indicates the presence of non-mixed populations from accreted objects.",
"The average metallicity out to the largest radii probed remains relatively high when compared to the values of the MW halo; metallicity gradients are also detected in half of the considered galaxies.",
"Surprisingly, and in contrast to the results presented by [127], the spiral galaxies in this sample do not show a strong correlation between the halo metallicity and the total mass of the galaxies, highlighting instead the stochasticity inherent to the halo formation process through accretion events (e.g., [28]).",
"The advantage of the GHOSTS dataset over the one from [127] is that the GHOSTS fields are deeper, there are several pointings per galaxy and they reach significantly larger galactocentric distances, thus offering a more global view of the haloes of the targets.",
"In an effort to increase the sample of nearby galaxies for which stellar haloes are resolved and characterized, several groups have individually targeted Local Volume objects with either ground-based or space-borne facilities: the low-mass spirals NGC 2403 ([6], with Subaru/SuprimeCam), NGC 300 ([185], with Gemini/GMOS), and NGC 55 ([171], with Subaru/SuprimeCam), the ellipticals NGC 3379 ([76], with HST) and NGC 3377 ([75], with HST), and the lenticular NGC 3115 ([138], with HST).",
"In most of these galaxies, a resolved faint halo (or at least an extended, faint and diffuse component) has been detected and is characterized by populations more metal-poor than the central/disk regions.",
"Most of these haloes also show signs of substructure, pointing at past accretion/merger events as predicted by a hierarchical galaxy formation model.",
"Even galaxies as far as the central elliptical of the Virgo cluster, M87, ($\\sim 16$ Mpc) are starting to be targeted with HST, although pushing its resolution capabilities to the technical limits ([13]).",
"While spectroscopically targeting individual RGB stars to obtain radial velocity and metallicity information is still prohibitive beyond the LG (see Sect.",
"REF ), some cutting-edge studies have pushed the limits of spectroscopy for dwarf galaxies within $\\sim 1.5$ Mpc (e.g., [98], and references therein).",
"At the same time, novel spectroscopic techniques are being developed to take full advantage of the information locked into galaxy haloes.",
"One example is the use of co-added spectra of individual stars, or stellar blends, to obtain radial velocities, metallicities and possibly gradients in galaxies within $\\sim 4$ Mpc, as robustly demonstrated by [176].",
"The development of new analysis methods and the advent of high-resolution spectrographs will soon allow for systematic spectroscopic investigations of nearby galaxy haloes which will importantly complement the available photometric studies, similarly to the studies of LG galaxies.",
"Besides the systematic studies presented here, which mostly involve deep space observations, an increasing effort is being invested in producing spatial density maps of outer haloes in some of the closest galaxies with ground-based observations, akin to the panoramic view of M31 offered by PAndAS.",
"In the following Section we describe some of these efforts."
],
[
"Panoramic Views of Individual Galaxies",
"Panoramic views of nearby galaxies can be obtained with the use of remarkable ground-based wide-field imagers such as Subaru/SuprimeCam and HyperSuprimeCam and CFTH/MegaCam in the northern hemisphere, and Magellan/Megacam, CTIO/DECam and VISTA/VIRCAM in the southern hemisphere.",
"Clearly, such CMDs cannot reach the depth of those obtained for M31; these studies nevertheless represent cornerstones for our investigation of global halo properties, and serve as precursor science cases for the next generation of telescopes that will open new perspectives for this kind of studies to be performed on a significantly larger sample of galaxies.",
"As mentioned in Sect.",
"REF , the haloes of low-mass galaxies are also starting to be systematically investigated, to gain a more complete picture of galaxy formation at all mass scales.",
"Here we further describe the few examples of spatially extended imaging obtained to date for some of the closest spiral and elliptical galaxies."
],
[
"NGC 891",
"Despite its relatively large distance ($\\sim 9$ Mpc, [143]), the “MW-twin” NCG 891 ([181]) is one of the first spirals to be individually investigated in resolved light.",
"Its high inclination and absence of a prominent bulge make it an appealing target for halo studies.",
"[129] exploit three HST pointings located approximately 10 kpc above the disk of NGC 891 to investigate the properties of this galaxy's halo.",
"The broad observed RGB indicates a wide range of metallicities in this population, with metal-rich peaks and extended metal-poor tails.",
"The three fields also show a decreasing mean metallicity trend as a function of increasing distance along the major axis.",
"The mean metallicity of this sample of RGB stars ([Fe/H]$\\sim -1$ ) falls on the halo metallicity-galaxy luminosity relation pointed out by [127]: this, together with the gradient mentioned before, is in contrast with the lower metallicities and absence of a gradient for non-rotating stars in the inner haloes of the MW and M31 ([21], [94]).",
"[129] thus suggest that not all massive galaxies' outskirts are dominated by metal-poor, pressure-supported stellar populations (because of the inclination and absence of a bulge, the studied RGB sample is thought to be representative of the true halo population).",
"A possible explanation is suggested with the presence of two separate populations: a metal-rich one that is present in the most massive galaxies' outskirts, and one constituting the metal-poor, pressure-supported halo, coming from the accretion of moderate-mass satellites.",
"For smaller-mass galaxies, the halo would instead be dominated by debris of small satellites with lower metallicities.",
"Follow-up analysis on the same HST dataset has been carried out by [83] and [145].",
"After careful accounting for the internal reddening of the galaxy, a mild metallicity gradient is confirmed in NGC 891's spheroidal component, which is surveyed out to $\\sim 20$ kpc (assuming elliptical radii), and suggested to arise from the presence of a distinct outer halo, similarly to the MW ([83]).",
"Most importantly, and for the first time, this refined analysis reveals a substantial amount of substructure not only in the RGB spatial distribution but also as metallicity fluctuations in the halo of NGC 891.",
"This evidence points at multiple small accretion events that have not fully blended into the smooth halo.",
"Motivated by these studies, [130] provide the first attempt to derive a PAndAS-like map of a MW-analogue beyond the LG: their wide-field map of NGC 891's halo is shown in Fig.",
"REF .",
"The panoramic survey, performed contiguously with Subaru/SuprimeCam, covers an impressive $\\sim 90\\times 90$ kpc$^2$ in the halo of NGC 891 with the $V$ and $i$ filters, reaching $\\sim 2$ mag below the TRGB.",
"Among the abundant substructures uncovered by the RGB map around NGC 891, a system of arcs/streams reaches out some $\\sim 50$ kpc into the halo, including the first giant stream detected beyond the LG with ground-based imaging.",
"The latter's shape does not rule out a single accretion event origin, but a possible progenitor cannot be identified as a surviving stellar overdensity.",
"These structures appear to be old, given the absence of corresponding overdensities in the luminous AGB (i.e., intermediate-age populations) maps.",
"Another surprising feature highlighted by the RGB map is a flattened, super-thick envelope surrounding the disk and bulge of NGC 891, which does not seem to constitute a simple extension of its thick disk but is instead believed to generate from the tidal disruption of satellites given its non-smooth nature ([83])."
],
[
"M81",
"Located at a distance of 3.6 Mpc ([143]) and with a dynamical mass inside 20 kpc of $\\sim 10^{11}\\,M_\\odot $ , M81 is one of the closest MW-analogues, and has thus been among the first targets for extended halo studies beyond the LG.",
"The earliest Hi imaging of the galaxy group dominated by this spiral unambiguously shows a spectacular amount of substructure, most prominently a bridge of gas between M81 and its brightest companions NGC 3077 and M82, located at a projected distance of $\\sim 60$ kpc ([180], [196]).",
"Given the high level of interaction and Hi substructure in a group that can be considered as a LG-analogue, it is natural to pursue the investigation of this complex environment even further.",
"The intergalactic gas clouds embedding this environment are traced by young stellar systems identified in resolved stellar studies ([48], [34], [38]).",
"Some of them are classified as tidal dwarf galaxies, such as Holmberg IX and the Garland ([107], [95], [152], [188]), characterized by a predominance of young stellar populations.",
"This type of galaxy has no counterpart in our own LG, and it is believed to be DM-free (see, e.g., [45]).",
"The first detailed look into the resolved populations in the outskirts of M81 is through the eye of HST: the predominantly old halo RGB stars show a broad range of metallicities and a radial gradient ([173], [128]).",
"The radial stellar counts (along several different directions) also reveal a break at a radius of $\\sim 25$ kpc, which is interpreted as the transition point between thick disk and halo ([173]).",
"In a similar fashion, the ground-based wide-field imager Subaru/SuprimeCam has been used to uncover a faint and extended component beyond M81's disk with a flat surface brightness profile extending out to $\\sim 0.5$ deg (or $\\sim 30$ kpc) to the north of M81 ([5]).",
"This low surface brightness feature ($\\sim 28$ mag arcsec$^{-2}$ ) traced by the brightest RGB star counts appears bluer than the disk, suggesting a metallicity lower than that of M81's main body, but its true nature remains unclear.",
"The authors suggest this component to have intermediate properties between the MW's halo and its thick disk, but the limited surveyed area ($0.3$ deg$^2$ ) precludes any robust conclusions.",
"As part of a campaign to obtain panoramic views of nearby galaxy haloes, [125] present a $0.9\\times 0.9$ deg$^2$ view of M81's surroundings obtained with the CFHT/MegaCam imager.",
"The images resolve individual RGB stars down to $\\sim 2$ mag below the TRGB, but this study focusses on the younger, bright populations such as massive main sequence stars and red supergiants, which reveal further young systems tracing the Hi tidal distribution between M81 and its companions.",
"These systems are younger than the estimated dynamical age of the large-scale interaction and do not have an old population counterpart, suggesting that they are not simply being detached from the main body of the primary galaxies but are instead formed within the Hi clouds.",
"[49] recently conducted a deeper, albeit spatially limited, HST study of a field at a galactocentric distance of $\\sim 20$ kpc.",
"This field reveals an [M/H]$\\sim -1.15$ population with an approximate old age of $\\sim 9$ Gyr.",
"This field thus contains the most metal-poor stars found in M81's halo to that date, which led the authors to the conclusion that they were dealing with an authentic halo component.",
"This study is extended by [119] with the HST GHOSTS dataset (see Sect.",
"REF ): they construct a colour profile out to a radius of $\\sim 50$ kpc, and this dataset does not show a significant gradient.",
"The mean photometric metallicity derived is [Fe/H]$\\sim -1.2$ , similarly to [49].",
"This result is found to be in good agreement with simulations and the authors suggest that the halo of M81 could have been assembled through an early accretion of satellites with comparable mass (e.g., [28], [57]).",
"As a further step in the investigation of M81's halo, the [5] and [125] ground-based imaging of M81 is being improved by means of the Subaru/HyperSuprimeCam.",
"The first $\\sim 2\\times 2$ deg$^2$ ($\\sim 100\\times 115$ kpc$^2$ ) resolved stellar maps from different subpopulations (upper main sequence, red supergiants, RGB and AGB stars) are presented in [135] and constitute a preview of an even wider-field effort to map the extended halo of this group.",
"These first maps (see Fig.",
"REF ) confirm a high degree of substructure, most interestingly: the youngest populations nicely trace the Hi gas content, confirming previous small FoV studies; the RGB distributions are smoother and significantly more extended than the young component, and show stream-like overlaps between the dominant group galaxies, e.g., M82's stars clearly being stripped by M81; a redder RGB distribution is detected for M81 and NGC 3077 compared to M82, indicating a lower metallicity in the latter; in addition, M82 and NGC 3077's outer regions present S-shaped morphologies, a smoking gun of the tidal interaction with M81 and typical of interacting dwarf galaxies with larger companions (e.g., [137]).",
"Not less importantly, the widest-field survey to date ($\\sim 65$ deg$^2$ ) of the M81 group has been performed by [23] with CFHT/MegaCam, although with only one filter.",
"The main goal of this survey was to identify new, faint dwarf galaxies and investigate the satellite LF in a highly interacting group environment as compared to the LG.",
"This is the first survey to systematically search for faint dwarfs beyond the LG.",
"Resolved spatial overdensities consistent with candidate dwarfs have been followed up with two-band HST/ACS and HST/WFPC2 observations.",
"Fourteen of the 22 candidates turned out to be real satellites of M81 based on their CMDs and TRGB distances, extending the previously known galaxy LF in this group by three orders of magnitude down to $M_r\\sim -9.0$ ([24]), with an additional possibly ultra-faint member at $M_r\\sim -7.0$ .",
"The measured slope of the LF in the M81 group appears to be flatter than cosmological predictions ($\\alpha \\sim -1.27$ , in contrast to the theoretical value of $\\alpha \\sim -1.8$ ), similar to what has been found for the MW and M31 satellites."
],
[
"NGC 253",
"Another obvious MW-mass spiral target for halo studies is NCG 253 ($\\sim 3.5$ Mpc, [143]).",
"Its role of brightest object within the loose Sculptor filament of galaxies makes it ideally suited to investigate the effects of external environment on the assembly of haloes.",
"As already apparent from old photographic plates, NGC 253's outskirts show faint perturbation signs, such as an extended shelf to the south of its disk ([108]), pointing at a possible accretion event.",
"This spiral galaxy, despite its relative isolation, is experiencing a recent starburst and a pronounced nuclear outflow: the latter is believed to host local star formation extending as high as $\\sim 15$ kpc above the disk in the minor axis direction (see [27], and references therein).",
"The resolved near-infrared study of [35] allowed them to detect bright AGB stars, but not RGB stars, extending out to $\\sim 13$ kpc from the disk plane in the south direction: these are interpreted as being expelled from the disk into the halo as consequence of a recent interaction.",
"Subsequently, [4] exploited a combination of HST data from the GHOSTS survey and ground-based Magellan/IMACS imaging, the former being deeper while the latter have a more extended FoV (out to $\\sim 30$ kpc in the halo NGC 253 in the south direction).",
"The authors are able to estimate NGC 253's halo mass as $\\sim 2\\times 10^9\\,M_\\odot $ , or 6% of the galaxy's total stellar mass: this value is broadly consistent with those derived from the MW and M31 but higher, reminiscent of the halo-to-halo scatter seen in simulations.",
"A power law is fit to the RGB radial profile which is found to be slightly steeper than that of the two LG spirals, and appears to be flattened in the same direction as the disk component.",
"This is the one of the few studies to date to quantitatively measure such parameters for a halo beyond the LG, and it sets the stage for the possibilities opened by similar studies of other nearby galaxies.",
"The RGB density maps derived in [4] from IMACS imaging confirm the early detection of a shelf structure, and uncover several additional kpc-scale substructures in the halo of this spiral.",
"A more recent wide-field study of NGC 253 is presented by [69], who exploit the near-infrared VISTA/VIRCAM imager to study the RGB and AGB stellar content of this galaxy out to $\\sim 40-50$ kpc, covering also the northern portion which was not included in previous studies.",
"This portion, in particular, reveals an RGB substructure symmetric (and likely connected) to the one in the south.",
"A prominent arc ($\\sim 20$ kpc in length) to the north-west of the disk is detected and estimated to arise from a progenitor with a stellar mass of roughly $\\sim 7\\times 10^6\\,M_\\odot $ .",
"The RGB radial density profile shows a break at a radius of $\\sim 25$ kpc, indicative of the transition from disk to halo.",
"The elongated halo component already discussed in [4] is confirmed here, but is considered to be an inner halo: an outer, more spherical and homogeneous component extends at least out to the galactocentric distances covered by this survey.",
"Intriguingly, the AGB density map reveals that 25% of this intermediate-age (i.e., up to a few Gyr old) population is spread out to $\\sim 30$ kpc above the disk: this component cannot easily be explained with either an in-situ or an accreted origin.",
"NGC 253 is also one of the two targets of the Panoramic Imaging Survey of Centaurus and Sculptor (PISCeS), recently initiated with the wide-field imager Magellan/Megacam.",
"This ambitious survey aims at obtaining RGB stellar maps of this galaxy and of the elliptical Centaurus A (Cen A; see next Section) out to galactocentric radii of $\\sim 150$ kpc, similarly to the PAndAS survey of M31.",
"Early results from this survey include the discovery of two new faint satellites of NGC 253, one of which is clearly elongated and in the process of being disrupted by its host ([158], [177])."
],
[
"NGC 5128 (Centaurus A)",
"It is important to target galaxies of different morphologies and environments to thoroughly investigate the assembly of haloes.",
"The closest ($\\sim 3.8$ Mpc; [74]) elliptical galaxy is Centaurus A (Cen A; technically speaking, Maffei 1 is slightly closer but it lies behind the Galactic disk and is thus heavily reddened, see [194]).",
"Cen A is the dominant galaxy of a rich and dense group, which also has a second subgroup component centred on the spiral M83 (e.g., [96]).",
"Despite having often been referred to as a peculiar galaxy, due to its pronounced radio activity, its central dust lanes, and a perturbed morphology, the luminosity of Cen A is quite typical of field elliptical galaxies: a recent ($<1$ Gyr) merger event is believed to be the culprit for its peculiar features (see [88], and references therein).",
"Besides this main merger event, [139] uncover a system of faint shells and an arc within $\\sim 25$ kpc of Cen A's centre from integrated light observations; the arc is believed to have been produced by the infall of a low-mass, star forming galaxy around $\\sim 300$ Myr ago.",
"This elliptical galaxy has been the subject of a systematic study conducted with HST/ACS and HST/WFPC2 throughout the past couple of decades: a number of pointings at increasingly large galactocentric radii (from a few out to $\\sim 150$ kpc) have been used to investigate the properties and gradients of Cen A's halo populations ([147], and references therein).",
"The considered pointings out to 40 kpc reveal metal-rich populations ([Fe/H]$>-1.0$ ), not dissimilar to what has been observed for the haloes of spiral galaxies.",
"The deepest CMD to date of this elliptical is presented by [146] for the HST field at 40 kpc: this study concludes that the vast majority of Cen A's halo population is old ($\\sim 12$ Gyr), with a younger ($\\sim 2-4$ Gyr) component accounting for $\\sim 20\\%$ of the total population.",
"The first wide-field study of Cen A was performed with the ground-based VLT/VIMOS imager, reaching out to $\\sim 85$ kpc along both minor and major axes ([31]).",
"Cen A's halo population seems to extend all the way out to this large radius.",
"This study confirms the relatively high metallicity for halo populations found by the HST studies, although with a considerable presence of metal-poor stars at all radii; the authors also highlight the absence of a strong metallicity gradient from a $\\sim 30$ kpc radius out to the most distant regions probed.",
"This study suggests that the outer regions of Cen A's halo show an increase in ellipticity as a function of radius, which could, however, be interpreted as the presence of substructure contaminating the observed fields.",
"A subsequent study exploits additional HST pointings out to a remarkably large radius of $\\sim 150$ kpc: the edge of Cen A's halo is not reached even by this study ([147]).",
"This dataset, analysed together with the previous HST pointings, confirms that a very mild metallicity gradient is present, with median metallicities remaining high out to the largest distances probed.",
"[147], however, also detect a significant pointing-to-pointing variation in both the RGB star counts and the median metallicity, which is likely indicative of non-mixed accreted populations.",
"Recently, the PISCeS survey (see previous Section) has sketched a PAndAS-like picture of Cen A's halo out to $\\sim 150$ kpc: the RGB stellar density map derived from a mosaic of Magellan/Megacam images is presented in Fig.",
"REF .",
"This map, very much like the ones obtained for M31 and NGC 891, uncovers a plethora of faint substructures, both in the inner regions of the target galaxy and in its outskirts.",
"The morphological variety of these features is reminiscent of that observed in PAndAS, with shells, plumes, an extended cloud and long tidal streams.",
"In particular, one of the newly discovered dwarf satellites of Cen A is clearly in the process of being disrupted, with $\\sim 2$ deg long tails: taking into account the stellar content of these tails, this galaxy's pre-disruption luminosity could have been similar to that of Sagittarius in the LG.",
"This survey also led to the discovery of nine (confirmed) dwarf satellites down to $M_V\\sim -7$ .",
"Their properties are consistent with those of faint LG satellites, but some of them lie at the faint/diffuse end of the LG luminosity/surface brightness/radius distribution: this indicates that we might be looking at previously unexplored physical regimes for these faintest satellites, which opens new exciting perspectives for future studies.",
"In a $\\rm \\Lambda $ CDM hierarchical model, all galaxies are predicted to have experienced mergers, of which many should be recognizable as debris/streams that make up for a large fraction of their haloes.",
"Haloes and their substructures thus provide a unique glimpse into the assembly history of galaxies, and can inform the models at the smallest galactic scales, where they still fall short in reproducing observations.",
"The time is now ripe for in-depth systematic studies of the resolved stellar populations in galaxy haloes, which will dramatically increase our understanding of galaxy evolution over the next decade.",
"The challenges for this type of studies are of a different nature: for our own Galaxy, state-of-the-art results on its halo shape, profile and mass inevitably suffer from assumptions on underlying density models and extrapolations of the available data to radii larger than observed.",
"The major current limitation of MW halo studies lies in observational biases due to small field-of-view samples, which preclude the identification of possible substructure contamination.",
"Future surveys hold the promise to advance the knowledge of our Galaxy by obtaining significantly larger samples of tracers, especially in areas so far not covered.",
"Most notably, the astrometric Gaia mission (which will provide unprecedented six-dimensional phase space information for two billion stars out to the inner MW halo) and the Large Synoptic Survey Telescope (LSST; designed to provide a southern sky counterpart to SDSS, and reaching $\\sim 4$ magnitudes fainter than its predecessor for a total sample of tens of billions of stars), are going to revolutionize our view of the MW.",
"At the same time, the current and future generation of high-resolution spectrographs will follow up these surveys from the ground, providing comprehensive kinematic and chemical information to assess the origin of halo stars and characterize their birthplaces (see also Figueras, this volume).",
"The pioneering studies of an increasing number of haloes beyond the LG, and across a range of masses, will soon be extended by the next generation of ground-based extremely large telescopes (E-ELT, GMT, TMT), as well as space-borne missions (JWST, Euclid, WFIRST).",
"The PAndAS survey of M31 has extensively demonstrated that only the synergy of wide-field ground-based observations, deep (but spatially limited) observations from space, and spectroscopy can return a truly global understanding of haloes made up of a complex mixture of in-situ and accreted populations.",
"The aforementioned facilities will open new perspectives with wide-field optical and infrared imagers in concert with high-resolution spectrographs, which will allow us to systematically survey hundreds of galaxies within tens of Mpc in the next decade or two.",
"For example, with the E-ELT/MICADO and JWST/NIRcam imagers (the former having higher resolving power and the latter a wider field-of-view), we should resolve stars down to the HB within $\\sim 10$ Mpc, thus identifying and characterizing the SFHs of streams and faint satellites; derive radial profiles, MDFs and stellar population gradients in haloes within 20 Mpc from the uppermost few magnitudes of the RGB; and trace the halo shape and possible overdensities down to $\\mu _V\\sim 33$ mag arcsec$^{-2}$ from the uppermost $\\sim 0.5$ mag of the RGB out to 50 Mpc ([70]).",
"These observational constraints will be crucial to inform increasingly sophisticated theoretical models, and ultimately answer intriguing open questions (as well as possibly unexpected ones that will likely be raised by these observations themselves), such as: Do all galaxies have haloes?",
"What are the relative fractions of in-situ versus accreted populations in galaxy haloes, and how does this depend on galactocentric distance, galaxy morphology, and environment?",
"What are the properties of the objects currently being accreted, i.e., mass, chemical content, SFH, orbital properties, and how do they relate to those of the present day low-mass satellites?",
"Do low-mass galaxies possess haloes/satellites of their own, and what is their fate and contribution upon infall onto a massive galaxy?",
"How extended really are the haloes of massive galaxies?",
"What is the shape and mass of the DM haloes underlying galaxies?",
"What is the relation between the outer halo and the bulge/disk of a galaxy?",
"What is the role of internal versus external processes in shaping a galaxy's properties, especially at the low-mass end of the galaxy LF?",
"What is the relation between the present-day haloes/satellites and their unresolved, high-redshift counterparts?",
"The era of resolved populations in galaxy haloes has just begun, and it holds the promise to be a golden one.",
"I would like to thank the organizers for a lively and stimulating conference.",
"I am indebted to S. Pasetto for his advice and support throughout the preparation of this contribution.",
"I acknowledge the hospitality of the Carnegie Observatories during the completion of this work."
]
] | 1612.05471 |
[
[
"Stagnant Shells in the Vicinity of the Dusty Wolf-Rayet-OB Binary WR 112"
],
[
"Abstract We present high spatial resolution mid-infrared images of the nebula around the late-type carbon-rich Wolf-Rayet (WC)-OB binary system WR~112 taken by the recently upgraded VLT spectrometer and imager for the mid-infrared (VISIR) with the PAH1, NeII\\_2, and Q3 filters.",
"The observations reveal a morphology resembling a series of arc-like filaments and broken shells.",
"Dust temperatures and masses are derived for each of the identified filamentary structures, which exhibit temperatures ranging from $179_{-6}^{+8}$ K at the exterior W2 filament to $355_{-25}^{+37}$ K in the central 3\".",
"The total dust mass summed over the features is $2.6\\pm0.4\\times10^{-5}$ $\\mathrm{M}_\\odot$.",
"A multi-epoch analysis of mid-IR photometry of WR~112 over the past $\\sim20$ yr reveals no significant variability in the observed dust temperature and mass.",
"The morphology of the mid-IR dust emission from WR~112 also exhibits no significant expansion from imaging data taken in 2001, 2007, and 2016, which disputes the current interpretation of the nebula as a high expansion velocity ($\\sim1200$ km s$^{-1}$) \"pinwheel\"-shaped outflow driven by the central WC-OB colliding-wind binary.",
"An upper limit of $\\lesssim120$ km s$^{-1}$ is derived for the expansion velocity assuming a distance of $4.15$ kpc.",
"The upper limit on the average total mass-loss rate from the central 3\" of WR~112 is estimated to be $\\lesssim8\\times10^{-6}$ $\\mathrm{M}_\\odot$ yr$^{-1}$.",
"We leave its true nature as an open question, but propose that the WR~112 nebula may have formed in the outflow during a previous red or yellow supergiant phase of the central Wolf-Rayet star."
],
[
"Introduction",
"Wolf-Rayet (WR) stars are primarily the descendants of massive O-stars that drive powerful winds with terminal speeds $\\gtrsim 1000$ km/s and high mass-loss rates $\\gtrsim 10^{-5}$ $\\mathrm {M}_\\odot \\,\\mathrm {yr}^{-1}$ .",
"Carbon-rich WR (WC) stars, which are identified by broad C emission lines, are unique since many of them are observed to be efficient dust-making factories ($\\sim 10^{-6}$ $\\mathrm {M}_\\odot \\,\\mathrm {yr}^{-1}$ in dust; Gerhz & Hackwell 1974; Williams et al.",
"1987; Crowther 2007), despite their hot temperatures ($\\mathrm {T}_\\mathrm {eff}\\sim 40,000$ K) and large radiative power output ($\\mathrm {L}_*\\sim 10^5$ $\\mathrm {L}_\\odot $ ).",
"A majority of these dusty WC stars are in binary systems with an OB-star companion: strong winds from the WC star collide with weaker winds from the companion and create dense regions in the wake of the companion's orbit that are shielded from the harsh radiation field and allow for dust to condense (e.g.",
"Tuthill et al.",
"1999).",
"The tell-tale signature of this dust formation process is a remarkable “pinwheel\" that appears to rotate in accordance with the orbital motion of the system (Monnier, Tuthill & Danchi 1999, 2002; Tuthill et al.",
"1999, 2006).",
"Dusty WC systems provide a unique laboratory for investigating the mass-loss history of WR binaries since the morphology of the nebula is linked to the orbital parameters of the binary system.",
"These studies can help form a clearer picture of how WR stars evolve in the current paradigm where a majority of massive stars are expected to exchange mass with a close binary companion (Sana et al.",
"2012).",
"Observations of these systems have largely been performed using aperture-masking interferometry on near-IR facilities, which probe hot ($\\mathrm {T}_\\mathrm {dust}\\sim 1000$ K) inner regions of the pinwheel at high angular resolution (e.g.",
"Tuthill et al.",
"1999, 2006; Monnier et al.",
"2007).",
"Near-IR observations, however, are not sensitive to cooler dust that has propagated further from the central system.",
"The late-type WC+OB binary WR 112 (Massey & Conti 1983; van der Hucht 2001) is one of the few dusty WC systems whose nebula has been detected and resolved in the mid-infrared (8-18 $\\mu $ m; Marchenko et al.",
"2002, hereafter referred to as M2002).",
"Others known are WR 48a (Marchenko & Moffat 2007) and WR 140 (Monnier, Tuthill, & Danchi 2002; Williams et al.",
"2009).",
"The well studied WR 140 is the only one of these systems with kinematically measured masses, hosting a 15 M$_\\odot $ WC7 star and a 36 M$_\\odot $ O5 star (Fahed et al.",
"2011; Monnier et al.",
"2011).",
"The resolved mid-IR emission from WR 112, which resembles a series of up to 5 successive broken shells and arc-like filaments, traces cooler dust and provides an opportunity to probe further into the mass-loss history of the central system.",
"Based on mid-IR imaging from Gemini/OSCIR on 2001 May 7, M2002 interpret the WR 112 nebula as a pinwheel produced by the central WC colliding-wind binary with an orbital period of 24.8 yr.",
"Figure: (A) False-color VLT/VISIR image of WR 112 overlaid with the identified filaments and features.",
"Blue, green, and red colors correspond to emission in the PAH1 (8.598.59 μ\\mu m), NeII_2 (13.0413.04 μ\\mu m), and Q3 (19.5019.50 μ\\mu m) filters that have been convolved to a common Gaussian FWHM of 0.5 '' 0.5^{\\prime \\prime }.",
"The overlaid black contours correspond to the flux levels at 1, 2, 4, 8, 16, 32, and 64 % of the peak flux in the NeII_2 filter.",
"(B 1 _{1} - D 1 _{1}) WR 112 observed in the PAH1, NeII_2, and Q3 filters.",
"The black circles in the lower left represent the FWHM of the standard stars observed on the same night with the same filter (B 2 _{2} - D 2 _{2}) Deconvolved images of WR 112 at the identical waveband as the upper row.",
"The overlaid white contours in each image correspond to the flux levels at 1, 2, 4, 8, 16, 32, and 64 % of the peak flux at each respective filter.",
"Black circles represent the FWHM of the Gaussian PSF used to reconvolve the image.",
"North is up and east is to the left.In this letter, we present high spatial resolution imaging of WR 112 using the recently upgraded VLT spectrometer and imager for the mid-infrared (VISIR; Lagage et al.",
"2004) to investigate the M2002 pinwheel interpretation and the dust properties of the nebula.",
"We perform a multi-epoch analysis from archival space-based IR photometry and spectroscopy of WR 112, and compare our imaging results with that of M2002 and mid-IR observation performed on 2007 May 7 with the Thermal-Region Camera Spectrograph (T-ReCs; De Buizer & Fisher 2005) on Gemini South (P. ID: GS-2007A-Q-38)."
],
[
"Mid-IR Imaging Observations and Reduction",
"Observations of WR 112 (P.ID: 097.D-0707(A); P.I.",
"R. Lau) were performed using VISIR (Lagage et al.",
"2004) at the Cassegrain focus of UT3.",
"VISIR offers diffraction-limited imaging at high sensitivity in three atmospheric windows: the M-band at 5 $\\mu $ m, the N-band between $7-14$ $\\mu $ m, and the Q-band between $17-25$ $\\mu $ m. The new AQUARIUS detector in VISIR provided a field of view of $38\\times 38^{\\prime \\prime }$ with a plate scale of $0.045^{\\prime \\prime }$ per pixel.",
"Images of WR 112 (R.A. 18:16:33.49, Dec. -18:58:42.3; Cutri et al.",
"2003) were acquired with the PAH1 ($\\lambda _\\mathrm {c}=8.59$ $\\mu $ m, $\\Delta \\lambda =0.42$ ), NeII_2 ($\\lambda _\\mathrm {c}=13.04$ $\\mu $ m, $\\Delta \\lambda =0.22$ ), and Q3 ($\\lambda _\\mathrm {c}=19.50$ $\\mu $ m, $\\Delta \\lambda =0.40$ ) filters on 2016 Jul 14, 2016 Aug 9, and 2016 Jul 14, respectively.",
"Chopping and nodding were used to remove the sky and telescope thermal backgrounds.",
"The extent of WR 112 ($<20^{\\prime \\prime }$ ) allowed for an on-chip $20^{\\prime \\prime }$ -amplitude perpendicular chop-nod observing configuration.",
"The total on-source integration times with the PAH1, NeII_2, and Q3 filters were 5.6, 5.4, and 23.7 min, respectively.",
"Raw image files were accessed and downloaded from the ESO Science Archive Facility and processed using the Modest Image Analysis and Reduction (MIRA) software written by Terry Herter to reduce and analyze mid-IR imaging data.",
"The final images of WR 112 were calibrated against mid-IR standard stars obtained within the same night and using the standard-star flux catalog for VISIR imaging filters based on Cohen et al.",
"(1999).",
"A $10\\%$ and $20\\%$ uncertainty is adopted for the N- and Q-band imaging, respectively, based on the observed long-term variability of the absolute photometric calibration using the VISIR standard stars (Dobrzycka & Vanzi 2008).",
"The mean point spread function (PSF) of selected standard stars observed in 2016 was used to deconvolve the PAH1 and NeII_2 images of WR 112 and by a Richardson-Lucy deconvolution routine with a maximum of 1000 iterations.",
"Due to the low signal-to-noise ratio and poor stability of the Q3 standards, an Airy function convolved with a Gaussian was instead used as an artificial PSF for deconvolving the Q3 image.",
"The deconvolved PAH1, NeII_2, and Q3 images were then reconvolved with a Gaussian PSF with FWHM of $0.27^{\\prime \\prime }$ , $0.36^{\\prime \\prime }$ , and $0.50^{\\prime \\prime }$ , respectively."
],
[
"Warm Dust Morphology",
"Thermal mid-IR emission from warm dust in WR 112 exhibits a morphology of discontinuous filaments and arcs that are asymmetric about the central flux peak (see Fig.",
"REF A) and resembles the shape of a pinwheel as first interpreted by M2002.",
"The morphology of both the bright central $\\sim 3^{\\prime \\prime }$ region and outer filaments is largely consistent across all three wavebands.",
"The central 3” contains $\\gtrsim 90\\%$ of the total emission at each waveband and exhibits a “U\"-shaped morphology.",
"Interestingly, the orientation and shape of the inner mid-IR emission resembles the structure revealed from high-resolution near-IR imaging of WR 112 (FWHM$\\sim $ 20 mas; Monnier et al.",
"2007) but on larger size scales.",
"A circular arc $\\sim 1^{\\prime \\prime }$ south of the peak is apparent in the deconvolved PAH1- and NeII_2-band image (Fig.",
"REF $\\mathrm {B}_2$ and C$_2$ ), which is likely an artifact from the deconvolution.",
"Additionally, the $\\sim 0.5^{\\prime \\prime }$ -sized linear horizontal feature located 1.4” east of the peak in the PAH1-band image is a detector artifact caused by the position of the bright peak falling on the edge between two detector outputs.",
"The outer regions of WR 112 consist of a series of broken asymmetric arcs and filaments.",
"These observed structures are referred to as E3, E2, E1, W1, and W2 in Fig.",
"REF A).",
"The brightest feature outside of the inner region is located 2.7” east and slightly north of the central peak and is referred to as the W1 Ridge.",
"The observed properties of the identified structures are summarized in Tab.",
"."
],
[
"Observed Dust Temperature, Mass, and Luminosity",
"The temperature of the emitting warm dust is derived for each morphological component using the fluxes measured from the PAH1- and Q3-band fluxes.",
"It is assumed the emission is optically thin and takes the form of $F_\\nu \\propto B_\\nu (T_d)\\nu ^{\\beta }$ , where $B_\\nu (T_d)$ is the Planck function at frequency $\\nu $ and dust temperature $T_d$ , and $\\beta $ is the index of the emissivity power-law.",
"A value of -1.5 is adopted for $\\beta $ , which is consistent with amorphous carbon grains that are believed to compose the nebula (Chiar & Tielens 2001; M2002).",
"The derived temperatures provided in Tab.",
"are remarkably consistent with the radial temperature profile derived by M2002: $T_0\\left(\\frac{r}{r_0}\\right)^{\\gamma }$ , where $T_0=320$ K, $r_0=1^{\\prime \\prime }$ , and $\\gamma =-0.4$ .",
"For example, the profile predicts a temperature of 222 K at the location of the W1 Ridge, which is consistent with the estimated value of $217^{+12}_{-9}$ K. This decreasing radial temperature profile indicates that the nebula is heated centrally by the WR 112 system.",
"Dust masses were derived from the Q3-band flux and dust temperatures of the features as indicated in Eq.",
"REF : $M_d=\\frac{(4/3)\\,a\\,\\rho _b\\,F_\\lambda \\,d^2}{Q_C(\\lambda ,a )\\,B_\\lambda (T_d)},$ where $a$ is the adopted grain size, $\\rho _b$ is the bulk density of the dust grains, $F_\\lambda $ is the measured flux, $d$ is the distance to WR 112, and $Q_C(\\lambda ,a)$ is the grain emissivity model for amorphous carbon (Zubko et al.",
"2004).",
"A bulk density of $\\rho _b=2.2$ $\\mathrm {gm}$ $\\mathrm {cm}^{-3}$ (e.g.",
"Draine & Li 2007) and a grain size of $a=0.5$ $\\mu $ m (M2002) is assumed for the emitting amorphous carbon.",
"The dust mass estimated for each feature is provided in Tab. .",
"The total dust mass summed over all of the features is $2.6\\pm 0.4\\times 10^{-5}$ $\\mathrm {M}_\\odot $ , which again shows a remarkable agreement with the dust mass estimated from M2002 ($2.8\\times 10^{-5}$ $\\mathrm {M}_\\odot $ ).",
"In Fig.",
"REF , archival spectroscopic and photometric IR observations of WR 112 from various platforms over the past $\\sim 20$ yr are presented.",
"The IR space-based data shown in Fig.",
"REF were acquired from the Infrared Space Observatory (ISO; Kessler et al.",
"1996) Short Wavelength Spectrometer (SWS; de Graauw et al.",
"1996), SPIRIT III on the Midcourse Space Experiment (MSX; Mill et al.",
"1994), and the Infrared Camera (IRC) All-Sky Survey (Ishihara et al.",
"2010) from AKARI (Murakami et al.",
"2007).",
"The photometry of WR 112 from MSX and AKARI received the highest quality indicator.",
"A total observed IR luminosity ($2.4-45.4$ $\\mu $ m) of L$_\\mathrm {IR}=5.6\\times 10^{4}$ L$_\\odot $ was determined from the ISO/SWS spectrum assuming a distance of 4.15 kpc.",
"The IR luminosity is consistent with re-radiating a significant fraction of WR 112's stellar luminosity of $L_*=9\\times 10^{4}$ L$_\\odot $ , which is derived from its absolute $V$ -band magnitude ($M_V=-4.62$ , van der Hucht 2001) and bolometric correction of BC$_\\mathrm {V}=3.0$ (Smith, Meynet, Mermilliod 1994).",
"Figure: Multi-epoch IR photometry and spectroscopy of WR 112.",
"The gray lines around the ISO/SWS spectrum correspond to the 1-σ1-\\sigma flux uncertainty.",
"Photometry with no visible error bars indicates that the errors are smaller than the size of the plot marker.Despite observations of variable near-IR (Williams et al.",
"2015) and radio (Monnier et al.",
"2002; Yam et al.",
"2015) emission, the mid-IR data do not indicate significant variations in the dust mass or temperature over time.",
"The orbital properties of the central colliding-wind binary system, which are presumably linked to the variable near-IR and radio emission, may therefore not be associated with the production of the observed dust in the nebula.",
"Importantly, the assumed 24.8 yr period of the WR 112 binary (M2002) is well-sampled by the IR observations taken at 1996 (ISO and MSX), 2001 (Gemini), 2006-07 (AKARI), and 2016 (VLT)."
],
[
"Comparison to the “Pinwheel\" Model",
"The current interpretation of the WR 112 nebula is that of a broken “Pinwheel\" produced by a central colliding-wind late-type WC9+OB binary system (M2002; Monnier et al.",
"2002, 2007) due to the regularly spaced features and their similar appearance.",
"Under this interpretation, one can estimate the orbital period of the central system and predict the appearance of the nebula assuming an expansion velocity.",
"Our mid-IR imaging data provides a sufficient temporal baseline against previous resolved mid-IR images (M2002) to test the validity of the pinwheel model.",
"Additionally, we include an intermediary mid-IR imaging epoch of WR 112 taken by Gemini/T-ReCsThe Gemini/T-ReCs image was deconvolved using the same routine as the VISIR images (Sec.",
"), except a Gaussian with a FWHM of $0.45^{\\prime \\prime }$ was adopted for the PSF.",
"on 2007 May 7 (P. ID: GS-2007A-Q-38) with the Si5 filter ($\\lambda _\\mathrm {eff}=11.6$ $\\mu $ m) in order to identify morphological variations on shorter timescales.",
"We adopt the high-eccentricity colliding-wind binary/pinwheel model fit by M2002 to their 2001 mid-IR images of WR 112 and compare their results against the 2007 and 2016 images (Fig.",
"REF A and B).",
"The model parameters assume an expansion velocity of 1200 km $\\mathrm {s}^{-1}$ and the following orbital parameters for the central the binary system: $i=35^\\circ $ , $\\Omega =130^\\circ $ , $\\omega =165^\\circ $ , $e=0.40$ , and $P = 23.5$ yr.",
"Predicted 2007 and 2016 pinwheel models overlaid in Fig.",
"REF A and B (blue dashed lines) are derived by propagating the 2001 model assuming a velocity of 1200 km $\\mathrm {s}^{-1}$ .",
"By comparing the pinwheel models and imaging data, we draw the following conclusions: 1.",
"The mid-IR nebula is not propagating in accordance with the colliding-wind binary/pinwheel model proposed by M2002.",
"2.",
"There is no apparent expansion of the nebula in the 6, 9, and 15 yr intervals between the 2001, 2007, and 2016 observations.",
"The“stagnant\" morphology implies an upper limit on the expansion velocity of $\\lesssim 120$ km $\\mathrm {s}^{-1}$ between 2007 and 2016 images assuming a $\\sim 1$ VISIR pixel (0.045”) centroid alignment uncertainty, a distance of 4.15 kpc, and an inclination of $i=35^\\circ $ .",
"The low velocity is unlikely due to geometric projection effects since a near edge-on inclination of $i\\sim 85^\\circ $ would be required to infer a velocity consistent with the predicted 1200 km s$^{-1}$ WR outflow.",
"Such a high inclination is inconsistent with the near face-on and circular appearance of the nebula.",
"The upper limit derived for the expansion velocity is almost an order of magnitude less than the predicted and observed velocities of dusty outflows produced in colliding-wind WR binaries like WR 140 (Williams et al.",
"2009) and WR 104 (Tuthill et al.",
"1999).",
"Additionally, the mid-IR fluxes and inferred dust properties of WR 112 (Tab. )",
"do not exhibit significant variability, whereas the orbitally-modulated dust production in WR 140 shows an order of magnitude variation in 8.75 and 12.5 $\\mu $ m emission (Williams et al.",
"2009).",
"It is therefore unlikely that the nebula originated from the high velocity outflows of the central WR binary.",
"An alternative dust production scenario is thus required to explain the presence of the nebula."
],
[
"On the Nature of the WR 112 Nebula: Mass-loss from a Previous Evolutionary Phase?",
"Here, we discuss a possible scenario that can account for stagnant or slowly expanding shells in the vicinity of the WR 112 system.",
"It is important to note that previous studies have spectroscopically verified the presence of a WC9 star at the center of WR 112 (Massey & Conti 1983; Figer, McLean, & Najarro 1997).",
"Additionally, a binary companion is inferred from variable non-thermal radio emission that is believed to originate from a central wind-collision zone (Chapman et al.",
"1999; Monnier et al.",
"2002).",
"The unresolved radio counterpart of WR 112 exhibits a peculiar motion of $-100\\pm 34$ km $\\mathrm {s}^{-1}$ in Galactic longitude (Yam et al.",
"2015), which is not consistent with the position angle of the shells nor the central “U\"-shaped region if they were instead bow shocks from the interstellar medium.",
"Larger-sized nebulae ($\\gtrsim 1$ pc) surrounding WR stars are commonly interpreted as ejecta from a previous phase of high mass-loss.",
"The ejecta can originate from the slow and dense winds of the star during its luminous blue variable (LBV; $v\\sim 100$ km $\\mathrm {s}^{-1}$ ) or red supergiant (RSG; $v\\sim 30$ km $\\mathrm {s}^{-1}$ ) phase, later interacting with the high velocity winds and hard radiation field from the subsequent WR phase (e.g.",
"Garcia-Segura & Mac Low 1995; Freyer et al.",
"2006; Toalá et al.",
"2015).",
"Another potential origin for the WR 112 nebula is episodic mass-loss during a short-lived post-RSG yellow supergiant (YSG) phase, where the star undergoes a blueward evolution to the WR phase (e.g.",
"de Jager 1998; Smith et al.",
"2004; Humphreys et al.",
"2013; Gordon et al.",
"2016).",
"We attempt to distinguish amongst an LBV, YSG, and RSG phase for the origin of the WR 112 nebula by estimating upper limits on the average mass-loss rate.",
"Assuming a gas-to-dust mass ratio of 100 and an expansion velocity $<120$ km $\\mathrm {s}^{-1}$ , the average mass-loss rate from the central 3” (0.06 pc) region is $\\dot{M}<8\\times 10^{-6}$ $\\mathrm {M}_\\odot $ yr$^{-1}$ .",
"This upper limit is several orders of magnitude lower than the observed mass-loss rates from dust-forming LBVs during giant eruptions ($\\gtrsim 10^{-3}$ $\\mathrm {M}_\\odot $ yr$^{-1}$ ; Kochanek 2011; Smith 2014); however, the mass-loss rate and total dust mass in the nebula is consistent with values derived from circumstellar material surrounding RSGs and YSGs ($\\dot{M}\\sim 10^{-5}-10^{-4}$ $\\mathrm {M}_\\odot $ yr$^{-1}$ , M$_\\mathrm {d}\\sim 10^{-5}-10^{-4}$ M$_\\odot $ ; Gordon et al.",
"2016).",
"Therefore, if the nebula formed prior to the WR phase it most likely originated from an RSG or post-RSG/YSG outflow, but we cannot decisively rule out an LBV origin.",
"We note that these evolved phases of massive stars typically exhibit an oxygen-rich chemistry, which conflicts with the inferred carbon-rich composition of the nebula (Chiar & Tielens 2001; M2002).",
"However, the presence of amorphous carbon was determined from spatially unresolved mid-IR spectroscopy dominated by emission from the central $\\sim 3^{\\prime \\prime }$ of WR 112.",
"Identifying chemical differentiation between the central and extended regions of the nebula would aid in testing the RSG/YSG hypotheses.",
"The YSG interpretation is interesting from an evolutionary standpoint because the proximity of the nebula to the central system and its low expansion velocity imply that the WR star has newly transitioned.",
"However, the carbon-rich photosphere of WR 112 indicates it is in the most evolved stage of the WR sequence (e.g.",
"Crowther 2007).",
"An estimate of the dynamical age of the W1 filament assuming an YSG-like outflow velocity of 50 km s$^{-1}$ implies that WR 112 may have transitioned within the past $\\sim 1000$ yr, close to the observationally estimated lifetime of YSGs ($\\sim 3000$ yr; Drout et al.",
"2009) but two orders of magnitude less than the predicted WR star lifetime ($\\sim 10^5$ yr; Meynet & Maeder 2005).",
"One possible explanation for the rapid evolution of WR 112 may be mass-transfer via interactions with a close binary companion (e.g.",
"Smith et al.",
"2011a).",
"Two notable examples of observed mass transfer likely leading to a stripped-envelope WR star are NaSt1 (Mauerhan et al.",
"2015) and RY Scuti (Smith et al.",
"2011b).",
"WR 112 may therefore be another such example of binary interaction influencing the evolution of massive stars.",
"Ultimately, we leave the true nature of WR 112 as an open question since we are only able to conclude that the stagnant nebula is inconsistent with dusty WC outflows."
],
[
"Future Work",
"High spatial resolution mid-IR spectroscopy of WR 112 would provide crucial information on the dust composition and chemical abundances throughout the nebula to verify the origin of their formation and the conditions of grain growth.",
"Given the sensitivity limitations of current ground-based platforms due to Earth's atmosphere, we require the combined sensitivity and spatial resolution that will be achievable with the Mid-Infrared Instrument (MIRI; e.g.",
"Wells et al.",
"2015) on the James Webb Space Telescope (JWST), which is expected to launch Oct 2018.",
"JWST will be the ideal platform for investigating the origin of the compact warm nebulae around dusty WC systems and deciphering the mass loss history and evolution of their central engines.",
"Acknowledgments.",
"This work is based on observations made with the VISIR instrument on the ESO VLT telescope (program ID.",
"097.D-0707A) and observations obtained at the Gemini Observatory (P. ID: GS-2007A-Q-38), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil).",
"This work was partially carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.",
"M.J.H.",
"acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No.",
"DGE-1144153.",
"J.S.B acknowledges that this work was partly supported by OPTICON, which is sponsored by the European Commission's FP7 Capacities programme (Grant number 312430).",
"AFJM is grateful for financial aid to NSERC (Canada) and FQRNT (Quebec).",
"R.S.",
"acknowledges funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n$^\\mathrm {o}$ [614922].",
"R.L.",
"would like to thank Sergey Marchenko for valuable feedback and comments, and Jim De Buizer and James Radomski for the helpful insight on T-ReCs imaging data.",
"R.L.",
"also thanks Astrid Lamberts for the enlightening conversations on colliding wind binaries.",
"Lastly, we thank the anonymous referee for the insightful suggestions and comments."
],
[
"Tables",
"cccccccc Observed fluxes and dust properties throughout the WR 112 nebula 0pt Region $l$ (”) $d$ (”) $\\mathrm {F}_\\mathrm {PAH1}$ $\\mathrm {F}_\\mathrm {NeII\\_2}$ $\\mathrm {F}_\\mathrm {Q3}$ $\\mathrm {T}_\\mathrm {d}$ (K) $\\mathrm {M}_\\mathrm {d}$ $(10^{-6}\\,\\mathrm {M}_\\odot )$ W1 3.7 2.51.162.442.31$216_{-9}^{+12}$$2.5_{-0.4}^{+0.4}$ W2 4.5 4.20.10.490.48$179_{-6}^{+8}$$1.1_{-0.2}^{+0.2}$ W1 Ridge 1.0 2.50.30.680.58$217_{-9}^{+12}$$0.6_{-0.1}^{+0.1}$ E1 3.61.81.933.172.82$233_{-10}^{+14}$$2.4_{-0.4}^{+0.4}$ E2 3.62.70.360.870.85$207_{-8}^{+11}$$1.1_{-0.2}^{+0.2}$ E3* 2.7 4.20.120.310.19 – – Center 3 $<1.5$ 179.3140.71.37$355_{-25}^{+37}$$18.6_{-3.8}^{+3.6}$ Fluxes given are provided in Jy.",
"The approximate length, $l$ , and distance from the central peak, $d$ , of each feature is given in arc seconds.",
"- Dust temperatures and masses were not derived for the E3 filament due to the low signal-to-noise ratio of the detection in the Q3 image.",
"cccccc Multi-Epoch IR Photometry and Derived Dust Properties of WR 112 0pt Date Observatory Band (Wavelength) Flux (Jy) $\\mathrm {T}_\\mathrm {d}$ (K) $\\mathrm {M}_\\mathrm {d}$ $(10^{-5}\\,\\mathrm {M}_\\odot )$ Aug 1996 MSX/SPIRIT III B1 $(4.29\\,\\mu \\mathrm {m})$ $107.2\\pm 9.3$ $335_{-7}^{+7}$ $2.2_{-0.1}^{+0.1}$ A $(8.28\\,\\mu \\mathrm {m})$ $152\\pm 6.2$ C $(12.13\\,\\mu \\mathrm {m})$ $136\\pm 6.8$ D $(14.65\\,\\mu \\mathrm {m})$ $120\\pm 7.3$ E $(21.34\\,\\mu \\mathrm {m})$ $59\\pm 3.6$ May 2006 - Aug 2007 AKARI/IRC S9W $(9\\,\\mu \\mathrm {m})$ $152.6\\pm 0.86$ $329_{-1}^{+1}$ $2.2_{-0.02}^{+0.02}$ L18W $(18\\,\\mu \\mathrm {m})$ $83.5\\pm 0.15$ Jul 2016 VLT/VISIR PAH1 $(8.59\\,\\mu \\mathrm {m})$ $194\\pm 19$ $331_{-21}^{+31}$ $2.8_{-0.6}^{+0.5}$ Aug 2016 NeII_2 $(13.04\\,\\mu \\mathrm {m})$ $161\\pm 16$ Jul 2016 Q3 $(19.50\\,\\mu \\mathrm {m})$ $92\\pm 18$ Dust temperatures, T$_\\mathrm {d}$ , were derived from the A/E, S9W/L18W, and PAH1/Q3 bands in for the observations provided by MSX, AKARI, and VLT, respectively.",
"Dust masses, M$_\\mathrm {d}$ , were then derived from Eq.",
"REF assuming identical dust properties that were used to provide the values in Tab."
]
] | 1612.05650 |
[
[
"Targeting Infeasibility Questions on Obfuscated Codes"
],
[
"Abstract Software deobfuscation is a crucial activity in security analysis and especially, in malware analysis.",
"While standard static and dynamic approaches suffer from well-known shortcomings, Dynamic Symbolic Execution (DSE) has recently been proposed has an interesting alternative, more robust than static analysis and more complete than dynamic analysis.",
"Yet, DSE addresses certain kinds of questions encountered by a reverser namely feasibility questions.",
"Many issues arising during reverse, e.g.",
"detecting protection schemes such as opaque predicates fall into the category of infeasibility questions.",
"In this article, we present the Backward-Bounded DSE, a generic, precise, efficient and robust method for solving infeasibility questions.",
"We demonstrate the benefit of the method for opaque predicates and call stack tampering, and give some insight for its usage for some other protection schemes.",
"Especially, the technique has successfully been used on state-of-the-art packers as well as on the government-grade X-Tunnel malware -- allowing its entire deobfuscation.",
"Backward-Bounded DSE does not supersede existing DSE approaches, but rather complements them by addressing infeasibility questions in a scalable and precise manner.",
"Following this line, we propose sparse disassembly, a combination of Backward-Bounded DSE and static disassembly able to enlarge dynamic disassembly in a guaranteed way, hence getting the best of dynamic and static disassembly.",
"This work paves the way for robust, efficient and precise disassembly tools for heavily-obfuscated binaries."
],
[
"Introduction",
"Context.",
"Obfuscation [1] is a prevalent practice aiming at protecting some functionalities or properties of a program.",
"Yet, while its legitimate goal is intellectual property protection, obfuscation is widely used for malicious purposes.",
"Therefore, (binary-level) software deobfuscation is a crucial task in reverse-engineering, especially for malware analysis.",
"A first step of deobfuscation is to recover the most accurate control-flow graph of the program (disassembly), i.e.",
"to recover all instructions and branches of the program under analysis.",
"This is already challenging for non-obfuscated codes due to tricky (but common) low-level constructs [2] like indirect control flow (computed jumps, jmp eax) or the interleaving of code and data.",
"But the situation gets largely worst in the case of obfuscated codes.",
"Standard disassembly approaches are essentially divided into static methods and dynamic methods.",
"On one hand, static (syntactic) disassembly tools such as IDA or Objdump have the potential to cover the whole program.",
"Nonetheless, they are easily fooled by obfuscations such as code overlapping [3], opaque predicates [4], opaque constants [5], call stack tampering [6] and self-modification [7].",
"On the other hand, dynamic analysis cover only a few executions of the program and might miss both significant parts of the code and crucial behaviors.",
"Dynamic Symbolic Execution (DSE) [8], [9] (a.k.a concolic execution) is a recent and fruitful formal approach to automatic testing, has recently been proposed has an interesting approach for disassembly [10], [11], [12], [13], [14], more robust than static analysis and covering more instructions than dynamic analysis.",
"Currently, only dynamic analysis and DSE are robust enough to address heavily obfuscated codes.",
"Problem.",
"Yet, these dynamic methods only address reachability issues, namely feasibility questions, i.e.",
"verifiying that certain events or setting can occur, e.g.",
"that an instruction in the code is indeed reachable.",
"Contrariwise, many questions encountered during reversing tasks are infeasibility questions, i.e.",
"checking that certain events or settings cannot occur.",
"It can be used either for detecting obfuscation schemes, e.g.",
"detecting that a branch is dead (i.e.",
"it cannot be taken) or to prove their absence, e.g.",
"proving that a computed jump cannot lead to an improper address.",
"These infeasibility issues are currently a blind spot of both standard and advanced disassembly methods.",
"Dynamic analysis and DSE do not answer the question because they only consider a finite number of paths while infeasibility is about considering all paths.",
"Also, (standard) syntactic static analysis is too easily fooled by unknown patterns.",
"Finally, while recent semantic static analysis approaches [15], [13], [16], [17] can in principle address infeasibility questions, they are currently neither scalable nor robust enough.",
"At first sight infeasibility is a simple mirror of feasibility, however from an algorithmic point of view they are not the same problem.",
"Indeed, since solving feasibility questions on general programs is undecidable, practical approaches have to be one-sided, favoring either feasibility (i.e.",
"answering “feasible” or \"I don't know”) or infeasibility (i.e.",
"answering \"I don't know” or “infeasible”).",
"While there currently exist robust methods for answering feasibility questions on heavily obfuscated codes, no such method exist for infeasibility questions.",
"Goal and challenges.",
"In this article, we are interested in solving automatically infeasibility questions occurring during the reversing of (heavily) obfuscated programs.",
"The intended approach must be precise (low rates of false positives and false negatives) and able to scale on realistic codes both in terms of size (efficient) and protection – including self-modification (robustness), and generic enough for addressing a large panel of infeasibility issues.",
"Achieving all these goals at the same time is particularly challenging.",
"Our proposal.",
"We present Backward-Bounded Dynamic Symbolic Execution (bb-dse), the first precise, efficient, robust and generic method for solving infeasibility questions.",
"To obtain such a result, we have combined in an original and fruitful way, several state-of-the-art key features of formal software verification methods, such as deductive verification [18], bounded model checking [19] or DSE.",
"Especially, the technique is goal-oriented for precision, bounded for efficiency and combines dynamic information and formal reasoning for robustness.",
"Contribution.",
"The contribution of this paper are the following: First, we highlight the importance of infeasibility issues in reverse and the urging need for automating the investigation of such problems.",
"Indeed, while many deobfuscation-related problems can be encoded as infeasibility questions (cf.",
"Section ) it remains a blind spot of state-of-the-art disassembly techniques.",
"Second, we propose the new Backward-Bounded DSE algorithm for solving infeasibility queries arising during deobfuscation (Section ).",
"The approach is both precise (low rates of false positives and false negatives), efficient and robust (cf.",
"Table REF ), and it can address in a generic way a large range of deobfuscation-related questions – for instance opaque predicates, call stack tampering or self-modification (cf.",
"Section ).",
"The technique draws from several separated advances in software verification, and combines them in an original and fruitful way.",
"We present the algorithm along with its implementation within the Binsec open-source platform http://binsec.gforge.inria.fr/ [20], [21].",
"Third, we perform an extensive experimental evaluation of the approach, focusing on two standard obfuscation schemes, namely opaque predicates and call stack tampering.",
"In a set of controlled experiments with ground truth based on open-source obfuscators (cf.",
"Section ), we demonstrate that our method is very precise and efficient.",
"Then, in a large scale experiment with standard packers (including self-modification and other advanced protections), the technique is shown to scale on realistic obfuscated codes, both in terms of efficiency and robustness (cf.",
"Section ).",
"Finally, we present two practical applications of Backward-Bounded DSE.",
"First, we describe an in-depth case-study of the government-grade malware X-Tunnel [22] (cf.",
"Section ), where bb-dse allows to identify and remove all obfuscations (opaque predicates).",
"We have been able to automatically extract a de-obfuscated version of functions – discarding almost 50% of dead and “spurious” instructions, and providing an insights into its protection schemes, laying a very good basis for further in-depth investigations.",
"Second, we propose sparse disassembly (cf.",
"Section ), a combination of Backward-Bounded DSE, dynamic analysis and standard (recursive, syntactic) static disassembly allowing to enlarge dynamic disassembly in a precise manner – getting the best of dynamic and static techniques, together with encouraging preliminary experiments.",
"Our implementation and experimental data will be made available if the paper is accepted for publication.",
"Discussion.",
"Several remarks must be made about the work presented in this paper.",
"First, while we essentially consider opaque predicates and call stack tampering, bb-dse can also be useful in other obfuscation contexts, such as flattening or virtualization.",
"Also self-modification is inherently handled by the dynamic aspect of bb-dse.",
"Second, while we present one possible combination for sparse disassembly, other combinations can be envisioned, for example by replacing the initial dynamic analysis by a (more complete) DSE [10] or by considering more advanced static disassembly techniques [2].",
"Finally, some recent works target opaque predicate detection with standard forward DSE [12].",
"As already pointed out, DSE is not tailored to infeasibility queries, while bb-dse is – cf.",
"Sections and .",
"Impact.",
"Backward-Bounded DSE does not supersede existing disassembly approaches, it complements them by addressing infeasibility questions.",
"Altogether, this work paves the way for robust, precise and efficient disassembly tools for obfuscated binaries, through the careful combination of static/dynamic and forward/backward approaches.",
"Table: Disassembly methods for obfuscated codes"
],
[
"Background",
"Obfuscation.",
"These transformations [1] aim at hiding the real program behavior.",
"While approaches such as virtualization or junk insertion make instructions more complex to understand, other approaches directly hide the legitimate instructions of the programs – making the reverser (or the disassembler) missing essential parts of the code while wasting its time in dead code.",
"The latter category includes for example code overlapping, self-modification, opaque predicates and call stack tampering.",
"We are interested here in this latter category.",
"For the sake of clarity, this paper mainly focuses on opaque predicates and call stack tampering.",
"An opaque predicate always evaluates to the same value, and this property is ideally difficult to deduce.",
"The infeasible branch will typically lead the reverser (or disassembler) to a large and complex portion of useless junk code.",
"Figure REF shows the x86 encoding of the opaque predicate $7y^2 - 1 \\ne x^2$ , as generated by O-LLVM [23].",
"This condition is always false for any values of ds:x, ds:y, so the conditional jump jz $<$ addr_trap$>$ is never going to be taken.",
"A (call) stack tampering, or call/ret violation, consists in breaking the assumption that a ret instruction returns to the instruction following the call (return site), as exemplified in Figure REF .",
"The benefit is twofold: the reverser might be lured into exploring useless code starting from the return site, while the real target of the ret instruction will be hidden from static analysis.",
"Figure: opaque predicate: 7y 2 -1≠x 2 7y^2 - 1 \\ne x^2Figure: Standard stack tamperingDisassembly.",
"We call legit, an instruction in a binary if it is executable in practice.",
"Two qualities expected for a disassembly are (1) soundness: does the algorithm recover only legit instructions?",
"and (2) completeness: does the algorithm recover all legit instructions?",
"Standard disassembly approaches essentially include (static) recursive disassembly, (static) linear sweep and dynamic disassembly.",
"Recursive disassembly consists in exploring the executable file from a given (list of) entry point(s), recursively following the possible successors of each instructions.",
"This technique may miss a lot of instructions, typically due to computed jums (jmp eax) or self-modification.",
"In addition, the approach is easily fooled into disassembling junk code obfuscated by opaque predicates or call stack tampering.",
"As such, the approach is neither safe nor complete.",
"Linear sweep consists in decoding linearly all possible instructions in the code sections.",
"The technique aims at being more complete than recursive traversal, yet it comes at the price of many additional misinterpreted code instructions.",
"Meanwhile, the technique can still miss instructions hidden by code overlapping or self-modification.",
"Hence the technique is unsafe, and incomplete on obfuscated codes.",
"Dynamic disassembly retrieves only legit instructions and branches observed at runtime on one or several executions.",
"The technique is safe, but potentially highly incomplete – yet, it does recover part of the instructions masked by self-modification, code overlapping, etc.",
"For example, while Objdump is solely based on linear sweep, IDA performs a combination of linear sweep and recursive disassembly (geared with heuristics).",
"Dynamic Symbolic Execution.",
"Dynamic Symbolic Execution (DSE) [9], [8] (a.k.a concolic execution) is a formal technique for exploring program paths in a systematic way.",
"For each path $\\pi $ , the technique computes a symbolic path predicate $\\Phi _{\\pi }$ as a set of constraints on the program input leading to follow that path at runtime.",
"Intuitively, $\\Phi _{\\pi }$ is the conjunction of all the branching conditions encountered along $\\pi $ .",
"This path predicate is then fed to an automatic solver (typically a SMT solver [24]).",
"If a solution is found, it corresponds to an input data exercising the intended path at runtime.",
"Path exploration is then achieved by iterating on all (user-bounded) program paths, and paths are discovered lazily thanks to an interleaving of dynamic execution and symbolic reasoning [25], [26].",
"Finally, concretization [25], [26], [27] allows to perform relevant under-approximations of the path predicate by using the concrete information available at runtime.",
"The main advantages of DSE are correctness (no false negative in theory, a bug reported is a bug found) and robustness (concretization does allow to handle unsupported features of the program under analysis without losing correctness).",
"Moreover, the approach is easy to adapt to binary code, compared to other formal methods [28], [8], [29], [30].",
"The very main drawback of DSE is the so-called path explosion problem: DSE is doomed to explore only a portion of all possible execution paths.",
"As a direct consequence, DSE is incomplete in the sense that it can only prove that a given path (or objective) is feasible (or coverable), but not that it is infeasible.",
"DSE is interesting for disassembly and deobfuscation since it enjoys the advantages of dynamic analysis (especially, safe disassembly and robustness to self-modification or code overlapping), while being able to explore a larger set of behaviors.",
"Yet, while on small examples DSE can achieve complete disassembly, it often only slightly improves coverage (w.r.t.",
"pure dynamic analysis) on large and complex programs."
],
[
"Motivation",
"Let us consider the obfuscated pseudo-code given in Figure REF .",
"The function <main> contains an opaque predicate in 1 and a call stack tampering in 2.",
"Figure: Motivating exampleGetting the information related to the opaque predicate and the call stack tampering would allow to: 1 to know that <fun1> is always called and reciprocally that <fun2> is never called.",
"As consequence b and d are dead instructions; 2 to know that the ret of <fun1> is tampered and never return to the caller.",
"As consequence a and c are dead instructions.",
"Such trick would also allow to hide the real payload located at <X>.",
"Hence the main motivation is not to be fooled by such infeasibility-based tricks that slow-down the program reverse-engineering and its global understanding.",
"Applications.",
"The main application is to improve a disassembly algorithm with such information, since static disassembly will be fooled by such tricks and dynamic disassembly will only cover a partial portion of the program.",
"Our goal is to design an efficient method for solving infeasibility questions.",
"This approach could then passes the original code annotated with infeasibility highlights to other disassembly tools, which could take advantage of this information – for example by avoiding disassembling dead instructions.",
"Such a view is depicted in Figure REF , and a throughout study of such combination is discussed in Section .",
"Figure: motivation schemaMoreover, such infeasibility related information could also be used in other contexts, for instance to obtain more accurate code coverage rates in software testing or to guide vulnerability analysis toward weak parts of the code."
],
[
"Backward-Bounded DSE",
"We present in this section the new Backward-Bounded DSE technique dedicated to solving infeasibility queries on binary codes.",
"Preliminaries.",
"We consider a binary-level program $P$ with a given initial code address $a_0$ .",
"A state $s \\triangleq (a,\\sigma )$ of the program is defined by a code address $a$ and a memory state $\\sigma $ , which is a mapping from registers and memory to actual values (bitvectors, typically of size 8, 32 or 64).",
"By convention, $s_0$ represents an initial state, i.e.",
"$s_0$ is of the form $(a_0,\\sigma )$ .",
"The transition from one state to another is performed by the $post$ function that execute the current instruction.",
"An execution $\\pi $ is a sequence $\\pi \\triangleq (s_0 \\cdot s_1 \\cdot ... \\cdot s_n)$ , where $s_{j+1}$ is obtained by applying the $post$ function to $s_j$ ($s_{j+1}$ is the successor of $s_j$ ).",
"Let us consider a predicate $\\varphi $ over memory states.",
"We call reachability condition a pair $c \\triangleq (a,\\varphi )$ , with $a$ a code address.",
"Such a condition $c$ is feasible if there exists a state $s \\triangleq (a,\\sigma )$ and an execution $\\pi _s \\triangleq (s_0 \\cdot s_1 \\cdot ... \\cdot s)$ such that $\\sigma $ satisfies $\\varphi $ , denoted $\\sigma \\models \\varphi $ .",
"It is said infeasible otherwise.",
"An feasibility (resp.",
"infeasibility) question consists precisely in trying to solve the feasibility (resp.",
"infeasibility) of such a reachability condition.",
"These definitions do not take self-modification into account.",
"They can be extended to such a setting by considering code addresses plus waves or phases [3].",
"Principles.",
"We build on and combine 3 key ingredients from popular software verification methods: backward reasoning from deductive verification, for precise goal-oriented reasoning; combination of dynamic analysis and formal methods (from DSE), for robustness; bounded reasoning from bounded model checking, for scalability and the ability to perform infeasibility proofs.",
"The initial idea of bb-dse is to perform a backward reasoning, similar to the one of DSE but going from successors to predecessors (instead of the other way).",
"Formally, DSE is based on the $post$ operation while bb-dse is based on its inverse $pre$ .",
"Perfect backward reasoning $pre^*$ (i.e.",
"fixpoint iterations of relation $pre$ , collecting all predecessors of a given state or predicate) can be used to check feasibility and infeasibility questions.",
"But this relation is not computable.",
"Hence, we rely on computable bounded reasoning, namely $pre^{k}$ , i.e.",
"collecting all the “predecessors in $k$ steps” ($k$ -predecessors) of a given state (or predicate).",
"Now symmetry does not hold anymore: while $pre^{k}$ can answer positively to infeasibility queries (if a predicate has no $k$ -predecessor, it has no $k^{\\prime }$ -predecessor for any $k^{\\prime }>k$ and cannot be reached), but cannot falsify them (because it could happen that a predicate is infeasible, for a reason beyond the bound $k$ ).",
"Moreover, it is efficient as the computation does not depend on the program size or trace length, but on the user-chosen bound $k$ .",
"In practice, checking whether $pre^{ k} = \\emptyset $ can be done in a symbolic way, like it is done in DSE: the set $pre^{ k}$ is computed implicitly as a logical formula (typically, a quantifier-free first-order formula over bitvectors and arrays), which is unsatisfiable iff the set if empty.",
"This formula is then passed to an automatic solver, typically a SMT solver [24] such as Z3.",
"Yet, backward reasoning is still very fragile at binary-level, since computing $pre$ in a perfect way may be highly complex because of dynamic jumps or self-modification.",
"The last trick is to combine this $pre^{k}$ reasoning with dynamic traces, so that the whole approach benefits from the robustness of dynamic analysis.",
"Actually, the $pre^{k}$ is now computed w.r.t.",
"the control-flow graph induced by a given trace $\\pi $ – in a dynamic disassembly manner.",
"We denote this sliced $pre^{k}$ by $pre_{\\pi }^{k}$ .",
"Hence we get robustness, yet since some real parts of $pre^{ k}$ may be missing from $pre_{\\pi }^{ k}$ , we now lose correctness: we may have false positive FP (because $pre_{\\pi }^{k}$ will be incomplete w.r.t $pre^{k}$ ), additionally to the false negative FN due to “boundedness” (because of too small $k$ ).",
"A picture of the approach is given in Figure REF .",
"Algorithm.",
"Considering a reachability condition $(a,\\varphi )$ , bb-dse starts with a dynamic execution $\\pi $ : if $\\pi $ reaches code address $a$ , then compute $pre_{\\pi }^{ k}((a,\\varphi ))$ as a formula and solve it if it is UNSAT, then the result is INFEASIBLE; if it is SAT, then the result is UNKOWN; if it is TO (timeout), then the result is TO; otherwise the result is UNKOWN.",
"As a summary, this algorithm enjoys the following good properties: it is efficient (depends on $k$ , not on the trace or program length) and as robust as dynamic analysis.",
"On the other hand, the technique may report both false negative (bound $k$ too short) and false positive (dynamic CFG recovery not complete enough).",
"Yet, in practice, our experiments demonstrate that the approach performs very well, with very low rates of FP and FN.",
"Experiments are presented in Sections , and .",
"By convenience, we will not distinguished anymore between the predicate $\\varphi $ and the reachability condition $(a,\\varphi )$ if $a$ is clear from context.",
"Figure: pre k pre^k schemaImplementation.",
"This algorithm is implemented on top of Binsec/se [21], a forward DSE engine inside the open-source platform Binsec [20] geared to formal analysis of binary codes.",
"The platform currently proposes a front-end from x86 (32bits) to a generic intermediate representation called DBA [31] (including decoding, disassembling, simplifications).",
"It also provides several semantic analyses, including the Binsec/se DSE engine [21].",
"Binsec/se features a strongly optimized path predicate generation as well as highly configurable search heuristics [21], [13] and C/S policies [27].",
"The whole platform amount for more than 40k of OCaml line of codeshttp://binsec.gforge.inria.fr/tools.",
"Binsec also makes use of two other components.",
"First, the dynamic instrumentation called Pinsec, based on Pin in charge to run the program and to record all runtime values along with self-modification layers.",
"Written in C++ it amounts for more than 3k lines of code.",
"Second, Idasec is an IDA plugin written in Python ($\\sim $ 13k loc) aiming at triggering analyzes and post-processing results generated by Binsec.",
"The bb-dse algorithm is tightly integrated in the Binsec/se component.",
"Indeed, when solving a predicate feasibility, Binsec/se DSE performs a backward pruning pass aiming at removing any useless variable or constraint.",
"bb-dse works analogously, but also takes into account the distance from the predicate to solve: any definition beyond the $k$ bound is removed.",
"In a second phase, the algorithm creates a new input variable for any variable used but never defined in the sliced formula.",
"The $k$ bound value is defined by the user and can be modulated as needed."
],
[
"Solving Infeasibility Questions with ",
"We show in this section how several natural problems encountered during deobfuscation and disassembly can be thought of as infeasibility questions, and solved with bb-dse."
],
[
"Opaque Predicates",
"As already stated in Section , an opaque predicate (OP) is a predicate always evaluating to the same value.",
"They have successfully been used in various domains [32], [1].",
"Recent works [12] identify three kinds of opaque predicates: invariant: always true/false due to the structure of the predicate itself, regardless of inputs values, contextual: opaque due to the predicate and its constraints on input values, dynamic: similar to contextual, but opaqueness comes from dynamic properties on the execution (e.g. memory).",
"Approach with bb-dse.",
"Intuitively, to detect an opaque predicate the idea is to backtrack all its data dependencies and gather enough constraints to conclude to the infeasibility of the predicate.",
"If the predicate is local (invariant), the distance from the predicate to its input instantiation will be short and the predicate will be relatively easy to break.",
"Otherwise (contextual, dynamic) the distance is linear with the trace length, which does not necessarily scale.",
"This is a direct application of bb-dse, where $ p=(a,\\varphi )$ is the pair address-predicate for which we want to check for opacity.",
"We call $\\pi $ the execution trace under attention (extension to a set of traces is straightforward).",
"Basically, the detection algorithm is the following: if $p$ is dynamically covered by $\\pi $ , then returns FEASIBLE; otherwise, returns bb-dse ($p$ ), where INFEASIBLE is interpreted as “opaque”.",
"The result is guaranteed solely for FEASIBLE, since bb-dse has both false positives and false negatives.",
"Yet, experiments (Sections , , ) show that these error ratios are very low in practice.",
"Concerning the choice of bound $k$ , experiments in Section demonstrates that a value between 10 and 20 is a good choice with invariant opaque predicates.",
"Interestingly, the X-Tunnel case study (Section ) highlights that such rather small bound values may be sufficient to detect opaque predicates with long dependency chains (up to 230 in the study, including contextual opaque predicates), since we do not always need to recover all the information in order to conclude on the infeasibility."
],
[
"Call Stack Tampering",
"Call stack tampering consists in altering the standard compilation scheme switching from function to function by associating a call and a ret and making the ret to return to the call next instruction.",
"The ret is tampered (a.k.a violated) if it does not return to the expected return site pushed on the stack at the call.",
"New taxonomy.",
"In this work we refine the definition of a stack tampering in order to characterize it better.",
"integrity: does ret return to the same address as pushed by the call?",
"It characterizes if the tampering takes place or not.",
"A ret is then either [genuine] (always returns to the caller) or [violated].",
"alignment: is the stack pointer (esp) identical at call and ret?",
"If so, the stack pointer is denoted [aligned], otherwise [disaligned].",
"multiplicity: in case of violation, is there only one possible ret target?",
"This case is noted [single], otherwise [multiple].",
"Approach with bb-dse.",
"The goal is to check several properties of the tampering using bb-dse.",
"We consider the following predicates on a ret instruction: $@[esp_{\\lbrace call\\rbrace }] = @[esp_{\\lbrace ret\\rbrace }]$ : Compare the content of the value pushed at call $@[esp_{\\lbrace call\\rbrace }]$ with the one used to return $@[esp_{\\lbrace ret\\rbrace }]$ .",
"If it evaluates to VALID, the ret cannot be tampered [genuine].",
"If it evaluates to UNSAT, a violation necessarily occurs [violated].",
"Otherwise, cannot characterize integrity.",
"$esp_{\\lbrace call\\rbrace } = esp_{\\lbrace ret\\rbrace }$ : Compare the logical esp value at the call and at ret.",
"If it evaluates to VALID, the ret necessarily returns at the same stack offset [aligned], if it evaluates to UNSAT the ret is [disaligned].",
"Otherwise cannot characterize alignment.",
"$\\mathcal {T} \\ne @[esp_{\\lbrace ret\\rbrace }]$ : Check if the logical ret jump target $@[esp_{\\lbrace ret\\rbrace }]$ can be different from the concrete value from the trace ($\\mathcal {T}$ ).",
"If it evaluates to UNSAT the ret cannot jump elsewhere and is flagged [single].",
"Otherwise cannot characterize multiplicity.",
"The above cases can be checked by bb-dse (for checking VALID with some predicate $\\psi $ , we just need to query bb-dse with predicate $\\lnot \\psi $ ).",
"Then, our detection algorithm works as follow, taking advantage of bb-dse and dynamic analysis: the dynamic analysis can tag a ret as: [violated], [disaligned], [multiple]; bb-dse can tag a ret as: [genuine], [aligned], [single] ([violated] and [disaligned] are already handled by dynamic analysis).",
"As for opaque predicates, dynamic results can be trusted, while bb-dse results may be incorrect.",
"Table REF summarizes all the possible situations.",
"Table: Call stack tampering detectionThis call stack tampering analysis uses bb-dse, but with a slightly non-standard setting.",
"Indeed, in this case the bound $k$ will be different for every call/ret pair.",
"The trace is analysed in a forward manner, keeping a formal stack of call instructions.",
"Each call encountered is pushed to the formal stack.",
"Upon ret, the first call on the formal stack is poped and bb-dse is performed, where $k$ is the distance between the call and the ret.",
"From an implementation point of view, we must take care of possible corruptions of the formal stack, which may happen for example in the following situations: Call to a non-traced function: because the function is not traced, its ret is not visible.",
"In our implementation these calls are not pushed in the formal stack; Tail call [2] to non-traced function: tail calls consists in calling functions through a jump instruction instead of call to avoid stack tear-down.",
"This is similar to the previous case, except that care must be taken in order to detect the tail call."
],
[
"Other deobfuscation-related infeasibility issues",
"Opaque constant.",
"Similar to opaque predicates, opaque constants are expressions always evaluating to a single value.",
"Let us consider the expression $e$ and a value $v$ observed at runtime for $e$ .",
"Then, the opaqueness of $e$ reduces to the infeasibility of $e \\ne v$ .",
"Dynamic jump closure.",
"When dealing with dynamic jumps, switch, etc., we might be interested in knowing if all the targets have been found.",
"Let us consider a dynamic jump jump eax for which 3 values $v_1, v_2, v_3$ have been observed so far.",
"Checking the jump closure can be done through checking the infeasibility of $eax \\ne v_1 \\wedge eax \\ne v_2 \\wedge eax \\ne v_3$ .",
"Virtual Machine & CFG flattening.",
"Both VM obfuscation and CFG flattening usually use a custom instruction pointer aiming at preserving the flow of the program after obfuscation.",
"In the case of CFG flattening, after execution of a basic block the virtual instruction pointer will be updated so that the dispatcher will know where to jump next.",
"As such, we can check that all observed values for the virtual instruction pointer have been found for each flattened basic block.",
"Thus, if for each basic block we know the possible value for the virtual instruction pointer and have proved it cannot take other values, we can ultimately get rid of the dispatcher.",
"A glimpse of conditional self-modification.",
"Self-modification is a killer technique for blurring static analysis, since the real code is only revealed at execution time.",
"The method is commonly found in malware and packers, either in simple forms (unpack the whole payload at once) or more advanced ones (unpack on-demand, shifting-decode schemes [33]).",
"The example in Figure REF (page REF ) taken from ASPack combines an opaque predicate together with a self-modification trick turning the predicate to true in order to fool the reverser.",
"Other examples from existing malwares have been detailed in previous studies (NetSky.aa [10]).",
"Dynamic analysis allows to overcome the self-modification as the new modified code will be executed as such.",
"Yet, bb-dse can be used as well, to prove interesting facts about self-modification schemes.",
"For example, given an instruction known to perform a self-modification, we can take advantage of bb-dse to know whether another kind of modification by the same instruction is possible or not (conditional self-modification).",
"Let us consider an instruction $mov\\ [addr],\\ eax$ identified by dynamic analysis to generate some new code with value $eax=v$ .",
"Checking whether the self modification is conditional reduces to the infeasibility of predicate $eax \\ne v$ .",
"As a matter of example, this technique has been used on the example of Figure REF to show that no other value than 1 can be written.",
"This self-modification is thus unconditional."
],
[
"Evaluation: Controlled Experiments",
"We present a set of controlled experiments with ground truth values aiming at evaluating the precision of bb-dse as well as giving hints on its efficiency and comparing it with DSE."
],
[
"Preliminary: Comparison with Standard DSE",
"We compare bb-dse with standard forward DSE, as well as with (unbounded) backward DSE.",
"We are interested in comparing their efficiencies and their adequacy to infeasibility questions – through the distribution of their results, between SAT, UNSAT and timeout.",
"The experiment is performed on a trace of 115000 instructions and we check at each conditional jump if the branch not taken is infeasible (UNSAT) or not (SAT), which is equal to checking if the branch is dead.",
"For bb-dse, we take the algorithm for opaque predicate detection described in Section , with bound values $k=100$ and $k=20$ .",
"We argue in latter experiments (Section REF ) that $k=20$ is a reasonable bound.",
"We use the forward DSE of Binsec/se, and backward DSE is obtained from bb-dse with a bound set to $\\infty $ .",
"Results are presented in Table REF .",
"While forward and backward DSE provide similar results, bb-dse clearly surpasses them in terms of efficiency, spending less than a second for every predicate without any timeout ($\\ge $ 2000 with DSE).",
"From a result point of view, bb-dse with k=16 returns very few UNSAT answers compared to the other methods (54 vs $\\ge $ 7000).",
"Actually, this was expected since DSE takes the whole path into account, and while dead branches are rare in normal code, dead paths are very common.",
"Table: Benchmark DSE versus bb-dseConclusion.",
"This preliminary experiment gives a clear demonstration on the advantages of bb-dse over DSE on infeasibility questions.",
"Indeed, besides the dramatic gap in efficiency (which was of course expected since DSE depends on the whole size trace), DSE reports far more infeasible branches – which would lead in practice to too many false positives.",
"These results were expected, as they are direct consequences of the design choices behind DSE and bb-dse.",
"On the opposite, bb-dse is not suitable for feasibility questions."
],
[
"Opaque Predicates evaluation",
"We consider here the bb-dse-based algorithm for opaque predicate detection.",
"We want to evaluate its precision, as well as to get insights on the choice of the bound $k$ .",
"Protocol and benchmark.",
"We consider two sets of programs: (1) all 100 coreutils without any obfuscation, as a genuine reference data set, and (2) 5 simple programs taken from the State-of-the-Art in DSE deobfuscation [10] and obfuscated with O-LLVM [23].",
"Each of the 5 simple programs was obfuscated 20 times (with different random seeds) in order to balance the numbers of obfuscated samples and genuine coreutils.",
"We have added some new opaque predicates in O-LLVM (which is open-source) in order to maximize diversity (Table REF ).",
"Table: OP implemented in O-LLVMIn total, 200 binary programs were used.",
"For each of them a dynamic execution trace was generated with a maximum length of 20.000 instructions.",
"By tracking where opaque predicates were added in the obfuscated files, we are able a priori to know if a given predicate is opaque or not, ensuring a ground truth evaluation.",
"Note that we consider all predicates in coreutils to be genuine.",
"The 200 samples sums up a total of 1,091,986 instructions trace length and 11,725 conditional jumps with 6,170 genuine and 5,556 opaque predicates.",
"Finally, experiments were carried using different values for the bound $k$ , and with a 5 second timeout per query.",
"Results.",
"Among the 11,725 predicates, 987 were fully covered by the trace and were excluded from these results, keeping 10,739 predicates (and 5,183 genuine predicates).",
"Figure REF and Table REF show the relation between the number of predicates detected as opaque (OP) or genuine (OK) as well as false positive (FP) and false negatives (FN) depending of the bound value $k$ .",
"The experiment shows a tremendous peak of opaque detection with $k=10$ .",
"Alongside, the number of false negative steadily decreases as the number of false positive grows.",
"An optimum is reached for $k=16$ , with no false negative, no timeout and a small number of false positive (293), representing 6.28% of all predicates marked opaque and only 3.17% of all predicates.",
"In that case, the detection method achieves 1.46 false positive per sample (very low).",
"Results are still very precise up to $k$ = 30, and very acceptable for $k$ = 50.",
"Table: Opaque predicate detection resultsFigure: Graph opaque predicate detection ratioA glimpse at efficiency.",
"Taking the same detection algorithm and queries, we want to evaluate predicate solving time.",
"Each predicate formula is solved using Z3http://github.com/Z3Prover/z3 with a 5 seconds timeout.",
"Table REF gives for each value of $k$ the total time taken for solving and the average time per query.",
"For $k$ = 16 the average time per query is 0.018s which tends to prove that this technique scales.",
"Previous works aiming at solving invariant opaque predicates [12] (based on forward DSE) reports an average of 0.49s per queries (min:0.09, max:0.79).",
"Hence, bb-dse seems to provides a very significant speed up.",
"Table: Solving time (10,739 queries)Conclusion.",
"bb-dse is very precise at detecting opaque predicates, and while a bound is of importance, the technique still performs very well if the bound is not too far from the optimal one ($k$ between 10 and 30).",
"Query solving time is also very low."
],
[
"Call Stack Tampering evaluation",
"We want to assess the precision of the bb-dse algorithm for stack tampering detection (Section REF ).",
"We consider two sets of programs: (1) the 5 samples used Section REF , obfuscated with the source-to-source obfuscator Tigress [34]; (2) 5 (randomly selected) coreutils without any obfuscation.",
"Files have been obfuscated with the AntiBranchAnalysis transformation that replace all conditional branches with calls and rets.",
"The two schemes used by the obfuscator are: push; call; ret; ret (the last ret jumps on pushed value) push; ret (idem) Results.",
"Results are given in Table REF .",
"The 5 obfuscated samples sums up 218 different ret.",
"Among them 77 are genuine and 141 violated (including 132 single-target).",
"Our detection method achieves here a perfect score, with no false positive nor false negative.",
"On The 5 coreutils, our method does not yield any false positive and most of the ret are proved genuine (149/156).",
"The few remaining unproved ret come from unhandled libc side-effects, making formulas wrongly UNSAT.",
"Table: Stack tampering resultsConclusion.",
"bb-dse performs very well here, with no false positive and a perfect score on obfuscated samples.",
"The technique recovers both genuine ret and single-source tampered ret.",
"Interestingly, no tampered ret were found on the few (randomly selected) coreutils, supporting the idea that such tampering is not meant to occur in legitimate programs."
],
[
"Conclusion",
"These different controlled experiments demonstrate clearly that bb-dse is a very precise approach for solving different kinds of infeasibility questions.",
"They also demonstrate that finding a suitable bound $k$ is not a problem in practice.",
"Finally, the approach seems to be scalable.",
"This last point will be definitely proved in Sections and ."
],
[
"Large-scale Evaluation on Packers",
"To validate the scalability of bb-dse on representative codes, in terms of both size and protection, we perform a large scale experiment on packers with the two detection algorithms already used in Section .",
"Context.",
"Packers are programs embedding other programs and decompressing/deciphering them at runtime.",
"Since packers are used for software protection, most of them contain several obfuscation schemes (including self-modification).",
"As a matter of fact, packers are also widely used by malware, and actually in many cases they are the only line of defense.",
"Hence, packers are very representative for our study, both in terms of malware protections and size, as packed programs tend to have huge execution traces.",
"Protocol.",
"We want to check if bb-dse is able to detect opaque predicates or call stack tampering on packed programs.",
"For that, a large and representative set of packers was chosen, ranging from free to commercial tools.",
"Then a stub binary (hostname) was packed by each packer.",
"Analyses are then triggered on these packed programs in a black-box manner, that is to say, without any prior knowledge of the internal working of the packers – we do not know which obfuscation are used.",
"For homogeneity, trace length are limited to 10M instructions and packers reaching this limit were not analysed."
],
[
"Results",
"Table REF shows the partial results on 10 packers.",
"The complete results are given in Table REF in appendix.",
"First, bb-dse is efficient and robust enough to pass on most of the packed programs, involving traces of several millions of instructions and advanced protections such as self-modification.",
"Second, over the 32 packers, 420 opaque predicates and 149 call/stack tampering have been found, and many functions have also been proved genuine.",
"All the results that have been manually checked appeared to be true positive (we did not checked them all because of time constraints).",
"Table: Packer experiment OP & Stack tampering"
],
[
"Other Discoveries",
"Opaque predicates.",
"Results revealed interesting patterns, for instance ACProtect tends to add opaque predicates by chaining conditional jumps that are mutually exclusive like: jl 0x100404c ; jge 0x100404c.",
"In this example the second jump is necessarily opaque since the first jump strengthens the path predicate, enforcing the value to be lower.",
"This example shows that our approach can detect both invariant and contextual opaque predicates, and should also detect dynamic opaque predicates since they are similar to contextual opaque predicates.",
"Many other variants of this pattern were found: jp/jnp, jo/jno, etc.",
"Similarly, the well-known opaque predicate pattern xor ecx, ecx; jnz was detected in Armadillo.",
"As a value xor(ed) by itself always return 0, the jnz is never taken.",
"The dynamic aspect of bb-dse allowed to bypass some tricks that would misled a reverser into flagging a predicate as opaque.",
"A good example is a predicate found in ASPack seemingly opaque but that turned not to be opaque due to a self-modification (Figure.",
"REF ).",
"Statically, the predicate is opaque since bl is necessarily 0 but it turns out that the second opcode bytes of the mov bl, 0x0 is being patched to 1 in one branch in order to take the other branch when looping back later on.",
"Figure: ASPack opaque predicate decoyCall/stack tampering.",
"From the call/stack tampering perspective and according to the taxonomy defined in Section , many different kinds of violations were detected.",
"The first two patterns found in ACProtect shown in Figures REF and REF are respectively detected as [violated], [single], [aligned] and [violated], [single], [disaligned].",
"Figures REF , REF and REF show three different kinds of violation found in ASPack.",
"In the first example (cf.",
"Figure REF ) the tampering is detected with labels [violated], [disaligned] since the stack pointer read the ret address at the wrong offset.",
"In the second example (cf.",
"Figure REF ), the return value is modified in place.",
"The tampering is detected with the [violated], [aligned], [single] tags.",
"The last example (cf.",
"Figure REF ), takes place between the transition of two self-modification layers and the ret is used for tail-transitioning to the packer payload (i.e., the original unpacked program).",
"This violation is detected with [violated], [disaligned], [single] since the analysis matches a call far upper in the trace which is disaligned.",
"Note that instruction push 0x10011d7 at address 10043ba is originally a push 0, but it is patched by instruction at address 10043a9, triggering the entrance in a new auto-modification layer when executing it.",
"This pattern reflects a broader phenomenon found in many packers like nPack, TELock or Upack having a single ret tampered: these packers perform their tail transition to the entrypoint of the original (packed) program with push; ret.",
"Thus, such analysis allows to find precisely that moment in the execution trace, where the payload is highly likely decompressed in memory.",
"Figure: ACProtect violation 1/2Figure: ACProtect violation 2/2Figure: ASPack violation 1/3Figure: ASPack violation 2/3Figure: ASPack violation 3/3"
],
[
"Conclusion",
"By detecting opaque predicates and call/stack tampering on packers with multi-million trace length, this experiment clearly demonstrates both the ability of bb-dse to scale to realistic obfuscated examples (without any prior-knowledge of the protection schemes) and its usefulness.",
"This study yields also a few unexpected and valuable insights on the inner working on the considered packers, such as some kinds of protections or the location of the jump to the entrypoint of the original unpacked program."
],
[
"Context & Goal",
"Context.",
"As an application of the previous techniques we focus in this section on the heavily obfuscated X-Tunnel malware.",
"X-Tunnel is a ciphering proxy component allowing the X-Agent malware to reach the command and control (CC) if it cannot reach it directly [22].",
"It is usually the case for machines not connected to internet but reachable from an internal network.",
"These two malwares are being used as part of target attack campaigns (APT) from the APT28 group also known as Sednit, Fancy Bear, Sofacy or Pawn Storm.",
"This group, active since 2006, targets geopolitical entities and is supposedly highly tight to Russian foreign intelligence.",
"Among alleged attacks, noteworthy targets are NATO [35], EU institutions [36], the White House [37], the German parliaments [38] and more recently the American Democrate National Comittee DNC [39] that affected the running of elections.",
"This group also makes use of many 0-days [40] in Windows, Flash, Office, Java and also operate other malwares like rootkits, bootkits, droppers, Mac 0SX malwares [41] as part of its ecosystem.",
"Goal.",
"This use-case is based on 3 X-Tunnel samplesWe warmly thank Joan Calvet for providing the samples.",
"covering a 5 month period (if timestamps are correct).",
"While Sample #0 is not obfuscated and can be straightforwardly analyzed, Samples #1 and #2 are, and they are also much larger than Sample #0 (cf.",
"Table REF ).",
"The main issue raised here is: G1: Is there new functionalities in the obfuscated samples?",
"Answering this question requires first to be able to analyse the obfuscated binaries.",
"Hence we focus here on a second goal: G2: Recover a de-obfuscated version of the obfuscated samples.",
"Table: Samples infosWe show in the latter how bb-dse can solve goal G2, and we give hints on what is to be done to solve G1.",
"Analysis context.",
"Obfuscated samples appeared to contain a tremendous amount of opaque predicates.",
"As a consequence, our goal is to detect and remove all opaque predicates in order to remove the dead-code and meaningless instructions to hopefully obtain a de-obfuscated CFG.",
"This deobfuscation step is a prerequisite for later new functionality finding.",
"The analysis here has to be performed statically: as the malware is a network component, it requires to connect to the CC server, which is not desirable; following the same line, many branching conditions are network-event based, thus unreliable and more hardly reproducible (and would also require infected clients for connection to X-Tunnel); X-Tunnel does not look to use any self-modification obfuscation or neatly tricks to hamper the disassembly.",
"Thus the whole disassembled code is available.",
"The only difference with previous experiments is the need to test the two branches for each conditional jumps."
],
[
"Analysis",
"OP detection.",
"The analysis performs a bb-dse on every conditional jumps of the program, testing systematically both branches.",
"Taking advantage of previous experiments, we set the the bound $k$ to 16.",
"The solver used is Z3 with a 6s timeout.",
"If both branches are unsat, the predicate is considered dead, as the unsatisfiability is necessarily due to path constraints indicating that the predicate is not reachable.",
"Code simplification.",
"We perform three additional computations in complement to the opaque predicate detection: predicate synthesis recovers the high-level predicate of an opaque predicate by backtracking on its logical operations.",
"The goal of this analysis is twofold: (1) indexing the different kind of predicates used and (2) identifying instruction involved in the computation of an OP denoted spurious instructions (in order to remove them); liveness propagation based on obfuscation-related data aims at marking instruction by theirs status, namely alive, dead, spurious; reduced CFG extraction extracts the de-obfuscated CFG based on the liveness analysis."
],
[
"Results",
"Execution time.",
"Table REF reports the execution time of the the bb-dse and predicate synthesis.",
"The predicate synthesis takes a non-negligible amount of time, yet it is still very affordable, and moreover our implementation is far from optimal.",
"Table: Execution timeOP diversity.",
"Each sample presents a very low diversity of opaque predicates.",
"Indeed, solely $7x^2 - 1 \\ne x^2$ and $ \\frac{2}{x^2 + 1} \\ne y^2 + 3$ were found.",
"Table REF sums up the distribution of the different predicates.",
"The amount of predicates and their distribution supports the idea that they were inserted automatically and picked randomly.",
"Table: Opaque predicates varietyDetection results.",
"As the diversity of opaque predicates is very low, we are able to determine, with quite a good precision, the amount of false negatives and false positives based on the predicate synthesized.",
"If a predicates matches one of the two OP and was detected OK, then we considered it false negative (respectively false positive).",
"Results are given in Table REF and Figure REF .",
"The detection rate is satisfactory as false negatives only represent 3% of all predicates.",
"Conversely, 8.4 to 8.6% of false positive are wrongly tagged opaque.",
"Table: Opaque predicates evaluationFigure: Graph of opacity distributionDependency evaluation.",
"As seen previously, a large $k$ bound can lead to false positive due to nested opaque predicates while in the meantime a low bound misses some predicates.",
"Finding the right balance is still an important issue, but results with 12138 OP detected against 1046 false negative tend to confirm that such a low bound is a good trade-off.",
"Across the two samples, the maximum distance between a predicate and its variable definition where 230 (Sample #1) and 148 (Sample #2).",
"Still, the average computed on all the OPs yield an average of 8.7.",
"Difference with O-LLVM.",
"Interesting differences with OP found in O-LLVM are to be emphasized.",
"Firstly, there is more interleaving between the payload and the OPs computation.",
"Some meaningful instructions are often encountered within the predicate computation.",
"Secondly, while O-LLVM OPs are really local to the basic block, there are here some code sharing between predicates.",
"As a consequence, predicates are not fully independent from one another.",
"Also, the obfuscator uses local function variables to store temporary results at the beginning of the function for later usage in opaque predicates.",
"This leads to increase the depth of the dependency chain and to complicate the detection.",
"Code simplification, Reduced CFG extraction.",
"Table REF shows the number of instructions re-classified based on their status.",
"The dead code represents 1/4 of all program instructions.",
"Computing the difference with the original non-obfuscated program shows a very low difference.",
"Therefore, the simplification pass allowed to retrieve a program which is roughly the size of the original one.",
"The difference is highly likely to be due to the false negatives or missed spurious instructions.",
"Finally, Figure REF shows a function originally (a), with the status tags (b), and the result after extraction (c) using tags (red:dead, orange:spurious, green:alive).",
"Although the CFG extracted still containing noise, it allows a far better understanding of the function behavior.",
"A demo video showing the deobfuscation of a X-Tunnel function with Binsec and Idasec is available as material for this paperhttps://youtu.be/Z14ab_rzjfA.",
"Table: Code simplification resultsFigure: Examples of CFG extraction"
],
[
"Conclusion",
"About the case-study.",
"We have been able to automatically detect opaque predicates in the two obfuscated samples of the X-Tunnel malware, leading a significant (and automatic) simplification of these codes – removing all spurious and dead instructions.",
"Moreover, we have gained insights (both strengths and weaknesses) into the inner working of X-Tunnel protections.",
"Hence, we consider that goal G2 has been largely achieved.",
"In order to answer to the initial question (G1), some similarity algorithms should now be computed between the non-obfuscated and simplified samples, in order to detect if some new functions have been added to the code.",
"Moreover, our analysis also pinpoints the protected functions (a small minority), and this information can surely be taken into account.",
"For now, this second analysis step is left as a future work.",
"About X-Tunnel protections.",
"The obfuscation found here are quite sophisticated compared with existing opaque predicates found in the state-of-the-art.",
"It successfully manages to spread the data dependency across a function so that some predicates cannot be solved locally at the basic block level.",
"Hopefully, this is not a general practice across predicates so that the bb-dse works very well in the general case.",
"The main issue of the obfuscation is the low diversity of opaque predicates in the way that some pattern matching can come in relay of symbolic approaches to classify a posteriori false positives and false negatives."
],
[
"Principles",
"As already explained, static and dynamic disassembly methods tend to have complementary strengths and weaknesses, and bb-dse is the only robust approach targeting infeasibility questions.",
"Hence, we propose sparse disassembly, an algorithm based on recursive disasssembly reinforced with a dynamic trace and complementary information about obfuscation (computed by bb-dse) in order to provide a more precise disassembly of obfuscated codes.",
"The basic idea is to enlarge and initial dynamic disassembly by a cheap syntactic disassembly in a guaranteed way, following information from bb-dse, hence getting the best of dynamic and static approaches.",
"The approach takes advantage of the two analyses presented in Sections REF and REF in the following way (cf.",
"FigureREF ): use dynamic values found in the trace to keep disassembling after indirect jump instructions; use opaque predicates found by bb-dse to avoid disassembling dead branches (thus limiting the number of recovered non legit instructions); use stack tampering information found by bb-dse to disassemble ret targets in case of violation, as well as not to disassemble the return site of the call in this case.",
"Figure: Sparse disassembly combinationImplementation.",
"A preliminary version of this algorithm has been integrated in Binsec, taking advantage of the existing recursive disassembly algorithm.",
"The bb-dse procedure sends OP and ret information to the modified recursive disassembler, which takes the information into account."
],
[
"Preliminary Evaluation",
"We report two sets of experiments, designed to assess the precision of the approach and its ability to enlarge an initial dynamic trace.",
"We compare our method mainly to the well-known disassembly tools IDA and Objdump.",
"IDA relies on a combination of recursive disassembly, linear sweep and dedicated heuristics.",
"Objdump performs only liner sweep.",
"Precision.",
"In the first evaluation, we compare these different tools on simple programs obfuscated either by O-LLVM (opaque predicates) or Tigress (stack tampering).",
"In each experiment, we compare the set of disassembled instructions with the set of legitimate instructions of the obfuscated program (i.e.",
"those instructions which can be part of a real execution).",
"It turns out on these small examples that all methods are able to find all the legitimate instructions, yet they may nor may not be lured into dead instructions introduced by obfuscation.",
"Tables REF and REF present our results.",
"We report for each program and each disassembly method the number of recovered instructions.",
"It turns out that this information is representative of the quality of the disassembly (the less instruction, the better), given the considered obfuscations and the fact that here all methods recover all legitimate instructions (actually, all results have been checked manually).",
"Table: Sparse disassembly opaque predicatesTable: Sparse disassembly stack tamperingIn both cases, sparse disassembly achieves a perfect score – recovering all but only legitimate instructions, performing better than IDA and Objdump.",
"Especially, when opaque predicates are considered, sparse disassembly recovers up to 32% less instructions than IDA.",
"Improvement over dynamic analysis.",
"We now seek to assess whether sparse disassembly can indeed enlarge a dynamic analysis in a significant yet guaranteed way, i.e.",
"without adding dead instructions.",
"We consider 5 larger coreutils programs obfuscated with O-LLVM.",
"We compare sparse disassembly to dynamic analysis (starting from the same trace).",
"Here again, the number of recovered instructions is a good metric of precision (the bigger, the better), since both methods report only legitimate instructions on these examples (we checked that bb-dse was able to find all inserted opaque predicates).",
"Results are reported in Table REF .",
"We also report the output of IDA and Objdump in order to give an upper-bound of the number of instructions, yet the two tools recover many dead instructions.",
"Table: Sparse disassembly coreutilsActually, these experiments demonstrate that sparse disassembly is an effective way to enlarge a dynamic disassembly, in a both significant and guaranteed manner.",
"Indeed, sparse disassembly recovers between 6x and 16x more instructions than dynamic disassembly, yet it still recovers much less than linear sweep – due to the focused approach of dynamic disassembly and the guidance of bb-dse.",
"Hence, sparse disassembly stays close to the original trace.",
"Conclusion.",
"The carried experiments showed very good and accurate results on controlled samples, achieving perfect disassembly.",
"From this stand-point, sparse disassembly performs better than combination of both recursive and linear like in IDA, with up to 30% less recovered instructions than IDA.",
"The coreutils experiments showed that sparse disassembly is also an effective way to enlarge a dynamic disassembly in a both significant and guaranteed manner.",
"In the end, this is a clear demonstration of infeasibility-based information used in the context of disassembly.",
"Yet, our sparse disassembly algorithm is still very preliminary.",
"It is currently limited by the inherent weaknesses of recursive disassembly (rather than sparse disassembly shortcomings), for example the handling of computed jumps would require advanced pattern techniques."
],
[
"Discussion: Security Analysis",
"From the attacker point of view, two main counter-measures can be employed to hinder our approach.",
"We present them as well as some possible mitigation.",
"The first counter-measure is to artificially spread the computation of the obfuscation scheme over a long sequence of code, hoping either to evade the “k” bound of the analysis (false negatives) or to force a too high value for k (false positives or timeouts).",
"Nevertheless, it is often not necessary to backtrack all the dependencies to prove infeasibility.",
"An example is given in X-Tunnel were many predicates have a dependency chain longer than the chosen bound (k=16, chain up to 230) but this value was most of the time sufficient to gather enough constraints to prove predicate opacity.",
"Moreover, a very good mitigation for these “predicates with far dependencies” is to rely on a more generic notion of the k bound, based for example on def-use chain length or some formula complexity criterias rather than a strict number of instructions.",
"The second counter-measure is to introduce hard-to-solve predicates (based for example on Mixed-Boolean Arithmetic [42] or cryptographic hashing functions) in order to lead to inconclusive solver responses (timeout).",
"As we cannot directly influence the solving mechanism of SMT solvers, there is no clear mitigation from the defender perspective.",
"Nonetheless, solving such hard formula is an active topic research and some progress can be expected in a middle-term.",
"Moreover, triggering a timeout is already a valuable information, since bb-dse with reasonable k bound usually does not timeout.",
"The defender can take advantage of it by manually inspecting the timeout root cause and deduce a hard-to-solve (in-)feasible pattern, which can now be detected through mere syntactic matching.",
"Finally, such counter-measures would greatly complicate the malware design (and its cost!)",
"and a careless insertion of such complex patterns could lead to atypical code structures prone to relevant malware signatures."
],
[
"Related Work",
"DSE and deobfuscation.",
"Dynamic Symbolic Execution has been used in multiple situations to address obfuscation, generally for discovering new paths in the code to analyze.",
"Recently, Debray at al.",
"[10], [11] used DSE against conditional and indirect jumps, VM and return-oriented programming on various packers and malware in order to prune the obfuscation from the CFG.",
"Mizuhito et al.",
"also addressed exception-based obfuscation using such techniques [43].",
"Recent work from Ming et al.",
"[12] used (forward) DSE to detect different classes of opaque predicates.",
"Yet, their technique has difficulties to scale due to the trace length (this is consistent with experiments in Section REF ).",
"Indeed, by doing it in a forward manner they needlessly have to deal with the whole path predicate for each predicate to check.",
"As consequence they make use of taint to counterbalance which far from being perfect brings additional problems (under-tainting/over-tainting).",
"DSE is designed to prove the reachability of certain parts of code (such as path, branches or instructions).",
"It is complementary to bb-dse in that it addresses feasibility queries rather than infeasibility queries.",
"Moreover, bb-dse scales very well, since it does not depend on the trace length but on the user-defined parameter $k$ .",
"Thus, while backward-bounded DSE seems to be the most appropriate way to solve infeasibility problems no researches have used this technique.",
"Backward reasoning.",
"Backward reasoning is well-known in infinite-state model checking, for example for Petri Nets [44].",
"It is less developed in formal software verification, where forward approaches are prevalent, at the notable exception of deductive verification based on weakest precondition calculi [18].",
"Interestingly, Charreteur et al.",
"have proposed (unbounded) backward symbolic execution for goal-oriented testing [45].",
"Forward and backward approaches are well-known to be complementary, and can often be combined with benefit [46].",
"Yet, purely backward approaches seem nearly impossible to implement at binary level, because of the lack of a priori information on computed jumps.",
"We solve this problem in bb-dse by performing backward reasoning along some dynamic execution paths observed at runtime, yet at the price of (a low-rate of) false positives.",
"Disassembly.",
"Standard disassembly techniques have already been discussed in Section .",
"Advanced static techniques include recursive-like approaches extended with patterns dedicated to difficult constructs [2].",
"Advanced dynamic techniques take advantage of DSE in order to discover more parts of the code [14], [28].",
"Binary-level semantic program analysis methods [15], [16], [17], [13], [47] does allow in principle a guaranteed exhaustive disassembly.",
"Even if some interesting case-studies have been conducted, these methods still face big issues in terms of scaling and robustness.",
"Especially, self-modification is very hard to deal with.",
"The domain is recent, and only very few work exist in that direction [48], [49].",
"Several works attempt to combine static analysis and dynamic analysis in order to get better disassembly.",
"Especially, Codisasm [3] take advantage of the dynamic trace to perform syntactic static disassembly of self-modifying programs.",
"Again, our method is complementary to all these approaches which are mainly based on forward reasoning [50].",
"Obfuscations.",
"Opaque predicates were introduced by Collberg [4] giving a detailed theoretical description and possible usages [51], [52] like watermarking.",
"In order to detect them various methods have been proposed [53], notably by abstract interpretation [49] and in recent work with DSE [12].",
"Issues raised by stack tampering and most notably non-returning functions are discussed by Miller [2].",
"Lakhotia [6] proposes a method based on abstract interpretation [6].",
"None of the above solutions address the problem in such a scalable and robust way as bb-dse does."
],
[
"Conclusion",
"Many problems arising during the reverse of obfuscated codes come down to solve infeasibility questions.",
"Yet, this class of problem is mostly a blind spot of both standard and advanced disassembly tools.",
"We propose Backward-Bounded DSE, a precise, efficient, robust and generic method for solving infeasibility questions related to deobfuscation.",
"We have demonstrated the benefit of the method for several realistic classes of obfuscations such as opaque predicate and call stack tampering, and given insights for other protection schemes.",
"Backward-Bounded DSE does not supersede existing disassembly approaches, but rather complements them by addressing infeasibility questions.",
"Following this line, we showed how these techniques can be used to address state-sponsored malware (X-Tunnel) and how to merge the technique with standard static disassembly and dynamic analysis, in order to enlarge a dynamic analysis in a precise and guaranteed way.",
"This work paves the way for precise, efficient and disassembly tools for obfuscated binaries."
]
] | 1612.05675 |
[
[
"Least reliable messages based early termination method for LT soft\n decoder"
],
[
"Abstract In this paper, we propose a new early termination method (ETM) for Luby transform (LT) belief propagation (BP) decoder.",
"The proposed ETM, which we call least reliable messages (LRM), observes only sign alterations of a small cluster in log-likelihood ratio (LLR) messages passing between nodes in BP decoder.",
"Simulation results and complexity analyzes show that LRM significantly lower computational complexity of early termination section in decoder without any performance degradation and decreases the average decoding iteration amounts compared to conventional ETMs in literature.",
"The method can be easily applied to code families which can be decoded by BP such as low density parity check (LDPC) codes, polar codes and Raptor codes."
],
[
"Introduction",
"Due to their capacity-approaching and unique rateless properties, there has been a particular interest in using Luby transform (LT) and Raptor codes, which are members of rateless codes family, over noisy channels [1], [2].",
"Message-passing algorithms such as belief propagation (BP) are used for decoding of rateless codes.",
"BP iterative decoder uses a pre-set fixed iteration number in order to stop decoding.",
"However, BP mostly converges to original data at an early stage of decoding.",
"Since the decoding continues up to pre-set fixed iteration number, decoder performs redundant processes which cause high computational complexity, decoding latency and energy dissipation.",
"To avoid the aforementioned negations, decoder should be supported by an early termination mechanism to detect convergence and stop decoding.",
"In literature, there are some early termination methods (ETMs) based on check-sum satisfaction ratio (CSR) for rateless codes [3]-[6].",
"CSR is a common success criterion for BP decoding algorithm to observe whether message estimation satisfies constraints imposed by check nodes.",
"Iterative BP decoding algorithm is performed through log-likelihood ratio (LLR) message-passing between nodes.",
"At the end of each iteration CSR decides output bits, re-encodes them and compare with input bits to determine successful convergence.",
"In this letter we propose a completely new ETM for LT BP decoder which we denote as least reliable messages (LRM) ETM.",
"Our method observes only sign alterations of a small cluster in passing LLR messages between BP nodes.",
"Results show that proposed LRM ETM significantly reduces the computational complexity of early termination section in decoder without any performance loss and also decreases the average iteration amounts compared to CSR."
],
[
"BP Decoder for LT Codes",
"Tanner graph representation of LT codes contains two types of nodes, check-node (CN) and variable-node (VN).",
"BP decoding algorithm is performed through LLR message-passing between these CNs and VNs iteratively.",
"After running LT decoder for a pre-set fixed iteration amount, decision process is done and decoding is completed [1], [2].",
"The updating equations of CN and VN in LT BP decoder are given as $m_{c \\rightarrow v}^{(l)}=sign\\left(m_{c}\\prod _{v^{\\prime } \\ne v}m_{v^{\\prime } \\rightarrow c}^{(l)} \\right)\\times \\ 2tanh^{-1} \\left[tanh\\left({|m_{c}|}/{2} \\right) \\prod _{v^{\\prime } \\ne v} tanh\\left({|m_{v^{\\prime } \\rightarrow c}^{(l)}|}/{2} \\right) \\right]$ and $m_{v \\rightarrow c}^{(l+1)}=\\sum _{c^{\\prime } \\ne c}m_{c^{\\prime } \\rightarrow v}^{(l)}$ , respectively.",
"Here, $m_{c}$ stands for LLR values of the codewords come from channel and is directly sent to corresponding CN $c$ , $m_{c \\rightarrow v}$ and $m_{v \\rightarrow c}$ is the outgoing LLR messages from the CN $c$ to VN $v$ and vice versa.",
"$tanh$ and $tanh^{-1}$ represent hyperbolic tangent and its inverse operations, respectively.",
"Superscript $l$ denotes iteration index.",
"Hard-decision process of BP is given as $m_{v}=m_{c \\rightarrow v}^{(l)}+m_{v \\rightarrow c}^{(l+1)}$ and $\\hat{m}_{v}=1$ if $m_{v} \\ge 0$ , $\\hat{m}_{v}=0$ , if $m_{v} < 0$ .",
"Here, $\\hat{m}_{v}$ represents hard value for corresponding VN $v$ ."
],
[
"CSR Early Termination Method",
"A common criterion for early termination of rateless decoding is observing if the estimated messages $\\hat{m}_{v}$ satisfy the constraints imposed by CNs [3], [4].",
"The criterion controls whether the equation $\\hat{m}_{c}\\oplus \\ \\left(\\bigoplus _{v}\\hat{m}_{v}\\right)$ is equal to zero for all CNs, where $\\hat{m}_{c}$ stands for hard decision of $m_{c}$ messages, $\\oplus $ represents modulo-2 addition and $\\bigoplus $ denotes the summation operator for modulo-2 addition.",
"Parenthetical expression represents re-encoding process and rest of it represents compare process.",
"After that, CSR test is calculated by $\\mu _{CSR}=s^{(l)}/N_{CN}$ , where $s^{(l)}$ is number of satisfied CNs at decoding iteration $l$ and $N_{CN}$ is total number of CNs.",
"The test is satisfied when inequality $\\mu _{CSR} \\ge \\Gamma _{CSR}$ is correct, where $\\Gamma _{CSR}$ is a user-defined threshold.",
"This method is known as CSR ETM.",
"LT BP decoder with CSR is presented in Algorithm .",
"LT BP decoder with CSR method: [1] Calculate $m_{c}$ ; Set $m_{c \\rightarrow v}^{(0)}$ and $m_{v \\rightarrow c}^{(0)}$ messages to zero, $l=0$ ; $(l < max\\_iter)$ and $(\\Gamma _{LC} \\ is \\ not \\ satisfied)$ CN update(); VN update(); Decision(); Calculate CSR and $\\Delta $ CSR; $l=l+1;$ In the algorithm, the difference between CSR values of two consecutive iterations denoted as $\\Delta $ CSR.",
"If $\\Delta $ CSR has a value of ”0” for $\\Gamma _{LC}$ amount of consecutive iterations, decoding is terminated [5].",
"$\\Gamma _{LC}$ is a user-defined integer value."
],
[
"Proposed LRM Early Termination Method",
"LRM ETM is based on observing sign alterations of a small cluster in $m_{v \\rightarrow c}$ messages.",
"Since the sign parts of the LLR values are utilized for hard-decision in the decision part of BP, observing sign alterations of $m_v$ during successive iterations can be used to determine whether estimated data bits change.",
"If the estimated data bits stop changing for a number of consecutive iterations ($\\Gamma _{LC}$ ) it can be assumed that decoder successfully converged.",
"To be able to get lowest average iteration amounts, $\\Gamma _{LC}$ value should be as low as possible.",
"Instead of $m_v$ messages, our proposed method observes sign alterations of $m_{v \\rightarrow c}$ messages that specify $m_v$ .",
"Therefore, our method doesn't require performing ”Decision()” at each decoding iteration.",
"On the other hand, proposed LRM method is basically based on the fact that $m_{v \\rightarrow c}$ messages with lower absolute LLR values are less reliable among entire $m_{v \\rightarrow c}$ messages [2] and they converge later than messages that have higher absolute LLR values.",
"Therefore, we observe only LRM which is a small cluster of LLR values to determine successful convergence.",
"This simplification also reduces the computational complexity of ETM section significantly.",
"Determination of LRM which means finding the smallest absolute LLR values in all $m_{v \\rightarrow c}$ messages, can be easily done by using a selection algorithm.",
"We use quickselect algorithm which has low computational complexity [7].",
"LRM ETM determines LRM to observe sign alterations after running decoder for a few iterations.",
"This is because LT BP decoder typically needs a few iterations to propagate initial channel LLR values.",
"We call these threshold for iteration numbers as determination condition of LRM (DC-LRM).",
"It is easy to see that larger DC-LRM value increases probability of choosing accurate LRM because better propagation occurs when iteration number increases.",
"On the other hand, DC-LRM shouldn't be larger than minimum iteration number that decoder converged to keep average iteration number as low as possible.",
"DC-LRM values are chosen as 45, 28, 22, 18 and 15 for 0.5, 1.0, 1.5, 2.0 and 2.5dB according to simulations, respectively.",
"DC-LRM values for different systems can be determined by simulations and previously loaded to a look-up table.",
"LT BP decoding process with proposed LRM method is presented in Algorithm .",
"LT BP decoder with LRM method: [1] Calculate $m_{c}$ ; Set $m_{c \\rightarrow v}^{(0)}$ and $m_{v \\rightarrow c}^{(0)}$ messages to zero, $l=0$ ; $(l < max\\_iter)$ and $(\\Gamma _{LC} \\ is \\ not \\ satisfied)$ CN update(); VN update(); $(l ==$ DC-LRM$)$ Quickselect(); $(l >$ DC-LRM$)$ Count sign changes in LRM; $l=l+1;$ Decision();"
],
[
"Complexity Analyzes",
"In this section, we analyze the computational complexities of CSR ETM and proposed LRM ETM.",
"We count up computational complexities of considered ETMs and illustrate the results in Table REF .",
"We assume $abs$ , $sign$ and $XOR$ operations have same complexities to simplify the comparison.",
"Table: Complexities of ETMs for single iterationIn the table, $N$ is coded packet length, $K$ is uncoded packet length, $\\lambda _{1}$ is the fraction of VNs of degree 1, $d_{c}$ is CN degree, $\\rho _{d_c}$ is the fraction of CNs of degree $d_c$ and $d_{c_{max}}$ is maximum CN degree.",
"$N_{B}$ symbolizes number of LRM determined by $N_{B}=B*N_{m_{v \\rightarrow c}}$ , where $B$ is the percentage to determine the amount of LRM, $N_{m_{v \\rightarrow c}}$ is number of all $m_{v \\rightarrow c}$ messages and calculated by $N_{m_{v \\rightarrow c}}=N\\Omega ^{\\prime }(1)$ , where $\\Omega ^{\\prime }(1)$ is average degree of degree distribution chosen for LT code [8].",
"As we mentioned above, LRM method performs quickselect algorithm only one time for whole decoding process to determine least reliable messages.",
"The quickselect uses less than $2N_{m_{v \\rightarrow c}}$ compare operations to find the smallest $N_{B}$ items of an array with length $N_{m_{v \\rightarrow c}}$ [7].",
"We add the average effect of quickselect to computational complexities for each iteration by $2N_{m_{v \\rightarrow c}}/l_{avg}$ comparisons in the table.",
"Here, $l_{avg}$ is average iteration number.",
"It should be also emphasized that all operations required for CSR method are performed in every decoding iteration until decoding is terminated, while the operations for LRM method start after decoder runs DC-LRM iterations which does not emphasised in Table REF ."
],
[
"Numerical results",
"In this section, we evaluate the bit error rate (BER) performances of LT BP decoding algorithm with and without ETMs over binary-input additive white Gaussian noise (BIAWGN) channel by simulation works.",
"Also, computational complexities of ETMs and average iteration amounts of BP algorithm with LRM and CSR ETMs are compared.",
"For all simulation works and complexity analyzes, we consider the following degree distribution, $\\Omega (x) = 0.008x +0.494x^{2}+0.166x^{3}+0.073x^{4}+0.083x^{5}+0.056x^{8}$ $+0.037x^{9}+0.056x^{19}+ 0.025x^{65}+0.003x^{66}$ [8], code rate of $1/2$ , data packet length of 4000 and fixed iteration number of 100.",
"Figure: BER curves of LT BP decoder with and without ETMsFig.",
"REF illustrates BER curves of LT BP decoder with CSR and proposed LRM ETMs.",
"Simulations are performed for various $N_B$ and $\\Gamma _{LC}$ values.",
"Since larger $\\Gamma _{LC}$ cause larger average iteration amount, $B=\\%5$ is chosen to make $\\Gamma _{LC}$ value as small as possible and only the results for $B=\\%5$ are illustrated for various $\\Gamma _{LC}$ .",
"We also provide BER curve for LT BP with 100 fixed iteration number without ETM as a benchmark.",
"This benchmark shows the best BER values that decoder can reach.",
"Differences between benchmark and other BER values indicate that ETMs stop decoding before decoder converges.",
"An ETM shouldn't cause BER performance degradation.",
"As it can be seen in the figure, LRM method with ($\\Gamma _{LC}=1$ and $B=\\%5$ ) and CSR with $\\Gamma _{LC}=5$ don't cause BER performance degradation.",
"Therefore, performance comparison between ETMs is made with same parameters above.",
"Table REF compares average iteration amounts of LT BP decoder with selected LRM and CSR methods.",
"Second column in Table REF called ”Decoder Convergence” is considered as benchmark.",
"LRM ETM has smaller average iteration amounts than CSR but it has slightly higher than benchmark values.",
"Table: Average iteration amounts of LT BP decoder with ETMs and LT BP decoder successfully convergedAverage computation times of ETMs for decoding a code block are compared in Table REF with considered simulation parameters (CSR with $\\Gamma _{LC}=5 $ and LRM with $\\Gamma _{LC}=1, B=\\%5 $ ).",
"Results show that required computation time of LRM method is significantly lower than CSR.",
"Note that timing results demonstrate only ETM section of decoding process.",
"Furthermore, decoder with proposed LRM method has small average iteration amounts compared to decoder with CSR as shown in Table REF .",
"This provides additional reduction in computation time of whole decoding process.",
"Table: Average computation time of EMTs for decoding a code block"
],
[
"Conclusion",
"In this paper, we developed a new early termination method for LT BP decoder to avoid redundant processes which cause high computational complexity, decoding latency and energy dissipation.",
"Simulation results and complexity analyzes show that proposed LRM method significantly lower complexity and computation time of early termination section in decoder without BER performance degradation and decreases the average iteration amounts compared to conventional CSR ETM.",
"The method can be easily applied to code families which can be decoded by BP such as low density parity check (LDPC) codes, polar codes and Raptor codes.",
"The best way to compare ETMs can be done by hardware implementation which will be held in future.",
"C. Albayrak, C. Simsek and K. Turk (Department of Electrical Electronics Engineering, Karadeniz Technical University, Trabzon 61080, TR) E-mail: [email protected]"
]
] | 1612.05461 |
[
[
"Two components of critical current in YBa$_2$Cu$_3$O$_{7-\\delta}$ films"
],
[
"Abstract Combined action of weak and strong pinning centers on the vortex lattice complicates magnetic behavior of a superconductor since temperature and magnetic field differently affect weak and strong pinning.",
"In this paper we show that contributions of weak and strong pinning into magnetization of the layered superconductor YBa$_2$Cu$_3$O$_{7-\\delta}$ can be separated and analyzed individually.",
"We performed a careful analysis of temperature behavior of the relaxed superconducting current $J$ in YBa$_2$Cu$_3$O$_{7-\\delta}$ films which revealed two components of the current $J = J_1 +J_2$.",
"A simple method of separation of the components and their temperature dependence in low magnetic fields are discussed.",
"We found that $J_1$ is produced by weak collective pinning on the oxygen vacancies in CuO$_2$ planes while $J_2$ is caused by strong pinning on the Y$_2$O$_3$ precipitates.",
"$J_1$ component weakly changes with field and quasi-exponentially decays with temperature, disappearing at $T \\simeq 30$--40~K.",
"Rapid relaxation of $J_1$ causes formation of the normalized relaxation rate peak at $T \\simeq 20$~K.",
"$J_2$ component is suppressed by field as $J_2\\propto B^{-0.54}$ and decays with temperature following to the power law $J_2\\propto(1 - T/T_\\mathrm{dp} )^\\alpha$ where $T_\\mathrm{dp}$ is the depinning temperature.",
"Detailed comparison of the experimental data with pinning theories is presented."
],
[
"Introduction",
"Pinning of vortices on defects in type-II superconductors leads to formation of a critical state and appearance of the critical current $J_c$ .",
"[1], [2], [3], [4] Thermal fluctuations reduce the pinning strength and activate jumps of vortices between pinning centers (defects).",
"[3], [5], [6], [4] High temperature superconductors (HTSC) have a small activation energy and a high probability of thermal activation of vortices motion which leads to a giant magnetic flux creep and decay of the superconducting current over time.",
"[3], [4], [7] As a result the measured current $J$ becomes lower than $J_c$ .",
"[7] For HTSC materials the maximal currents are achieved in a highly textured YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"2G-tapes with a metal base and the superconducting YBa$_2$ Cu$_3$ O$_{7-\\delta }$ layer were developed for high-current applications.",
"[8], [9] Great work was done on studying and optimization of the defects landscape in 2G-tapes to obtain high currents in external magnetic fields and at high temperatures, see Ref.",
"Foltyn-NM-2007 for review.",
"As a result, several manufactures produce now long-length tapes with $J$ of several MA/cm$^2$ at liquid nitrogen temperature.",
"[9] Nevertheless the task of improving performance of the tapes by combinations of artificial and natural pinning centers[11] is still actual.",
"Investigations of pinning on natural defects in standard YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films play a major role in that work.",
"There is a large variety of defects in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films such as vacancies, substituting or extra atoms, dislocations, non-superconducting inclusions and so on.",
"[10] The latter two act as strong pinning centers.",
"The dislocations[12], [13], [14] are induced close to the substrate-film interface and develop up to the film surface.",
"[13] The nano-sized Y$_2$ O$_3$ precipitations[15], [16], [17], [18], [19], [20], [11], [21] are spontaneously formed during deposition of YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"It was shown that increase of the inclusion density rises the superconducting current[16] while increase of the dislocation density reduces suppression of the current by magnetic field $H$ .",
"[12], [13] Point defects, mainly oxygen vacancies in superconducting CuO$_2$ planes, act as weak pinning centers.",
"Weak pinning affects magnetic behavior of YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films at temperatures below 30–40 K. For example, the exponential dependence $J\\propto \\exp (-T/T_0)$ was observed in the range $ T< 60$ K[22], [23], [9], [24], [25], [26] while at high temperatures the current decay follows a power law $J\\propto [1-(T/T_c)^n]^\\alpha $ .",
"[27], [28], [29], [20], [30], [21].",
"Here $\\alpha =1.2$ –2, $n=1$ (Refs.",
"Fedotov-LTP-2002, Pashitskii-LTP-2001, Djupmyr-PRB-2005, Albrecht-JP:CM-2007) or 2 (Refs.",
"Ijaduola-PRB-2006, Miura-PRB-2011) and $T_0\\simeq 17$ –32 K.[22], [23], [9], [24], [25], [26] The magnetic flux creep also changes at low temperatures.",
"A peak of the normalized relaxation rate of the current $S=|d\\ln J/d\\ln t|$ was observed in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films at $ T\\simeq 20$ K[11], [31], [32] and the quantum creep was found below 1 K.[7], [33], [34], [35], [36] The quantum creep and a crossover to two-dimensional superconducting behavior observed at $ T<80$ K[37] revealed an importance of layered structure for superconductivity in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ .",
"The analysis of the critical state in HTSC is complicated by presence of weak and strong pinning and the layered structure of HTSC materials.",
"If pinning is weak, the elastic forces of the vortex lattice dominates over the pinning forces.",
"[2], [3], [4], [38] In this case the concerted action of many weak pins on the elastic vortex lattice is described by the collective pinning theory (CP theory).",
"[3] The collective pinning depends only slightly on parameters of individual pinning centers therefore CP theory is easy to generalize.",
"If pinning is strong, defects acts individually and introduce plastic deformations in the vortex system.",
"[2], [3], [38] In this case pinning depends on parameters of the defects so many various models were developed for different kinds of defects.",
"[3], [5], [39], [6], [40], [27], [14] Strong pinning in a layered superconductor, which contains point defects in the superconducting planes and three-dimensional defects in the bulk, was analyzed by Ovchinnikov and Ivlev[39] (OI theory).",
"They found that the critical current $J_c$ of such superconductor consists of two components produced by in-plane and in-volume pinning.",
"Further developing the OI theory, van der Beek et al.",
"considered in-volume pinning and calculated the dependence of $J_c$ on film thickness $d$ and temperature.",
"[19] The dependence $J(d)$ for thin YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films was successfully described[19], [20] in the frame of extended OI theory[19] by pinning on Y$_2$ O$_3$ inclusions.",
"At the same time the extended theory agree with experimental data on $J(T)$ and $H^*(T)$ only for $T>30$ K[19], [20] since the in-plane pinning was completely ignored by van der Beek et al.",
"Here $H^*$ is the crossover field above which $J$ becomes field dependent.",
"There exists the model describing strong pinning on edge dislocations at low-angle boundaries of crystallites in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films (EDP model).",
"[14], [28], [41] For some samples this model approximates well the field dependence of the current in a wide range of fields.",
"[14], [41], [42] At the same time the EDP model is not universal for all YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films, some conditions are necessary for its correct application.",
"[42] Restriction of the EDP model may be caused by neglecting of weak pinning which may influence $J(H)$ behavior at low temperatures.",
"$J(T)$ behavior that follows from the EDP model haven't been tested yet.",
"To clarify the role of weak and strong pinning we performed a careful analysis of temperature behavior of the relaxed superconducting current in different magnetic fields in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"The analysis allowed us to separate and describe the current components produced by weak and strong pinning.",
"The paper is organized as follows.",
"Samples and details of $J(T)$ measurements are discussed in Sec. .",
"Experimental results are presented in Sec. .",
"At first we show that $J(T)$ behavior observed in our experiments is common for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"Then analyzing experimental data we separate currents produced by in-plane and in-volume pinning.",
"The separated current components are analyzed in Sec.",
"We discuss a relationship between low-temperature peak of the relaxation rate and component of the current produced by weak pinning.",
"Then we show that this component is caused by single-vortex collective pinning in Cu-O$_2$ planes and try to describe it in the frame of CP theory.",
"At the end we consider the component produced by strong pinning and show that is well described by OI theory extended for strong pinning on Y$_2$ O$_3$ inclusions.",
"Our conclusions are presented in Sec. .",
"The critical current following from OI theory for magnetic field applied along a normal to the superconducting planes is calculated in Appendix."
],
[
"Experimental details",
"Thin epitaxial films of YBa$_2$ Cu$_3$ O$_{7-\\delta }$ were prepared by pulsed laser deposition technique using KrF excimer laser.",
"Disk-shaped single crystal plates of SrTiO$_3$ (100) were used as substrates.",
"The deposition took place at substrate temperature about 750$^\\circ $ C, the oxidizer pressure (N$_2$ O or O$_2$ ) varied from 400 to 800 mtorr in different experiments.",
"The velocity filter was used to select the fine part of the ablation plume (atoms and clusters of small size) and obtain better quality of the film surface.",
"[43], [44] The film structure was analyzed by XRD at D8 Discover diffractometer (Bruker) using Cu-$K_\\alpha $ radiation.",
"The study confirmed that films were epitaxial and $c$ -oriented.",
"No additional phase was detected.",
"The peaks (002), (005) and (007) were used to determine the $c$ lattice parameter and estimate the values of coherent scattering regions and microdeformation.",
"The $c$ lattice parameter was in the range 11.70–11.73 Å, the rocking curve widths $\\omega $ of the (005) Bragg peak for best samples was less than 0.2 degree.",
"The oxygen content varied depending on oxidation condition and brought about the variation of $c$ -parameter.",
"As follows from the values of the structure parameters presented in Table REF , the films had high-quality crystalline structure with small microdeformation and disorientation.",
"The critical temperatures of the superconducting transition $T_c=90$ –91 K were obtained in resistivity measurements performed on witness-samples made in the same deposition process.",
"The samples demonstrated sharp transitions with width of about 1 K. SQUID-magnetometry and study of the magnetic susceptibility in an alternating magnetic field were used to measure temperature of the magnetic transition $T_c^M$ .",
"Obtained $T_c$ and $T_c^M$ values are presented in the Table REF .",
"Table: Parameters of the YBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } thin film samples.The (002), (005) and (007) Bragg peaks were used to obtain the lattice parameter cc, size ofthe coherent scattering regions (CDB) and the microdeformation ε micro \\varepsilon _\\text{micro}.The full width on half maximum (FWHM) and the rocking curve width ω\\omega weremeasured for the (005) Bragg peak.T dp T_\\mathrm {dp} and JJ was measured in field H=910H=910 Oe for samples Y1–Y3 and1530 Oe for sample Y4.JJ was taken at T=4.21T=4.21 K. J 1 J_1 and J 2 J_2 are presented for zero temperature.D 𝑖𝑧 D_\\mathit {iz}, n i n_i and n i * n_i^* were calculated for D i D_i and n i d 𝑖𝑧 9/4 =n i (D 𝑖𝑧 /ξ 0 ) 9/4 n_i d_\\mathit {iz}^{9/4} =n_i(D_\\mathit {iz}/\\xi _0)^{9/4} obtained via fit of experimental curves.〈L〉\\langle L\\rangle was calculated for B=910B=910 G.Measurements of a persistent current induced in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ film under change of magnetic field were performed using home-built SQUID magnetometer.",
"[45], [46] During the measurements a sample was placed inside a copper tube isolated from the LHe bath by a vacuum jacket.",
"Temperature of the sample varied in the range from 4.21 to 300 K via heating of the tube filled with exchange-helium gas.",
"Magnetic field was produced by a NbTi tube enclosed in NbTi solenoid.",
"To apply a field the solenoid was supplied by current and the tube was warmed above $T_c$ by a short heat pulse.",
"After freezing of the field in the tube the current was withdrawn out the solenoid to minimize noises.",
"Magnetic field up to 2100 Oe can be frozen in the tube of 0.3 mm wall thickness.",
"High fields were applied step by step to prevent overheating of the superconducting films by the current induced under the abrupt change of the external magnetic field.",
"[47] At each step the field increment twice exceeded the characteristic field for flux penetration into the film[48], [49], [4] to make sure that the induced current is high enough to create the critical state throughout the sample.",
"When measurements were performed in zero applied field, a high field was applied at first and the sample was maintained several minutes in this field.",
"Then the field was decreased step by step and at last step the solenoid was warmed together with the tube to remove a magnetic flux frozen in its wire.",
"Due to strong demagnetization effect a self demagnetizing field is produced by current flowing in a superconducting film when field is applied perpendicular to the film plane.",
"[50], [51], [52] In the critical state this self field exists even after complete removal of external field.",
"The measurements were performed for applied fields of 910 and 1530 Oe and in self-field after removing field of 2090 Oe.",
"These fields were enough to form the homogeneous critical state in samples at all temperatures.",
"Figure: (Color online) Relaxation curves of remanent moment measured for sample Y1 at low andhigh temperatures.",
"The moment decays by about 5% and 14% in time-window of the relaxationmeasurements.A method of SQUID magnetometry with motionless sample[53], [46], [54], [55], [56] was used in our experiments.",
"The measurements were performed as follows.",
"The film locked in one of pick-up coils of a superconducting flux transformer was warmed above $T_c$ and cooled in zero field to a desired temperature (ZFC procedure).",
"After that a magnetic field was applied perpendicular to the film plane and a signal caused by change of film magnetic moment with time $\\delta M(t)$ due to relaxation was measured for one hour.",
"Then magnetometer indications were reset and the sample was warmed above $T_c$ in order to record its residual moment $M_*$ .",
"Combining $M_*$ with data on $\\delta M(t)$ we precisely obtained the time dependence of the magnetic moment $M(t)$ .",
"[57], [52] Examples of $M(t)$ curves obtained at low and high temperatures are shown in Figure REF .",
"As seen, the noise of the curves is considerably lower than the change of the moment due to relaxation.",
"Preliminary results of the relaxation experiments were published elsewhere.",
"[57], [58], [52] In the present work we analyze the current obtained from the $M_*(T)$ dependences so let us consider this issue in detail.",
"Figure: (Color online) Top: Temperature dependences of remanent momentmeasured for sample Y2 after relaxation for 1 hour.",
"Amplitude of the residual moment M * M_*obtained for T=24.8T=24.8 K is shown by arrow.Bottom: J(T)J(T) dependences obtained from the relaxed moments M * M_*(triangles) and measured under warming of films during temperature sweep (continuous curves).The sample was heated at the rate of 5 K/min up to $T=95$ K during warming and a signal produced by the magnetometer background and the film moment was recorded in steps of 0.1 K. The background signal measured without sample was subtracted from the total one to separate the signal produced by the film only.",
"Inaccuracy of obtaining the film magnetic moment $M$ due to the subtraction did not exceed 0.5% at $T\\simeq 80$ K and was considerably smaller at low temperatures.",
"$M(T)$ dependences obtained in such a manner are shown in top Fig.",
"REF .",
"$M(T)$ curves begin with a plateau caused by preceding relaxation of the moment.",
"$M$ values at the plateaus are equal to the residual moments $M_*(T)$ .",
"In bottom Fig.",
"REF we also presented temperature dependences of the current density calculated as[48], [49], [4] $J=24Mc/(\\pi D^3d)$ where $c$ is the light velocity, $D$ and $d$ are the film's diameter and thickness.",
"Two types of $J(T)$ curves are shown for comparison.",
"The first type, obtained from the $M_*(T)$ values, corresponds to a long-time relaxed current.",
"The second one, recorded under film warming immediately after magnetic field removal, presents a short-time relaxed current.",
"As seen in the bottom Fig.",
"REF , shapes of the short- and long-time relaxed curves slightly differ each other.",
"At some temperatures the long-time relaxed current for sample Y1 is greater than the short-time relaxed one while it obviously should be smaller.",
"This artifact is caused by fast temperature sweep during $M(T)$ recording.",
"[59] Since temperature measurement error can affect the $J(T)$ dependences recorded under film warming they are presented below mainly as illustrations.",
"At the same time the $J(T)$ curves obtained from the residual moments correspond to equilibrium temperatures at which the sample was kept more than hour.",
"These temperatures were stabilized and measured with accuracy better than 0.05 K"
],
[
"Results",
"We start from comparison of measured $J(T)$ dependences with published data to clarify which features of $J(T)$ behavior are common for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"However the current $J$ itself should be elucidated first.",
"The critical current $J_c$ determined by pinning theories cannot be measured directly because of huge Joule heat dissipated in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"[60] Therefore either a current $J_T$ measured in transport experiments or a current induced by applied magnetic field are used to characterize the superconducting current.",
"$J_T$ is maintained during measurement by a current source so it does not relax.",
"On the contrary the persistent current $J$ is affected by creep, therefore it is lower than $J_T$ .",
"[61] Moreover, dependences of $J$ and $J_T$ on $T$ and $H$ can differ especially at high temperatures and fields.",
"Therefore only data on the persistent current measured in self-field[27], [28], [29], [30], [20], [21], [62], [22], [23], [9], [26], [24], [25] were chosen for verification.",
"[63] Figure: (Color online) Temperature dependences of the current density ofYBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } films in different scales.",
"The curves for sample Y2 are shifted(multiplied by factor 2) to avoid a crossing with ones for sample Y1.",
"Symbols:J(T)J(T) obtained after relaxation for 1 hour inself-field (triangles) and in field of 910 Oe (squares).",
"Curves with small dots were measured inself-field under warming the samples at sweep rate of 5 K/min.Panel a: The lines are approximations J∝(1-T/T c ) α J \\propto (1-T/T_c)^\\alpha withα=2.2\\alpha =2.2 (solid), 1.2 (dashed) and 1.55 (dash-dotted).Panel b: Solid line is an approximation J∝τ - α J \\propto \\tau _-^\\alpha with α=1.52\\alpha = 1.52.",
"Dashed and dash-dotted lines are calculated for strongpinning on large J c ∝τ - 3/2 τ + 1/2 J_c \\propto \\tau _-^{3/2} \\tau _+^{1/2} and small J c ∝τ - 5/2 τ + -1/2 J_c \\propto \\tau _-^{5/2}\\tau _+^{-1/2} defects.Panel c: Dashed lines are fits by J∝e -T/T 0 J \\propto e^{-T/T_0}with T 0 =33T_0=33, 23.5 K for Y2, Y1 in self-field and T 0 =17T_0=17 K for H=910H=910 Oe.Solid lines are fits by the dependence (): T w =18T_w=18, 24 K for Y1, Y2 andT s =52T_s=52 K for both samples in self-field; T w =14T_w=14 K and T s =43T_s=43 K for H=910H=910 Oe.Representative $J(T)$ curves measured for our samples are shown in Fig.",
"REF in different scales to display $J$ behavior in different temperature ranges.",
"To illustrate a field influence, the curves obtained in self and external field of 910 Oe are shown for sample Y1.",
"A power law $J_c\\propto (1-T/T_c)^\\alpha $ is expected for pinning of vortices on boundaries between crystallites in the films.",
"[40], [27] Three temperature ranges with different $\\alpha $ values were found for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"[28], [27], [29], [30] The powers $\\alpha \\simeq 1.2$ –2, 0.9–1.2 and 1.4–2.5 were obtained respectively at high[27], [28] $T\\gtrsim 77$ K, elevated[29], [30] $36\\text{ K} \\lesssim T\\lesssim 72$ K and lower[29], [30] $12\\text{ K} \\lesssim T\\lesssim 35$ K temperatures.",
"As shown in Fig.",
"REF (a), our results well agree with the published data.",
"Fitting curves for sample Y2 demonstrate good approximation by the power law with $ \\alpha =1.55$ , 1.2 and 2.2 in the above mentioned ranges.",
"Relaxation slightly affects $J(T)$ at low and elevated temperatures and increases the power at high $T$ .",
"External field smoothes $J(T)$ and rises the powers in all ranges.",
"Summing up we conclude that our results are consistent with published data.",
"Since pinning parameters depend on the penetration depth $\\lambda (T) = \\lambda _0/\\sqrt{1-(T/T_c)^4}= \\lambda _0/\\sqrt{\\tau _+\\tau _-}$ and the coherence length $\\xi (T) =\\xi _0\\sqrt{(1+(T/T_c)^2)/(1-(T/T_c)^2)} = \\xi _0\\sqrt{\\tau _+/\\tau _-}$ of the superconductor[3] one can assume that their temperature change determine $J(T)$ behavior.",
"Here we denoted $\\tau _+=1+(T/T_c)^2$ , $\\tau _-=1-(T/T_c)^2$ and $\\lambda _0 = \\lambda (0)$ , $\\xi _0 = \\xi (0)$ .",
"In the frame of CP model[3] Griessen et al.",
"obtained that $J_c \\propto \\tau _-^{7/6}\\tau _+^{5/6}$ for $\\delta T_c$ pinning and $J_c \\propto \\tau _-^{5/2}\\tau _+^{-1/2}$ for $\\delta \\ell $ pinning.",
"[65], [66] Similar expressions were calculated by Klaassen et al.",
"for strong pinning on inclusions of large, $J_c \\propto \\tau _-^{3/2} \\tau _+^{1/2}$ , and small, $J_c \\propto \\tau _-^{5/2} \\tau _+^{-1/2}$ , size.",
"[64] The dependence $J\\propto \\tau _-^\\alpha $ with $\\alpha =1.2$ –1.4 in self-field at $T\\gtrsim 50$ K was observed experimentally in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"[20] In field of 100 Oe for $T\\gtrsim 40$ K the power $\\alpha =1.53$ was found while at high temperatures $T\\gtrsim 83$ K a more rapid decay of $J$ was observed.",
"[21] Our results presented in Fig.",
"REF (b) are consistent with the published data.",
"For example, for sample Y2 in self-field for $T\\gtrsim 40$ K we obtained $\\alpha =1.52$ .",
"The range in which a rapid decay of $J$ is observed shifts to lower temperatures in external field.",
"Thus we conclude again that our results well agree with published data.",
"The exponential decay $J\\propto \\exp (-T/T_0)$ was observed at $T< 50$ –60 K in HTSC single crystals,[67], [68], [69], [70] YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films[62], [22], [23] and 2G-tapes.",
"[9] Such behavior was attributed to oxygen vacancies acting as weak pinning centers.",
"The scaling temperature depends on field, for example $T_0=25$ –32 K in self-field[23], [9] and $T_0=17$ –25 K in $H=1$ –200 kOe.",
"[62], [23], [9] Both increase[9] and decrease[23], [9] of $T_0$ was observed in lower fields.",
"As shown in Fig.",
"REF (c), our results are in good agreement again with published temperatures.",
"In the range $ T < 60$ K we obtained $ T_0 = 33 $ , 23.5 K for self-field and 17 K for $H=910$ Oe.",
"Studying magnetization of solidified YBa$_2$ Cu$_3$ O$_{7-\\delta }$ -Y$_2$ BaCuO$_5$ composites Martínez et al.",
"[70] in the range $40\\text{ K} \\leqslant T\\leqslant 80 $ K found the dependence $J\\propto \\exp [-3(T/T^*)^2]$ caused by strong pinning on nonsuperconducting Y$_2$ BaCuO$_5$ precipitates.",
"Here $T^*$ is a characteristic temperature.",
"Authors also concluded that at low temperatures both weak and strong pinning centers were effective.",
"[70] Following this conclusion Plain et al.",
"[71] proposed the approximation $J=J_w \\exp (-T/T_w) + J_s \\exp \\left[-3(T/T_s)^2\\right]$ for the current in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"Here $w$ and $s$ mark the current components produced by weak and strong pinning.",
"This expression extends the range of the exponential approximation for $J(T)$ to $T\\lesssim 75$ K. The temperatures $T_w=8$ –13 K, $T_s=78$ –93 K were found for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"[26], [24], [25] The dependence $J_c\\propto \\exp [-3(T/T^*)^2]$ was calculated in theory of strong pinning on columnar pins (line correlated disorder).",
"[6] We found that the current $J_c\\propto (T/T^*)^2 \\exp \\left[-(T/T^*)^3\\right]$ , calculated for compact pins (point correlated disorder),[5] gives a better approximation for standard YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"As shown in Fig.",
"REF (c), in the range $T\\lesssim 75$ K the $J(T)$ curves are well fitted by the dependence $J_c=J_w \\exp (-T/T_w) + J_s \\left(\\frac{T}{T_s}\\right)^2 \\exp \\left[-(T/T_s)^3\\right].$ We obtained $T_w=18$ and 24 K and $T_s=52$ K in self-field.",
"Approximation of the curves by Eq.",
"(REF ) gave lower $T_w=8$ –10 K and higher $T_s=85$ –93 K values which excellently agree again with published data.",
"The above analysis confirms validity of all proposed earlier approximations for $J(T)$ for our samples in restricted temperature ranges.",
"The common features of this behavior are a slow quasi exponential decay at low temperatures and a more rapid power-law decay at high ones.",
"We assumed that at least two components are needed to describe $J(T)$ in the whole temperature range.",
"Thermal fluctuations must also be taken into account at high temperatures since they reduce the effective pinning strength and lead to depinning of vortices at some temperature $T_\\mathrm {dp}$ which is less than $T_c$ .",
"[3], [72] Above the depinning temperature $T_\\mathrm {dp}$ the critical state is destroyed, the persistent current disappears and its relaxation rate becomes zero.",
"We supposed that in vicinity of $T_\\mathrm {dp}$ the current depends on difference $(T_\\mathrm {dp}-T)$ or its powers and found the depinning temperatures for our samples to check this point.",
"Figure: (Color online) |dJ/dlnt|𝑣𝑠T|dJ/d\\ln t|\\textit { vs } T forYBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } films obtained after relaxation for 1 hour in fields of 1530 Oe(pentagons and triangles down), 910 Oe (squares and diamonds) and in self-field(triangles).",
"The dashed lines extrapolate RR to zero temperature.",
"The continuous lines are fitsR=ℛln β (T dp /T)R=\\mathcal {R}\\ln ^\\beta (T_\\mathrm {dp}/T).",
"Inset: RR vsln(T dp /T)\\ln (T_\\mathrm {dp}/T) in logarithmic scales.",
"See text for details.The relaxation rate $R\\equiv |dJ/d\\ln t|$ for our films is presented in Fig.",
"REF .",
"$R(T)$ curves demonstrate a well-known maximum at low temperatures[7] behind which they smoothly decrease down to zero.",
"We found that above 30 K the rate is well fitted by the dependence $R =\\mathcal {R}\\ln ^\\beta (T_\\mathrm {dp}/T)$ .",
"To obtain the fitting parameters $\\mathcal {R}$ , $T_\\mathrm {dp}$ and $\\beta $ we plotted $R$ vs $\\ln (T_\\mathrm {dp}/T)$ in logarithmic scales and varied $T_\\mathrm {dp}$ to straighten the curves as shown in insets of Fig.",
"REF .",
"Then $\\mathcal {R}$ and $\\beta $ were got as shifts and inclination factors of the fitting lines.",
"Obtained values of $T_\\mathrm {dp}$ and $\\beta $ slightly depend on magnetic field.",
"For sample Y1 we found $T_\\mathrm {dp}=84.5 \\pm 0.5$ K, $\\beta =1.2 \\pm 0.1$ in self-field and $T_\\mathrm {dp}=84 \\pm 0.5$ K, $\\beta =1.4 \\pm 0.1$ for $H=910$ and 1530 Oe.",
"For sample Y2 the same $T_\\mathrm {dp}=88\\pm 0.5$ K and $\\beta =1.0\\pm 0.05$ were found for all fields.",
"The obtained depinning temperatures are presented in Table REF .",
"While the critical temperatures are of the same order for all samples, their $T_\\mathrm {dp}$ strongly differ.",
"For example, $T_\\mathrm {dp}\\simeq T_c^M$ for sample Y2 but $T_c^M-T_\\mathrm {dp}\\simeq 14$ K for Y4.",
"To be sure in $T_\\mathrm {dp}$ evaluation we checked their maximal and minimal values by direct measurements of the magnetization thermal hysteresis.",
"[72] Figure: (Color online) M(T)M(T) measured in field of 910 Oe under warming (ZFC) and cooling(FC) of samples Y4 and Y2 (inset).",
"Dashed lines mark depinning and critical temperatures.",
"Toavoid a faulty hysteresis the low sweep rate of 3 K/min was used inmeasurements.$M(T)$ curves measured in field of 910 Oe for samples Y4 and Y2 are presented in Fig.",
"REF .",
"The data were obtained after ZFC procedure under warming and subsequent cooling of samples in field of 910 Oe.",
"A thermal hysteresis caused by pinning of vortices is observed below the depinning temperature in Fig.",
"REF .",
"A reversible magnetization is distinctly seen above $T_\\mathrm {dp}$ up to the critical temperature $T_c^M$ for sample Y4 while for Y2 it is indiscernible because of small difference between $T_\\mathrm {dp}$ and $T_c^M$ .",
"The measured depinning temperatures coincide with $T_\\mathrm {dp}$ obtained by fit of $R(T)$ dependences.",
"Taking obtained $T_\\mathrm {dp}$ we found that at high temperatures the current density follows the power law $J_2= J_2(0)(1-T/T_\\mathrm {dp})^\\alpha ,$ therefore we plotted $J$ vs $1-T/T_\\mathrm {dp}$ in logarithmic scales and additionally fit $T_\\mathrm {dp}$ as it was done for the relaxation rate.",
"$T_\\mathrm {dp}$ obtained by $R(T)$ and $J(T)$ fits mostly coincided or differed in the error range of 0.5 K. Figure: (Color online) J𝑣𝑠1-T/T dp J \\textit { vs } 1-T/T_\\mathrm {dp} for YBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } films.Dashed lines are fits by Eq. ().",
"The curves in bottom panel were measured infield of 910 Oe for samples Y1–Y3 and 1530 Oe for Y4.",
"See Fig.",
", for α\\alpha and T dp T_\\mathrm {dp} values.$J$ vs $1-T/T_\\mathrm {dp}$ dependences are shown in Fig.",
"REF in logarithmic scales.",
"The curves demonstrate a pronounced linear part at high temperatures.",
"Top panel of Fig.",
"REF shows that the curve obtained in self-field differs from ones measured in external fields which are quite similar.",
"The curves obtained in fields of 910 and 1530 Oe for sample Y1 are approximated by the same $T_\\mathrm {dp}=84$ K, difference of $\\alpha \\simeq 2\\pm 0.1$ is within the error and only $J_2(0)$ values differ by 19% (see Fig.",
"REF ).",
"In the self-field the current demonstrates a weaker temperature dependence with $\\alpha =1.3 \\pm 0.1$ and slightly higher $T_\\mathrm {dp}=84.5$ K. Analyzing $J(T)$ obtained for different samples we found that $T_\\mathrm {dp}$ and $\\alpha $ values do not correlate with each other, at the same time the higher current densities $J_2(0)$ correspond to the lower powers $\\alpha $ (see Fig.",
"REF ).",
"It can be seen in bottom panel of Fig.",
"REF where the curves measured in external field are presented for all samples.",
"For example, for samples Y1 and Y4 the fitting lines demonstrate the same inclination $ \\alpha =2.0\\pm 0.1$ , but $T_\\mathrm {dp}$ differ by 9 K (see Table REF and Fig.",
"REF ).",
"Figure: (Color online) Temperature dependences of the current density of theYBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } film (sample Y1) indifferent magnetic fields.",
"Top: current components J 1 =J-J 2 J_1=J-J_2 (left)and J 2 =J-J 1 J_2=J-J_1 (right).",
"Dashed lines are fits bydependences () and ().Bottom: Curves J(T)J(T) are shown in standard and semilogarithmic scales in orderto bring out both low and high temperature behavior of JJ.",
"The curves are shifted (multiplied byshown factors) to avoid a crowding.",
"Continuous lines are sums of fitting curves presented in toppanels.Figure: (Color online) Temperature dependences of the current density ofYBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } films in magnetic field of 910 Oe for samples Y1-Y3 and 1530 Oe forY4.Top: current components J 1 =J-J 2 J_1=J-J_2 (left) and J 2 =J-J 1 J_2=J-J_1 (right).",
"Dashed linesare fits by dependences ()and ().Bottom: Curves J(T)J(T) are shown in standard and semilogarithmic scales in orderto bring out both low and high temperature behavior of JJ.",
"The curves are shifted (multiplied byshown factors) to avoid a crossing.",
"Continuous lines are sums of fitting curves presented in toppanels.Below 30 K the measured current deviates from the power law (REF ).",
"Following to Ovchinnikov and Ivlev[39] we supposed that the current consists of two components and subtracted the dependence (REF ) from the experimental data to analyze a low-temperature behavior.",
"Results of subtraction are shown in left top panels of Figures REF and REF .",
"As seen, the current $J_1=J-J_2$ strongly changes in the range $T\\lesssim 30 $ K. At low temperatures $J_1(T)$ dependence moderates and above 20 K the current gradually falls down to zero at $T\\sim 40$ K. We found that low-temperature component of the current can be approximated by an empiric dependence $J_1=\\frac{J_1^*}{1+\\exp (T/T_1)/2T_1},$ where the parameter $T_1$ is in Kelvins in the exponent power and dimensionless in its divisor.",
"As seen in Figures REF and REF the exponential law (REF ) well fits $J_1(T)$ dependences.",
"In low field the parameter $J_1^*$ is field-independent and $T_1$ slightly decreases with $H$ (see left top panel in Fig.",
"REF ).",
"The current component $J_2=J-J_1$ is also plotted for comparison in right top panels of Figures REF and REF .",
"The ratio of the components $J_1$ and $J_2$ is sample dependent and changes with temperature.",
"For example, $J_2>J_1$ at all temperatures for samples Y2 and Y3 while $J_1$ becomes more than $J_2$ for Y1 and Y4 at low temperatures.",
"The ratio determines temperature behavior of total current.",
"Though $J$ is higher in samples Y2 and Y3 at elevated temperatures, at $T\\lesssim 15$ K it becomes higher in Y1 and Y4 because of rapid increase of large $J_1$ component.",
"$J(T)$ curves and sum of $J_1(T)$ and $J_2(T)$ fits are presented in bottom panels of Figures REF and REF .",
"As seen, the current change by about three orders of magnitude is well approximated.",
"Thus analysis of the separated current components allows us empirically describe $J(T)$ at all temperatures."
],
[
"Discussion",
"Let us consider a relation between the components and relaxation of the current.",
"Comparison of the relaxation rates $R(T)$ with $J_1$ components in Figs.",
"REF and REF demonstrates a correlation: the larger $J_1$ the larger $R(T)$ maxima.",
"$J_1$ rapidly decays with both time and temperature therefore it is evidently produced by weak pinning on point defects having a small pinning energy.",
"The normalized relaxation rate $S\\equiv R/J$ vs $T$ is plotted in Fig.",
"REF .",
"At low temperatures $S$ rises due to $R$ increase and $J$ decrease.",
"When temperature rises $R$ passes though maximum and then decreases.",
"Since $J$ also decreases, the well known $S(T)$ plateau[7] is observed.",
"At elevated temperatures $J$ decreases more rapidly than $R$ so $S$ rises again.",
"Before the plateau a maximum of $S(T)$ is often observed for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films[32], [11] in the temperature range where the $J_1$ component exists.",
"As seen in Figs.",
"REF and REF , the $S(T)$ peak is pronounced for sample Y4 having large $J_1$ and small $J_2$ but it is absent for sample Y3 having an inverse ratio of the components.",
"Therefore we suppose that the peak is caused by fast relaxation of the $J_1$ component.",
"Because of field suppression of both $R$ and $J_2$ (see Figs.",
"REF and REF ), value of $S \\simeq R/J_2$ at the plateau depends on $H$ and proves to be smallest in self-field.",
"At the same time, $H$ slightly affects $J_1$ and amplitude of $R(T)$ maximum therefore field influence on $S(T)=R/(J_1+J_2)$ is reduced in the temperature range of the peak location.",
"As a result, the peak is more pronounced in self-field as illustrated in bottom panel of Fig.",
"REF .",
"In Ref.",
"Maiorov-NM-2009 the peak was attributed to a synergetic combination of two types of pinning centers present in the films, namely artificial columnar BaZrO$_3$ inclusions aligned along $c$ axis and the Y$_2$ O$_3$ nanoparticles horizontally aligned in $ab$ plane.",
"There are no artificial inclusions in our films.",
"As discussed below, pinning in our samples is apparently produced by the Y$_2$ O$_3$ participates and oxygen vacancies.",
"Therefore we suppose that the $S(T)$ peak is caused by combination of these pinning centers inherent to YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"As seen in Figures REF and REF , the relaxation rates $R$ and $S$ extrapolated to zero temperature don't vanish.",
"The extrapolated $S$ well agrees with the value obtained for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ single crystal at $T<1$ K.[33] The nonzero rate is caused by the quantum tunneling of vortices which occurs in layered superconductors.",
"[3], [7] In YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films the quantum creep affects vortices dynamics at temperatures up to $5-10$ K.[34], [35], [36] Precise torque measurements of YBa$_2$ Cu$_3$ O$_{7-\\delta }$ single crystal, which revealed a crossover to two-dimensional superconducting behavior at $T<80$ K,[37] as well as the quantum creep testify importance of layered structure for superconductivity in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ .",
"Let's consider now the theoretical basis for two-component current in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"In Appendix we reduced the general solution of OI theory[39] for the case of magnetic field applied normally to the superconducting planes and calculated the critical current density $J_c=J_{c1}+J_{c2}, \\nonumber \\\\J_{c1}=J_p\\left[ 1-\\exp \\left\\lbrace -a_1 \\frac{(J_p/J_0)^{5/4}n_p s^{5/4}\\xi ^{3/4}}{[\\varepsilon ^2 b\\ln (1/b)]^{5/8}}\\right\\rbrace \\right], \\\\J_{c2} = a_2 J_0 \\frac{(F_v/\\varepsilon _0)^{9/4}n_v\\xi ^3}{[\\varepsilon ^2 b\\ln (1/b)]^{5/8}},\\\\J_0= J_0(0)\\tau _-^{3/2}\\tau _+^{1/2},\\qquad b=\\xi ^2B/\\Phi _0, \\nonumber \\\\J_0(0)=\\frac{c\\Phi _0}{12\\sqrt{3}\\pi ^2\\xi _0\\lambda _0^2}\\simeq 300\\text{ MA/cm}^2.",
"\\nonumber $ The current component $J_{c1}$ is produced by pinning in the superconducting layers.",
"Here $J_{p} \\equiv cF_p/\\Phi _0 s$ is the characteristic in-plane current density, $F_p$ and $n_p$ are values of maximal pinning force and concentration of point pinning centers, $s$ is a distance between the planes, $\\Phi _0$ is the magnetic flux quantum, and $a_1=0.5203$ is the numerical factor.",
"The component $J_{c2}$ is caused by an anisotropic pinning in the superconductor volume.",
"Here $F_v$ and $n_v$ are values of maximal pinning force and concentration of pinning centers, $\\varepsilon _0 =(\\Phi _0/4\\pi \\lambda )^2$ determines the self-energy of the vortex-lines, and $a_2=0.9273$ is the numerical factor.",
"Both components depend on the depairing current density $J_0$ (REF ), dimensionless reduced magnetic field $b$ and the anisotropy parameter $\\varepsilon $ .",
"Using $\\lambda _0 =1400$ Å,[73], [74] $\\xi _0=17.2$ Å[75] we estimated $J_0$ at $T=0$ for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ .",
"As mentioned above, the field dependence of the $J_1$ component of the measured current is weak.",
"For example, at $T=0$ for sample Y1 we obtained the same value $J_1(0)=8.0$ MA/cm$^2$ with accuracy of 0.6% for all fields.",
"Let us estimate $J_{c1}$ from its field dependence using the ratio $J_{c1}(B_1) / J_{c1}(B_2) \\lesssim 1.01$ of the order of $J_1(0)$ uncertainty for fields $B_1=910$ G and $B_2=1530$ G. From (REF ) we obtained $J_{c1}(B_1) / J_{c1}(B_2) =\\lbrace 1-\\exp [-xf(b_1)]\\rbrace / \\lbrace 1-\\exp [-xf(b_2)]\\rbrace $ where $f(b)=[b\\ln (1/b)]^{-5/8}$ and $x=a_1n_p\\xi ^2(sJ_p/\\varepsilon \\xi J_0)^{5/4}$ .",
"Taking the above ratio we calculated $x\\gtrsim 0.0851$ .",
"For oxygen deficiency $\\delta \\gtrsim 0.03$ in our films the concentration of randomly distributed vacancies in CuO$_2$ planes is estimated as $n_p =(4/7)\\delta /ab \\simeq 1.15\\cdot 10^{-3}\\text{ Å}^{-2}$ where $a \\simeq 3.82$ Å and $b \\simeq 3.89$ Å are the orthorhombic lattice cell parameters.",
"[76] Two distances separate pairs of CuO$_2$ planes in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ : the intra-pair distance $s_p=3.37$ Å and the inter-pair one of 8.32 Å.",
"[76] Using $s=8.32$ Å and $\\varepsilon = 1/6.5$ (Ref.",
"Bosma-PRB-2011) we estimated lower limits for both the ratio $J_p/J_0 \\gtrsim 0.177$ and the current $J_{c1}\\gtrsim 53$ MA at $T=0$ .",
"As follows from the estimation, a weak field dependence of $J_{c1}$ is realized for large currents which are much more than $J_1(0)$ .",
"$J_{c1}$ is really more than $J_1$ due to the quantum creep, but at $T\\rightarrow 0$ the relaxation rate is small (see Fig.",
"REF ) and a difference between $J_{c1}(0)$ and $J_1(0)$ must also be small.",
"Thus the field dependence of $J_{c1}$ following from expression (REF ) contradicts to that of $J_1$ .",
"The failure is caused by high concentration of defects in the CuO$_2$ planes.",
"Since a number of defects per square of vortex core exceeds unity, $N_p \\simeq \\pi \\xi ^2n_p \\simeq 1.07$ , the core contains a defect at any site.",
"In such conditions only fluctuations of defect density pin vortices and the pinning becomes collective[3], [38] while expression (REF ) is obtained for strong pinning.",
"In the case of the collective pinning the current is independent of field in the single-vortex pinning regime which is realized if $s<L_c^c<\\varepsilon a_0$ .",
"In magnetic field directed along the $c$ axis a length of the collective pinning segment for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ is estimated as $L_c^c\\simeq 10\\varepsilon \\xi _0 \\simeq 26.5$ Å.",
"[3] Both an inter-vortex distance $a_0 \\simeq (2\\Phi _0/\\sqrt{3}B)^{1/2}\\gtrsim 1100$ Å and its product $\\varepsilon a\\gtrsim 170$ Å exceeded $L_c^c$ in our experiments so conditions for field independence of the current were fulfilled.",
"Therefore we suppose that in-plane pinning is produced by the collective action of oxygen vacancies.",
"Let us compare $J_1$ with $J_c$ obtained in CP theory for a layered superconductor.",
"Figure: (Color online) J 1 (T)J_1(T) obtained in self-field for sample Y1 (triangles) and itsapproximations by Eq.",
"() — dashed line, J c in J_c^\\text{in}Eq.",
"() — continuous lines (left for β=0.62\\beta =0.62 and right forβ=0.1\\beta =0.1), J c c J_c^c Eqs.",
"() and () — dotted lines forδT c \\delta T_c pinning and dash-dotted lines for δℓ\\delta \\ell pinning.",
"The currents are magnifiedin inset to illustrate J c c (T)J_c^c(T) following from Eqs.",
"().In field applied along normal to superconducting planes the critical current coincides for layered and anisotropic superconductors.",
"For single vortex collective pinning it is expressed as[3] $J_c^c&=&J_0\\left[\\frac{\\delta _m}{\\varepsilon }\\right]^{2/3}{-22mu}=J_0(0)\\left[\\frac{\\delta _m(0)}{\\varepsilon }\\right]^{2/3}{-16mu}\\tau _+^{-1/2}\\tau _-^{5/2} \\quad \\delta \\ell \\text{ pin.",
"}\\quad \\\\J_c^c&=&J_0\\left[\\frac{\\delta _\\alpha }{\\varepsilon }\\right]^{2/3}{-22mu}=J_0(0)\\left[\\frac{\\delta _\\alpha (0)}{\\varepsilon }\\right]^{2/3}{-16mu}\\tau _+^{5/6}\\tau _-^{7/6} \\quad \\text{ $\\delta T_c$ pin.",
"}$ The dimensionless pinning parameters for oxygen vacancies in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ are estimated as $\\delta _m(0)/\\varepsilon \\simeq (0.2-1) 10^{-2}$ for $\\delta \\ell $ pinning and $\\delta _\\alpha (0)/\\varepsilon \\simeq 10^{-3}$ for $\\delta T_c$ pinning,[3] and the corresponding currents are $J_c^c(0) \\simeq (5-14)$ MA/cm$^2$ and $J_c^c(0) \\simeq 3$ MA/cm$^2$ .",
"The dependence $J_1(T)$ obtained in self-field for sample Y1 as well as fitting curves for Eqs.",
"() are presented in Fig.",
"REF .",
"As seen $J_1(T)$ disagree with $J_c^c(T)$ curves, moreover for $\\delta T_c$ pinning the current $J_c^c(0)$ is about two times lower than $J_1(0)$ .",
"Eqs.",
"() does not take into account thermal fluctuations suppressing the critical current at high temperatures[3] $J_c^c = \\frac{c(k_\\mathrm {B}T)^2}{\\Phi _0\\varepsilon _0\\xi ^3}\\exp \\left[-\\frac{3w}{2\\delta _{\\alpha ,m}}\\left(\\frac{k_\\mathrm {B}T}{\\varepsilon _0\\xi }\\right)^3\\right].$ Here $w$ is a factor of the order of unity.",
"Selecting temperature dependences of quantities[65] we write $J_c^c &= J_c^c(0)\\frac{\\tau ^2\\tau _-^{1/2}}{\\tau _+^{5/2}}\\exp \\left[-\\frac{3w}{2}\\left(\\frac{T}{T^*}\\right)^3\\frac{1}{f_c^c(T)}\\right],\\\\f_c^c(T) &={\\left\\lbrace \\begin{array}{ll}\\tau _+^3\\tau _-^3 &\\text{for } \\delta \\ell \\text{ pinning},\\\\\\tau _+^5\\tau _- &\\text{for } \\delta T_c\\text{ pinning},\\\\\\end{array}\\right.}",
"\\nonumber \\\\J_c^c(0) &=\\frac{3\\sqrt{3}}{4}J_0(0) \\left[\\frac{k_\\mathrm {B}T_c}{\\varepsilon _0(0)\\xi _0}\\right]^2\\simeq 1.07 \\text{ MA/cm}^2, \\nonumber \\\\T^* &= \\frac{\\varepsilon _0(0)\\xi _0\\delta _{\\alpha ,m}^{1/3}(0)}{k_\\mathrm {B}}\\simeq {\\left\\lbrace \\begin{array}{ll}(198-116)\\text{ K} & \\delta \\ell \\text{ pin.",
"},\\\\92\\text{ K} & \\delta T_c \\text{ pin}.\\\\\\end{array}\\right.}",
"\\nonumber $ Due to fluctuations the current is strongly suppressed at temperatures above the depinning temperature which is calculated from the equation $T_\\mathrm {dp}^3 ={T^*}^3{f_c^c}(T_\\mathrm {dp})$ .",
"[3] The temperatures $T_\\mathrm {dp}\\simeq 89$ K and 71–79 K calculated for $\\delta T_c$ and $\\delta \\ell $ pinning are considerably higher than temperatures at which $J_1$ disappears.",
"In Fig.",
"REF dependences (REF ) are shown.",
"As seen, a magnitude of $J_1(T)$ a lot more than maximal values of $J_c^c(T)$ and the curves lie in different temperature ranges.",
"Thus we conclude that neither Eqs.",
"() nor (REF ) describe $J_1$ component of the measured current.",
"In magnetic field parallel to a superconducting layers the intrinsic pinning takes place in a layered superconductor.",
"[3], [78], [79] Kinks of vortices[80], [81] also lead to the intrinsic pinning.",
"Though our experiments were performed in a transverse field, a demagnetizing effect, which was strong because of low fields and large demagnetizing factor of films, produced a tangential component of field[82] directed along superconducting layers in the films.",
"Therefore the critical current produced by the intrinsic pinning should be also considered.",
"Its temperature dependence has the form[3], [79] $J_c^\\mathrm {in} = J_0\\left(\\frac{8\\varepsilon \\xi }{s}\\right)^2\\left(1-\\frac{B}{B_{c2}}\\right)\\exp \\left[-8\\left(\\frac{\\varepsilon \\xi }{s}\\right)^2\\right].$ Neglecting the field dependence, since $B\\ll B_{c2}$ in our experiments, we write it as $J_c^\\mathrm {in} =64\\beta J_0(0)\\tau _+^{3/2}\\tau _-^{1/2}\\exp [-8\\beta \\tau _+/\\tau _-],\\\\\\beta = (\\varepsilon \\xi _0/s)^2 = 0.62-0.1.\\nonumber $ Here we estimated $\\beta $ for the above mentioned distances between CuO$_2$ planes.",
"$J_c^\\mathrm {in}(T)$ curves calculated for $\\beta =0.62$ and 0.1 and normalized by factors 0.094 and 0.0092 respectively are presented in Fig.",
"REF .",
"While quasi-exponential shape of $J_c^\\mathrm {in}(T)$ is more appropriate to $J_1(T)$ , the intrinsic current decreases more slowly and disappears at higher temperatures.",
"In addition at low temperatures $J_c^\\mathrm {in}$ is one or two orders more than $J_1$ .",
"Summing up we conclude that $J_1$ component of the current is caused by the collective pinning of vortices on oxygen vacancies in the single vortex pinning regime.",
"However we failed to find an appropriate approximation for $J_1(T)$ in the frame of CP theory.",
"Apparently, because of smallness of pinning energy, $J_1$ rapidly relaxes and its temperature dependence is strongly affected by creep.",
"Turning to volume pinning we begin with a remark about pinning on the dislocations.",
"In EDP model[14] the critical current depends on an average size of the crystallites, i. e. CDB size in the diffraction experiments, as well as on a misorientation angle $\\omega $ between them.",
"[42] As seen in Table REF , in our samples the CDB size changes by one order and the angle $\\omega $ varies almost seven times, but these parameters do not correlate with $J_2$ .",
"Despite absence of the correlation, we compare below $J_2(T)$ with $J_c(T)$ calculated for pinning on both non-superconducting inclusions and the edge dislocations for analysis to be comprehensive.",
"In low fields the critical current of YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films is independent of field.",
"[12], [64], [19], [23], [10], [21] The field-independent current produced by pinning on the edge dislocations is calculated as[14] $J_{cd}(T,0)\\simeq \\frac{3\\sqrt{3}}{16\\sqrt{2}}\\frac{c\\varepsilon _0}{\\Phi _0}\\frac{r_d^2}{\\xi ^3}=J_{cd}^0\\left(\\frac{r_d}{\\xi _0}\\right)^2\\frac{\\tau _-^{5/2}}{\\tau _+^{1/2}},\\\\J_{cd}^0 = \\frac{27}{64\\sqrt{2}}J_0(0)=89.5 \\text{ MA/cm}^2,\\nonumber $ where $r_d$ is the radius of dislocation normal core.",
"Dependences $J_2(T)$ obtained in self-field and their fits by Eqs.",
"(REF ) are presented in Fig.",
"REF .",
"From the fits we obtained $r_d\\simeq 4.6$ , 5.0, 5.3 and 4.6 Å respectively for samples Y1–Y4.",
"Eq.",
"(REF ) provides the same temperature dependence for all samples scaled by $r_d$ values while $J_2(T)$ curves differ for different samples.",
"The dependence $J_{cd}(T,0)$ satisfactory approximates $J_2(T)$ only for sample Y1 while for other ones a discrepancy of the fitting curves and the experimental data is clearly seen at $T/T_c \\gtrsim 0.5$ .",
"The field-independent current caused by pinning on inclusions[19] $J_{ci}(T,0) \\simeq \\frac{3J_0}{4}\\sqrt{\\frac{}{}}{3n_i}{\\pi \\varepsilon ^2}\\left(\\frac{F_i\\xi }{\\varepsilon _0}\\right)^{3/2}$ depends on inclusion density $n_i$ and the pinning force $F_i$ approximated as[3], [19] ${\\begin{array}{c}\\frac{F_i\\xi }{\\varepsilon _0} \\simeq \\frac{D_\\mathit {iz}}{4}\\mathcal {F}(T,d_i),\\\\\\mathcal {F}(T,d_i)=\\ln \\left(1+\\frac{D_i^2}{2\\xi ^2}\\right)=\\ln \\left(1+\\frac{d_i^2\\tau _-}{2\\tau _+}\\right).\\end{array}}$ Here $D_i$ is an average extent of an inclusion, $D_\\mathit {iz}$ is its extent along the field direction and $d_i=D_i/\\xi _0$ .",
"Selecting temperature dependences of quantities we write the current in the form $J_{ci}(T,0)\\simeq J_{ci}^0[\\mathcal {F}(T,d_i)\\tau _-]^{3/2}\\tau _+^{1/2},\\\\J_{ci}^0 \\simeq \\frac{3\\sqrt{3} }{32\\sqrt{\\pi }}\\frac{J_0(0)}{\\varepsilon }\\sqrt{n_iD _\\mathit {iz}^3}\\simeq \\sqrt{n_iD _\\mathit {iz}^3} \\cdot 179\\text{ MA/cm}^2.\\nonumber $ Figure: (Color online) Top: J 2 (T)J_2(T) for YBa 2 _2Cu 3 _3O 7-δ _{7-\\delta } filmsobtained in self-field.",
"Experimental curve for sample Y4 was recorded under warming of filmright after magnetic field removing.",
"Dotted and continuous lines are fits bydependences () and ().",
"Bottom: Scaled J 2 (T)J_2(T)dependences obtained in fields 1530 Oe (triangles) and 910 Oe (other symbols).Dashed and dash-dotted lines are fits by dependences () and ().The curves are shifted (multiplied by shown factors) to avoid a crossing.",
"See text for details.$J_2(T)$ curves were fitted by the dependence (REF ) via parameters $J_{ci}^0$ and $D_i$ .",
"The currents $J_{ci}^0=2.7$ , 1.1 0.88 and 25 MA/cm$^2$ were respectively obtained for samples Y1–Y4.",
"$D_i$ values are presented in Table REF .",
"The fitting curves shown in Fig.",
"REF well agree with $J_2(T)$ for samples Y1 and Y3 though for Y1 the measured current decays more slowly at $T/T_c \\gtrsim 0.75$ .",
"For sample Y4 the measured and fitting curves coincide up to $T\\sim T_\\mathrm {dp}$ .",
"Figure: Temperature dependence of the parameter b=Bξ 2 /Φ 0 =Bξ 0 2 τ + /Φ 0 τ - b=B\\xi ^2/\\Phi _0 =B\\xi _0^2\\tau _+/\\Phi _0\\tau _- calculated for B=2B=2 kG and ξ 0 =17.2\\xi _0=17.2 Å.Inset: function f=[bln(1/b)] -5/8 f = [b\\ln (1/b)]^{-5/8} (continuous line) and itsapproximations f=0.5552·b -0.537 f = 0.5552 \\cdot b^{-0.537} (dashed), f=0.25·b -5/8 f = 0.25 \\cdot b^{-5/8}(dash-dot).Thus pinning on inclusions well describes $J_2(T)$ of YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films in self-field.",
"Interaction of vortices suppresses the critical current when a vortex density $\\tilde{n}\\simeq B/\\Phi _0$ increases.",
"Let us proceed with analysis of field dependence of $J_{c2}$ which is determined by the function $f = [b\\ln (1/b)]^{-5/8}$ , see Eq. ().",
"Because of large $B_{c2}$ in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ the parameter $b=B/2\\pi B_{c2}$ is small.",
"In Figure REF we plotted $b(T)$ for $B=2$ kG exceeding maximum field in our experiments.",
"As seen, $b$ is less than 0.005 for $T/T_c\\lesssim 0.95$ .",
"In the inset of Fig.",
"REF the function $f$ is plotted in the range up to $b=1/2\\pi $ corresponding to $B=B_{c2}$ .",
"For small $b$ it follows a power law and in the range $10^{-4}\\le b\\le 0.005$ is approximated as $f = 0.5552 \\cdot b^{-0.537}$ with the accuracy of $\\pm 1.2$ %.",
"The dependence $J\\propto B^{-\\alpha }$ with $\\alpha \\simeq 0.4$ –0.8 was often observed for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films[12], [64], [19], [23], [10], [21], [52] and 2G-tapes.",
"[9] As seen in Fig.",
"REF , at $b\\gtrsim 0.01$ the function $f(b)$ moderates and should be approximated by $f\\propto b^{-\\alpha }$ with a lower $\\alpha $ .",
"However strengthening $J(B)$ dependence in high fields was reliably established for YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films in numerous experiments.",
"[12], [64], [19], [28], [23], [20], [11], [21] Expression () is valid only if the lateral displacement of the vortex lines $u_0$ is small in comparison with the inter-vortex distance $a_0$ .",
"In high fields when $a_0\\propto B^{-1/2} $ becomes larger than $u_0$ a more strong suppression of the current $J_{c2}\\propto B^{-1}$ is expected.",
"[19] According to Eq.",
"() the curves $J_{c2}[b\\ln (1/b)]^{5/8}$ should be independent of field.",
"Indeed, as seen in bottom Fig.",
"REF , the data obtained for samples Y1 and Y2 in different fields are joined into common curves under such scaling.",
"Therefore we compare $J_c(T,B)[b\\ln (1/b)]^{5/8}$ dependences with the scaled data collected for different fields.",
"In the EDP model the field-dependent critical current[14] $J_{cd}(T,B)= J_{cd}(T,0)\\frac{\\tilde{n}_p}{\\tilde{n}},\\\\\\frac{\\tilde{n}_p(T,B)}{\\tilde{n}(B)} = 1-\\frac{[\\Gamma (\\nu ,\\eta )-\\eta \\Gamma (\\nu -1,\\eta )]^2}{\\Gamma ^2(\\nu )},\\nonumber \\\\\\nu =\\left[\\frac{\\langle L\\rangle }{\\sigma }\\right]^2,\\quad \\eta (T,B)=\\frac{r_d}{\\langle L\\rangle } \\frac{2\\nu }{\\xi _0}\\sqrt{\\frac{\\Phi _0}{B}\\frac{\\tau _-}{\\tau _+}}, $ is determined by a relative number of pinned vortices $\\tilde{n}_p/\\tilde{n}$ expressed via complete and incomplete Euler's gamma functions $\\Gamma (x)$ and $\\Gamma (x,y)$ .",
"[83] Here $\\sigma $ is the dispersion of the crystallite size distribution function around the mean value $\\langle L\\rangle $ .",
"The scaled currents were fitted by $J_{cd}(T,B)[b\\ln (1/b)]^{5/8}$ via $J_{cd}(0,0)$ , $\\nu $ and $\\eta $ using $B=910$ G as parameter.",
"The fitting curves, shown in bottom Fig.",
"REF , agree with experimental data for samples Y1–Y3 though a systematic deviation to a lower current is observed at low temperatures.",
"At the same time the fit badly approximates data for sample Y4.",
"From the fit we obtained $\\eta \\simeq 2$ for all samples, $\\nu \\simeq 7$ for Y2, Y3 and $\\nu \\simeq 1$ for Y1, Y4.",
"Then from () for $B=910$ G the ratio $r_d/\\langle L\\rangle $ was estimated as $1.1\\cdot 10^{-2}$ for samples Y2, Y3 and $1.6\\cdot 10^{-3}$ for Y1,Y4.",
"From $J_{cd}(0,0)$ values we calculated $r_d$ and then obtained $\\langle L\\rangle $ presented in Table REF .",
"As seen, the fit gives a correct order for $r_d$ and $\\langle L\\rangle $ values, however lengths $\\langle L\\rangle $ does not correlate with average sizes of crystallites CDB obtained in the diffraction experiments.",
"Neither $r_d$ nor $\\langle L\\rangle $ correlate with the measured current $J$ or its components.",
"For pinning on inclusions the field-dependent current calculated from Eqs.",
"() and (REF ) takes the form[84] $J_{ci}(T,B) \\simeq \\frac{J_{ci}^B}{[ b\\ln (1/b)]^{5/8}}\\mathrm {F}(T,d_i), \\\\\\mathrm {F}(T,d_i)=\\ln ^{9/4}\\left(1+\\frac{d_i^2\\tau _-}{2\\tau _+}\\right)\\tau _-^{9/8}\\tau _+^{7/8},\\\\J_{ci}^B = \\frac{3^{3/4}J_0(0)n_iD_\\mathit {iz}^{9/4}\\xi _0^{3/4}}{16\\cdot 2^{3/4}\\pi ^{5/8}\\varepsilon ^{5/4}}\\simeq n_i D_\\mathit {iz}^{9/4}\\xi _0^{3/4}\\cdot 129 \\text{ MA/cm}^2,\\nonumber $ The scaled currents were fitted by the dependence $J_{ci}^B\\mathrm {F}(T,d_i)$ via $J_{ci}^B$ and $D_i$ .",
"Since size of inclusions is independent of field, we used the same $D_i$ to fit $J_2$ by both (REF ) and ().",
"An effective density of inclusions $n_id_\\mathit {iz}^{9/4} = n_i(D_\\mathit {iz}/\\xi _0)^{9/4}$ was calculated from $J_{ci}^B$ values.",
"The fitting curves $J_{ci}^B\\mathrm {F}(T,d_i)$ , presented in bottom Fig.",
"REF , agree with experimental data for all samples.",
"Obtained $D_i$ and $n_id_\\mathit {iz}^{9/4}$ values are presented in Table REF .",
"The average extent of inclusions $D_i$ varying in the range $2-14$ nm well agrees with size of Y$_2$ O$_3$ precipitates in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"[15], [16], [17], [18] $D_i$ and $n_id_\\mathit {iz}^{9/4}$ values correlate with $J_2$ component of the current.",
"The larger inclusion extent the more $J_2$ .",
"For inclusions with the same $D_i$ the current rises with increase of the effective inclusion density $n_id_\\mathit {iz}^{9/4}$ .",
"As follows form Eqs.",
"(REF ) and (), the extent of inclusion along the field direction can be obtained from the ratio ${(J_{ci}^0})^2/J_{ci}^B = (D_\\mathit {iz}/\\xi _0)^{3/4}\\cdot 248$ A/cm$^2$ .",
"Then $n_i$ is simply calculated from $J_{ci}^B$ or $J_{ci}^0$ .",
"Values of $D_\\mathit {iz}$ and $n_i$ found in such a procedure are presented in Table REF .",
"Parameters of pinning centers obtained for our films well agree with $D_i=15$ nm and $n_i = (1-3)\\cdot 10^{15}$ cm$^{-3}$ found in magnetic experiments in Ref. vanderBeek-PRB-2002.",
"At the same time a lower density of inclusions was found in direct measurements by means of the electron microscopy.",
"[15], [16], [17], [18] Among microstructure defects in YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films[10], [17], [18] the precipitates[15], [16], [17], [18] [001]-Y$_2$ 0$_3$ and [110]-Y$_2$ 0$_3$ have dimensions close to our estimations of $D_i$ .",
"The [110]-Y$_2$ 0$_3$ precipitates are small cubes or rectangles with sides ranging from 3 to 5 nm.",
"[18] The [001]-Y$_2$ 0$_3$ precipitates have extension of 10 to 20 nm in the $ab$ -plane and about 6 to 8 nm along the $c$ axis.",
"[15], [16], [17], [18] There are no data on density and shape of inclusions with size smaller than 2 nm since such small inclusions are hard to recognize even in high-resolution electron microscopy (HREM) micrographs.",
"[17], [18] Our results for sample Y4 demonstrate presence of such inclusions which we classified as the [110]-Y$_2$ 0$_3$ precipitates.",
"Taking $D_\\mathit {iz}=D_i$ for samples Y1, Y4 and $D_\\mathit {iz}=6$ nm for Y2, Y4, from the effective density of inclusions obtained above we estimated densities $n_i^*$ presented in Table REF .",
"For samples Y1–Y3 estimated $n_i^*$ values are only twice less than that observed by direct HREM method in laser-deposited YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"[18] Since a density of Y$_2$ 0$_3$ precipitates depends on both method and conditions of the deposition process[17], [18], [10], [11] such agreement seems quite satisfactory.",
"Note also that the measured relaxed persistent current is less than $J_c$ so a lower limit for the inclusion densities was estimated in our experiment.",
"Summing up we conclude that the $J_2$ component of the measured current is well described by pinning on the Y$_2$ 0$_3$ inclusions.",
"The pinning is strong and its efficiency rises with increase of inclusions size."
],
[
"Summary",
"In this paper we confirmed experimentally that the critical current of laser-deposited YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films consists of two components caused by in-plane pinning of vortices by oxygen vacancies in superconducting CuO$_2$ planes and by anisotropic pinning on the Y$_2$ 0$_3$ precipitates in the superconductor volume.",
"[39] We proposed a simple method to separate the current components and found their temperature dependences (REF ) and (REF ).",
"Analysis of the current components led us to the following conclusions.",
"The component produced by the in-plane pinning is described as single-vortex collective pinning however we failed to find an appropriate theoretical dependence to approximate its temperature behavior.",
"This component slightly depends on field and rapidly relaxes.",
"The in-plane pinning is substantial only at low temperatures $T\\lesssim 30$ K but in this temperature range its contribution into the critical current and vortices dynamics should not be neglected.",
"The component produced by the volume pinning is well described in the frame of OI theory[39] further developed by van der Beek et al.",
"[19] We confirmed that in laser-deposited YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films the strong anisotropic volume pinning is produced by the nano-size Y$_2$ O$_3$ precipitates.",
"Varying inclusion sizes in different films causes difference in the depinning temperatures and parameters of $J(T)$ dependence.",
"Rather low magnetic field of about 1 kOe applied normally to the film plane affects this current component.",
"Different ratio of the current components and variation of size of the Y$_2$ O$_3$ inclusions lead to a wide variety of $J(T)$ dependences in standard YBa$_2$ Cu$_3$ O$_{7-\\delta }$ films.",
"Addition of artificial defects further complicates $J(T,\\mathbf {B})$ behavior.",
"Nevertheless films produced by different techniques demonstrate some common features discussed in beginning Sec. .",
"While in-plane and volume defects act simultaneously and provide additive components of the current, combining several types of volume defects is not simply additive.",
"Therefore engineering the pinning landscape in 2G/3G-tapes is a very complex problem.",
"[10], [11] We hope that separation and correct analysis of the additive components demonstrating strongly different temperature, field and angle behavior help in solving this actual problem.",
"*"
],
[
"An inhomogeneous layered superconductor with an axial anisotropy and the mass anisotropy ratio $\\varepsilon ^2 = m/M = \\lambda /\\lambda _c$ was considered by Ovchinnikov and Ivlev.",
"[39] Here $m$ and $\\lambda $ are the effective mass of carriers and the penetration depth in isotropic superconducting planes and $M,\\;\\lambda _c$ are the corresponding parameters along normal to the planes.",
"General case of magnetic field $B$ applied along the direction forming an angle $\\theta $ with the planes was analyzed and the critical current density was calculated.",
"We simplify the results obtained by Ovchinnikov and Ivlev for the case $\\theta =\\pi /2$ , when the field is directed normally to the planes, and rewrite the values in the notations of Ref.",
"Blatter-RMP-1994 commonly used at present.",
"As shown by Ovchinnikov and Ivlev, the critical current consists of two parts $J_c = J_{c1} + J_{c2}$ caused by in-plane pinning on point defects in the superconducting planes and by anisotropic pinning of vortices by “macro-defects” in the superconductor volume.",
"The anisotropic component of the current is written as follows [39] $J_{c2} = \\frac{cn_vF_v^2}{\\Phi _0\\alpha ^2\\sqrt{\\epsilon _{xx}C_{xx}}} \\left[\\frac{128 F_v\\xi ^3}{27\\sqrt{\\epsilon _{yy}C_{yy}}}\\right]^{1/4},$ where $F_v$ and $n_v$ are maximum values of pinning force and concentration of pinning centers in volume of the superconductor, $\\xi $ is the in-plane coherent length, $\\Phi _0$ is the magnetic flux quantum.",
"Taking into account the units $\\hbar =c=1$ used by Ovchinnikov and Ivlev [39] we multiplied the right hand side by the light velocity $c$ .",
"The function $\\alpha ^2 = \\sin ^2\\theta + \\varepsilon ^2\\cos ^2\\theta $ is equal to unity in our case.",
"The quantities $C_{yy} = \\alpha ^2C_{xx} = C_{xx} = \\frac{\\Phi _0 B}{8\\pi \\lambda ^2}=\\frac{2\\pi b\\varepsilon _0}{\\xi ^2}$ we express via the energy $\\varepsilon _0=(\\Phi _0/4\\pi \\lambda )^2$ which determines the self-energy of the vortex-lines[3] and the parameter $b=\\xi ^2B/\\Phi _0$ .",
"The quantities $\\epsilon _{xx}$ and $\\epsilon _{yy}$ are written as $\\epsilon = \\frac{\\Phi _0^2}{8\\pi ^2\\lambda ^2}\\ln \\left[ \\frac{\\sqrt{\\alpha \\Phi _0/B}}{d\\cos \\theta +\\alpha \\xi }\\right]\\tilde{\\epsilon }= \\varepsilon _0\\ln (1/b)\\tilde{\\epsilon }.$ From bulky but simple expressions for $\\tilde{\\epsilon }_{xx}(\\theta ,\\varepsilon )$ and $\\tilde{\\epsilon }_{yy}(\\theta ,\\varepsilon )$ in Ref.",
"Ovchinnikov-PRB-1991 we calculated that in our case $\\tilde{\\epsilon }_{xx} =\\tilde{\\epsilon }_{yy} =\\tilde{\\epsilon } =\\varepsilon ^2/2$ .",
"Substituting all obtained values in (REF ), and using expression for the depairing current density[3] $J_0 = \\frac{4}{3\\sqrt{3}}\\frac{c\\varepsilon _0}{\\Phi _0\\xi },$ after simple algebraic transformations we get the anisotropic component of the critical current density in the form ${\\begin{array}{c}J_{c2} = a_2 J_0 \\left(\\frac{F_v}{\\varepsilon _0}\\right)^{9/4}\\frac{n_v\\xi ^3}{[\\varepsilon ^2 b\\ln (1/b)]^{5/8}},\\\\a_2 = \\frac{3^{3/4}}{2^{1/4}\\pi ^{5/8}} =0.9373.\\end{array}}$ The current component caused by the layered structure of superconductor is written as[39] $J_{c1} = \\frac{cF_p}{\\Phi _0 s}\\left[1-\\exp \\left\\lbrace -\\frac{n_pF_p}{\\sin \\theta \\sqrt{\\epsilon _{xx}C_{xx}}}\\left(\\frac{128 F_p\\xi ^3}{27\\sqrt{\\epsilon _{yy}C_{yy}}}\\right)^{1/4}\\right\\rbrace \\right],$ where $F_p$ and $n_p$ are maximum values of pinning force and concentration of pinning centers in the superconducting planes, $s$ is a distance between the planes.",
"Substituting all values obtained above, taking into account that $\\sin \\theta =1$ in our case and denoting $J_{p}=cF_p/(\\Phi _0s)$ , after simple algebraic transformations we rewrite (REF ) as $J_{c1} = J_{p}\\left[ 1-\\exp \\left\\lbrace - a_1 \\left(\\frac{J_{p}}{J_0}\\right)^{5/4}\\frac{n_ps^{5/4}\\xi ^{3/4}}{[\\varepsilon ^2 b\\ln (1/b)]^{5/8}}\\right\\rbrace \\right], \\qquad a_1 =\\frac{16\\cdot 2^{1/4}}{9\\cdot (3\\pi )^{5/8}}=0.5203.$ The authors would like to thank O. A.",
"Krymskaya for the structure measurements, O.",
"A Churkin for technical assistance and A. P. Menushenkov for fruitful discussions.",
"The work was supported by Russian Science Foundation grant 14-22-00098."
]
] | 1612.05454 |
[
[
"Tverberg plus minus"
],
[
"Abstract We prove a Tverberg type theorem: Given a set $A \\subset \\mathbb{R}^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\\in \\{0,1,\\ldots,r-1\\}$, there is a partition of $A$ into $r$ sets $A_1,\\ldots,A_r$ with the following property.",
"The unique $z \\in \\bigcap_1^r \\mathrm{aff} A_j$ can be written as an affine combination of the element in $A_j$: $z = \\sum_{x \\in A_j} \\alpha(x)x$ for every $p$ and exactly $k$ of the coefficients $\\alpha(x)$ are negative.",
"The case $k=0$ is Tverberg's classical theorem."
],
[
"Introduction and main result",
"Assume $A=\\lbrace a_1,\\ldots ,a_n\\rbrace \\subset \\mathbb {R}^d$ where $n=(r-1)(d+1)+1$ and $r\\ge 2$ , $d\\ge 1$ are integers.",
"Suppose further that the coordinates of the $a_i$ (altogether $dn$ real numbers) are algebraically independent.",
"A partition $\\mathcal {A}=\\lbrace A_1,\\ldots ,A_r\\rbrace $ of $A$ is called proper if $1\\le |A_j|\\le d+1$ for every $j \\in [r]$ .",
"Here and in what follows $[r]$ stands for the set $\\lbrace 1,\\ldots ,r\\rbrace $ .",
"We will show later (Proposition REF ) that in this case the intersection of the affine hull of the $A_j$ s is a single point $z$ , that is, $\\lbrace z\\rbrace =\\bigcap _{j=1}^r \\mathrm {aff\\;}A_j$ .",
"Equivalently, the following system of linear equations has a unique solution: $z=\\sum _{x\\in A_j}\\alpha (x)x \\quad \\mbox{ and } \\quad 1=\\sum _{x\\in A_j}\\alpha (x) \\quad \\mbox{ for all }j \\in [r].$ One form of Tverberg's classical theorem [9] puts extra conditions on the coefficients $\\alpha (x)$ (consult [5] and the references therein for an introduction to the subject).",
"Theorem 1.1 (Tverberg's theorem) Under the above conditions there is a proper partition of $A$ into sets $A_1,\\ldots ,A_r$ such that $\\alpha (x)\\ge 0$ for all elements $x \\in A$ .",
"In other words, $\\lbrace z\\rbrace =\\bigcap _{j=1}^r\\mathrm {conv\\;}A_j$ .",
"This means that the unique solution to (REF ) has $\\alpha (x)>0$ for all $x \\in A$ .",
"Can we require here that exactly one (or two or more) of the $\\alpha (x)$ are negative?",
"A partial answer comes from the following theorem, which is the main result of this paper.",
"Theorem 1.2 Assume $k \\in \\lbrace 0,1,\\ldots ,r-1\\rbrace $ .",
"Under the conditions of Theorem REF there is a (proper) partition of $A$ into $r$ parts so that in the unique solution to (REF ) $\\alpha (x) <0$ for exactly $k$ elements $x \\in A$ .",
"Of course the same holds for any set $A$ of $n$ points in $\\mathbb {R}^d$ , we only have to relax the condition $\\alpha (x) <0$ to $\\alpha (x)\\le 0$ for $k$ elements $x \\in A$ and $\\alpha (x)\\ge 0$ for the rest.",
"Actually, the general position condition (on $A$ ) is used in order to avoid cases when $\\alpha (x)=0$ for some $x \\in A$ .",
"It is not clear for what other values of $k$ , $k\\in [n]$ , the theorem holds.",
"Since the sum of the coefficients for each $A_j$ is one, at least one is positive.",
"This implies the upper bound $k\\le n-r$ .",
"The case $d=1$ is very simple.",
"Then $n=2r-1$ and there is no $r$ -partition with $r$ or more negative coefficients, so the trivial bound $k\\le n-r=r-1$ is tight.",
"In the case $d=2,\\;r=3$ and $n=7$ Theorem REF gives a suitable partition for $k=0,1,2$ .",
"A careful case analysis shows that the statement holds for $k=3$ as well, and an extensive computer aided search did not find any example where it fails to hold for $k=4$ .",
"The case of $r=2$ , that is, Radon (plus minus) partitions can be checked directly.",
"Then $|A|=d+2$ and the outcome is that for any $k\\in \\lbrace 0,1,\\ldots ,\\lfloor \\frac{d+2}{2}\\rfloor \\rbrace $ there is a partition with exactly $k$ negative $\\alpha (x)$ .",
"Further, there are examples showing that this does not hold for $k>\\lfloor \\frac{d+2}{2}\\rfloor $ .",
"In this case everything is governed by the unique affine dependence of the vectors in $A$ , just as in the proof of Radon's theorem.",
"We omit the details.",
"We will see in Corollary REF in Section 3 that, for a strange reason, if both $d$ and $r$ are even, then Theorem REF holds with $k=\\frac{1}{2} [(r-1)(d+1)+1]$ as well.",
"This makes us wonder if Theorem REF holds for all integers $k\\le \\frac{1}{2} [(r-1)(d+1)+1]$ .",
"We are going to prove Theorem REF in a stronger form: to some extent we can prescribe the subset of $A$ where the coefficients in (REF ) are negative.",
"Theorem 1.3 Under the conditions of Theorem REF let $M\\subset A$ be a set of size at most $r-1$ such that $\\mathrm {conv\\;}M \\cap \\mathrm {conv\\;}(A\\setminus M)=\\emptyset $ .",
"Then there is a partition $\\mathcal {A}=\\lbrace A_1,\\ldots ,A_r\\rbrace $ of $A$ such that in (REF ) $\\alpha (x)<0$ if and only if $x \\in M$ .",
"We prove this theorem in Section 3 where we state a slightly stronger result whose proof is in Section 6.",
"Examples showing the necessity of the condition on $M$ are given in Section 2.",
"In Section we discuss coloured variations of Theorem REF .",
"In Section 5 we prove the following fact.",
"Proposition 1.1 Assume $A=\\lbrace a_1,\\ldots ,a_n\\rbrace \\subset \\mathbb {R}^d$ , the coordinates of the $a_i$ are algebraically independent and $r\\ge 2$ , $d\\ge 1$ are integers.",
"If the partition $\\mathcal {A}=\\lbrace A_1,\\ldots ,A_r\\rbrace $ of $A$ is proper and $n=(r-1)(d+1)+1$ , then $\\bigcap _{j=1}^r \\mathrm {aff\\;}A_j$ is a single point.",
"If $n\\le (r-1)(d+1)$ , then $\\bigcap _{j=1}^r \\mathrm {aff\\;}A_j=\\emptyset $ .",
"The last statement holds even if the partition is not proper.",
"The first part must be known, see for instance [6] or [4] for similar statements.",
"The second part is proved in [9].",
"We give a simple proof in Section ."
],
[
"The condition on $M$",
"The condition on $M$ in Theorem REF simply says that $M$ and $A \\setminus M$ can be separated by a hyperplane.",
"Example 1.",
"We give an example showing the necessity of this condition.",
"Let $V=\\lbrace v_1,\\ldots ,v_{d+1}\\rbrace $ be the set of vertices of a regular simplex $\\Delta $ , and let $c$ be the centre of $\\Delta $ and write $F_h$ for its facet opposite to $v_h$ .",
"For every $h \\in [d+1]$ let $U_h\\subset v_h+\\varepsilon B$ be an $(r-1)$ -element set.",
"Here $\\varepsilon >0$ is small and $B$ is the Euclidean unit ball in $\\mathbb {R}^d$ centred at the origin.",
"Define $A=\\lbrace c\\rbrace \\cup \\bigcup _{h=1}^{d+1}U_h$ .",
"We assume $A$ is in general position which can be clearly reached by choosing the sets $U_h$ suitably.",
"Set $M=\\lbrace c\\rbrace $ so the separation condition fails.",
"We claim that there is no proper $r$ -partition of $A$ such that in (REF ) only $\\alpha (c)$ is negative, $\\alpha (x)>0$ for all other $x \\in A$ .",
"Assume the contrary and let $A_1,\\ldots ,A_r$ be a proper partition with $z \\in \\bigcap _{j=1}^r \\mathrm {aff\\;}A_j$ so that $\\alpha (c) < 0$ and $\\alpha (x)> 0$ for all other $x \\in A$ in (REF ).",
"Given a convex compact set $C$ in $\\mathbb {R}^d$ and a point $u \\in \\mathbb {R}^d \\setminus C$ we let $S(u,C)$ denote the shadow of $C$ from $u$ which is the set of point $\\lbrace tu+(1-t)c: c\\in C,\\; t\\le 0\\rbrace $ , see Figure REF .",
"Figure: Construction of S(u,C)S(u,C).",
"Notice that C⊂S(u,C)C \\subset S(u,C).For every $h \\in [d+1]$ there is a $j=j(h) \\in [r]$ such that $A_{j(h)}$ and $U_h$ are disjoint, simply because each $U_h$ has $r-1$ elements and the number of sets $A_j$ is $r$ .",
"It follows that $A_{j(h)}\\subset F_h + \\varepsilon B \\subset S(c, F_h+\\varepsilon B)$ if $c \\notin A_{j(h)}$ , and then $z \\in \\mathrm {conv\\;}A_{j(h)} \\subset S(c, F_h+\\varepsilon B)$ .",
"If $c\\in A_{j(h)}$ , then in the equation $z=\\sum _{x\\in A_{j(h)}}\\alpha (x)x$ only the coefficient $\\alpha (c)$ is negative, so $z \\in S(c, F_h+\\varepsilon B)$ .",
"Therefore $z \\in \\bigcap _{h=1}^{d+1} S(c,F_h+\\varepsilon B).$ However, $ \\bigcap _{h=1}^{d+1} S(c,F_h+\\varepsilon B)=\\emptyset $ as long as $\\varepsilon < \\frac{\\mathrm {diam}\\Delta }{2d}$ .",
"See Figure REF for an illustration.",
"This follows from the fact that the shadows $S(c,F_h+\\varepsilon B)$ for $h \\in [d+1]$ are convex and their union covers the boundary of $\\Delta $ .",
"If they had a point in common, then their union would cover $\\Delta $ .",
"Since none of them contains $c$ , this is impossible.",
"This gives us the contradiction we were seeking.",
"Figure: The shadows of the sides of a triangle from its centre do not intersect.The proof above does not use the fact that $c$ is the only point near the centre of the simplex.",
"If we consider $U_0 \\subset c + \\varepsilon B$ any set and declare $U_0 = M$ , the same arguments as above show that there is no partition with the desired properties.",
"Therefore, the condition $\\mathrm {conv\\;}M \\cap \\mathrm {conv\\;}(A \\setminus M)= \\emptyset $ cannot be removed even if we allow $A$ to have more than $(r-1)(d+1)+1$ points.",
"Example 2.",
"This example shows a construction where $M$ satisfies the separation condition, $|M|=r$ and the conclusion of the theorem fails.",
"We work with the same simplex $\\Delta $ and $U_h\\subset v_h+\\varepsilon B$ is the same $(r-1)$ -element set as before for $h\\in \\lbrace 2,\\ldots ,d+1\\rbrace $ but for $h=1$ it is an $r$ -element set in $v_1+\\varepsilon B$ .",
"This time $A=\\bigcup _{h=1}^{d+1}U_h$ and $M=U_1$ .",
"We assume of course that $A$ is in general position.",
"Now $M$ is separated from $A \\setminus M$ and has exactly $r$ elements.",
"We claim that $A$ has no partition into $r$ parts with the required properties.",
"Proof.",
"Assume the contrary and let $A_1,\\ldots ,A_r$ be a proper partition with $z \\in \\bigcap _{j=1}^r \\mathrm {aff\\;}A_j$ such that in (REF ) $\\alpha (x) < 0$ if $x \\in M$ an $\\alpha (x)>0$ if $x \\notin M$ .",
"For $h \\in \\lbrace 2,\\dots ,d+1\\rbrace $ let $G_h$ be the convex hull of $V\\setminus \\lbrace v_1,v_h\\rbrace $ ; this is a $(d-2)$ -face of $\\Delta $ .",
"Set $\\beta =\\sum _{x \\in A_j\\cap U_1}\\alpha (x)$ , and note that $\\beta >0$ if $A_j \\cap U_1$ is nonempty.",
"Define $u_j=\\frac{1}{\\beta } \\sum _{x \\in A_j\\cap U_1}\\alpha (x)x$ if $A_j \\cap U_1$ is nonempty, and $u_j=v_1$ otherwise.",
"Note again that for each $h\\in \\lbrace 2,\\ldots ,d+1\\rbrace $ there is a $j(h)\\in [r]$ such that $U_h$ and $A_{j(h)}$ are disjoint.",
"Then $A_{j(h)}\\setminus U_1 \\subset G_h+\\varepsilon B$ .",
"This and the sign condition in (REF ) imply that for every $h>1$ with $A_{j(h)}\\cap U_1\\ne \\emptyset $ , $z \\in S(u_{j(h)},G_h+\\varepsilon B).$ This holds even if $A_{j(h)}\\cap U_1= \\emptyset $ since then $u_{j(h)}=v_1$ and $z \\in \\mathrm {conv\\;}A_{j(h)}\\subset G_h+\\varepsilon B \\subset S(v_1,G_h+\\varepsilon B)$ .",
"Thus $z \\in \\bigcap _{h=2}^{d+1} S(u_{j(h)},G_h+\\varepsilon B).$ But again, the shadows on the right hand side have no point in common, as one can check easily.$\\Box $"
],
[
"Proof of Theorem ",
"We are going to use the colourful Carathéodory theorem [1].",
"It says that given sets $S_1,\\ldots ,S_{n+1} \\subset \\mathbb {R}^n$ with the condition that $0 \\in \\bigcap _{i=1}^{n+1} \\mathrm {conv\\;}S_i$ , there is a transversal, that is, a choice $s_i \\in S_i$ for every $i \\in [n+1]$ , such that $0 \\in \\mathrm {conv\\;}\\lbrace s_1,\\ldots ,s_{n+1}\\rbrace $ .",
"Proof of Theorem REF .",
"We use a modification of Sarkaria's argument [7], in the form given by [3].",
"It starts with an artificial tool: let $v_1,\\ldots ,v_r$ be the vertices of a regular simplex in $\\mathbb {R}^{r-1}$ centred at the origin.",
"The important property here is that, apart from scalar multiples, their unique linear dependence is $v_1+\\ldots +v_r=0$ .",
"Assume $A=\\lbrace a_1,\\ldots ,a_n\\rbrace $ where $n=(r-1)(d+1)+1$ .",
"Recall that $M \\subset A$ , $|M|=k < r$ , and $M$ and $A \\setminus M$ are separated by a hyperplane.",
"Define $b_i=(a_i,1) \\mbox{ if } a_i \\notin M \\mbox{ and } b_i=(-a_i,-1) \\mbox{ if } a_i \\in M.$ where $(a_i,1)\\in \\mathbb {R}^{d+1}$ is vector $a_i$ appended with an $(d+1)$ -th coordinate equal to one, and similarly for $(-a_i,-1)$ .",
"For $i \\in [n]$ we set $S_i=\\lbrace v_1\\otimes b_i,v_2\\otimes b_i,\\ldots ,v_r\\otimes b_i\\rbrace .$ Here $v_j\\otimes b_i$ is the usual tensor product, which is the same as the matrix product of the $(r-1)$ -dimensional column vector $v_j$ and the $(d+1)$ -dimensional row vector $b_i^T$ : $v_j b_i^T$ , where we consider our vectors as vertical matrices.",
"So this product is an $(r-1)\\times (d+1)$ matrix, or equivalently a vector in $\\mathbb {R}^{n-1}$ .",
"Observe that $0 \\in \\mathrm {conv\\;}S_i$ for every $i$ , so the colourful Carathéodory theorem applies and gives a transversal $s_1,\\ldots ,s_n$ whose convex hull contains the origin, that is, there are non-negative coefficients $\\beta _1,\\ldots ,\\beta _n$ whose sum is 1 such that $\\sum _{i=1}^n\\beta _is_i=0$ .",
"Here each $s_i$ is of the form $v_j\\otimes b_i$ for a unique $j=j(i) \\in [r]$ .",
"We define $A_j=\\lbrace a_i\\in A: j(i)=j\\rbrace $ for all $j \\in [r]$ .",
"The sets $A_1,\\ldots ,A_r$ form an $r$ -partition of $A$ .",
"With the new notation $0&=&\\sum _{i=1}^n\\beta _is_i=\\sum _{i=1}^n\\beta _i v_{j(i)}\\otimes b_i\\\\&=&\\sum _{j=1}^r\\sum _{a_i \\in A_j}\\beta _iv_j\\otimes b_i=\\sum _{j=1}^rv_j\\otimes \\left(\\sum _{a_i \\in A_j}\\beta _i b_i\\right).$ Define now $\\alpha _i=-\\beta _i$ if $a_i \\in M$ and $\\alpha _i=\\beta _i$ otherwise.",
"The last equation becomes $0=\\sum _{j=1}^rv_j\\otimes \\left(\\sum _{a_i \\in A_j}\\alpha _i (a_i,1)\\right).$ There is a vector $u \\in \\mathbb {R}^{r-1}$ , orthogonal to $v_3,v_4,\\ldots ,v_r$ with $\\langle u, v_1\\rangle =1$ , where $\\langle \\cdot , \\cdot \\rangle $ denotes the dot product.",
"The condition $v_1+\\ldots +v_r=0$ implies that $\\langle u, v_2\\rangle =-1$ .",
"As (REF ) is a matrix equation, multiplying it from the left by the $(r-1)$ -dimensional row vector $u^T$ gives $\\sum _{a_i \\in A_1}\\alpha _i (a_i,1)=\\sum _{a_i \\in A_2}\\alpha _i (a_i,1)$ .",
"By symmetry we have $z:=\\sum _{a_i \\in A_1}\\alpha _i (a_i,1)=\\sum _{a_i \\in A_2}\\alpha _i (a_i,1)=\\ldots =\\sum _{a_i \\in A_r}\\alpha _i (a_i,1).$ There are two cases to be considered.",
"Case 1: when $A_j=\\emptyset $ for some $j \\in [r]$ .",
"Then $z=0$ and some $A_h$ , say $A_1$ , is nonempty and not all coefficients $\\alpha _i$ with $a_i \\in A_1$ are zero.",
"Thus $\\sum _{a_i \\in A_1}\\alpha _i (a_i,1)=0$ .",
"Then $\\alpha _i\\le 0$ for all $a_i \\in A_1\\cap M$ and $\\alpha _i\\ge 0$ for all $a_i \\in A_1\\setminus M$ .",
"Setting $\\gamma :=\\sum _{a_i \\in A_1\\cap M}\\alpha _i=\\sum _{a_i \\in A_1\\setminus M}-\\alpha _i,$ it follows that $\\gamma >0$ .",
"Consequently $\\mathrm {conv\\;}(A_1\\cap M)$ and $\\mathrm {conv\\;}(A_1\\setminus M)$ have a point in common, namely $\\frac{1}{\\gamma }\\sum _{a_i \\in A_1\\cap M}\\alpha _i a_i=\\frac{1}{\\gamma }\\sum _{a_i \\in A_1\\setminus M}(-\\alpha _i) a_i,$ contradicting the separation assumption.",
"Case 2: when $A_j$ is nonempty for all $j \\in [r]$ .",
"Reading the last coordinate of (REF ) gives that $\\gamma := \\sum _{a_i \\in A_1}\\alpha _i=\\sum _{a_i \\in A_2}\\alpha _i =\\ldots =\\sum _{a_i \\in A_r}\\alpha _i.$ Since $|M|<r$ , there is a $j \\in [r]$ such that $\\alpha _i>0$ for all $a_i \\in A_j$ , implying that $\\gamma >0$ .",
"Then the point $\\frac{1}{\\gamma }z$ is in the affine hull of every $A_j$ .",
"The construction guarantees that $\\alpha _i<0$ if and only if $a_i \\in M$ .",
"$\\Box $ Actually, this proof gives a stronger statement.",
"In Case 2, the positivity of $\\gamma $ is guaranteed by the condition $|M|=k<r$ .",
"Not assuming $k<r$ , $\\gamma $ can be negative or zero.",
"When $\\gamma <0$ equation (REF ) implies again that $\\frac{1}{\\gamma }z$ is in the affine hull of every $A_j$ , but this time $\\alpha (x)>0$ exactly when $x \\notin M$ .",
"We will exclude the case $\\gamma =0$ using the general position condition.",
"The proof of this is given in Section because it uses the content of Section .",
"So we have the following result.",
"Theorem 3.1 Under the conditions of Theorem REF let $M$ be a subset $A$ such that $\\mathrm {conv\\;}M \\cap \\mathrm {conv\\;}(A\\setminus M)=\\emptyset $ .",
"Then there is a partition $\\mathcal {A}=\\lbrace A_1,\\ldots ,A_r\\rbrace $ of $A$ such that in (REF ) either $\\alpha (x)<0$ if and only if $x \\in M$ , or $\\alpha (x)<0$ if and only if $x \\notin M$ .",
"The second example in Section shows that in some cases only the second alternative holds.",
"Corollary 3.1 Assume $r,d$ are both even and positive integers, $A\\subset \\mathbb {R}^d$ is in general position, $|A|=(r-1)(d+1)+1$ , and $k=\\frac{1}{2} [(r-1)(d+1)+1]$ .",
"Then $A$ has a proper $r$ -partition such that in (REF ) exactly $k$ of the coefficients $\\alpha (x)$ are negative.",
"The proof is easy.",
"Under the above conditions there is a subset $M$ of $A$ of size $k$ that is separated from $A\\setminus M$ .",
"According to Theorem REF , $A$ has an $r$ -partition such that in (REF ) either $\\alpha (x)<0$ if and only if $x \\in M$ , or $\\alpha (x)<0$ if and only if $x \\in A\\setminus M$ .",
"In both cases, exactly $k$ coefficients are negative.$\\Box $ Remark.",
"The same result can be proved using Tverberg's original method of moving the points.",
"The main idea is to start with a set of points which have a partition with the required conditions.",
"Then, as one moves one point continuously, if the partition stops working, one can show that points may be swapped in the partition in order to still satisfy the conclusion of the theorem.",
"The proof given above is shorter and simpler."
],
[
"Colourful Tverberg plus minus",
"Once a Tverberg type theorem with conditions on the signs of coefficients of the affine combinations has been established, it becomes natural to try to extend it to the coloured versions of Tverberg's theorem, as in [2].",
"Given disjoint sets $F_1, \\ldots , F_n$ of $r$ points each in $\\mathbb {R}^d$ , considered as colour classes, we say that $A_1, \\ldots , A_r$ is a colourful partition of them if $|F_i \\cap A_j| = 1$ for all $i\\in [n]$ , $j \\in [r]$ .",
"In such a case, we can denote the points by $x_{i,j} = F_i \\cap A_j$ .",
"The coloured Tverberg theorem is concerned about the existence of colourful partitions for which the convex hulls of the sets $A_j$ intersect.",
"In other words, we seek a colourful partition and a point $z\\in \\mathbb {R}^d$ for which there is a solution to the equations $z & = & \\sum _{i=1}^n \\alpha (x_{i,j})x_{i,j} \\mbox{ for all }j \\in [r] \\\\&&\\mbox{subject to } 1 = \\sum _{i=1}^n \\alpha (x_{i,j}) \\mbox{ for all } j \\in [r], \\ \\mbox{and} \\\\& & \\alpha ({x_{i,j}}) \\ge 0 \\mbox{ for all } i \\in [n], j \\in [r].",
"$ The question then becomes, given $M \\subset [n]$ , find a solution where we exchange () for $\\alpha ({x_{i,j}}) \\le 0 & & \\mbox{ for all } i \\in M, j \\in [r], \\mbox{and} \\\\\\alpha ({x_{i,j}}) \\ge 0 & & \\mbox{ for all } i \\in [n]\\setminus M, j \\in [r].$ In other words, we aim to prescribe negative coefficients, but we also require that the same restrictions hold accross the colour classes.",
"We obtain a partial result in this direction.",
"Theorem 4.1 Let $n=(r-1)d+1$ and $F_1, \\ldots , F_n$ be disjoint subsets of $\\mathbb {R}^d$ whose union is algebraically independent, each of cardinality $r$ and $M \\subset [n]$ .",
"Then, there is a colourful partition of $F_1, \\ldots , F_n$ into $r$ sets and solutions to equations (REF ) and () such that either $\\alpha (x_{i,j}) > 0$ for all $i \\in M$ and $\\alpha (x_{i,j})< 0$ for all $i \\in [n]\\setminus M$ , or $\\alpha (x_{i,j}) < 0$ for all $i \\in M$ and $\\alpha (x_{i,j})> 0$ for all $i \\in [n]\\setminus M$ .",
"Moreover, the affine combinations use the same coefficients for the colour classes.",
"In other words, for all $i \\in [n]$ and $j, j^{\\prime } \\in [r]$ , $\\alpha (x_{i,j}) = \\alpha (x_{i,j^{\\prime }})$ .",
"Proof.",
"We use the main result of [8].",
"It says that for $n = (r-1)d+1$ and the sets $F_1, \\ldots , F_n$ , there are solutions to equations (REF ), () and () where $\\alpha (x_{i,j}) = \\alpha (x_{i,j^{\\prime }})$ for all $i \\in [n], j \\in [r], j^{\\prime } \\in [r]$ .",
"Then, we apply this result to the sets $G_i = {\\left\\lbrace \\begin{array}{ll}F_i & \\mbox{if } i \\in M \\\\-F_i & \\mbox{otherwise.}\\end{array}\\right.",
"}$ Let $B_1, \\ldots , B_r$ be the colourful partition we obtain of $G_1, \\ldots , G_n$ , with $y_{i,j} = G_i \\cap B_j$ for all $i,j$ .",
"We denote by $\\beta (y_{i,j})$ the coefficients we obtain satisfying equations (REF ), () and ().",
"We rename them as $\\beta (y_{i,j}) = \\beta _i$ , since they do not depend on $j$ .",
"Let $x_{i,j} = \\pm y_{i,j}$ and $\\alpha _i = \\pm \\beta _i$ where the sign is positive (negative) if $i \\in M$ ($i \\notin M$ ), respectively.",
"Let $\\gamma = \\sum _{i=1}^n \\alpha _i$ .",
"Let us see what happens if $\\gamma \\ne 0$ .",
"By construction, if we consider $A_1, \\ldots , A_r$ the partition induced by the points $x_{i,j}$ and $\\alpha (x_{i,j}) = \\alpha _i / \\gamma $ for all $i,j$ , they satisfy all the requirements for the conclusion of the theorem.",
"The two cases in Theorem REF correspond to the possibilities for the sign of $\\gamma $ .",
"We have to verify that the general condition assumption we have on $F_1, \\ldots , F_n$ implies $\\gamma \\ne 0$ .",
"This part of the proof is technical, and it relies on the modification of Sarkaria's trick from [8].",
"Given a set $F= \\lbrace z_1, \\ldots , z_r\\rbrace \\subset \\mathbb {R}^d$ , a permutation $\\sigma : [r] \\rightarrow [r]$ and $v_1, \\ldots , v_r \\in \\mathbb {R}^{r-1}$ as in Section , we can define $F \\otimes \\sigma = \\sum _{j=1}^r z_j \\otimes v_{\\sigma (j)} \\in \\mathbb {R}^{n-1} ,\\qquad S(F) = \\lbrace F \\otimes \\sigma : \\sigma \\ \\mbox{is a permutation}\\rbrace $ The existence of $\\beta _1, \\ldots , \\beta _n$ follows from applying the colourful Carathéodory theorem to the sets $S(G_1), \\ldots , S(G_n)$ in $\\mathbb {R}^{n-1}$ .",
"However, if the original set of points $\\cup _{i=1}^n F_i$ is algebraically independent, then no transversal to $S(F_1), \\ldots , S(F_n)$ would have a non-trivial affine dependence in $\\mathbb {R}^{n-1}$ , so $\\gamma \\ne 0$ , as required.",
"$\\Box $ As mentioned, Theorem REF has equal coefficients accross the colour classes.",
"Removing this condition leads to the following problem.",
"Open problem 4.1 If we remove the equal coefficients condition, does Theorem REF hold with $n=d+1$ ?",
"The answer is affirmative with $r=2$ .",
"If $M = \\emptyset $ this is the main conjecture from [2]."
],
[
"Proof of Proposition ",
"We write the equation (REF ) in matrix form $M\\alpha =b$ .",
"The $(n+d)\\times (n+d)$ matrix $M$ is made up of blocks.",
"The block corresponding to $A_j$ is a $d\\times |A_j|$ matrix $N_j$ whose columns are the vectors in $A_j$ .",
"The row immediatley below block $N_j$ has a 1 in each column containing a vector from $A_j$ and zeroes everywhere else.",
"There are $r$ further blocks, each one is $-I_d$ , the negative $d \\times d$ identity matrix.",
"They are in the last $d$ columns of $M$ , with a row of zeroes between them.",
"These submatrices are arranged in $M$ as shown on Table 1.",
"All other entries of $M$ are zeroes.",
"The $i$ th column of $M$ corresponds to the vector $a_i$ .",
"Note that $M=M(\\mathcal {A})$ depends on $A$ and on the partition $\\mathcal {A}=\\lbrace A_1,\\ldots ,A_r\\rbrace $ as well.",
"Table: The matrix MM, the empty regions indicate zerosThe variables are $\\alpha =(\\alpha _1,\\ldots ,\\alpha _n,z_1,\\ldots ,z_d)^T \\in \\mathbb {R}^{n+d}$ and the right hand side vector is $b\\in \\mathbb {R}^{n+d}$ that has coordinate zero everywhere except in positions $d+1,2(d+1),\\ldots ,r(d+1)$ where it has one.",
"The original system (REF ) is the same as $M\\alpha =b.$ As we have seen, $\\bigcap _{j=1}^r \\mathrm {aff\\;}A_j$ is a single point if and only if the linear system (REF ), or what is the same, the equation (REF ) has a unique solution which happens if and only if $\\det M \\ne 0$ .",
"Here $\\det M$ is a polynomial with integral coefficients in the coordinates of the $a_i$ .",
"If this polynomial is zero at some algebraically independent points $a_1,\\ldots ,a_n$ , then it is identically zero.",
"So it suffices to show one example where it is non-zero or, what is the same, one example where $\\bigcap _1^r \\mathrm {aff\\;}A_j$ is a single point.",
"The example is simple.",
"Suppose $|A_j|=d+1-m_j$ for all $j\\in [r]$ and $m_1\\ge m_2\\ge \\ldots \\ge m_r$ .",
"As $\\mathcal {A}$ is a proper partition, $0\\le m_j\\le d$ .",
"Let $H_j$ be the subspace of $\\mathbb {R}^d$ defined by equations $x_i=0$ for $i=\\sum _{h=1}^{j-1}m_h+1,\\ldots ,\\sum _{h=1}^jm_h$ .",
"Since $n=(r-1)(d+1)+1$ , $\\sum _1^r m_j=d$ , implying that $\\bigcap _1^r H_j$ is a single point, namely the origin.",
"For each $p\\in [r]$ choose $|A_j|$ affinely independent points in $H_j$ .",
"Their affine hull is exactly $H_j$ , finishing the proof of the first part.",
"For the second part we can assume that $A_j$ is nonempty for all $j$ , and also that $|A_j|\\le d+1$ as otherwise one can delete some elements of $A_j$ while keeping its affine hull the same.",
"We suppose further that $n=(r-1)(d+1)$ by adding extra (and algebraically independent) points to some suitable $A_j$ s. Then $\\bigcap _{j=1}^r \\mathrm {aff\\;}A_j\\ne \\emptyset $ if and only if the corresponding linear system (REF ) has a solution.",
"Now $M$ is an $(n+1)\\times n$ matrix.",
"Adding $b$ to $M$ as a last column we get a matrix that we denote by $M^*$ .",
"The system (REF ) has a solution if and only if $\\textrm {rank\\;}M=\\textrm {rank\\;}M^*$ .",
"The previous argument shows that $\\textrm {rank\\;}M=n-1$ and so we have that, as a polynomial, $\\det M^*$ is identically zero.",
"Again it suffices to give a single example where $\\bigcap \\mathrm {aff\\;}A_j=\\emptyset $ .",
"We use the same example as before except that this time $\\sum _{j=1}^r m_j=d+1$ so we can add the equation $\\sum _{i=1}^dx_i=1$ to the ones defining $H_1$ if $m_1<d$ and then $\\bigcap H_j=\\emptyset $ , indeed.",
"If $m_1=d$ then $H_1=0$ and $m_2=1$ and we define $H_2$ by the single equation $x_1+x_2=1$ , and again $\\bigcap H_j=\\emptyset $ .",
"The sets $A_j$ are constructed the same way as above.$\\Box $"
],
[
"Proof of Theorem ",
"Proof.",
"As we have seen we only have to show that $\\gamma \\ne 0$ .",
"Assume $\\gamma =0$ .",
"This happens if and only if the homogeneous version of equation (REF ), that is $A\\alpha =0$ has a nontrivial solution, which happens again if and only if $\\det M=0$ .",
"This is impossible if the partition is proper (as we have seen in the previous section).",
"Note that $z\\ne 0$ and no $A_j$ is the emptyset, this follows from Case 1 of the proof of Theorem REF .",
"So assume the partition is not proper.",
"Then $A_j$ has more than $d+1$ elements for some $j$ .",
"Assume that $|A_1|>d+1$ , say.",
"This means that $\\sum _{x \\in A_1}\\alpha (x)(x,1)=(z,0).$ The vectors $(x,1)$ , $x \\in A_1$ are affinely dependent, implying that there is a non-trivial affine dependence $\\sum _{x \\in A_1}\\beta (x)(x,1)=(0,0)$ .",
"Then for all $t \\in \\mathbb {R}$ $\\sum _{x \\in A_1}(\\alpha (x)+t\\beta (x))(x,1)=(z,0),$ We choose here $t=t_0$ so that $\\alpha (x)+t_0\\beta (x)=0$ for some $x=x_0\\in A_1$ .",
"Set $\\alpha ^{\\prime }(x)=\\alpha (x)+t_0\\beta (x)$ when $x \\in A_1$ and $\\alpha ^{\\prime }(x)=\\alpha (x)$ otherwise.",
"We change now the partition $A_1,\\ldots ,A_r$ to another one $A_1^{\\prime }\\ldots ,A_r ^{\\prime }$ as follows.",
"Set $A_1^{\\prime }=A_1\\setminus \\lbrace x_0\\rbrace $ and choose some $A_j$ with $|A_j|\\le d$ and set $A_j^{\\prime }=A_j\\cup \\lbrace x_0\\rbrace $ .",
"All other $A_h$ remain the same.",
"Let $M^{\\prime }$ be the corresponding matrix.",
"We claim now that $\\det M^{\\prime }=0$ .",
"The linear system (REF ) has a nontrivial solution, namely $\\alpha (x)=\\alpha ^{\\prime }(x)$ with $z\\ne 0$ unchanged.",
"So indeed, $\\det M^{\\prime }= 0$ .",
"Repeating this step finitely many times gives a proper partition such that (REF ) has a nontrivial solution.",
"But the previous section shows that for a proper partition, (REF ) has no non-trivial solution.$\\Box $ Acknowledgements.",
"This material is partly based upon work supported by the National Science Foundation under Grant No.",
"DMS-1440140 while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.",
"Support from Hungarian National Research Grants no K111827 and K116769 is acknowledged.",
"We are also indebted to Attila Pór and Manfred Scheucher for useful discussions, and to an anonymous referee for careful reading and valuable comments."
]
] | 1612.05630 |
[
[
"Space-Wave Routing via Surface Waves Using a Metasurface System"
],
[
"Abstract We introduce the concept of a metasurface system able to route space wave via surface waves.",
"This concept may be used to laterally shift or modulate the beam width of scattered waves.",
"We propose two corresponding synthesis techniques, one that is exact but leads to practically challenging material parameters and one that is approximate but leads to simpler material parameters.",
"The concept is experimentally verified in an electromagnetic periscope.",
"Additionally, we propose two other potential applications namely a beam expander and a multi-wave refractor."
],
[
"Introduction",
"Metasurfaces are thin electromagnetic films composed of flat scatterers and represent the two-dimensional counterparts of volume metamaterials [1], [2].",
"Over recent years, they have attracted tremendous attention due to their unprecedented capabilities to control electromagnetic waves conjugated with their ease of fabrication, low loss and high compactness.",
"The vast majority of metasurface designs and applications reported to date have been restricted to isolated metasurfaces, i.e.",
"single metasurface structures performing specific electromagnetic transformations.",
"In order to extend the range of these transformations, we propose here the concept of a metasurface system, namely a combination of several metasurface structures collectively exhibiting properties that would be unattainable with a single metasurface.",
"Specifically, we present a metasurface system composed of three juxtaposed metasurfaces, that routes space-wave beams, between different locations, via surface waves.",
"Such a system may be used, for instance, to laterally shift or modulate the beam width of scattered waves.",
"This paper is organized as follows.",
"Section introduces the concept of space-wave routing via surface waves in a metasurface system.",
"Then, Sec.",
"discusses two synthesis techniques for the design of such a system.",
"Based on the second proposed synthesis method, we experimentally demonstrate system routing in an “electromagnetic periscope,” whose design and measurements are presented in Sec. .",
"Finally, Sec.",
"proposes two additional potential applications, namely a compact beam expander and a multi-wave refractor.",
"Conclusions are given in Sec.",
"."
],
[
"Space-Wave Routing Concept",
"The fundamental idea, which is depicted in Fig.",
"REF , consists in converting an incoming space wave into a surface wave, propagating this surface wave between two points along a desired path, and then converting it back, with possible other transformations, into an outgoing space wave.",
"This concept may be used to laterally shift reflected or transmitted waves (electromagnetic periscope), modulate the width of beams, or enable multiple refraction, in a very compact fashions, as will be discussed thereafter.",
"Figure: Concept of metasurface system performing the operations of space-wave routing via surface waves for pp-polarized beams and generalized refraction for ss-polarized beams.In the system depicted in Fig.",
"REF , the metasurface is assumed to be monoanisotropic diagonal, and hence birefringent, allowing for the independent control of $s$ and $p$ polarizations.",
"The metasurface may be designed, for instance, to route $p$ -polarized waves and refract (or perform any another transformation on) $s$ -polarized waves.",
"We shall now describe the space-wave routing concept in more details.",
"Let us consider the optical system depicted in Fig.",
"REF , which consists of a dielectric waveguide with two prisms placed at different locations above it.",
"This system may be used to perform the routing operation described in Fig.",
"REF .",
"Assume that an input beam $\\Psi _\\text{in}$ is impinging on the left prism at an angle $\\theta > \\theta _\\text{c}$ , where $\\theta _\\text{c}$ is the angle of total internal reflection.",
"An evanescent wave with wavenumber $k_x$ , corresponding to that of the incident wave, is formed between the prism and the waveguide due to total internal reflection.",
"This evanescent wave then couples to a waveguide mode with matched $k_x$ , and the resulting wave propagates along the waveguide in the $+x$ -direction.",
"The amount of coupling between the incident space wave and the guided wave is proportional to the distance $d$ between the prism and the waveguide, and is usually less than unity, leading to a non-zero reflected wave $\\Psi _\\text{r}$ .",
"Farther along the waveguide, the guided wave is transformed back into an output space wave $\\Psi _\\text{out}$ by the second prism by the reverse mechanism.",
"Figure: Representations of two optical systems performing the same wave routing operation.",
"(a) Combination of two prisms and a dielectric waveguide.",
"(b) Composite metasurface, including two spatially modulated metasurfaces placed at the ends of a guiding metasurface.We introduce here the metasurface system depicted in Fig.",
"REF to perform the same operation in a much more compact (purely planar) and (ideally) perfectly reflection-less fashion.",
"This system consists of three different metasurfaces juxtaposed to each other.",
"The input space wave is coupled into a guided surface wave by a spatially modulated metasurface.",
"The middle metasurface is a surface-wave guiding structure that propagates the guided wave in the $+x$ -direction.",
"Finally, another spatially modulated metasurface transforms the guided wave back into a space wave at the other end of the system."
],
[
"Exact Synthesis Based on GSTCs",
"The metasurface system introduced above can be rigorously synthesized so as to provide the exact medium parameters performing the transformation depicted in Fig.",
"REF .",
"For mathematical convenience, the metasurface is assumed to be of zero thickness.",
"This assumption allows one to describe it as a spatial electromagnetic discontinuity, for which rigorous continuity conditions, the generalized sheet transition conditions (GSTCs), are available [3], [4].",
"A complete presentation of this synthesis method was given in [5], [6].",
"In the case of a monoanisotropic metasurface lying in the $xy$ -plane at $z=0$ , the GSTCs The time dependence $e^{j\\omega t}$ is omitted throughout the paper.",
"read $\\hat{z}\\times \\Delta H&=j\\omega \\epsilon _0\\overline{\\overline{\\chi }}_\\text{ee}E_\\text{av},\\\\\\Delta E\\times \\hat{z}&=j\\omega \\mu _0 \\overline{\\overline{\\chi }}_\\text{mm}H_\\text{av},$ where $\\overline{\\overline{\\chi }}_\\text{ee}$ and $\\overline{\\overline{\\chi }}_\\text{mm}$ are the electric and magnetic susceptibilities of the metasurface, respectively, $\\Delta E$ and $\\Delta H$ are the difference of the electric and magnetic fields on both sides of the metasurface, and $E_\\text{av}$ and $H_\\text{av}$ are the arithmetic averages of these fields.",
"In the problem considered here, no rotation of polarization is required and therefore the monoanisotropic susceptibility tensors are purely diagonal.",
"Moreover, it is assumed that the metasurface is not polarizable in its longitudinal direction which would otherwise lead to more complicated GSTC relations than (REF ) [5].",
"Solving (REF ) to express the susceptibilities as a function of the specified fields yields the closed-form relations $\\chi _{\\text{ee}}^{xx}&=\\frac{-\\Delta H_{y}}{j\\omega \\epsilon _0 E_{x,\\text{av}}},\\quad \\chi _{\\text{mm}}^{yy}=\\frac{-\\Delta E_{x}}{j\\omega \\mu _0 H_{y,\\text{av}}},\\\\\\chi _{\\text{ee}}^{yy}&=\\frac{\\Delta H_{x}}{j\\omega \\epsilon _0 E_{y,\\text{av}}},\\quad \\chi _{\\text{mm}}^{xx}=\\frac{\\Delta E_{y}}{j\\omega \\mu _0 H_{x,\\text{av}}},$ which describe a birefringent metasurface able to independently control and transform $x-$ and $y-$ polarized waves.",
"Let us now synthesize the space-wave to surface-wave transformation performed by the first metasurface in Fig.",
"REF .",
"Let us assume a $p$ -polarized wave ($E\\in xz$ -plane and $H\\parallel \\hat{y}$ ), to be routed (Fig.",
"REF ), which corresponds to the synthesis relations (REF ).",
"In this case, the tangential electromagnetic fields, at $z=0$ , are $E_x^{\\text{a}}=A^{\\text{a}}\\frac{k_z^{\\text{a}}}{k_0}e^{-jk_x^{\\text{a}}x}\\quad \\text{and} \\quad H_y^{\\text{a}}=A^{\\text{a}}e^{-jk_x^{\\text{a}}x}/\\eta _0,$ where $A$ is a complex constant, $k_x$ and $k_z$ are the tangential and longitudinal wavenumbers, respectively, $\\eta _0$ and $k_0$ are the impedance and wavenumber of free-space, respectively, and the superscript a $=$ i, r, t denotes the incident, reflected and transmitted waves, respectively.",
"We shall consider the transformation of an incident space wave with $A^\\text{i}=1$ and $k_x^\\text{i}=k_0\\sin {\\theta ^\\text{i}}$ , where $\\theta ^\\text{i}=45^\\circ $ is the incidence angle, into a surface wave with $A^\\text{t}=0.728$ and $k_x^\\text{t}=1.2k_0$ .",
"The corresponding longitudinal $k$ -component is found as $k_z^\\text{a}=\\sqrt{k_0^2 - (k_x^\\text{a})^2}$ which, in the case of the transmitted wave, is an imaginary quantity corresponding to wave evanescence perpendicular to the metasurface and surface-wave propagation in the $+x$ -direction.",
"The value 0.728 was derived to ensure a purely passive (although lossy) reflection-less ($A^\\text{r}=0$ ) metasurface This proviso was found by inserting (REF ) into (REF ) with specified parameters $A^\\text{i}$ , $k_x^\\text{a}$ and $k_z^\\text{a}$ and solving for $A^\\text{t}$ such that $\\Im (\\chi _{\\text{ee}}^{xx}),\\Im (\\chi _{\\text{mm}}^{yy})<0$ .",
"This shows the space wave cannot be transformed into a surface wave without dissipation.. Finite-difference frequency domain (FDFD) simulations [9] are used to analyse the response of the synthesized metasurface.",
"Figure REF shows how an obliquely incident Gaussian beam is transformed into a surface wave on the transmit side of the metasurface.",
"Note that the transformation in Fig.",
"REF is “perfect” in the sense that no parasitic diffraction order is present.",
"However, the surface wave only exists in the region where the Gaussian beam illuminates the metasurface and does not propagate farther along the structure.",
"This is because the metasurface was synthesized assuming an incident plane wave illuminating the entire structure and not just a small portion of it, as is the case with a Gaussian beam.",
"Consequently, the surface wave is restricted to the region within the waist of the incident beam and cannot propagate beyond its excitation zone as it is not an eigen-mode of this metasurface.",
"Figure: Finite-difference frequency-domain (FDFD) simulations showing the real part of H y H_y in the case of (a) the conversion of a space wave into a localized surface wave, (b) the coupling of the surface wave into a guided wave that propagates along a juxtaposed metasurface, and (c) the propagating surface wave is then transformed back into a space wave.In order to propagate the surface wave farther along the surface, it is necessary to introduce a discontinuity in the metasurface or, in other words, to place a second metasurface next to the first one, as shown in Fig.",
"REF .",
"Thus, the first metasurface is synthesized as a space-wave to surface-wave transformer to convert the incident wave into a guided wave, while the second metasurface is synthesized as a surface-wave guiding structure, to route the wave along the overall structure.",
"This operation is achieved, considering the juxtaposed (first and second) metasurface system as a composite metasurface, by specifying the fields as follows.",
"The incident field is defined with the parameters $A^\\text{i}=H(-x-5\\lambda _0)$ and $k_x^\\text{i}=k_0\\sin {\\theta ^\\text{i}}$ ($\\theta ^\\text{i}=45^\\circ $ ), where $H(x)$ is the Heaviside function.",
"The Heaviside function is used here to create a discontinuity in the incident field at the position $-5\\lambda _0$ on the metasurface.",
"Additionally, we set $A^\\text{t}=0$ and $A^\\text{r}=0.728$ and $k_x^\\text{r}=1.2k_0$ In this second example, the surface wave is placed on the reflection side of the metasurface to later permit an easier design of the third metasurface.. Inserting these field specifications into (REF ) and performing an FDFD simulation for an incident Gaussian beam impinging on the metasurface at the position $-7\\lambda _0$ yields the result presented in Fig.",
"REF .",
"As can be seen, the surface wave effectively couples into the second metasurface where it now propagates as a surface wave.",
"Note that the presence of the discontinuity between the two metasurfaces introduces some spurious scattering of the incident wave, which could be avoided using a smooth transition.",
"Finally, the energy carried by the surface wave is extracted and transformed back into a space wave by the third metasurface in Fig.",
"REF upon specifying a non-zero transmitted wave with parameters $A^\\text{t}=0.6\\cdot \\text{exp}[4(x-6.5\\lambda _0)^2/5]$ and $k_x^\\text{t}=k_0\\sin {\\theta ^\\text{t}}$ , where the transmission angle is chosen here to be $\\theta ^\\text{t}=0$ .",
"The simulation result is shown in Fig.",
"REF .",
"To synthesize the second metasurface for refraction of the $s$ -polarized wave, as shown in Fig.",
"REF , one would simply need to insert the $s$ -counterpart of (REF ) into (), as conventionally done for generalized refractive metasurfaces [5].",
"In this case, assuming a non-zero transmission angle, the metasurface becomes globally nonuniform in the $x$ -direction, although it is seen as perfectly uniform to the $p$ -polarized wave (birefringence).",
"The electric and magnetic susceptibilities of the composite metasurface corresponding to the simulation in Figs.",
"REF , are plotted in Fig.",
"REF and REF , respectively.",
"The conversion from space wave to surface wave occurs in the portion of the metasurface where $x<-5\\lambda _0$ .",
"In this region, the metasurface (first metasurface) is spatially varying and exhibits nonuniform loss as is evidenced by the oscillating negative imaginary parts of the susceptibilities.",
"From $x=-5\\lambda _0$ to approximatively $x=0$ , the metasurface (second metasurface) supports the propagation of a surface wave.",
"It is interesting to note that, in this region, the metasurface is perfectly uniform, passive and lossless, with susceptibilities given by the following simple relations $\\chi _\\text{ee}^{xx}=2j/k_z^\\text{r}$ and $\\chi _\\text{mm}^{yy}=2jk_z^\\text{r}$ , where $k_z^\\text{r}$ is the purely imaginary propagation constant of the surface wave in the longitudinal direction.",
"Figure: Electric (a) and magnetic (b) susceptibilities corresponding to the transformation presented in Fig. .",
"The solid blue lines are the real parts while the dashed red lines are the imaginary parts.Finally, starting from $x>0$ , the metasurface (third metasurface) becomes spatially varying again, allowing part of the energy conveyed by the surface wave to progressively leak out to form the space wave.",
"Here, the susceptibilities have values oscillating between positive and negative imaginary parts.",
"This indicates that the metasurface is successively varying between gain and loss.",
"The presence of loss, as in the first part of the metasurface, is generally required to suppress undesired diffraction orders.",
"The presence of active regions, corresponding to gain in the last part of the metasurface, is due to the way the fields were specified in the synthesis.",
"Indeed, the surface wave (reflected wave) was specified with constant amplitude over the entire metasurface, including in the third region, and it is therefore not surprising that gain is required in the region where the transmitted space wave is generated and where it draws power from the surface wave.",
"The metasurface could be made perfectly passive by specifying a surface wave with progressively decreasing amplitude as its energy is being leaked out.",
"In that case, the third metasurface would actually act as a leaky-wave antenna."
],
[
"Simplified Synthesis Based on Leaky-Wave and Guiding-Wave Structures",
"The GSTCs-based metasurface synthesis technique [5], used in the previous section, yields the exact susceptibilities performing the specified transformation.",
"However, the resulting susceptibilities may, in some situations, be difficult to realize.",
"For instance, the metasurface described by the susceptibilities in Figs.",
"REF presents spatially varying electric and magnetic losses which may be challenging to implement.",
"Moreover, the generation of the transmitted space wave also requires gain as is evidenced by the positive imaginary parts of the susceptibilities on the right-hand side of Figs.",
"REF .",
"For these reasons, we next propose an alternative synthesis method for space-wave to surface-wave transformations performed by the two end metasurfaces in Fig.",
"REF .",
"This method will result in a slightly different design that will be much easier to realize while sacrificing little efficiency.",
"The waveguiding structure (second metasurface in Fig.",
"REF ) will be realized using the susceptibilities found in the previous section since these susceptibilities are exact and easy to realize, as seen in Fig.",
"REF .",
"However, the structure will require some optimization to account for deviations from the ideal response due to its non-zero thickness.",
"This may be achieved by following design procedures routinely used in the implementation of slow-wave structures [11], [12], as will be discussed thereafter.",
"In order to transform the incident space wave into a surface wave with a specific propagation constant along the metasurface, we will use here a simple phase gradient structure.",
"Let us consider the generalized law of refraction [13], that can be expressed, using the transverse wavenumber of the incident and refracted waves and the effective wavenumber of the phase gradient structure $K$ , as $k_x^\\text{t} = k_x^\\text{i} + K,$ where $K=2\\pi /P$ with $P$ being the phase-gradient period of the metasurface.",
"This period is designed such that the specified incident wave is refracted at a specified angle, i.e.",
"$P=\\lambda _0/(\\sin {\\theta ^\\text{t,spec}}-\\sin {\\theta ^\\text{i,spec}})$ .",
"From (REF ), we express the normalized transverse wavenumber of the transmitted wave as a function of $K$ and the incidence angle as $\\frac{k_x^\\text{t}}{k_0} = \\sin {\\theta ^\\text{i}} + \\frac{K}{k_0},$ which allows to determine the transverse wavenumber for any incidence angle $\\theta ^\\text{i}$ .",
"As an illustration, relation (REF ) is plotted in Fig.",
"REF as a function of the incidence angle for the specified angles $\\theta ^\\text{i,spec}=0$ and $\\theta ^\\text{t,spec}=45°$ .",
"Figure: Normalized transverse wavenumber of the transmitted wave versus incidence angle in the phase-gradient metasurface for the specified angles θ i,spec =0\\theta ^\\text{i,spec}=0 and θ t,spec =45°\\theta ^\\text{t,spec}=45° [Eq.",
"()].The region in blue, where $|k_x^\\text{t}/k_0|<1$ , corresponds to space-wave modes.",
"Outside of this region, $|k_x^\\text{t}/k_0|$ is larger than 1 and the longitudinal wavenumber $k_z^\\text{t}=\\sqrt{k_0^2 - (k_x^\\text{t})^2}$ is therefore imaginary, corresponding to a $z$ -evanescent or surface-wave mode.",
"This shows that a simple phase gradient metasurface can be used as a converter between a space wave and a surface wave when the metasurface wavenumber $K$ and the incidence angle $\\theta ^\\text{i}$ are properly chosen [14], [15], [16].",
"The three-metasuface system in Fig.",
"REF may therefore be realized as follows.",
"The first metasurface is designed as a phase-gradient metasurface with increasing phase in the $+x$ -direction; this positive phase ramp increases the momentum of the incident wave (in the $x$ -direction) so as to transform it into a surface wave.",
"The second metasurface is designed to support the propagation of a surface wave with the same wavenumber.",
"Finally, the third metasurface is again designed as a phase-gradient but this time with increasing phase in the $-x$ -direction, which reduces the momentum of the surface wave and hence transforms it back into a space wave."
],
[
"Realization",
"For the realization of the metasurface system, and particularly the realization of the space-wave - surface-wave converters (metasurfaces 1 and 3), we use, for simplicity, the approximate synthesis technique presented in Sec.",
"REF , rather than the exact but more problematic technique based on GSTCs presented in Sec.",
"REF .",
"A schematic of the metasurface system is presented in Fig.",
"REF .",
"The figure shows the conceptual operation of the structure with momentum “push” ($K_1$ ) and momentum “pull” ($K_2$ ) induced by the first and last metasurfaces, respectively, and the surface-wave guidance in the middle metasurface.",
"We shall now design the metasurface system for the following specifications: input angle $\\theta ^\\text{in}=30^\\circ $ and output angle $\\theta ^\\text{out}=-7.2^\\circ $ .",
"Figure: Schematic representation of the electromagnetic periscope metasurface system.The first phase-gradient metasurface, transforming the input $p$ -polarized space wave into a surface wave, will be implemented with a supercell of 8 unit cells of size $\\lambda _0/5$ with transmission phases ranging from 0 to $2\\pi $ .",
"The corresponding metasurface wavenumber is $K_1=2\\pi /P_1=2\\pi /(8\\lambda _0/5)=5k_0/8$ .",
"Then, for the specified input wave of $\\theta ^\\text{in}=30^\\circ $ , corresponding to $k_x^\\text{in}=k_0/2$ , one finds, using (REF ) with $k_x^\\text{i}=k_x^\\text{in}$ and $K=K_1$ , the surface-wave wavenumber to be $k_x^\\text{sw}=k_x^\\text{t}=9k_0/8$ , which corresponds to the $x$ -wavenumber across the entire metasurface system.",
"Upon this basis, the third metasurface is designed as follows.",
"The output angle of $\\theta ^\\text{out} = -7.2^\\circ $ corresponds to $k_x^\\text{out}=-k_0/8$ .",
"We apply (REF ) with $k_x^\\text{i}=k_x^\\text{sw}=9k_0/8$ and $k_x^\\text{t}=k_x^\\text{out}=-k_0/8$ , which yields $K_2=-5k_0/4$ .",
"Since $|K_2/K_1|=2$ , $P_2=P_1/2$ , and hence, still assuming $\\lambda _0/5$ unit cells, the supercell includes now 4 unit cells.",
"This metasurface may for instance be identical to the first metasurface where every two unit-cell rows have been removed.",
"The $p$ -polarization surface-wave guiding structure, in the middle of the metasurface system, may also be realized as a metasurface, for compatibility with its phase-gradient neighbours, instead of as a traditional waveguiding structure.",
"In addition, to allow the $s$ -polarization generalized refraction operation depicted in Fig.",
"REF , this structure must be completely transparent, and could therefore not be implemented in the form of a conventional waveguide.",
"As explained in REF , the metasurface is designed using the $p$ -polarization susceptibilities already found with the exact synthesis [Eqs.",
"(REF )], namely $\\chi _\\text{ee}^{xx}=2j/k_z^\\text{sw}$ and $\\chi _\\text{mm}^{yy}=2jk_z^\\text{sw}$ , where $k_z^\\text{sw}=\\sqrt{k_0^2-(k_x^\\text{sw})^2}$ with the value $k_x^\\text{sw}=9k_0/8$ found above.",
"In the current design, we consider the particular case of $s$ -polarization normal transmission, leading to global uniformity.",
"Figure: Dispersion curve and magnetic field distribution (absolute value at 10 GHz) for the fundamental mode of the waveguiding metasurface.",
"The separate inset represents the excited fields in the surrounding phase-gradient metasurfaces (also at 10 GHz).The overall metasurface system, composed of the three juxtaposed metasurfaces, is implemented as a multilayer structure with three metallization layers and two dielectric spacers.",
"The overall thickness of the structure is $\\lambda _0/10$ .",
"In each metasurface, the unit cell has a transverse size of $\\lambda _0/5\\times \\lambda _0/5$ and includes in each layer a metallic scatterer in the form of a Jerusalem cross with specific geometric parameters [17], [18], [6], [19].",
"The exact dimensions of the Jerusalem crosses are found by numerical simulations using a commercial software.",
"Each unit cell is simulated individually assuming periodic boundary conditions as approximate boundaries for smoothly varying patterns.",
"The resulting scattering parameters, obtained from the simulations, are optimized by varying the dimensions of the crosses until the expected response is achieved [17], [18], [6], [19].",
"For the two phase-gradient metasurfaces, the scattering parameters of each unit cell are assumed to simply consist of a phase transmission coefficient, $T=e^{j\\phi }$ , where the phase shift $\\phi $ depends on the unit cell position within the supercell.",
"For the waveguiding metasurface, the susceptibilities given above are first converted into scattering parameters following the procedure given in [5].",
"Because this metasurface is uniform (as seen by a $p$ -polarized wave), in contrast to the phase-gradient metasurfaces, only one unit cell has to be designed.",
"Once the dimensions of the Jerusalem crosses corresponding to the susceptibilities have been found, the unit cell is optimized using an eigenmode solver with the goal to achieve a wavenumber of $k_x^\\text{sw}=9k_0/8$ at the operation frequency set here to $f=10$ GHz.",
"The dispersion curve for the fundamental mode of the optimized waveguiding structure is plotted in Fig.",
"REF .",
"Note that the horizontal axis represents the $x$ -wavenumber normalized to the free-space wavenumber, so that the figure shows only the slow-wave region ($k_x^\\text{sw}/k_0>1$ ).",
"Comparing the two insets in the figure shows that the field distribution of this fundamental mode is essentially identical, and hence compatible, with the field distributions of the two phase-gradient metasurfaces.",
"Since the metasurfaces have been in addition designed to all exhibit the same polarization and wavenumber, it may be inferred that the coupling between them is maximized, as desired.",
"The realized metasurface system is shown in Fig.",
"REF .",
"Due to limitation of our fabrication process, the three metasurfaces have been realized separately rather than as a single entity and have then been screwed to a plastic frame (at the back and hence not visible in Fig.",
"REF ) to form the overall metasurface system.",
"Each metasurface is made of $24\\times 24$ unit cells, corresponding to a size of $4.8\\lambda _0\\times 4.8\\lambda _0$ .",
"The dimensions of the system are 45 cm$\\times $ 15 cm$\\times 3$ mm.",
"Figure: Fabricated metasurface system corresponding to Fig. .",
"The metasurfaces from the left to the right perform the following operations on the pp-polarized wave: space-wave to surface-wave transformation, surface-wave propagation, and surface-wave to space-wave transformation.",
"At the same time, the central metasurface is perfectly transparent to ss-polarized waves.",
"The difference between the phase-gradients of the two end metasurfaces is clearly visible."
],
[
"Experiment",
"The measurement of the metasurface system was performed using the experimental setup depicted in Fig.",
"REF .",
"The input side of the metasurface system is covered everywhere by absorbers except for a small aperture allowing the illumination of the first metasurface on the left.",
"A high-gain X-band horn antenna illuminates the structure at the input side while a waveguide probe scans the metasurface system at the output side in the near-field region.",
"The near-field is measured in the middle of the metasurface system in Fig.",
"REF along the $x$ -direction.",
"The measured near-field will be first Fourier-transformed to compute the spatial ($k$ -domain) spectrum, and hence identify the modes excited at the output side of the system, and next propagated in the $xz$ -plane by the angular spectrum technique [20], so as to verify the periscope operation of the system.",
"Figure: Side view of the metasurface system measurement setup.The modes excited along the overall structure, as the probe scans the entire $x-$ dimension of the system, are revealed in Fig.",
"REF , which plots the normalized $x$ -Fourier transform of the output near-field measured along the $x$ -direction using the setup of Fig.",
"REF .",
"The mode excited at the output of the metasurface system with the highest amplitude is a surface wave of wavenumber $k_x^\\text{sw}=9k_0/8$ corresponding to the wavenumber of the specified surface-wave mode.",
"The reason why this mode is dominant is because it is excited along the entire structure, being first generated on the first metasurface, next guided by the second one and eventually radiated by the third one.",
"The mode excited with the next higher intensity is the space-wave mode at $k_x^\\text{t}=k_0/2$ , which corresponds to the input wave impinging the metasurface at $\\theta ^\\text{in}=30^\\circ $ .",
"The third largest peak lies in the negative side of the horizontal axis and corresponds to the specified transmitted space wave with wavenumber $k_x^\\text{t}=-k_0/8$ generated by the third metasurface.",
"Figure: Normalized xx-Fourier transform (k x k_x-domain) of the output near-field measured along the xx-direction at 1 cm from the metasurface in the zz-direction (Fig. ).",
"(a) Scanning across the entire metasurface system.",
"(b) Scanning only across the third metasurface.",
"The regions highlighted in blue correspond to the radiation region.Figure REF shows the modes excited only at the output side of the third metasurface, when the near-field probe scans only on that part of the system.",
"As expected, the two strongest modes correspond to the specified transmitted space wave with $k_x^\\text{t}=-k_0/8$ and the specified surface wave with $k_x^\\text{t}=9k_0/8$ .",
"Next, we compute the field scattered from the metasurface system by applying the angular spectrum propagation technique [20] to the near-field measured along the entire structure.",
"To clearly see the propagation of the expected transmitted space wave with $k_x^\\text{t}=-k_0/8$ , we ignore the contribution of the input wave, which generates important spurious scattering, as is visible in Fig.",
"REF around $k_x^\\text{t}/k_0=0.5$ .",
"This is achieved by first taking the Fourier transform of the near-field, yielding the data in Fig.",
"REF , and next setting to zero all the modes excited in the region $0.2< k_x^\\text{t}/k_0 <0.8$ in Fig.",
"REF to remove the contributions from the input wave.",
"Then, the field is propagated along the $z$ -direction following the usual procedure of the angular spectrum propagation technique.",
"The resulting scattered field is plotted in Fig.",
"REF , where the metasurface system lies at $z=0$ and extends from $x=-22.5$ cm to $x=22.5$ cm.",
"In this figure, we can see the presence of a strong surface wave near the structure close to $z=0$ .",
"In the region around $x=-10$ cm, we see some scattering which is due to the discontinuity between adjacent metasurfaces.",
"In the region around $x=10$ cm, we see a beam emerging from the metasurface system and being deflected towards the left.",
"This beam corresponds to the specified transmitted space wave with $\\theta ^\\text{out}=-7.2^\\circ $ .",
"Figure: Absolute value of the transmitted electric field (E x E_x component) obtained by angular spectrum propagation.",
"The metasurface system is at z=0z=0 and extends from x=-22.5x=-22.5 cm to x=22.5x=22.5 cm.In order to better understand the result shown in Fig.",
"REF , we next analyze the spatial power distributions of the surface wave ($k_x^\\text{sw} = 9k_0/8$ ) and of the transmitted space wave ($k_x^\\text{out} = -k_0/8$ ) along the metasurface system.",
"From the data plotted in Figs.",
"REF , it is possible to extract the power distribution of the different modes over the metasurface system.",
"This is achieved by first isolating the modes of interest in the data of Fig.",
"REF by setting to zero everything except the relevant regions (appropriate peaks) – for example leaving only the peak centered at $k_x^\\text{t} = 9k_0/8$ to isolate the surface wave – and then taking the inverse Fourier transform to generate the spatial distribution of the mode.",
"The results are presented in Fig.",
"REF , where the power distribution of the surface wave is represented by the solid black line and that of the space wave by the dashed red line.",
"Figure: Normalized power distribution of the surface-wave mode (solid black line) and of the transmitted wave (dashed red line) over the metasurface system.",
"The two vertical dashed black lines indicate the separation between the three metasurfaces.As one moves along the $x$ -axis, the power distribution of the surface wave (solid black curve) first increases, following the power distribution of the exciting horn antenna, which points at the junction between the first and second metasurfaces.",
"At this point, it reaches a corresponding maximum.",
"Then, it decreases as the wave propagates along the waveguiding metasurface while experiencing metallic and dielectric dissipation losses.",
"Finally, it further decreases on the third metasurface due to combined dissipation and radiation losses.",
"The power level of the transmitted space wave (dash red curve) is relatively high at the junction between the first and second metasurfaces, which is explained by spurious scattering of the incident wave at this discontinuity, similarly to the undesired scattering apparent in Figs.",
"REF and REF .",
"Then, this power rapidly decreases along the second metasurface, as expected from the fact that this surface does not radiate.",
"Along the third metasurface, the power of the space wave progressively increases as it is progressively generated in terms of leaky-wave radiation by the interaction between the surface wave and the phase-gradient of the metasurface.",
"The experimental results presented above are in perfect agreement with the expected response of the metasurface system, with the exception of a relatively low efficiency of about $10\\%$ .",
"This low efficiency is due to a combination of effects that include surface-wave dissipation loss, scattering at each of the two metasurface discontinuities, the limited coupling of the incident wave which is effectively converted to a surface wave, and the imperfect conversion between space wave and surface wave (and vice-versa) due to the simplified synthesis technique used for the implementation of the phase-gradient metasurfaces.",
"Several of these issues may be addressed by further optimization."
],
[
"Other Potential Applications",
"The concept of space-wave via surface-wave routing may lead to a diversity of other potential applications.",
"As an illustration, we will discuss two of them in this section."
],
[
"Compact Beam Expander",
"An optical beam expander is a device that is used in telescopes or microscopes: it increases (or decreases) the lateral size of the incoming beam.",
"The simplest way to realize such a device is to cascade two thin lenses of different focal lengths.",
"We propose here an alternative beam expanding system, based on the concept of space-wave via surface-wave routing.",
"Compared to the lens system, this routing system presents two significant advantages.",
"First, it uses a single (composite) metasurface instead of two lenses.",
"Second, in contrast to the lens system, it does not require any separation distance, where such a distance at optical frequencies represents several thousands of wavelengths, and hence it leads a very compact system.",
"We present here two different beam expander designs, both increasing the beam width by a factor 3.",
"One performs a direct conversion (without any lateral shift) while the other one performs an offset (laterally shifted) beam expansion.",
"The direct beam expander is made of three metasurfaces, the middle one transforming the incident beam into two contra-propagating surface waves that are then both transformed back into space waves by the two end metasurfaces.",
"The simulation showing this direct expansion is presented in Fig.",
"REF .",
"As may be seen in this plot, the presence of the two metasurface discontinuities induces important spurious scattering.",
"The simulation of the offset beam expander is shown in Fig.",
"REF .",
"The system is identical to that of Fig.",
"REF except that both the incident and transmitted angles are now normal to the surface.",
"For the two structures in Figs.",
"REF , the beam expansion of the transmitted wave is about three times that of the incident wave.",
"Consequently, the amplitude of the transmitted wave is also three times less.",
"Figure: FDFD simulation of a beam expander with (a) direct transformation and (b) offset transformation.",
"The metasurface system is designed to increase the beamwidth of the incident wave by a factor 3."
],
[
"Multi-wave Refractor",
"The capability to route beams via surface waves may also be leveraged to implement a multi-wave refractor, i.e.",
"system performing several refractive transformations with a single metasurface system, in contrast to a conventional metasurface that can only perform two independent refraction transformations, one for an $x$ -polarized wave and one for a $y$ -polarized wave (or up to 4 refractions by leveraging nonreciprocity and making use of gain and loss, as discussed in [21]).",
"The proposed system is realized by inserting a metasurface at the Fourier plane of an optical 4-$f$ system.",
"A 4-$f$ system is generally used as a spatial filter where a mask is placed at the Fourier plane to filter out certain spatial components of the incident wave [22].",
"Here, the metasurface placed at the Fourier plane is not used to filter out spatial components but, instead, to shift the spatial components of the incident waves to another region of the plane, which effectively changes the direction of propagation of the transmitted waves.",
"The concept is depicted in Fig.",
"REF , where two input beams, $\\Psi _1$ and $\\Psi _2$ , are transformed in terms of their spectral contents in the 4-$f$ system.",
"Figure: Multi-wave refractor consisting of a 4-f-f system, with 2 routing metasurface systems in its Fourier plane, which refracts the input waves Ψ 1 \\Psi _1 and Ψ 2 \\Psi _2 into different directions.Figure: Representations of the metasurface at the Fourier plane of Fig.",
"with the two optical routes shifting the waves Ψ 1 \\Psi _1 and Ψ 2 \\Psi _2 to different locations in the Fourier plane.The first lens focalizes the two beams at different locations in the Fourier plane, where a metasurface system is placed.",
"This metasurface system consists of two “optical routes”, as shown in Fig.",
"REF , each composed of three different metasurfaces successively transforming the incident space wave into a surface wave, guiding this surface wave along the the Fourier plane to the appropriate $(k_x,k_y)$ point, and transforming it back into a space wave in the desired direction.",
"In this example, the two beams have been shifted along the $-x$ -direction in the Fourier plane.",
"Their respective momenta along $x$ have therefore been decreased.",
"Consequently, the two beams exit the system, collimated by the second lens, with transmission angle depending on the points to which they have been shifted in the Fourier plane.",
"Such a metasurface system might be populated with several additional “optical routes” so as to achieve even more refraction transformations."
],
[
"Conclusion",
"In this work, we have introduced the concept of space-wave via surface-wave routing system composed of several juxtaposed metasurfaces.",
"We have presented two synthesis techniques to design such a routing system.",
"One is exact and most efficient but also complex to realize in practice due to the presence of spatially varying electric and magnetic losses.",
"The other one is an alternative simplified approach that consists in using phase-gradient metasurfaces, to generate the surface wave, and on dispersion engineering, to guide the surface wave along the structure.",
"This alternative synthesis technique is based on the generalized law of refraction, which is approximate, and therefore leads to a less efficient structure, due to undesired scattering, but offers the advantage of being easier to realize.",
"As a proof of concept, we have presented a metasurface system acting as an “electromagnetic periscope”.",
"This system spatially shifts an incident beam impinging the structure under a given angle and then reradiates it under a small angle.",
"This structure produces the expected result, but suffers from a relatively low efficiency, which may be improved by further optimization.",
"To illustrate the capabilities of the proposed concept, we have also presented two other potential applications, namely a compact beam expander and a multi-wave refractor that may be used as a spatial coupler between multiple inputs and outputs."
],
[
"Acknowledgment",
"This work was accomplished in the framework of the Collaborative Research and Development Project CRDPJ 478303-14 of the Natural Sciences and Engineering Research Council of Canada (NSERC) in partnership with the company Metamaterial Technology Inc."
]
] | 1612.05576 |
[
[
"Experimental verification of orbital engineering at the atomic scale:\n charge transfer and symmetry breaking in nickelate heterostructures"
],
[
"Abstract Epitaxial strain, layer confinement and inversion symmetry breaking have emerged as powerful new approaches to control the electronic and atomic-scale structural properties in complex metal oxides.",
"Nickelate heterostructures, based on RENiO$_3$, where RE is a trivalent rare-earth cation, have been shown to be relevant model systems since the orbital occupancy, degeneracy, and, consequently, the electronic/magnetic properties can be altered as a function of epitaxial strain, layer thickness and superlattice structure.",
"One such recent example is the tri-component LaTiO$_3$-LaNiO$_3$-LaAlO$_3$ superlattice, which exhibits charge transfer and orbital polarization as the result of its interfacial dipole electric field.",
"A crucial step towards control of these parameters for future electronic and magnetic device applications is to develop an understanding of both the magnitude and range of the octahedral network's response towards interfacial strain and electric fields.",
"An approach that provides atomic-scale resolution and sensitivity towards the local octahedral distortions and orbital occupancy is therefore required.",
"Here, we employ atomic-resolution imaging coupled with electron spectroscopies and first principles theory to examine the role of interfacial charge transfer and symmetry breaking in a tricomponent nickelate superlattice system.",
"We find that nearly complete charge transfer occurs between the LaTiO$_3$ and LaNiO$_3$ layers, resulting in a Ni$^{2+}$ valence state.",
"We further demonstrate that this charge transfer is highly localized with a range of about 1 unit cell, within the LaNiO$_3$ layers.",
"The results presented here provide important feedback to synthesis efforts aimed at stabilizing new electronic phases that are not accessible by conventional bulk or epitaxial film approaches."
],
[
"Introduction",
"For many technologically-relevant materials systems, in particular transition-metal oxides (TMOs), the orbital structure (relative energies, filling, etc.)",
"directly correlates to the material's resulting properties.",
"[1], [2], [3], [4], [5] For example, systems such as the manganites (colossal magnetoresistance),[6] the cobaltates (spin-state transitions),[7], [8] and the cuprates (high-temperature superconductivity)[9], [10] owe their behaviors to specific configurations of the electronically active transition-metal cation $d$ orbitals, which, for near-cubic symmetry, are split into the (lower energy) t$_{2g}$ and (higher energy) e$_g$ orbitals.",
"The development of atomically precise growth techniques for oxides has opened up the possibility of controlling orbital configurations via layered heterostructures.",
"Ni$^{3+}$ ($d^7$ ) is a $d$ orbital open-shell system, with fully occupied t$_{2g}$ orbitals and a single electron occupying the twofold-degenerate e$_g$ orbital.",
"LaNiO$_3$ (LNO), possessing a pseudocubic perovskite structure, is a material recently explored in the context of orbital engineering, with the goal of breaking its orbital degeneracy and emulating the single-band structure of the cuprates.",
"[11], [12], [13] A recent publication on a LaTiO$_3$ -LaNiO$_3$ -LaAlO$_3$ (LTNAO) superlattice demonstrated the successful breaking of this orbital degeneracy by using atomic-layer synthesis to alter its symmetry and filling; an approximately 50% change in the occupation of the Ni $d$ orbitals was reported[14] and verified via X-ray absorption spectroscopy and ab initio theory, confirming the creation of an electronic configuration which approaches a single-band Fermi surface.",
"The three-component superlattice, where 1 unit cell (uc) of LaTiO$_3$ (LTO) and 2 uc of LNO are sandwiched between 3 uc of LaAlO$_3$ (LAO) develops a large orbital polarization as a result of an inherent inversion symmetry breaking, internal charge transfer, and the resultant ionic polarization.",
"[15], [14] The principle is based on the transfer of a single electron from the LTO layer (Ti$^{3+}$ ) to the LNO layer (Ni$^{3+}$ ) due to the mismatch in electronegativity of the two ions.",
"The electron transfer creates a charge imbalance and, hence, a dipole field which leads to large polar distortions with polarization pointing towards the NiO$_2$ layer of the LNO.",
"The combination of these polar distortions and the symmetry breaking of the superlattice about the LNO results in asymmetric stretching of the NiO$_6$ oxygen octahedra, leading to a large crystal field splitting and a polarization in the orbital occupations which resembles the arrangement in the high-temperature superconducting cuprates.",
"A previous study[14] infers the charge transfer from the spatially-averaged X-ray absorption spectroscopy (XAS) measurements on the Ti $L$ -edgeand Ni $L$ -edge.",
"In this work, we focus on a specific superlattice system, consisting of 1 uc of LaTiO$_3$ (LTO), 2 uc of LNO and 3 uc of LaAlO$_3$ (LAO).",
"We aim to directly quantify the proposed charge transfer mechanism and determine its range using this superlattice structure.",
"More specifically, we utilize aberration-corrected scanning transmission electron microscopy (STEM) coupled with both energy dispersive X-ray (EDX) and electron energy loss (EEL) spectroscopies to quantify the orbital manipulation in a nickelate heterostructure; specifically, charge transfer and symmetry breaking at the atomic scale.",
"We directly map the charge transfer with STEM EELS/EDX, providing direct evidence for the key driving force of orbital polarization in the three-component system.",
"Furthermore, we detect the signatures of orbital polarization in this LTNAO superlattice with atomic resolution, as previously suggested by sample-averaged experiments.",
"Additionally, we perform first principles density functional theory (DFT) simulations within the local density approximation (LDA) to simulate and verify basic aspects of the electronic structure of these heterostructures using a $c\\left(2\\times 2\\right)$ interfacial unit cell that includes rotations and tilts of oxygen octahedra.",
"[16], [17], [18] It should be noted here that, until recent instrumentation and software advances, an analysis with the spatial and chemical resolution as presented here would not be have been possible, since it requires an imaging probe which has a high enough current density to generate appreciable X-rays, yet is also able to achieve sub-Å resolution.",
"Indeed, electron microscopy has a rich history in the advanced characterization of oxides, for example, in the atomic-scale imaging of composition, bonding, electron spatial distribution at interfaces, valence determination, etc.",
"[19], [20], [21], [22], [23], [24], [25], [26] By taking advantage of the numerous imaging and spectroscopy modes on advanced aberration-corrected instruments, it is feasible to locally conduct a complete chemical, structural, and electronic characterization at the atomic scale.",
"The high/low angle annular dark field (H/LAADF) and annular bright field (ABF) STEM signals can be simultaneously acquired, resulting in images which are sensitive to atomic number, strain, and light element contrast, respectively.",
"[27], [28], [29], [30] In terms of spectroscopy, both EDX and EEL signals can be simultaneously acquired, thereby providing atomically-resolved chemical and electronic information.",
"The tricomponent superlattice is grown on LaAlO$_3$ (001) single crystal substrates using oxygen plasma assisted molecular beam epitaxy.",
"The layering sequence for superlattice is [(LaTiO$_3$ )$_1$ -(LaNiO$_3$ )$_2$ -(LaAlO$_3$ )$_3$ ]$\\times $ 12 with a total film thickness of $\\approx $ 30 nm.",
"Each layer is grown via co-deposition of the respective elements.",
"The growth is monitored in situ by reflection high energy electron diffraction (RHEED).",
"Post-growth RHEED images display sharp narrow streaks indicative of coherent epitaxy.",
"Ex situ atomic force microscopy reveals low surface roughness ($\\approx $ 1-2 Å) and unit cell high ($\\approx $ 4 Å) steps post-growth.",
"More details on the thin film growth can be found in Ref.",
"Disa15PRL"
],
[
"X-ray Absorption Spectroscopy",
"XAS measurements shown in Figures REF and REF were carried out at beamline $U4B$ at the National Synchrotron Light Source.",
"Spectra were recorded in total electron yield mode and normalized by the incident flux as measured by an upstream Au mesh.",
"The energy of the Ti $L$ - and O $K$ -edges were calibrated with reference to a simultaneously measured TiO$_2$ powder.",
"A linear background is subtracted from the data by fitting to the pre-edge region $\\approx $ 5-10 eV below the edge."
],
[
"First-Principles Modeling",
"We performed first principles calculations using Density Functional Theory (DFT)[16], [17] with ultrasoft pseudopotentials.",
"[31], [32], [33] To approximate the effects of exchange-correlation, we used the local density approximation (LDA)[18] as it has been proven to be the best approach for describing bulk LNO bulk from first principles.",
"[34] $k$ -point sampling of the Brillouin zone employed a mesh equivalent to a $12\\times 12\\times 12$ mesh for a 5-atom pseudocubic bulk unit cell.",
"Band occupations were Gaussian broadened with width 0.03 eV.",
"The plane wave cutoff was 35 Ryd for the wave functions and 280 Ryd for the electron density.",
"Structural relaxations were terminated when all components of atomic forces are below 0.03 eV/Åin magnitude.",
"The simulated superlattices were periodic in all directions and biaxially strained to the theoretically computed pseudocubic lattice parameter of LaAlO$_3$ at 3.71 Å. Superlattices with $c\\left(2\\times 2\\right)$ interfacial unit cells were simulated allowing for octahedral rotations and tilts."
],
[
"Scanning Transmission Electron Microscopy",
"Combined atomic EELS and EDS data were acquired using a cold-field emission gun JEOL GrandARM 60 – 300 kV, operated at 160 kV with a beam current of about 85 pA.",
"The microscope is equipped with dual large solid angle SDD detectors for the acquisition of EDS data and a Gatan GIF Quantum ER for the acquisition of EELS data.",
"EELS data was acquired in DualEELS mode where both the the low- and core-loss spectra were acquired simultaneously.",
"The zero-loss peak, present in the low-loss spectra, can be used to correct and remove all the effects of energy drift allowing a more accurate measurement of any chemical shift.",
"The EELS spectrometer was setup with a dispersion of 0.1 eV / channel resulting in an energy resolution of 0.5 eV that was needed in order to resolve all the spectral features present in the EELS spectrum moving across the super lattice layers.",
"For simultaneous high-angle annular dark field (HAADF) and annular bright field (ABF) imaging, a probe-convergence angle of 25 mrad was used with a inner detector angle and angular range of 12 mrad for ABF and an inner detector angle of 55 mrad for HAADF imaging."
],
[
"Results and Discussion",
"Figure REF presents a STEM overview of the superlattice structure, following a focused ion beam (FIB) preparation; note that not all the superlattice repeats are visible, as the topmost section of the sample has been milled away in order to render the remainder of the sample sufficiently thin for STEM analysis.",
"Figures REF a,b) provide a low magnification view of the structure in both LAADF and ABF modes, which reveal some regions of localized strain (anomalously bright/dark in LAADF/ABF), likely from the presence of occasional dislocations; this localized strain is in addition to the strain associated with the superlattice, evidenced by the layering in both the LAADF and ABF images.",
"In general, it is very difficult to discern the identity of the individual layers exclusively via imaging, as there is very little $Z$ -contrast gradient across the interfaces of LAO-LTO-LNO (Figure REF c); thus, chemical spectroscopy is required.",
"Indeed, as LAADF/ABF are considerably less sensitive to atomic number contrast, there is no reason other than strain (the forced constraint on the lattice parameters) for the obvious contrast between layers in Figures REF a,b,d).",
"Chemical and electronic analyses have made great strides in recent years, owing largely to high-area silicon drift EDX detectors and high-speed/high-sensitivity EEL spectrometers.",
"[35][36] Simultaneous acquisition of both signals allows one to avoid the high energy edges in EELS (La, Ni, and Al in this case) in favor of a higher energy dispersion, and to rely on EDX to identify the remaining elements.",
"The higher energy dispersion in EELS then enables the detailed near-edge fine structure analysis of relevant energy loss, including the Ti $L$ - and O $K$ -edge.",
"Several integrated signals are presented in Figure REF , coming from both EELS and EDX.",
"The O $K$ -edge is normalized to the La signal, which is expected to remain relatively constant.",
"Finally, RGB images of various combinations detail the chemical makeup of the LTNAO superlattice structure.",
"The remaining figures present the relevant EELS fine structure results, beginning with the Ti $L$ -edge (Figure REF ), integrated from the indicated Ti column.",
"The clear presence of four peaks is the classic signature of Ti$^{3+}$ , resulting from the splitting of the degenerate $3d$ final states into the t$_{2g}$ and e$_g$ levels for each of the Ti $L_2$ and $L_3$ edges.",
"[20] This is in contrast to bulk LTO, where the Ti is in a 3+ ($d^1$ ) state which leads to less well-defined t$_{2g}$ -e$_g$ splitting, and a markedly different Ti $L$ -edge signature all of which is easily identified via EELS.",
"[19] That the Ti in the LTNAO superlattice is 4+ is the first piece of direct evidence of the desired donation of an $e^-$ from Ti.",
"The 4+ state of Ti in the superlattice is consistent with the previously reported XAS data, which averages over the entire superlattice film (Figure REF ); XAS has the advantage of a considerably higher energy resolution than EELS but provides little spatial resolution.",
"Analysis of the Ni $L$ -edge to describe the Ni valence in the LNO layers is not possible here due to the nearly complete overlap between the La $M$ - and the Ni $L$ -edges.",
"We want to further emphasize here that the observed change in Ti valence is not due to film stoichiometry, i.e.",
"oxygen vacancies.",
"Figure REF shows the integrated O $K$ -edge intensity normalized to the La concentration for the superlattice.",
"It can be seen that the oxygen stoichiometry for all three layers does not vary, and the EEL spectra from the LaAlO$_3$ layers (which can be considered as a bulk reference in this context) show the fine structure expected for stoichiometry LaAlO$_3$ .",
"Therefore, it appears that all layers in the superlattice are stoichmetric and the observed changes in the valence and EELS fine-structure are associated with interfacial charge transfer.",
"In the following, we will therefore focus on the O $K$ -edge analysis to extract and quantify the interfacial charge transfer.",
"Various O $K$ -edge spectra are provided for the Ti and Ni columns contained within the spectroscopic region of interest, again integrating a number of rows, along the respective columns (Figure REF ).",
"While there are some difference in the absolute intensity of the main peak, we will be focusing on the pre-peak of the O $K$ -edge, which results from electronic transitions into the hybridized O $2p$ - transition metal (TM) $3d$ orbitals.",
"[37] What is visible in all the spectra is a pre-peak centered around 530 eV (labeled $A$ ), and an additional peak near 528 eV ($B$ ) which is only present in the spectra from the Ni columns.",
"These peaks are readily explained by comparison to known bulk EEL spectra, shown in Figure REF .",
"For example, looking at bulk LTO (TM valence is 3+), we see merely a slight pre-shoulder on the main peak, due to its $3d^1$ configuration, due to a decreased number of unoccupied states.",
"In SrTiO$_3$ , where the Ti $d$ -orbitals are completely empty, a strong pre-peak intensity, $A$ , is seen.",
"Peak $A$ in the Ti column of the LTNAO superlattice is analogous to that seen in the STO as opposed to the LTO bulk reference, suggesting that Ti is in a 4+ valence state.",
"Examining a reference spectrum for bulk LNO (Figure REF ), we see a pre-peak at 528 eV, shifted lower in energy with respect to the LTO and STO pre-peaks located at an edge onset energy of 530 eV.",
"In the LNO superlattice, the lower energy pre-peak is manifested as peak $B$ seen in the Ni columns, though of a reduced intensity compared to the bulk reference spectra for LNO, due to the acceptance of an e- from the Ti layer and hence a reduced Ni valence.",
"Again, we expect a diminished pre-peak[38] in this case because the additional $e^-$ into the Ni layers reduces the number of empty hybridized O $2p$ - TM $3d$ orbitals probed by the incident electrons.",
"Indeed, XAS of the O $K$ -edge confirms the presence of both of these pre-peaks, $A$ and $B$ .",
"We reiterate that XAS is a spatially averaging spectroscopic technique and cannot tell us the specific spatial origin of these peaks.",
"Looking carefully at peak $A$ in all of the spectra shown in Figures REF and (c), it is apparent that there is still significant spectral weight in both Ni columns.",
"Given that there is some Ti present in the first Ni column of each superlattice (based on the EELS/EDX spectroscopy), some of this intensity in Peak $A$ can potentially be explained by remnant Ti contributions.",
"However, when examining the O $K$ -edge fine-structure of La$_2$ NiO$_4$ (Figure REF ), where the Ni valence state is expected to be 2+, we do not find any sign of the pre-peak $B$ at 528 eV (as seen for Ni$^{3+}$ in LaNiO$_3$ reference), but instead a shoulder at $\\approx $ 530 eV, which coincides with the position of peak $A$ in Figure REF .",
"The peak $A$ intensity in the spectra taken from the LNO layers is therefore not completely due to the remnant Ti contributions but also due to the contribution of Ni$^{2+}$ .",
"It is interesting to note that the intensity of peak $A$ is significantly higher in the layer closest to LTO (i.e.",
"Ni col1a and Ni col2a) and decreases in the layers closest to LAO (i.e.",
"Ni col1b and Ni col2b).",
"Without further insights from theoretical modeling, it is impossible to disentangle the contributions to this peak stemming from remnant Ti$^{3+}$ in the LNO layer closest to LTO and from the increasing Ni valence in the layer closest to the LAO layers.",
"The authors acknowledge that with the electron probe in a channeling condition (i.e., a zone axis orientation) as is the case here, one must be cautious when attempting quantitative EELS and EDX measurements, as these experiments can be convoluted by elastic and thermal diffuse scattering of the incident electrons.",
"[39], [40] However, in the case of the present fine structure analysis, we are simply looking at clear trends in the spectra, which appear and disappear rapidly, generally within a single unit cell, as shown in Figure REF ; we furthermore note the good agreement the XAS data, which is insensitive to channeling.",
"For our DFT calculations, we simulated the LTNAO system strained to LaAlO$_3$ .",
"In addition, we consider bulk NiO (with nominal Ni$^{2+}$ valence), bulk LaNiO$_3$ (nominally Ni$^{3+}$ ), bulk LaTiO$_3$ (nominally Ti$^{3+}$ ), and bulk SrTiO$_3$ (nominally Ti$^{3+}$ ).",
"We calculate the relaxed LDA atomic-scale structure as well as the orbital occupancies and the O $K$ -edgespectra using both the $Z$ and $Z+1$ approximations.",
"The most useful comparison between theory and experiment comes from the calculated charge in the Ni $d$ orbitals, seen in Table REF .",
"By comparing to the $d$ occupancy of NiO and LaNiO$_3$ , we determine an interpolated nominal charge on the Ni atoms in the two LNO layers of the LTNAO superlattices.",
"This analysis gives a nominal charge Ni$^{2.3+}$ for the layer closest to LTO (i.e., Ni col1a and Ni col2a) and Ni$^{2.77+}$ for the layer closest to LAO (i.e., Ni col1b and Ni col2b).",
"This result shows that the electron transfer from Ti is primarily limited to the LNO layers directly adjacent to LTO.",
"Due to the difference in charge, we would expect the O $K$ -edge spectra to differ for the two Ni layers in the LTNAO.",
"As a reminder, by comparison to bulk references (Figure REF ), we determine that the energy of pre-peak $A$ ($\\approx $ 530 eV) primarily corresponds to Ti$^{4+/3+}$ states plus Ni$^{2+}$ states and the energy of pre-peak $B$ ($\\approx $ 528 eV) corresponds primarily to Ni$^{3+}$ states.",
"Thus, from the calculated Ni charge, we would expect the intensity ratio of peaks $A$ and $B$ to be larger for the LNO adjacent to LTO (i.e., Ni col1a and Ni col2a)) than for the second LNO layer (Ni col1b and Ni col2b), reflecting the larger amount of Ni$^{2+}$ character.",
"This prediction is in agreement with the experimental measurements and confirmed by integrating peak $A$ and $B$ intensities for the LTO, two Ni-a, and two Ni-b columns.",
"The average $A/B$ -peak ratio decreases as expected: 5.24 (Ti), 3.26 (Ni-a), and 2.56 (Ni-b), demonstrating the larger amount of Ni$^{2+}$ character in the Ni column adjacent to LTO.",
"It is worth noting that, despite much effort, a direct comparison of EELS data to theoretically calculated O $K$ -edge spectra did not produce good agreement of the energies or intensities of the peaks for bulk LNO or LTNAO.",
"Hence we do not rely on them in our theoretical analysis.",
"Given that DFT is a ground state theory, it can, in principle, correctly compute the mean electron density and orbital occupations.",
"However, as is well known, using DFT to predict electronic excitations (such as EELS) is much more problematic.",
"Furthermore, localized dynamical electronic correlations are not included in DFT band structures, further degrading comparisons to experiments in such correlated complex oxides.",
"We have, therefore, focused on using the observables that should be predicted correctly by DFT: the electronic density and mean occupancy of orbitals."
],
[
"Conclusions",
"In summary, by combining atomically-resolved energy-loss and X-ray data with first principles DFT calculations, direct evidence is provided of the charge transfer from LTO into LNO in tricomponent superlattices.",
"Using the high spatial sensitivity of STEM imaging and electron spectroscopies, we confirm previous XAS measurements, which have reported a $\\approx $ 50 % change in the orbital occupation that is significantly higher (by a factor of 2-3) compared to previous results.",
"[41] Furthermore, we demonstrate that this interfacial charge transfer from the LTO to the LNO layers is highly localized in real space and is limited to the layers directly adjacent to each other.",
"The range of the interfacial charge transfer is of the order of 1 unit-cell or about 4 Å.",
"These types of results and analysis provide crucial feedback for future orbitally-selective synthesis methods, where the magnitude and range of charge transfer and orbital polarization will be used to stabilize novel electronic phases inaccessible by conventional epitaxial methods."
],
[
"Acknowledgements",
"PJP and RFK acknowledge funding from the National Science Foundation (NSF) via Grant Number DMR-1408427 to support this work.",
"Work at Yale supported by NSF MRSEC DMR-1119826 (CRISP) and AFOSR under grant number FA95501510472.",
"Use of the National Synchrotron Light Source at Brookhaven National Lab was supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No.",
"DE-AC02-98CH10886."
]
] | 1612.05610 |
[
[
"Dynamic Spectrum Leasing with Two Sellers"
],
[
"Abstract This paper studies dynamic spectrum leasing in a cognitive radio network.",
"There are two spectrum sellers, who are two primary networks, each with an amount of licensed spectrum bandwidth.",
"When a seller has some unused spectrum, it would like to lease the unused spectrum to secondary users.",
"A coordinator helps to perform the spectrum leasing stage-by-stage.",
"As the two sellers may have different leasing period, there are three epochs, in which seller 1 has spectrum to lease in Epochs II and III, while seller 2 has spectrum to lease in Epochs I and II.",
"Each seller needs to decide how much spectrum it should lease to secondary users in each stage of its leasing period, with a target at revenue maximization.",
"It is shown that, when the two sellers both have spectrum to lease (i.e., in Epoch II), the spectrum leasing can be formulated as a non-cooperative game.",
"Nash equilibria of the game are found in closed form.",
"Solutions of the two users in the three epochs are derived."
],
[
"Introduction",
"Cognitive radio has been considered as a promising solution to the spectrum shortage problem in the near future.",
"In cognitive radio, if a primary user (a licensed user with some licensed spectrum bandwidth) has some unused spectrum for a certain amount of time, it may lease the unused spectrum to secondary users.",
"By this method, the spectrum opportunities are exploited, and the primary user can earn extra payment from secondary users.",
"Spectrum leasing has been well investigated in the literature, in the modes of monopoly spectrum leasing (in which there is one spectrum seller) and oligopoly spectrum leasing (in which multiple spectrum sellers exist).",
"In either mode, the research focus is on how to set the spectrum price.",
"In monopoly spectrum leasing, the major target is to achieve the maximal revenue of the seller.",
"In the work of [1], there are a spectrum provider, a broker, and a number of secondary users.",
"By a Stackelberg game modeling, the broker optimally decides on the number of channels it should purchase from the spectrum provider as well as the price it should use to sell the purchased spectrum to secondary users.",
"The work in [2] also considers a broker.",
"It is assumed that for a given spectrum price, the amount of spectrum demand from secondary users is random.",
"The work in [3] considers the impact of spectrum leasing on primary user performance (such as possible extra interference to the primary system).",
"An optimal solution is given for the primary user, which strikes a balance between the earned revenue and the cost.",
"In oligopoly spectrum leasing, the major target is to achieve an equilibrium in the competition among multiple spectrum sellers.",
"Two brokers are assumed in [4].",
"Each broker decides on the amount of spectrum that it should purchase from spectrum providers and on the spectrum price that it should announce to secondary users, with a target at profit maximization.",
"The work in [5] also considers two brokers, by assuming that the leased spectrum may be shared by multiple secondary users simultaneously.",
"Therefore, interference among secondary users needs to be taken into account.",
"The works in [6], [7], [8] consider a duopoly market, in which the price competition of two spectrum sellers is investigated by using game theoretical approaches.",
"The work in [9] discusses the case with multiple sellers.",
"By using an evolutionary game model, a solution is given for secondary users for their spectrum selection and for sellers for price setting.",
"The work in [10] considers multiple sellers as well as one broker, in which the impact of spectrum leasing on sellers' performance (i.e., service quality degradation) is taken into account.",
"The work in [11] considers heterogenous secondary users, i.e., different secondary users may have different criteria on their spectrum leasing decisions.",
"In all above works, spectrum leasing is performed only once, and the price is fixed for the whole spectrum leasing duration, referred to as static spectrum leasing.",
"On the other hand, dynamic spectrum leasing, in which the spectrum price may change over time, is more appropriate for the cases that the secondary users may need spectrum at different time instants.",
"In [12], dynamic pricing in monopoly spectrum leasing is performed over infinite time horizon.",
"The spectrum price is set dynamically, with a target of long-term average revenue maximization.",
"In [13], dynamic pricing in monopoly spectrum leasing is performed over a finite duration.",
"The finite duration is divided into a number of stages, and the price in each stage is set up so as to maximize the overall revenue.",
"To the best of our knowledge, there is no research in the literature on dynamic pricing with more than one spectrum seller.",
"Figure: Leasing periods of the two sellers.To fill the research gap, in this paper, we study dynamic spectrum leasing problem in a duopoly market with two sellersWe consider two sellers (i.e., a duopoly spectrum market) for the following reasons.",
"1) A duopoly spectrum market is a typical and popular scenario for cognitive radio, and has been adopted by many research efforts in the literature [4], [5], [6], [7], [8].",
"2) Sufficient insights can be provided by the duopoly scenario into the spectrum leasing, and our method in this paper can be extended to the scenarios with more spectrum sellers, with increased complexity in analysis and presentation.",
"For ease of analysis and presentation, we consider a duopoly scenario.. As the two sellers may have different leasing periods, the system has three epochs, in which seller 1 has spectrum to lease in Epochs II and III, while seller 2 has spectrum to lease in Epochs I and II.",
"The main contributions in this paper are summarized as follows.",
"1) We show that, the spectrum leasing problems of the sellers in Epoch I and Epoch III are convex optimization problems.",
"For Epoch II, we formulate spectrum leasing of the two sellers as a non-cooperative game.",
"We derive closed-form expressions for the Nash equilibria of the non-cooperative game.",
"2) The amount of spectrum that seller 1 would like to lease in Epoch III affects the non-cooperative game in Epoch II, and thus, affects the total revenues of the two sellers.",
"By analyzing properties of seller 1's revenue in Epoch II and Epoch III, we propose a method that finds the optimal amount of spectrum that seller 1 should lease to secondary users in Epoch III.",
"The rest of this paper is organized as follows.",
"In Section , the system model is presented, and the spectrum leasing problems for the two sellers are formulated.",
"In Section , Nash equilibria of the non-cooperative game in Epoch II are derived.",
"Section discusses how seller 1 should distribute its spectrum to be leased in Epoch II and Epoch III.",
"Numerical results are given in Section , and finally the paper is concluded in Section ."
],
[
"System Model and Problem Formulation",
"Consider two spectrum sellers (seller 1 and seller 2), one coordinator, and multiple secondary users.",
"The coordinator is to help the two sellers to lease spectrum to secondary users.",
"The two sellers are primary networks with a certain amount of licensed spectrum bandwidth.",
"For each seller, when the data traffic from its own users is light, the seller may partition its spectrum bandwidth into two portions: primary portion and secondary portion.",
"The primary portion will be assigned to the seller's own users, and the secondary portion can be leased to secondary users.",
"In specific, consider that seller 1 and seller 2 have bandwidth $Q_1$ and $Q_2$ in their secondary portion, respectively.",
"For each seller, the bandwidth in its secondary portion can be leased to secondary users for a duration (called leasing period).",
"Consider that the two sellers' leasing periods are not identical,If the two leasing periods are identical, it is a special case of the problem considered in this paper.",
"and overlap with each other.",
"Without loss of generality, we assume that the leasing period of seller 2 starts earlier than the leasing period of seller 1.",
"We also assume that the leasing period of seller 2 ends earlier than that of seller 1.Note that the method in this paper can be straightforwardly extended to deal with the case when the leasing period of seller 2 ends later than that of seller 1.",
"An illustration of the two leasing periods is given in Fig.",
"REF .",
"Here the union of the two leasing periods contains $N$ fixed-length stages.",
"For presentation simplicity, the last stage of seller 1's leasing period is called stage 1, while the first stage of seller 2's leasing period is called stage $N$ .",
"Seller $i$ ($i=1,2$ ) would distribute its spectrum bandwidth $Q_i$ to be leased in the stages of its leasing period.",
"In other words, it needs to decide on the amount of spectrum bandwidth to be leased in each stage in its leasing period, with a constraint that the total amount of leased spectrum bandwidth in the stages is bounded by $Q_i$ .",
"For seller $i$ , denote the amount of spectrum bandwidth it would like to lease to secondary users in stage $n$ as $d_{n,i}$ .",
"At the beginning of stage $n$ , seller $i$ should report to the coordinator the information of $d_{n,i}$ .",
"At the beginning of stage $n$ , after the coordinator gets the information of $d_{n,1}$ and $d_{n,2}$ , it would set up a spectrum unit price (the price per unit bandwidth per stage) and lease the spectrum bandwidth $(d_{n,1}+d_{n,2})$ to secondary users.",
"In other words, the coordinator should set up the unit price to attract $(d_{n,1}+d_{n,2})$ spectrum bandwidth demand from secondary users.",
"Denote the price $p$ to attract $d$ spectrum bandwidth demand as $P(d)$ , which is a function of $d$ .",
"Economics analysis [14], [15] has shown that price and demand typically follow a linear model, and thus, price $p$ and spectrum bandwidth demand $d$ satisfy the following feature: $ p=P\\left(d\\right)=C_0 - C_1 \\cdot d$ in which $C_0$ and $C_1$ are coefficients.",
"$P(d)$ is a decreasing function of $d$ .",
"In addition, $d\\cdot P(d)$ should be an increasing function of $d$ (as the total revenue for more leased spectrum bandwidth should be higher), based on which we have $ C_0 > 2 C_1 \\left(Q_1+Q_2\\right).$ From Fig.",
"REF , the union of the two sellers' leasing periods can be divided into three epochs: In Epoch I, only seller 2 has spectrum to lease; in Epoch II, both sellers have spectrum to lease; and in Epoch III, only seller 1 has spectrum to lease.",
"Denote the set of stages in Epoch I, II, and III as $\\mathcal {N}_{\\text{I}}$ , $\\mathcal {N}_{\\text{II}}$ , and $\\mathcal {N}_{\\text{III}}$ , respectively.",
"Denote the set of stages in the leasing period of seller 1 and seller 2 as $\\mathcal {N}_1$ and $\\mathcal {N}_2$ , respectively.",
"Thus, we have $\\mathcal {N}_1 = \\mathcal {N}_{\\text{II}} \\cup \\mathcal {N}_{\\text{III}}$ and $\\mathcal {N}_2 = \\mathcal {N}_{\\text{I}} \\cup \\mathcal {N}_{\\text{II}}$ .",
"Seller $i$ $(i\\in \\lbrace 1,2\\rbrace )$ aims at maximizing its total revenue over all the stages by deciding on $d_{n,i}, n\\in \\mathcal {N}_i$ .",
"Next, the spectrum leasing problem in each epoch is discussed."
],
[
"Spectrum Leasing Problem in Epoch I",
"In Epoch I, only seller 2 has spectrum to lease, and it does not know when seller 1 will join the spectrum leasing market and how much spectrum bandwidth seller 1 will offer for spectrum leasing.",
"So seller 2 assumes a monopoly market.",
"At a stage in Epoch I, once an amount of spectrum is leased to secondary users, the spectrum can be used by secondary users until the last stage of seller 2's leasing period.",
"Seller 2's collected revenue at stage $n$ is $ \\left(C_0-C_1 d_{n,2}\\right) d_{n,2} \\left(n-|\\mathcal {N}_{\\text{III}}| \\right)$ , in which $|\\cdot |$ means cardinality of a set.",
"To maximize its overall revenue, primary use 2 should solve the following optimization problemIn Epoch I, seller 2 does not know the value of $|\\mathcal {N}_{\\text{III}}|$ .",
"However, it knows the value of $\\left(n-|\\mathcal {N}_{\\text{III}}| \\right)$ (the length from stage $n$ until the end of seller 2's leasing period).",
"Thus, in Problem REF , we use notation $\\left(n-|\\mathcal {N}_{\\text{III}}| \\right)$ , for consistence of the formulated spectrum leasing problems in the three epochs.",
": Problem 1 $\\begin{array}{cll}\\mathop {\\max } \\limits _{\\lbrace d_{n, 2}|n \\in \\mathcal {N}_{2}\\rbrace } & \\sum \\limits _{n \\in \\mathcal {N}_{2}} { \\left(C_0-C_1 d_{n,2}\\right) d_{n,2} \\left(n-|\\mathcal {N}_{\\text{III}}| \\right)} \\\\\\text{s.t.}",
"& \\sum \\limits _{n \\in \\mathcal {N}_{2}} d_{n,2} \\le Q_{2} \\\\& d_{n,2} \\ge 0, \\forall n \\in \\mathcal {N}_{2}.\\end{array}$ Problem REF is a convex optimization problem.",
"Thus the global optimal solution of Problem REF can be achieved by existing numerical optimization methods."
],
[
"Spectrum leasing Problem in Epoch II",
"At Epoch II's first stage (denoted as stage $l$ ), seller 1 has available spectrum bandwidth $Q_1$ , while we denote the remaining spectrum bandwidth of seller 2 as $Q_2^\\text{II}$ (in other words, spectrum bandwidth with amount $(Q_2 - Q_2^\\text{II})$ has been leased out by seller 2 in Epoch I).",
"At the beginning of stage $l$ , each seller does not know the presence of the other seller, and thus, assumes a monopoly spectrum leasing.",
"So each seller reports to the coordinator the amount of spectrum bandwidth it would like to lease to secondary users in the stage.",
"In specific, seller 1 first solves the following convex optimization problem: $\\begin{array}{cll}\\mathop {\\max } \\limits _{\\lbrace d_{n, 1}|n \\in \\mathcal {N}_{1}\\rbrace } & \\sum \\limits _{n \\in \\mathcal {N}_{1}} { \\left(C_0-C_1 d_{n,1}\\right) d_{n,1} n} \\\\\\text{s.t.}",
"& \\sum \\limits _{n \\in \\mathcal {N}_{1}} d_{n,1} \\le Q_{1} \\\\& d_{n,1} \\ge 0, \\forall n \\in \\mathcal {N}_{1}\\end{array}$ and reports to the coordinator the values of $d_{l,1}$ ($d_{l,1}$ is from the optimal solution of the above problem) and $|\\mathcal {N}_1|$ (the leasing duration for the $d_{l,1}$ spectrum bandwidth).",
"On the other hand, seller 2 reports to the coordinator the values of $d_{l,2}$ and ($|\\mathcal {N}_2|-|\\mathcal {N}_\\text{I}|$ ) (which is the length of seller 2's remaining leasing period), in which $d_{l,2}$ is from the optimal solution of the following convex optimization problem: $\\begin{array}{cll}\\mathop {\\max } \\limits _{\\lbrace d_{n, 2}|n \\in \\mathcal {N}_{2}\\backslash \\mathcal {N}_\\text{I} \\rbrace } & \\sum \\limits _{n \\in \\mathcal {N}_{2}\\backslash \\mathcal {N}_\\text{I}} { \\left(C_0-C_1 d_{n,2}\\right) d_{n,2} \\left(n-|\\mathcal {N}_{\\text{III}}| \\right)} \\\\\\text{s.t.}",
"& \\sum \\limits _{n \\in \\mathcal {N}_{2}\\backslash \\mathcal {N}_\\text{I}} d_{n,2} \\le Q_{2}^\\text{II} \\\\& d_{n,2} \\ge 0, \\forall n \\in \\mathcal {N}_{2}\\backslash \\mathcal {N}_\\text{I}.\\end{array}$ Then the coordinator feeds back to the two sellers by telling 1) that now two sellers have spectrum to lease, 2) how much spectrum bandwidth each seller offers in this stage, and 3) how long the leasing period is for each seller.",
"From $d_{l,1}$ and $|\\mathcal {N}_1|$ in the feedback information, seller 2 can find out the available stock of seller 1, by searching the value of $Q_1$ (using bisection search) that makes $d_{l,1}$ be in the optimal solution of the problem in (REF ).",
"Similarly, seller 1 can also find out the available stock of seller 2.",
"Based on stock information of the other seller, each seller adjusts the amount of offered spectrum bandwidth ($d_{l,1}$ or $d_{l,2}$ ) and resubmits to the coordinator, and the coordinator decides on a unit price based on (REF ) with total spectrum demand $(d_{l,1}+d_{l,2})$ .",
"In each subsequent stage (say stage $n$ ) in Epoch II, by knowing the existence of the other seller, each seller reports to the coordinator the amount of offered spectrum bandwidth ($d_{n,1}$ and $d_{n,2}$ ), and the coordinator decides on a unit price based on (REF ) with total spectrum demand $(d_{n,1}+d_{n,2})$ .",
"In every stage in Epoch II, once an amount of spectrum bandwidth of a seller is leased to secondary users, the spectrum can be used by secondary users until the last stage of the corresponding seller's leasing period.",
"A decision that seller 1 should make in Epoch II is the amount $Q_1^\\text{III}$ of spectrum bandwidth it reserves for Epoch III, where $Q_1^\\text{III} \\in [0, Q_1]$ .",
"In other words, seller 1 would like to lease spectrum bandwidth ($Q_1-Q_{1}^{\\text{III}}$ ) in Epoch II.",
"In Epoch II, the announced unit price at each stage (say stage $n$ ) depends on the sum of $d_{n,1}$ and $d_{n,2}$ .",
"Thus, there is a non-cooperative game between the two sellers.",
"In this game, the two players are seller 1 and seller 2, and the strategy of seller 1 and seller 2 are $\\mathcal {S}_{1} \\triangleq \\lbrace d_{n, 1}|n\\in \\mathcal {N}_{\\text{II}}\\rbrace $ and $\\mathcal {S}_{2} \\triangleq \\lbrace d_{n,2}|n\\in \\mathcal {N}_{\\text{II}}\\rbrace $ , respectively.",
"The payoff function of seller 1 and seller 2 can be expressed as ${R}_{1} \\left(\\mathcal {S}_{1}, \\mathcal {S}_{2} \\right) \\triangleq \\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} { \\left(C_0-C_1 \\left( d_{n,1} + d_{n,2}\\right)\\right) d_{n,1} n}$ and ${R}_{2} \\left(\\mathcal {S}_{1}, \\mathcal {S}_{2} \\right) \\triangleq \\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} { \\left(C_0-C_1\\left(d_{n,1} + d_{n,2}\\right)\\right) d_{n,2} \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)},$ respectively.",
"Define the feasible region of seller 1's strategy as $\\mathcal {F}_{1}\\left(y\\right) = \\left\\lbrace \\lbrace d_{n,1}|n\\in \\mathcal {N}_{\\text{II}} \\rbrace \\Big |\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n,1} \\le y, d_{n,1} \\ge 0\\right\\rbrace $ when seller 1 would like to lease to secondary users spectrum bandwidth amount $y$ in Epoch II, and define the feasible region of seller 2's strategy as $\\mathcal {F}_{2}\\left(z\\right) = \\left\\lbrace \\lbrace d_{n,2} |n\\in \\mathcal {N}_{\\text{II}} \\rbrace \\Big |\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n,2} \\le z, d_{n,2} \\ge 0\\right\\rbrace $ when seller 2 would like to lease to secondary users spectrum bandwidth amount $z$ in Epoch II.",
"The objective of seller 1 is to solve the following optimization problem Problem 2 $\\begin{array}{cll}\\mathop {\\max } \\limits _{\\mathcal {S}_{1}} & {R}_{1} \\left(\\mathcal {S}_{1}, \\mathcal {S}_{2} \\right) \\\\\\text{s.t.}",
"& \\mathcal {S}_{1} \\in \\mathcal {F}_{1}\\left(Q_1 - Q_{1}^{\\text{III}}\\right)\\end{array}$ and the objective of seller 2 is to solve the following optimization problem Problem 3 $\\begin{array}{cll}\\mathop {\\max } \\limits _{\\mathcal {S}_{2}} & {R}_{2} \\left(\\mathcal {S}_{1}, \\mathcal {S}_{2} \\right) \\\\\\text{s.t.}",
"& \\mathcal {S}_{2} \\in \\mathcal {F}_{2}\\left(Q_{2}^{\\text{II}}\\right).\\end{array}$ For the non-cooperative game of the two sellers, a Nash equilibrium defines a strategy pair $(\\mathcal {S}_{1}, \\mathcal {S}_{2} )$ that a seller cannot earn more revenue by deviating from its strategy while keeping the other seller's strategy unchanged.",
"In other words, a Nash equilibrium should be a joint optimal solution of Problem REF and Problem REF .",
"Since the objective functions of Problem REF and Problem REF are continuous and concave, and the feasible regions of the two sellers' strategies are convex, closed, bounded, and uncoupledWhen the two feasible regions are independent from each other, we say that the two feasible regions are uncoupled., there exists at least one Nash equilibrium [16]."
],
[
"Spectrum Leasing Problem in Epoch III",
"In Epoch III, only seller 1 is active in the spectrum market, and thus, monopoly spectrum leasing is performed.",
"Once an amount of spectrum bandwidth is leased to secondary users, the spectrum can be used by secondary users until the end of Epoch III.",
"To maximize the revenue of seller 1 in Epoch III, the following optimization problem should be solved.",
"Problem 4 $ \\begin{array}{cll}V\\left(Q_{1}^{\\text{III}}\\right) \\triangleq \\mathop {\\max } \\limits _{\\lbrace d_{n, 1}|n \\in \\mathcal {N}_{\\text{III}}\\rbrace } & \\sum \\limits _{n \\in \\mathcal {N}_{\\text{III}}} { \\left(C_0-C_1 d_{n, 1}\\right) d_{n,1} n} \\\\\\text{s.t.}",
"& \\sum \\limits _{n \\in \\mathcal {N}_{\\text{III}}} d_{n, 1} \\le Q_{1}^{\\text{III}} \\\\& d_{n,1} \\ge 0, \\forall n \\in \\mathcal {N}_{\\text{III}}.\\end{array}$ It can be seen that Problem REF is a convex optimization problem, and thus, can be solved by existing numerical optimization methods.",
"From the perspective of seller 1, it can adjust $Q_{1}^{\\text{III}}$ .",
"For a specific $Q_{1}^{\\text{III}}$ , the two sellers need to follow a Nash equilibrium in the non-cooperative game in Epoch II.",
"Thus, the strategy of seller 1 can be written as $Q_{1}^{\\text{III}}$ and $\\mathcal {S}_{1}$ , while the strategy of seller 2 can be written as $\\mathcal {S}_{2}$ .",
"When seller 1 reserves spectrum bandwidth $Q_{1}^{\\text{III}}$ for Epoch III, it means that seller 1 would like to lease spectrum bandwidth ($Q_1-Q_{1}^{\\text{III}}$ ) in Epoch II.",
"Accordingly, we denote the revenue of seller 1 in Epoch II as $U(Q_1-Q_{1}^{\\text{III}})$ , a function of ($Q_1-Q_{1}^{\\text{III}}$ ).",
"Then for seller 1 to maximize its overall revenue, the following optimization problem should be solved Problem 5 $\\begin{array}{cll}\\mathop {\\max } \\limits _{Q_{1}^{\\text{III}}} & U\\left(Q_1 - Q_{1}^{\\text{III}}\\right) + V\\left(Q_{1}^{\\text{III}}\\right) \\\\\\text{s.t.}",
"& 0 \\le Q_{1}^{\\text{III}} \\le Q_1.\\\\\\end{array}$ In the following, in Section we find out Nash equilibria in Epoch II for a specific $Q_{1}^{\\text{III}}$ , and in Section we select the value of $Q_{1}^{\\text{III}}$ for seller 1."
],
[
"Uniqueness of Nash Equilibrium in the Non-Cooperative Game in Epoch II when $|\\mathcal {N}_{\\text{III}}| \\le 12$",
"We have the following theorem.",
"Theorem 1 When $|\\mathcal {N}_{\\text{III}}| \\le 12$ , there is only one Nash equilibrium for the non-cooperative game in Epoch II.",
"Define the vectorized strategy of seller 1 and seller 2 in Epoch II as $\\mathbf {x}_1=[d_{|\\mathcal {N}_{\\text{III}}|+ |\\mathcal {N}_{\\text{II}}|, 1},$ $d_{|\\mathcal {N}_{\\text{III}}|+ \\left(|\\mathcal {N}_{\\text{II}}|-1\\right), 1},$ $...,$ $d_{|\\mathcal {N}_{\\text{III}}|+ 1, 1}]^{T}$ and $\\mathbf {x}_2=\\left[d_{|\\mathcal {N}_{\\text{III}}|+ |\\mathcal {N}_{\\text{II}}|, 2}, d_{|\\mathcal {N}_{\\text{III}}|+ \\left(|\\mathcal {N}_{\\text{II}}|-1\\right), 2}, ..., d_{|\\mathcal {N}_{\\text{III}}|+ 1, 2}\\right]^{T}$ , respectively, in which $[\\cdot ]^T$ means transpose operation.",
"The payoff function of seller 1 and seller 2 can be rewritten as $R_1\\left(\\mathcal {S}_1, \\mathcal {S}_2\\right)=R_1\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)$ and $R_2\\left(\\mathcal {S}_1, \\mathcal {S}_2\\right)=R_2\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)$ , respectively.",
"Denote $\\mathbf {x}=\\left(\\mathbf {x}_1^T, \\mathbf {x}_2^T\\right)^T$ and define $\\sigma (\\mathbf {x})= R_1\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right) + R_2\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right).$ Then the pseudo-gradient of $\\sigma (\\mathbf {x})$ can be given as $\\mathbf {k}\\left(\\mathbf {x}\\right)=\\left[\\begin{array}{c}\\nabla _1 R_1\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right) \\\\ \\nabla _2 R_2\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)\\end{array}\\right]$ where $|\\mathcal {N}_{\\text{II}}|\\times 1$ matrix $\\nabla _1 R_1\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)$ is the gradient of $R_1\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)$ with respect to vector $\\mathbf {x}_1$ , and $|\\mathcal {N}_{\\text{II}}|\\times 1$ matrix $\\nabla _2 R_2\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)$ is the gradient of $R_2\\left(\\mathbf {x}_1, \\mathbf {x}_2\\right)$ with respect to vector $\\mathbf {x}_2$ .",
"According to Theorem 2 and Theorem 6 of [16], the Nash equilibrium of the non-cooperative game in Epoch II is unique if the $2|\\mathcal {N}_{\\text{II}}|\\times 2|\\mathcal {N}_{\\text{II}}|$ symmetric matrix $\\mathbf {L}(\\mathbf {x})=-\\left[\\mathbf {K}(\\mathbf {x})+\\mathbf {K}^T(\\mathbf {x})\\right]$ is positive definite, where $\\mathbf {K}(\\mathbf {x})$ is the Jacobian of $\\mathbf {k}(\\mathbf {x})$ with respect to $\\mathbf {x}$ .",
"After some math manipulation, the matrix $\\mathbf {L}(\\mathbf {x})$ can be written as the following form $\\mathbf {L}(\\mathbf {x}) = \\left[\\begin{array}{c}\\mathbf {L}_{11}(\\mathbf {x}) \\\\ \\mathbf {L}_{21}(\\mathbf {x})\\end{array} \\ \\begin{array}{c}\\mathbf {L}_{12}(\\mathbf {x}) \\\\ \\mathbf {L}_{22}(\\mathbf {x})\\end{array}\\right]$ where $\\mathbf {L}_{11}(\\mathbf {x})=\\text{Diag}\\big (4C_1(|\\mathcal {N}_{\\text{III}}|+|\\mathcal {N}_{\\text{II}}|), 4C_1(|\\mathcal {N}_{\\text{III}}|+|\\mathcal {N}_{\\text{II}}|-1), ..., 4C_1(|\\mathcal {N}_{\\text{III}}|+1)\\big )$ , $\\mathbf {L}_{12}(\\mathbf {x})=\\mathbf {L}_{21}(\\mathbf {x})=\\text{Diag}\\big (C_1(|\\mathcal {N}_{\\text{III}}|+2|\\mathcal {N}_{\\text{II}}|), C_1(|\\mathcal {N}_{\\text{III}}|+2|\\mathcal {N}_{\\text{II}}|-2), ..., C_1(|\\mathcal {N}_{\\text{III}}|+2)\\big )$ , and $\\mathbf {L}_{22}(\\mathbf {x})=\\text{Diag}\\big (4C_1|\\mathcal {N}_{\\text{II}}|, 4C_1(|\\mathcal {N}_{\\text{II}}|-1), ..., 4C_1\\big )$ .",
"Here $\\text{Diag}(\\cdot \\cdot \\cdot )$ means a diagonal matrix with all diagonal elements listed in $(\\cdot \\cdot \\cdot )$ .",
"The matrix $\\mathbf {L}(\\mathbf {x})$ can be guaranteed to be positive definite, if the leading principal minors are all positive [17], i.e., the determinant of $m\\times m$ upper-left submatrix of $\\mathbf {L}(\\mathbf {x})$ is larger than 0 for $m=1, 2, ..., 2|\\mathcal {N}_{\\text{II}}|$ .",
"Since there is $\\text{Det} \\left(\\left[\\begin{array}{c}\\mathbf {A} \\\\ \\mathbf {C}\\end{array} \\ \\begin{array}{c}\\mathbf {B} \\\\ \\mathbf {D}\\end{array}\\right] \\right)= \\text{Det}\\left(\\mathbf {A}\\right)\\text{Det}\\left(\\mathbf {D} - \\mathbf {C} \\mathbf {A}^{-1} \\mathbf {B}\\right)$ when matrix $\\mathbf {A}$ is invertible [18], the determinant of $m\\times m$ upper-left submatrix of $\\mathbf {L}(\\mathbf {x})$ is larger than 0 for $m=1, 2, ..., 2|\\mathcal {N}_{\\text{II}}|$ when the following inequalities hold $12\\left(|\\mathcal {N}_{\\text{II}}|-k\\right)^2+ 12 |\\mathcal {N}_{\\text{III}}|\\left(|\\mathcal {N}_{\\text{II}}| - k\\right) - |\\mathcal {N}_{\\text{III}}|^2 > 0, \\forall k=0, 1, ..., \\left(|\\mathcal {N}_{\\text{II}}|-1\\right),$ i.e., when $ \\frac{\\left(|\\mathcal {N}_{\\text{II}}|-k\\right)}{|\\mathcal {N}_{\\text{III}}|} > \\left(-\\frac{1}{2}+ \\frac{1}{\\sqrt{3}}\\right), \\forall k \\in 0, 1, ..., \\left(|\\mathcal {N}_{\\text{II}}|-1\\right).$ The inequalities in (REF ) hold if $|\\mathcal {N}_{\\text{III}}|< \\frac{1}{-\\frac{1}{2}+ \\frac{1}{\\sqrt{3}}}=12.9282.$ This completes the proof.",
"As the number of stages in Epoch III is normally limited, it is very likely that the value of $|\\mathcal {N}_{\\text{III}}|$ is bounded by 12, and thus, Nash equilibrium of the non-cooperative game in Epoch II is unique.",
"Nevertheless, in next subsection, we show how to find Nash equilibria in the non-cooperative game in Epoch II without constraint $|\\mathcal {N}_{\\text{III}}|\\le 12$ (i.e., Nash equilibrium may or may not be unique)."
],
[
"Finding Nash Equilibria in the Non-Cooperative Game in Epoch II",
"As aforementioned, a Nash equilibrium of the non-cooperative game in Epoch II is a joint optimal solution of Problem REF and Problem REF .",
"As both Problem REF and Problem REF are convex problems and satisfy the Slater's condition, KKT condition is a sufficient and necessary condition for optimal solution for each problem [19], [20], [21].",
"For the ease of presentation, we denote $Q_{1}^{\\text{II}_c} = Q_1 - Q_{1}^{\\text{III}}$ as the spectrum bandwidth amount that seller 1 would like to lease to secondary users in Epoch II.",
"For Problem REF , the KKT condition is $2 C_1 n d_{n,1} \\!",
"-\\!",
"\\left(C_0 \\!-\\!",
"C_1 d_{n,2}\\right) n + \\lambda -\\mu _n =0, ~\\forall n\\in \\mathcal {N}_{\\text{II}} \\\\\\lambda \\left(\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 1} - Q_{1}^{\\text{II}_c} \\right)=0 \\\\\\mu _n d_{n,1}=0, ~\\forall n\\in \\mathcal {N}_{\\text{II}} \\\\\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 1} \\le Q_{1}^{\\text{II}_c} \\\\d_{n,1} \\ge 0, \\forall n \\in \\mathcal {N}_{\\text{II}} \\\\\\lambda \\ge 0; \\mu _n \\ge 0, ~\\forall n\\in \\mathcal {N}_{\\text{II}}$ where $\\lambda $ and $\\mu _n$ are Lagrange multipliers associated with the constraints $\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 1} \\le Q_{1}^{\\text{II}_c}$ and $d_{n,1} \\ge 0$ , respectively.",
"For Problem REF , the KKT condition is $2 C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)d_{n,2} - \\left(C_0 - C_1 d_{n,1}\\right) \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) \\\\+ \\zeta -\\nu _n =0, ~\\forall n\\in \\mathcal {N}_{\\text{II}} \\\\\\zeta \\left(\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 2} - Q_{2}^{\\text{II}} \\right)=0 \\\\\\nu _n d_{n,2}=0, ~\\forall n\\in \\mathcal {N}_{\\text{II}} \\\\\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 2} \\le Q_{2}^{\\text{II}} \\\\d_{n,2} \\ge 0, \\forall n \\in \\mathcal {N}_{\\text{II}} \\\\\\zeta \\ge 0; \\nu _n \\ge 0, ~\\forall n\\in \\mathcal {N}_{\\text{II}}$ where $\\zeta $ and $\\nu _n$ are Lagrange multipliers associated with the constraints $\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 2} \\le Q_{2}^{\\text{II}}$ and $d_{n,2} \\ge 0$ , respectively.",
"To get Nash equilibrium of the non-cooperative game in Epoch II, the equations (REF ) and (REF ) should be solved jointly.",
"We have two properties for the joint optimal solution: Property 1: Equality should hold in () and () (in other words, we have $\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 1} = Q_{1}^{\\text{II}_c}$ and $\\sum \\limits _{n \\in \\mathcal {N}_{\\text{II}}} d_{n, 2} = Q_{2}^{\\text{II}}$ ).",
"Property 2: If $d_{n,1}>0$ ($n \\in \\mathcal {N}_{\\text{II}}$ ), then we have $\\mu _n=0$ ; if $d_{n,2}>0$ , then we have $\\nu _{n}=0$ .",
"Property 1 is due to the facts that the objective function of Problem REF is a monotonically increasing function of $d_{n,1}$ ($n\\in \\mathcal {N}_{\\text{II}}$ ) and that the objective function of Problem REF is a monotonically increasing function of $d_{n,2}$ ($n\\in \\mathcal {N}_{\\text{II}}$ ).",
"Property 2 can be obtained directly from the equalities () and ().",
"Next, we try to find the expressions of $d_{n,1}$ and $d_{n,2}$ by solving (REF ) and (REF ).",
"From the equalities (REF ) and (REF ), $d_{n,1}$ and $d_{n,2}$ for $n \\in \\mathcal {N}_{\\text{II}}$ can be expressed as $ d_{n,1}= \\frac{\\left(C_0 -C_1 d_{n,2}\\right) n -\\lambda + \\mu _n }{2 C_1 n },$ $ d_{n,2} = \\frac{\\left(C_0 -C_1 d_{n,1}\\right) \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) - \\zeta + \\nu _n }{2C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)},$ from which we have $ d_{n,1} = \\frac{2\\left(C_0 n - \\lambda + \\mu _n \\right)}{3C_1n} -\\frac{C_0 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) -\\zeta + \\nu _n }{3 C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)},$ $ d_{n,2} = -\\frac{C_0 n - \\lambda + \\mu _n }{3C_1n} + \\frac{2 \\left( C_0 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) -\\zeta + \\nu _n \\right) }{3 C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)}.$ Define $\\mathcal {Z}_{1}=\\lbrace n| d_{n,1}>0, d_{n,2}>0, n \\in \\mathcal {N}_{\\text{II}}\\rbrace $ , $\\mathcal {Z}_{2}=\\lbrace n| d_{n,1}>0, d_{n,2}=0, n\\in \\mathcal {N}_{\\text{II}}\\rbrace $ , $\\mathcal {Z}_{3}=\\lbrace n| d_{n,1}=0, d_{n,2}>0, n\\in \\mathcal {N}_{\\text{II}}\\rbrace $ and $\\mathcal {Z}_{4}=\\lbrace n| d_{n,1}=0, d_{n,2}=0, n\\in \\mathcal {N}_{\\text{II}}\\rbrace $ .",
"Then {$\\mathcal {Z}_1$ , $\\mathcal {Z}_2$ , $\\mathcal {Z}_3$ , $\\mathcal {Z}_4$ } constitutes a decomposition of the set $\\mathcal {N}_{\\text{II}}$ , which means that $\\mathcal {Z}_1\\bigcup \\mathcal {Z}_2\\bigcup \\mathcal {Z}_3\\bigcup \\mathcal {Z}_4=\\mathcal {N}_{\\text{II}}$ and $\\mathcal {Z}_i\\bigcap \\mathcal {Z}_j= \\emptyset $ for $i \\ne j$ and $i,j \\in \\lbrace 1,2,3,4\\rbrace $ .",
"Totally there are $2^{2|\\mathcal {N}_\\text{II}|}$ decompositions.",
"Next we find out the expressions of $d_{n,1}$ and $d_{n,2}$ for a specific decomposition {$\\mathcal {Z}_1$ , $\\mathcal {Z}_2$ , $\\mathcal {Z}_3$ , $\\mathcal {Z}_4$ }.",
"From Property 1, we have $\\sum _{n\\in \\mathcal {Z}_1} d_{n,1} + \\sum _{n\\in \\mathcal {Z}_2} d_{n,1} = Q_{1}^{\\text{II}_c},$ $\\sum _{n\\in \\mathcal {Z}_1} d_{n,2} + \\sum _{n\\in \\mathcal {Z}_3} d_{n,2} = Q_{2}^{\\text{II}}.$ In the two equations, substituting the expressions of $d_{n,1}$ and $d_{n,2}$ in (REF ) and (REF ) for $n \\in \\mathcal {Z}_1$ , substituting the expressions of $d_{n,1}$ and $d_{n,2}$ in (REF ) and (REF ) for $n \\in \\mathcal {Z}_2$ and $n\\in \\mathcal {Z}_3$ , and using Property 2, we have the following equations: $ \\begin{array}{ll}& -A_{11} \\lambda + A_{12} \\zeta = Q_1^{\\text{II}_c} - \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{C_0}{3C_1} - \\sum \\limits _{n \\in \\mathcal {Z}_2} \\frac{C_0}{2C_1} \\\\& A_{21} \\lambda - A_{22} \\zeta = Q_2^{\\text{II}} - \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{C_0}{3C_1} - \\sum \\limits _{n \\in \\mathcal {Z}_3} \\frac{C_0}{2C_1} \\\\\\end{array}$ where $ A_{11} = \\sum \\limits _{n\\in \\mathcal {Z}_1} \\frac{2}{3C_1 n} + \\sum \\limits _{n \\in \\mathcal {Z}_2} \\frac{1}{2C_1 n},$ $ A_{12} = \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{1}{3 C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)},$ $ A_{21} = \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{1}{3 C_1 n},$ $ A_{22} = \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{2}{3C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)} + \\sum \\limits _{n \\in \\mathcal {Z}_3} \\frac{1}{2C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)}.$ Note that $A_{11}$ , $A_{12}$ , $A_{21}$ and $A_{22}$ are all larger than zero.",
"According to the equations in (REF ), the Lagrange multipliers $\\lambda $ and $\\zeta $ can be expressed as $ \\begin{array}{lr}\\lambda =& -\\frac{A_{22}}{A_{11}A_{22} - A_{21}A_{12}} \\left(Q_1^{\\text{II}_c} - \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{C_0}{3C_1} - \\sum \\limits _{n \\in \\mathcal {Z}_2} \\frac{C_0}{2C_1}\\right) \\\\& - \\frac{A_{12}}{A_{11}A_{22} - A_{21}A_{12}} \\left( Q_2^{\\text{II}} - \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{C_0}{3C_1} - \\sum \\limits _{n \\in \\mathcal {Z}_3} \\frac{C_0}{2C_1}\\right),\\end{array}$ $ \\begin{array}{lr}\\zeta =& -\\frac{A_{21}}{A_{11}A_{22} - A_{21}A_{12}} \\left(Q_1^{\\text{II}_c} - \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{C_0}{3C_1} - \\sum \\limits _{n \\in \\mathcal {Z}_2} \\frac{C_0}{2C_1}\\right) \\\\& - \\frac{A_{11}}{A_{11}A_{22} - A_{21}A_{12}} \\left( Q_2^{\\text{II}} - \\sum \\limits _{n \\in \\mathcal {Z}_1} \\frac{C_0}{3C_1} - \\sum \\limits _{n \\in \\mathcal {Z}_3} \\frac{C_0}{2C_1}\\right).\\end{array}$ With the aid of Property 2 and using equations (REF ), (REF ), (REF ), and (REF ), the closed-form expressions of $d_{n,1}$ and $d_{n,2}$ for $n \\in \\mathcal {N}_{\\text{II}}$ are given as follows: $ d_{n,1}={\\left\\lbrace \\begin{array}{ll}\\frac{2\\left(C_0 n - \\lambda \\right)}{3C_1 n} -\\frac{C_0 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) -\\zeta }{3 C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)} &\\text{if } n \\in \\mathcal {Z}_1 \\\\\\frac{C_0 n -\\lambda }{2 C_1 n} &\\text{if } n \\in \\mathcal {Z}_2 \\\\0 &\\text{if } n \\in \\mathcal {Z}_3 \\bigcup \\mathcal {Z}_4\\\\\\end{array}\\right.",
"}$ $ d_{n,2}={\\left\\lbrace \\begin{array}{ll}-\\frac{C_0 n - \\lambda }{3C_1 n} + \\frac{2 \\left( C_0 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) -\\zeta \\right) }{3 C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)} &\\text{if } n \\in \\mathcal {Z}_{1} \\\\\\frac{C_0 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right) - \\zeta }{2C_1 \\left(n - |\\mathcal {N}_{\\text{III}}|\\right)} &\\text{if } n \\in \\mathcal {Z}_3 \\\\0 &\\text{if } n \\in \\mathcal {Z}_2 \\bigcup \\mathcal {Z}_4\\\\\\end{array}\\right.",
"}$ where $\\lambda $ and $\\zeta $ are given in (REF ) and (REF ), respectively.",
"By now, given the decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ , expressions of $d_{n,1}$ and $d_{n,2}$ for $n \\in \\mathcal {N}_{\\text{II}}$ are derived.",
"To guarantee that every equality or inequality in (REF ) and (REF ) is satisfied, a feasibility check is further required, which is given as follows: $\\lambda $ and $\\zeta $ , which can be calculated from (REF ) and (REF ), are non-negative.",
"$d_{n,1}$ and $d_{n,2}$ , which are calculated from (REF ) and (REF ), are non-negative for $n\\in \\mathcal {N}_{\\text{II}}$ .",
"$\\mu _n$ and $\\nu _n$ , which can be calculated from (REF ) and (REF ) given the obtained $d_{n,1}$ , $d_{n,2}$ , $\\lambda $ and $\\zeta $ , are non-negative for $n \\in \\mathcal {N}_{\\text{II}}$ .",
"If the above feasibility check passes, the decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is said to be feasible, and the derived $d_{n,1}$ and $d_{n,2}$ expressions in (REF ) and (REF ) for $n \\in \\mathcal {N}_{\\text{II}}$ given the decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is a Nash equilibrium of the non-cooperative game in Epoch II.",
"For the set $\\mathcal {N}_{\\text{II}}$ , there are $2^{2|\\mathcal {N}_\\text{II}|}$ possible decompositions.",
"To find all Nash equilibria of the game in Epoch II, an exhaustive search of all $2^{2|\\mathcal {N}_\\text{II}|}$ decompositions is required.",
"As the number of stages in Epoch II is normally very limited, and the calculations in checking feasibility of each decomposition are simple, an exhaustive search of all $2^{2|\\mathcal {N}_\\text{II}|}$ decompositions is considered to be acceptable.",
"In addition, the following theorem is helpful in reducing the complexity in the exhaustive search.",
"Theorem 2 For a feasible decomposition, if there exists a stage (say stage $n$ ) in $\\mathcal {Z}_4$ (i.e., $d_{n,1}=d_{n,2}=0$ ), then all stages with a lower index in Epoch II should belong to $\\mathcal {Z}_4$ .",
"We use the proof by contradiction.",
"In the Nash equilibrium of the decomposition, suppose there is $n\\dag $ satisfying $n\\dag < n, n\\dag \\in \\mathcal {N}_{\\text{II}} \\backslash \\mathcal {Z}_4$ .",
"We first assume that $n\\dag \\in \\mathcal {Z}_2$ , which indicates that $d_{n\\dag ,1}>0, d_{n\\dag ,2}=0$ .",
"Then the total revenue collected over stage $n$ and stage $n\\dag $ by seller 1 is $\\left(C_0 -C_1 d_{n\\dag ,1}\\right)d_{n\\dag ,1}n{\\dag }$ .",
"By interchanging seller 1's offered spectrum bandwidth amounts in stage $n$ and stage $n\\dag $ , the total revenue that seller 1 collects in stages $n$ and $n\\dag $ becomes $\\left(C_0 -C_1 d_{n\\dag ,1}\\right)d_{n\\dag ,1}n$ , which is larger than $\\left(C_0 -C_1 d_{n\\dag ,1}\\right)d_{n\\dag ,1}n\\dag $ since $n\\dag < n$ .",
"This contradicts the definition of Nash equilibrium.",
"Similarly, $n\\dag \\in \\mathcal {Z}_1$ or $n\\dag \\in \\mathcal {Z}_3$ also leads to a contradiction.",
"This completes the proof.",
"Remark: Theorem REF shows that in a feasible decomposition, if $\\mathcal {Z}_4$ is not empty, then it contains consecutive stages until the end of Epoch II.",
"Therefore, in the exhaustive search of all possible decompositions, we can skip those decompositions in which $\\mathcal {Z}_4$ contains non-consecutive stages or does not last until the end of Epoch II.",
"Thus, the number of decompositions that we should check reduces from $2^{|2\\mathcal {N}_{\\text{II}}|}$ to $\\sum _{i=0}^{|\\mathcal {N}_{\\text{II}}|} 3^i$ .",
"So far all Nash equilibria of the non-cooperative game in Epoch II have been found.",
"If there exists only one unique Nash equilibrium (e.g., when $|\\mathcal {N}_{\\text{III}}| \\le 12$ ), then both sellers follow the unique Nash equilibrium.",
"If there are two or more Nash equilibria, the two sellers need to select one Nash equilibrium to follow.",
"Here it is assumed that the two sellers agree to follow the Nash equilibrium that maximizes the minimum unit-bandwidth revenue of the two sellers.",
"Here for seller 1, its unit-bandwidth revenue is the ratio of its total revenue in Epoch II to $Q_1^{\\text{II}_c}$ ; for seller 2, its unit-bandwidth revenue is the ratio of its total revenue in Epoch II to $Q_2^{\\text{II}}$ ."
],
[
"Total Revenue Maximization for seller 1",
"In the previous section, we have found the strategies of the two sellers in Epoch II with a specific $Q_{1}^{\\text{III}}$ (the bandwidth that seller 1 reserves for Epoch III).",
"Now, we try to solve Problem REF , i.e., find out the optimal value of $Q_{1}^{\\text{III}}$ that maximizes seller 1's total revenue.",
"A method could be: 1) for each possible value of $Q_{1}^{\\text{III}}$ , search all possible Nash equilibria, find the Nash equilibrium that maximizes the minimum unit-bandwidth revenue of the two sellers, and calculate the revenue that seller 1 can earn during its leasing period; 2) compare the revenue values that seller 1 can earn during its leasing period with different $Q_{1}^{\\text{III}}$ , and select the optimal $Q_{1}^{\\text{III}}$ that makes seller 1 earn the most revenue.",
"However, the complexity of the method is huge, due to the infinite number of values of $Q_{1}^{\\text{III}}\\in [0, Q_1]$ .",
"Thus, we target at an approximation method to select $Q_{1}^{\\text{III}}$ .",
"When $Q_{1}^{\\text{III}}=x$ , $U(Q_1-x)$ and $V(x)$ given in (REF ) are the revenue of seller 1 in Epoch II and Epoch III, respectively.",
"To select $x$ (i.e., $Q_{1}^{\\text{III}}$ ), we need to evaluate how $V\\left(x\\right)$ and $U\\left(Q_1-x\\right)$ change when $x$ varies.",
"Lemma 1 The function $V(x)$ is an increasing and concave function with $x$ .",
"The proof follows a similar procedure to the proof of Lemma 6 of [22].",
"Now we evaluate function $U(Q_1-x)$ when $x$ varies.",
"To evaluate $U(Q_1-x)$ for a specific decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ , we need to know $d_{n,1}$ and $d_{n,2}$ ($n\\in \\mathcal {N}_\\text{II}$ ) in the Nash equilibrium corresponding to the decomposition.",
"Therefore, next we show how $d_{n,1}$ and $d_{n,2}$ change when $x$ varies.",
"Consider a decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ .",
"Consider two $Q_{1}^{\\text{II}_c}$ values (recalling that $Q_{1}^{\\text{II}_c} = Q_1 - Q_{1}^{\\text{III}}$ ): $Q^{\\dag }$ and $Q^{\\ddag }$ , with $Q^{\\dag } \\le Q^{\\ddag }$ .",
"Assume the decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ is feasible for both $Q_{1}^{\\text{II}_c} $ values.",
"For the decomposition, denote the corresponding Nash equilibrium when $Q_{1}^{\\text{II}_c} =Q^{\\dag }$ as $\\left(\\mathcal {S}_1^{\\dag }, \\mathcal {S}_2^{\\dag }\\right) \\triangleq \\left(\\left\\lbrace d_{n,1}^{\\dag }|n \\in \\mathcal {N}_{\\text{II}}\\right\\rbrace , \\left\\lbrace d_{n,2}^{\\dag }|n \\in \\mathcal {N}_{\\text{II}}\\right\\rbrace \\right),$ and the corresponding Nash equilibrium when $Q_{1}^{\\text{II}_c} =Q^{\\ddag }$ as $\\left(\\mathcal {S}_1^{\\ddag }, \\mathcal {S}_2^{\\ddag }\\right) \\triangleq \\left(\\left\\lbrace d_{n,1}^{\\ddag }|n \\in \\mathcal {N}_{\\text{II}}\\right\\rbrace , \\left\\lbrace d_{n,2}^{\\ddag }|n \\in \\mathcal {N}_{\\text{II}}\\right\\rbrace \\right).$ Then the following lemmas can be expected.",
"Lemma 2 For seller 1, $d_{n,1}^{\\dag } \\le d_{n,1}^{\\ddag }$ for $n \\in \\mathcal {Z}_2$ , and $d_{n,1}^{\\dag } = d_{n,1}^{\\ddag }=0$ for $n \\in \\mathcal {Z}_3 \\bigcup \\mathcal {Z}_4$ .",
"By the definitions of set $\\mathcal {Z}_3$ and $\\mathcal {Z}_4$ , seller 1 does not offer spectrum bandwidth to be leased in stages in $\\mathcal {Z}_3$ and $\\mathcal {Z}_4$ , and thus, $d_{n,1}^{\\dag } = d_{n,1}^{\\ddag }=0$ for $n \\in \\mathcal {Z}_3 \\bigcup \\mathcal {Z}_4$ .",
"For $n \\in \\mathcal {Z}_2$ , with the aid of (REF ) and (REF ), we have $\\begin{array}{lll}d_{n, 1}^{\\dag } - d_{n, 1}^{\\ddag } &= \\frac{C_0 A_{22}}{2C_1 n \\left(A_{11}A_{22} - A_{21}A_{12}\\right)} \\left(Q^{\\dag } -Q^{\\ddag }\\right) \\\\& \\le 0\\end{array}$ in which the inequality comes from $A_{22} \\ge 0$ , $Q^{\\dag } \\le Q^{\\ddag }$ , and $\\left(A_{11}A_{22} - A_{21}A_{12}\\right) >0$ according to (REF ), (REF ), (REF ), and (REF ).",
"This completes the proof.",
"Lemma 3 For seller 2, $d_{n,2}^{\\dag } \\le d_{n,2}^{\\ddag }$ for $n \\in \\mathcal {Z}_3$ , and $d_{n,2}^{\\dag } = d_{n,2}^{\\ddag }=0$ for $n \\in \\mathcal {Z}_2 \\bigcup \\mathcal {Z}_4$ .",
"The proof is similar to the proof for Lemma REF , and thus, is omitted here.",
"Theorem 3 If a decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ is feasible when $Q_{1}^{\\text{III}}=x \\in \\mathcal {I}$ where $\\mathcal {I} \\subseteq [0,Q_1] $ is an interval, then when the Nash equilibrium corresponding to the decomposition is followed by the two sellers in Epoch II, seller 1's revenue $U(Q_1-x)$ in Epoch II can be written as $U(Q_1-x)=G(x)-H(x)$ where $G(x)$ and $H(x)$ are monotonically increasing functions with respect to $x\\in \\mathcal {I}$ .",
"Suppose the Nash equilibrium corresponding to the decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ is $(\\left\\lbrace d_{n,1}|n \\in \\mathcal {N}_{\\text{II}}\\right\\rbrace $ , $\\left\\lbrace d_{n,2}|n \\in \\mathcal {N}_{\\text{II}}\\right\\rbrace )$ .",
"Then $U(Q_1-x)$ can be written as $ \\begin{array}{l}U(Q_1-x) \\\\=\\sum \\limits _{n\\in \\mathcal {N}_\\text{II}} \\left(C_0-C_1 \\left(d_{n,1} + d_{n,2} \\right)\\right) d_{n,1} n \\\\\\overset{(a)}{=} \\sum \\limits _{n \\in \\mathcal {Z}_1} \\left(C_0-C_1 \\left(d_{n,1} + d_{n,2} \\right)\\right) d_{n,1} n \\\\~~~~~~~~ + \\sum \\limits _{n \\in \\mathcal {Z}_2} \\left(C_0-C_1 d_{n,1} \\right) d_{n,1} n \\\\\\overset{(b)}{=} \\sum \\limits _{n \\in \\mathcal {Z}_1} \\left(\\frac{C_0}{3} + \\frac{\\zeta }{3 \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)} + \\frac{\\lambda }{3 n}\\right) \\Big (\\frac{\\zeta }{3 C_1 \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)}-\\frac{2 \\lambda }{3 C_1 n}\\\\~~~~~~~~+\\frac{C_0}{3 C_1}\\Big ) n+ \\sum \\limits _{n \\in \\mathcal {Z}_2} \\left(C_0-C_1 d_{n,1} \\right) d_{n,1} n \\\\= \\sum \\limits _{n \\in \\mathcal {Z}_1} \\left(\\frac{\\zeta ^2}{9 C_1 \\left(n-|\\mathcal {N}_{\\text{III}}|\\right){}^2} +\\frac{2 C_0 \\zeta }{9 C_1 \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)}+\\frac{C_0^2}{9 C_1}\\right) n\\\\~~~~~~~~- {\\sum \\limits _{n \\in \\mathcal {Z}_1}} \\left(\\frac{\\zeta \\lambda }{9 C_1 n \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)} + \\frac{2 \\lambda ^2}{9 C_1 n^2} + \\frac{C_0 \\lambda }{9 C_1 n}\\right) n \\\\~~~~~~~~+ \\sum \\limits _{n \\in \\mathcal {Z}_2} \\left(C_0-C_1 d_{n,1} \\right) d_{n,1} n\\end{array}$ where $(a)$ holds since $d_{n,1}=0$ for $n \\in \\mathcal {Z}_3 \\bigcup \\mathcal {Z}_4$ and $d_{n,2}=0$ for $n \\in \\mathcal {Z}_2$ , and $(b)$ can be obtained by substituting $d_{n,1}$ and $d_{n,2}$ according to (REF ) and (REF ).",
"As the decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ is feasible, $\\lambda $ and $\\zeta $ are non-negative.",
"Additionally, from (REF ) and (REF ), it can be seen that $\\lambda $ and $\\zeta $ are monotonically decreasing with $Q_1^{\\text{II}_c}$ , i.e., $(Q_1-x)$ .",
"So in the expression (REF ), both the term $\\sum \\limits _{n \\in \\mathcal {Z}_1} \\left(\\frac{\\zeta ^2}{9 C_1 \\left(n-|\\mathcal {N}_{\\text{III}}|\\right){}^2} +\\frac{2 C_0 \\zeta }{9 C_1 \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)}+\\frac{C_0^2}{9 C_1}\\right) n$ and the term ${\\sum \\limits _{n \\in \\mathcal {Z}_1}} \\left(\\frac{\\zeta \\lambda }{9 C_1 n \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)} + \\frac{2 \\lambda ^2}{9 C_1 n^2} + \\frac{C_0 \\lambda }{9 C_1 n}\\right) n$ are monotonically decreasing with $Q_1^{\\text{II}_c}$ , and thus, are monotonically increasing with $x$ (as $Q_1^{\\text{II}_c}=Q_1-x$ ).",
"It can be also checked that the term $\\sum \\limits _{n \\in \\mathcal {Z}_2} \\left(C_0-C_1 d_{n,1} \\right) d_{n,1} n$ in (REF ) is a monotonically increasing function with respect to $Q_1^{\\text{II}_c}$ (since the function $\\left(C_0 -C_1y\\right)y$ is monotonically increasing with $y$ and $d_{n,1}$ grows with $Q_1^{\\text{II}_c}$ [from Lemma REF ]), and thus, is a monotonically decreasing function with respect to $x$ .",
"Define $G(x)= \\!",
"\\sum \\limits _{n \\in \\mathcal {Z}_1} \\!\\left(\\!\\frac{\\zeta ^2}{9 C_1 \\left(n\\!-\\!|\\mathcal {N}_{\\text{III}}|\\right){}^2} \\!+\\!\\frac{2 C_0 \\zeta }{9 C_1 \\left(n\\!-\\!|\\mathcal {N}_{\\text{III}}|\\right)}+\\frac{C_0^2}{9 C_1}\\right)\\!",
"n$ and $H(x) = {\\sum \\limits _{n \\in \\mathcal {Z}_1}} \\left(\\frac{\\zeta \\lambda }{9 C_1 n \\left(n-|\\mathcal {N}_{\\text{III}}|\\right)} + \\frac{2 \\lambda ^2}{9 C_1 n^2} + \\frac{C_0 \\lambda }{9 C_1 n}\\right) n - \\sum \\limits _{n \\in \\mathcal {Z}_2} \\left(C_0-C_1 d_{n,1} \\right) d_{n,1} n.$ It can be seen that $U(Q_1-x)=G(x)-H(x)$ , and both $G(x)$ and $H(x)$ monotonically increase with $x$ .",
"This completes the proof.",
"In Lemma REF , Lemma REF , and Theorem REF , it is assumed that the decomposition $\\left\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\right\\rbrace $ is feasible for $x=Q_1-Q^{\\dag }$ , $x=Q_1-Q^{\\ddag }$ or $x \\in \\mathcal {I}$ .",
"The next theorem will answer the following question: If a decomposition is feasible for a specific value of $x$ , will it continue to be feasible if $x$ increases or decreases?",
"Theorem 4 Assume a decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is feasible for $x=x_0\\in [0, Q_1]$ .",
"If $x$ increases from $x_0$ , then there exists a point denoted $x_1\\in [x_0, Q_1]$ such that the decomposition is always feasible in interval $[x_0,x_1]$ , and is always infeasible in interval $(x_1, Q_1]$ .",
"If $x$ decreases from $x_0$ , then there exists a point denoted $x_2\\in [0, x_0]$ such that the decomposition is always feasible in interval $[x_2,x_0]$ , and is always infeasible in interval $[0, x_2)$ .",
"Here we only prove the case when $x$ increases, as the case when $x$ decreases can be proved similarly.",
"For an $x$ (i.e., $Q_1^{\\text{III}}$ ) value, the feasibility of decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is checked as follows: calculate $\\lambda $ and $\\zeta $ based on (REF ) and (REF ), calculate $d_{n,1}$ and $d_{n,2}$ based on (REF ), (REF ), and the calculated $\\lambda $ and $\\zeta $ values, and calculate $\\mu _n$ and $\\nu _n$ based on (REF ), (REF ), and the calculated $d_{n,1}$ , $d_{n,2}$ , $\\lambda $ and $\\zeta $ values.",
"If all the values of $\\lambda $ , $\\zeta $ , $d_{n,1}$ , $d_{n,2}$ , $\\mu _n$ , and $\\nu _n$ ($n\\in \\mathcal {N}_\\text{II}$ ) are non-negative, then the decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is feasible; otherwise, it is infeasible.",
"Expressions (REF ) and (REF ) show that $\\lambda $ and $\\zeta $ are linear functions of $x$ (i.e., $Q_1^{\\text{III}}$ ).",
"Expressions (REF ) and (REF ) show that $d_{n,1}$ and $d_{n,2}$ are linear functions of $\\lambda $ and $\\zeta $ , and thus, are linear functions of $x$ .",
"Expressions (REF ) and (REF ) show that $\\mu _n$ and $\\nu _n$ are linear functions of $\\lambda $ , $\\zeta $ , $d_{n,1}$ , and $d_{n,2}$ , and thus, are linear functions of $x$ .",
"Overall, $\\lambda $ , $\\zeta $ , $d_{n,1}$ , $d_{n,2}$ , $\\mu _n$ , and $\\nu _n$ ($n\\in \\mathcal {N}_\\text{II}$ ) are all linear functions of $x$ (i.e., $Q_1^{\\text{III}}$ ).",
"When $x=x_0$ , as the decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is feasible, all the $\\lambda $ , $\\zeta $ , $d_{n,1}$ , $d_{n,2}$ , $\\mu _n$ , and $\\nu _n$ ($n\\in \\mathcal {N}_\\text{II}$ ) are non-negative.",
"When $x$ increases from $x$ , values of $\\lambda $ , $\\zeta $ , $d_{n,1}$ , $d_{n,2}$ , $\\mu _n$ , and $\\nu _n$ linearly change accordingly.",
"If at one point, say $x=x_1$ , one of $\\lambda $ , $\\zeta $ , $d_{n,1}$ , $d_{n,2}$ , $\\mu _n$ , and $\\nu _n$ decreases to value 0, then we can see that for $x\\in [x_0,x_1]$ , the decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is always feasible, and for $x\\in (x_1,Q_1]$ , the decomposition is always infeasible.As an extreme case, if $\\lambda $ , $\\zeta $ , $d_{n,1}$ , $d_{n,2}$ , $\\mu _n$ , and $\\nu _n$ all keep non-negative when $x$ increases from $x_0$ to $Q_1$ , then we have $x_1=Q_1$ .",
"This completes the proof.",
"Remark: Theorem REF shows that if a decomposition $\\lbrace \\mathcal {Z}_1, \\mathcal {Z}_2, \\mathcal {Z}_3, \\mathcal {Z}_4\\rbrace $ is feasible for $x=x_0$ , then there exists an interval of $x$ containing $x_0$ such that the decomposition is feasible inside the interval, and infeasible outside the interval.",
"Based on Lemma REF , Theorem REF , and Theorem REF , we propose that seller 1 uses the following Algorithm 1 to select $x$ (i.e., $Q_1^{\\text{III}}$ ).",
"[H] Searching procedure for $x$ (i.e., $Q_1^{\\text{III}}$ ).",
"[1] Set $x^*=0$ , and $R^*=0$ .",
"Set $x^\\dag =0$ For $x=x^\\dag $ , find out all feasible Nash equilibria, and pick up the Nash equilibrium that maximizes the minimal unit-bandwidth revenue of the two sellers.",
"Find (using bisection search) a point denoted $x_1$ such that the Nash equilibrium picked in Step 3 is feasible for $x\\in [x^\\dag , x_1]$ , and infeasible for $x\\in (x_1, Q_1]$ .",
"Set $x^\\ddag = x_1 $ .",
"The Nash equilibrium picked in Step 3 is feasible for $x\\in [x^\\dag , x^\\ddag ]$ .",
"For complexity reduction, approximately seller 1 considers that the Nash equilibrium picked in Step 3 is followed by both sellers when $x\\in [x^\\dag , x^\\ddag ]$ .",
"The revenue of seller 1 is $U(Q_1-x) + V(x)$ .",
"Here $U(Q_1-x)$ is the difference of two monotonically increasing functions of $x$ (from Theorem REF ), while $V(x)$ is an increasing function of $x$ (from Lemma REF ).",
"Thus, $U(Q_1-x) + V(x)$ can be viewed as the difference of two monotonically increasing functions of $x\\in [x^\\dag , x^\\ddag ]$ .",
"To maximize the difference of two monotonically increasing functions, a polyblock method can be used (please refer to [23], [24] for details).",
"Denote the optimal point as $\\hat{x}$ and the corresponding revenue $U(Q_1-\\hat{x}) + V(\\hat{x})$ of seller 1 as $\\hat{R}$ .",
"If $\\hat{R}>R^*$ , then set $x^*=\\hat{x}$ and $R^*=\\hat{R}$ .",
"If $x^\\ddag =Q_1$ , then terminate the algorithm, and output $x^*$ .",
"Set $x^\\dag =x^\\ddag $ , and proceed to Step 3.",
"Table: The number of decompositions without and with the aid of Theorem .In the algorithm, $x^*$ denotes the selection of seller 1 for $x$ , and $R^*$ denotes the corresponding overall revenue of seller 1.",
"For $x=x^\\dag =0$ , in Step 3 we first select the Nash equilibrium that maximizes the minimal unit-bandwidth revenue of the two sellers.",
"In Steps 4 and 5, we find the interval of $x$ , denoted $[x^\\dag , x^\\ddag ]$ , such that the selected Nash equilibrium is feasible inside the interval and infeasible when $x>x^\\ddag $ .",
"We approximately consider that the Nash equilibrium is followed by both sellers for the interval $x\\in [x^\\dag , x^\\ddag ]$ .If $|\\mathcal {N}_{\\text{III}}| \\le 12$ , then according to Theorem REF , the Nash equilibrium is the unique Nash equilibrium for $x\\in [x^\\dag , x^\\ddag ]$ , and thus, is always followed by both sellers when $x\\in [x^\\dag , x^\\ddag ]$ .",
"Then for $x\\in [x^\\dag , x^\\ddag ]$ , seller 1's revenue $U(Q_1-x) + V(x)$ can be shown as the difference of two monotonically increasing functions of $x$ .",
"Existing methods in the literature (such as a polyblock algorithm) can be used to find the optimal value of $x\\in [x^\\dag , x^\\ddag ]$ , denoted $\\hat{x}$ , such that the overall revenue of seller 1 is maximized.",
"Then the $\\hat{x}$ is a candidate for seller 1's selection of $x$ .",
"Then we set $x^\\dag =x^\\ddag $ in Step 9 and repeat the above procedure, and find other candidates for seller 1's selection of $x$ .",
"Among all the candidates, the one that has the maximal overall revenue of seller 1 is eventually selected by seller 1.",
"Overall, the strategies of the two sellers are as follows.",
"In Epoch I, seller 2 derives its optimal strategy by solving Problem REF .",
"At the beginning of Epoch II, seller 1 uses Algorithm 1 to find the value of $x$ , denoted $x^*$ , Then in the non-cooperative game in Epoch II with $Q_1^{\\text{III}}=x^*$ , both sellers follow the Nash equilibrium that maximizes the minimal unit-bandwidth revenue of the two sellers.",
"In Epoch III, seller 1 can derive its optimal strategy by solving Problem REF with $Q_1^{\\text{III}}=x^*$ ."
],
[
"Verification of the Analysis",
"We use numerical results by Matlab to verify the theoretical analysis in this paper.",
"Since the spectrum leasing problem in Epoch I and Epoch III are both convex optimization problems, here we focus on Epoch II.",
"At the beginning of Epoch II, seller 1 has spectrum bandwidth with amount $Q_1=100$ , while seller 2 has available spectrum bandwidth with amount $Q_2^{\\text{II}}=60$ .",
"We take $C_0=480$ and $C_1=1$ .",
"The number of stages in Epoch III is $|\\mathcal {N}_{\\text{III}}|=3$ .",
"In this subsection, the effectiveness of Theorem REF in complexity reduction is verified.",
"Table REF lists the number of all possible decompositions and the number of decompositions that should be checked for feasibility with the aid of Theorem REF .",
"It is clear that using Theorem REF can significantly reduce the number of decompositions that should be checked."
],
[
"Verification of Lemma ",
"In this subsection, Lemma REF is verified.",
"Fig.",
"REF plots the function $V(x)$ (the revenue of seller 1 in Epoch III) as $x$ (i.e., $Q_1^{\\text{III}}$ ) grows from 0 to 100.",
"From Fig.",
"REF , it can be seen that the function $V(x)$ is an increasing and concave function with respect to $x$ , which is consistent with Lemma REF .",
"Note that the reference line in Fig.",
"REF is a straight line connecting points $(0, V(0))$ and $(100, V(100))$ , which helps to observe the concavity of function $V(x)$ .",
"Figure: V(x)V(x) versus xx (i.e., Q 1 III Q_1^{\\text{III}}).Figure: d n,1 d_{n,1} and d n,2 d_{n,2} versus nn for 𝒵 1 ={7,8}\\mathcal {Z}_1=\\lbrace 7,8\\rbrace , 𝒵 2 ={6}\\mathcal {Z}_2=\\lbrace 6\\rbrace , 𝒵 3 =∅\\mathcal {Z}_3=\\emptyset , 𝒵 4 ={4,5}\\mathcal {Z}_4=\\lbrace 4,5\\rbrace , and Q 1 II c =70,80,90Q_1^{\\text{II}_c}=70, 80, 90.Figure: d n,1 d_{n,1} and d n,2 d_{n,2} versus nn for 𝒵 1 ={8}\\mathcal {Z}_1=\\lbrace 8\\rbrace , 𝒵 2 =∅\\mathcal {Z}_2=\\emptyset , 𝒵 3 ={7}\\mathcal {Z}_3=\\lbrace 7\\rbrace , 𝒵 4 ={4,5,6}\\mathcal {Z}_4=\\lbrace 4,5,6\\rbrace , and Q 1 II c =2,5,8Q_1^{\\text{II}_c}=2, 5, 8."
],
[
"Verification of Lemma ",
"In this subsection, Lemma REF and Lemma REF are verified.",
"Consider $\\mathcal {N}_\\text{II}=\\lbrace 4,5,6,7,8\\rbrace $ .",
"Fig.",
"REF plots $d_{n,1}$ and $d_{n,2}$ versus the stage index $n$ for a feasible decomposition in which $\\mathcal {Z}_1=\\lbrace 7,8\\rbrace $ , $\\mathcal {Z}_2=\\lbrace 6\\rbrace $ , $\\mathcal {Z}_3=\\emptyset $ , and $\\mathcal {Z}_4=\\lbrace 4,5\\rbrace $ when $Q_1^{\\text{II}_c}$ is set to be 70, 80, and 90.",
"Fig.",
"REF plots $d_{n,1}$ and $d_{n,2}$ versus the stage index for a feasible decomposition in which $\\mathcal {Z}_1=\\lbrace 8\\rbrace $ , $\\mathcal {Z}_2=\\emptyset $ , $\\mathcal {Z}_3=\\lbrace 7\\rbrace $ , and $\\mathcal {Z}_4=\\lbrace 4,5,6\\rbrace $ when $Q_1^{\\text{II}_c}$ is set to be 2, 5, and 8.",
"From Fig.",
"REF and Fig.",
"REF , it can be seen that, when $Q_1^{\\text{II}_c}$ changes, $d_{n,1}$ and $d_{n,2}$ vary in the same way as Lemma REF and Lemma REF describe."
],
[
"Verification of Theorem ",
"In this subsection, the characteristic of $U(Q_1-x)$ described in Theorem REF is verified.",
"Still consider $\\mathcal {N}_\\text{II}=\\lbrace 4,5,6,7,8\\rbrace $ .",
"Two decompositions are investigated, which are listed in Table REF .",
"Consider two intervals of $x$ : $[0,30]$ and $[40,70]$ , in which the two decompositions are feasible, respectively.",
"Fig.",
"REF and Fig.",
"REF plot the function $U(Q_1- x)$ as well as $G(x)$ and $H(x)$ (from Theorem REF ) for the two decompositions over the two corresponding intervals, respectively.",
"It can be seen that both the functions $G(x)$ and $H(x)$ are monotonically increasing for each decomposition in the corresponding interval of $x$ , which is consistent with Theorem REF .",
"Table: The Decompositions used when verifying Theorem Figure: Functions U(Q 1 -x)U(Q_1- x), G(x)G(x) and H(x)H(x) (x∈[0,30]x\\in [0,30]) for the first decomposition in Table .Figure: Functions U(Q 1 -x)U(Q_1- x), G(x)G(x) and H(x)H(x) (x∈[40,70]x\\in [40,70]) for the second decomposition in Table .Figure: The revenue of the two sellers in our proposed scheme and the cooperative scheme."
],
[
"Comparison with a cooperative scheme",
"Now we compare with other schemes.",
"As there is no research in the literature on dynamic pricing for more than one seller, we compare with a cooperative scheme.",
"The difference of the cooperative scheme from our proposed scheme is as follows.",
"When the two sellers know the existence of each other (i.e., at the beginning of Epoch II), the two sellers cooperate to jointly maximize the total revenue of them over Epoch II and III, by solving the optimization problem shown in (REF ).",
"$\\begin{array}{cll}\\mathop {\\max } \\limits _{\\lbrace d_{n, 1}|n \\in \\mathcal {N}_\\text{II}\\cup \\mathcal {N}_\\text{III}\\rbrace ,\\lbrace d_{n, 2}|n \\in \\mathcal {N}_\\text{II}\\rbrace } & \\sum \\limits _{n \\in \\mathcal {N}_\\text{II}} \\left(C_0-C_1 (d_{n,1}+d_{n,2})\\right) d_{n,1} n + \\sum \\limits _{n \\in \\mathcal {N}_\\text{III}} \\left(C_0-C_1 d_{n,1}\\right) d_{n,1} n \\\\&+\\sum \\limits _{n \\in \\mathcal {N}_\\text{II}} \\left(C_0-C_1 (d_{n,1}+d_{n,2})\\right) d_{n,2} (n- |\\mathcal {N}_{\\text{III}}|) \\\\\\text{s.t.}",
"& \\sum \\limits _{n \\in \\mathcal {N}_\\text{II}\\cup \\mathcal {N}_\\text{III}} d_{n,1} \\le Q_{1} \\\\& \\sum \\limits _{n \\in \\mathcal {N}_\\text{II}} d_{n,2} \\le Q_{2}^\\text{II}\\\\& d_{n,1} \\ge 0, \\forall n \\in \\mathcal {N}_\\text{II}\\cup \\mathcal {N}_\\text{III} \\\\& d_{n,2} \\ge 0, \\forall n \\in \\mathcal {N}_\\text{II}.\\end{array}$ For performance comparison, the simulation is set up as follows.",
"Since the cooperative scheme and our proposed scheme perform the same in Epoch I, we set $\\cal {N}_\\text{I}=\\emptyset $ .",
"And $\\mathcal {N}_\\text{II}=\\lbrace 6,5,4,3\\rbrace $ , $\\mathcal {N}_\\text{III}=\\lbrace 2,1\\rbrace $ .",
"We fix the sum of $Q_1$ and $Q_2$ to be 200, and consider three configurations of $(Q_1,Q_2)$ : $(50,150), (100,100)$ , and $(150,50)$ .",
"Fig.",
"REF shows the achieved revenue of the two sellers in our proposed scheme and the cooperative scheme.",
"It can be seen that each seller's revenue in our proposed non-cooperative scheme is very close to that in the cooperative scheme, thus verifying the efficiency of our proposed scheme."
],
[
"Conclusions",
"In this paper, we investigate spectrum leasing with two sellers.",
"In Epoch II, the two sellers both have spectrum to lease, and competition between the two sellers exists.",
"Thus, the spectrum leasing in Epoch II is formulated as a non-cooperative game.",
"Nash equilibria of the game are derived in closed form by jointly solving two optimization problems.",
"By analyzing the choices of seller 1, solutions of the two sellers in the spectrum leasing are developed.",
"The analysis and solutions in this work should help design oligopoly spectrum leasing strategies in future cognitive radio networks."
]
] | 1612.05702 |
[
[
"Phase Diagram of Boron Carbide With Variable Carbon Composition"
],
[
"Abstract Boron carbide exhibits intrinsic substitutional disorder over a broad composition range.",
"The structure consists of 12-atom icosahedra placed at the vertices of a rhombohedral lattice, together with a 3-atom chain along the 3-fold axis.",
"In the high carbon limit, one or two carbons can replace borons on the icosahedra while the chains are primarily of type C-B-C. We fit an interatomic pair interaction model to density functional theory total energies to investigate the substitutional carbon disorder.",
"Monte Carlo simulations with sampling improved by replica exchange and augmented by 2d multiple histogram analysis, predicts three phases.",
"The low temperature, high carbon composition monoclinic Cm \"tilted polar\" structure disorders through a pair of phase transitions, first via an Ising-like transition to a \"bipolar\" state with space group C2/m, then via a first order 3-state Potts-like transition to the experimentally observed \"nonpolar\" \\bar{R}3m symmetry."
],
[
"Introduction",
"The experimentally reported phase diagram of boron-carbon [1], [2] displays three phases: elemental boron and graphite, each in coexistence with boron carbide.",
"The carbon concentration of boron carbide ranges approximately from 9-19.2$\\%$ at high temperature.",
"Crystallographically [3], [4], [5], [6], [7], boron carbide has a 15-atom primitive cell, consisting of an icosahedron and a 3-atom chain, in a rhombohedral lattice with symmetry $R\\bar{3}m$ .",
"Icosahedra are primarily boron with some carbon substitution, and chains are usually of type C-B-C.",
"The energetically favored electron precise [8], [9] structure, with all bonding orbitals occupied, has stochiometry B$_4$ C with 20 atomic $\\%$ carbon, slightly outside the experimentally observed range.",
"The icosahedral carbon preferentially occupies a “polar\" site (see Fig REF ).",
"Icosahedra are connected along edges of the rhombohedral lattice, which pass through the polar sites.",
"For other compositions, the icosahedra can be B$_{12}$ , B$_{11}$ C$^p$ , or even B$_{10}$ C$_2^p$ (the bi-polar defect [10]), and the chain can be C-B-C, C-B-B, B-B$_2$ -B or B-V-B (V means vacancy) [11], [12], [13].",
"Based on density functional theory (DFT) study [7], [10], [14], besides the stable monoclinic B$_4$ C at the high carbon limit, rhombohedral B$_{13}$ C$_2$ is stable with 13.3$\\%$ carbon, where every icosahedron is of type B$_{12}$ and every chain is of type C-B-C.",
"Figure: Primitive cell of boron carbide showing C-B-C chain at center along the 3-fold axis.",
"The icosahedron (not to scale) occupies the cell vertex.",
"Equatorial sites of the icosahedron are shown in red, labeled “e”, the north polar sites are shown in green and labeled p 0 p_0, p 1 p_1 and p 2 p_2, while the south polar sites are shown in cyan and labeled as p 0 ' p_0^{\\prime }, p 1 ' p_1^{\\prime } and p 2 ' p_2^{\\prime }.",
"Equatorial borons bond to chain carbons, while north polar atoms bond to south polar atoms on a neighboring icosahedron.In our previous study [15], we built a pair interaction model at 20$\\%$ carbon where each icosahedron contains one polar carbon.",
"The only degree of freedom was the placement of the polar carbon among six polar sites in each icosahedron.",
"We predicted two phases transitions.",
"The low temperature monoclinic “tilted polar\" phase with space group $Cm$ transformed to the “polar\" phase with space group $R3m$ through a first order 3-state Potts-like phase transition and then to the experimentally observed “nonpolar\" rhombohedral phase with space group $R\\bar{3}m$ via a continuous Ising-like phase transition.",
"In this work, we consider variable carbon composition by including B$_{12}$ , B$_{11}$ C$^p$ and B$_{10}$ C$_2^p$ icosahedra.",
"More than two carbons in a single icosahedron is quite rare since it introduces a nearest neighbor C-C bond within the icosahedron that is energetically unfavorable.",
"Similarly, if the global average number of polar carbons exceeds 1, the excess electrons must occupy energetically costly antibonding orbitals [9] or mid-gap states.",
"Since every chain is C-B-C, we can relate the total carbon composition $x_{\\rm C}$ to the mean number of polar carbons per icosahedron $x_p$ as $x_{\\rm C}=(2+x_p)/15$ .",
"On energetic grounds we shall constrain $0\\le x_p\\le 1$ .",
"We build a new pair interaction model with variable carbon composition based on DFT total energies.",
"Simulations on a $2d$ grid of temperature T and chemical difference $\\mu \\equiv \\mu _{\\rm C}-\\mu _{\\rm B}$ use $2d$ replica exchange to better reach equilibrium and overcome large energy barriers between states.",
"We analyze our simulation results with a 2d multiple histogram method [16], [17].",
"Similar to our previous study [15], we predict three phases and two phase transitions.",
"However, the intermediate state is now the “bipolar\" state with space group $C2/m$ instead of the “polar\" state with space group $R3m$ , signifying the importance of including the bipolar defect [10] B$_{10}$ C$_2^p$ which was not considered in the old study.",
"We predict the phase diagram at various carbon compositions and characterize the features of these two phase transitions.",
"As in our previous study we note the relaxed total energy is a function solely of the initial assignment of carbon atoms to polar sites.",
"In principle we could fit this function with a cluster expansion [18] that is a linear combination of 1-, 2-, and many-body interactions.",
"Here we use the pair interactions of the cluster expansion, while adding a nonlinear (cubic) function of carbon composition which models the concave shape of energies above convex hull between B$_{13}$ C$_2$ and B$_4$ C, as seen in Fig.",
"REF .",
"We call this the “poly-pair interaction model\" [19].",
"Our model $E(N_1,\\dots ,N_m)=E_0+\\sum _{i=1}^{23} a_k N_k+\\beta _0x_p+\\beta _1x_p^2+\\beta _2x_p^3$ includes 23 types of polar atom pairs ranging from $R_1=1.72$ , through the rhombohedral lattice constant $R_9=5.17$ up to $R_{23}=6.58$ Å.",
"As in our previous study [15], we use the density functional theory-based Vienna ab initio simulation package (VASP) [20], [21], [22], [23] to calculate the total energies, of about 600 structures of supercells from 2x2x2 (120 atoms) to 4x4x4 (960 atoms).",
"Our fitting procedure minimizes the weighted mean-square deviation of model energy from calculated DFT energy, taking an exponential weight related to the energy $\\Delta E$ above the convex hull for each structure, so we weight low energy structures more heavily.",
"Five-fold cross validation shows weighted training error around 0.28 meV/atom and weighted test error around 0.31 meV/atom, which corresponds to 15$\\times $ 0.31=4.65 meV/cell, or $k_BT$ per degree of freedom at T=54K.",
"Fig REF illustrates the comparison between DFT and model in one five-fold cross validation.",
"Figure: Energy EE above convex hull calculated from DFT and poly-pair interaction model.",
"Note concave shape with respect to polar carbon concentration x p x_p.",
"Inset shows cross validation of the model with respect to DFT-calculated total energies."
],
[
"Symmetry and order parameters",
"Landau theory allows two symmetry-breaking paths [15], $R\\bar{3}m \\rightarrow R3m \\rightarrow Cm$ and $R\\bar{3}m \\rightarrow C2/m \\rightarrow Cm$ , that can be distinguished on the basis of site occupations.",
"Define $m_0,m_1,m_2,m_{0^{\\prime }},m_{1^{\\prime }},m_{2^{\\prime }}$ as the mean carbon occupancy at polar site $p_0,p_1,p_2,p_0^{\\prime },p_1^{\\prime },p_2^{\\prime }$ , respectively.",
"The polar carbon composition $x_p=\\sum _i (m_i+m_{i^{\\prime }})$ .",
"We introduce two order parameters.",
"The longitudinal polarization $P_z=m_0+m_1+m_2-m_{0^{\\prime }}-m_{1^{\\prime }}-m_{2^{\\prime }}$ transforms as the one-dimensional irreducible representation $A_{2u}$ of group $D_{3d}$ , which breaks inversion symmetry, while preserving rotation and reflection.",
"The pair of functions $P_{x}=(m_0+m_{0^{\\prime }})-\\frac{1}{2}(m_{1^{\\prime }}+m_{2^{\\prime }}+m_1+m_2), ~~~P_{y}={\\sqrt{3}\\over 2}(m_1+m_{1^{\\prime }}-m_2-m_{2^{\\prime }})$ transform as the irrep $E_g$ , which breakes rotational symmetry while preserving inversion.",
"To create a rotationally invariant measure of rotational symmetry breaking we define $P_{xy}=\\sqrt{P_{x}^2+P_{y}^2}$ .",
"Both symmetry-breaking paths begin with the fully disordered “nonpolar\" state of highest symmetry $R\\bar{3}m$ in which all $m_i$ and $m_{i^{\\prime }}$ are equal (e.g.",
"$m_i=m_{i^{\\prime }}=1/6$ in the high carbon limit) and $P_z=P_{xy}=0$ .",
"Then, in the first path, $m_i\\ne m_{i^{\\prime }}$ (e.g.",
"$m_i=1/3$ and $m_{i^{\\prime }}=0$ in the high carbon limit) gives $P_z>0$ and $P_{xy}=0$ , characterizing the “polar\" state $R3m$ .",
"Completing the symmetry breaking so that a single polar site is distinguished (e.g.",
"$m_0=1$ while all others vanish in the high carbon limit), we reach $P_z>0$ and $P_{xz}>0$ , characterizing the lowest symmetry “tilted polar\" state, with symmetry $Cm$ .",
"The intermediate state in the second path distinguishes a particular axis $i$ while maintaining the $i\\rightarrow i^{\\prime }$ inversion symmetry (e.g.",
"$m_0=m_{0^{\\prime }}=1/2$ while other $m$ 's vanish in the high carbon limit) so that $P_z=0$ and $P_{xy}>0$ , characterizing the “bipolar\" phase $C2/m$ .",
"Our previous study [15], where no bi-polar defects were allowed, followed the first path."
],
[
"Monte Carlo simulation and 2d multi-histogram method",
"We perform Metropolis Monte Carlo simulations in $L\\times L\\times L$ supercells of the rhombohedral primitive cell, with $L$ ranging from 3 to 8.",
"Our simulation includes two types of move: (1) randomly pick a polar site and change the species; (2) randomly interchange a polar carbon site and a polar boron.",
"In view of the high energy cost for occupying antibonding orbitals, we reject moves leading to $x_p>1$ .",
"For each move we calculate the energy change $\\Delta E$ and the change in the number of carbon atoms $\\Delta N$ .",
"Moves are accepted or rejected according to the Boltzmann factor $\\exp (-(\\Delta E-\\mu \\Delta N)/k_BT)$ .",
"Following an equilibration period, we begin recording the total energy $E$ and the occupations $m_i$ ($i=0, 1, 2, 0^{\\prime }, 1^{\\prime }, 2^{\\prime }$ ) of the polar sites for each subsequent configuration.",
"To enhance sampling efficiency we perform $2d$ replica exchange.",
"Consider a set of simulation trajectories; suppose the $i^{th}$ one is at temperature $T_i$ and chemical potential $\\mu _i$ , with total energy $E_i$ and polar carbon number $N_i$ , and similarly for the $j^{th}$ one.",
"The probability of occurence of these two trajectories is proportional to the corresponding Boltzmann factor $P_1=P(E_i,N_i;T_i,\\mu _i)P(E_j,N_j;T_j,\\mu _j)\\propto e^{-(E_i-N_i\\mu _i)/k_BT_i}e^{-(E_j-N_j\\mu _j)/k_BT_j}$ If we swap these two trajectories, so that trajectory $i$ is now at temperature $T_j$ and $\\mu _j$ , and trajectory $j$ is at temperature $T_i$ and $\\mu _i$ , then the probability is $P_2=P(E_i,N_i;T_j,\\mu _j)P(E_j,N_j;T_i,\\mu _i)\\propto e^{-(E_i-N_i\\mu _j)/k_BT_j}e^{-(E_j-N_j\\mu _i)/k_BT_i}.$ Detailed balance requires that the acceptance probability to interchange these two trajectories is $P=P_2/P_1=e^{\\Delta \\beta \\Delta E-\\Delta (\\beta \\mu )\\Delta N}$ where we define $\\Delta \\beta =1/(k_BT_i)-1/(k_BT_j)$ , $\\Delta E=E_i-E_j$ , $\\Delta _N=N_i-N_j$ and $\\Delta (\\beta \\mu )=\\mu _i/(k_BT_i)-\\mu _j/(k_BT_j)$ .",
"We analyze the Monte Carlo results using $2d$ histograms, similar to the $1d$ analysis in our previous work [15].",
"At a given simulation temperature $T_s$ and chemical potential $\\mu _t$ , a 2d histogram $H_{T_s,\\mu _t}(E,x)$ of configuration energy $E$ and polar carbon composition $x$ , can be converted into a density of states [16] $W(E,x)=H_{T_s,\\mu _t}(E,x)\\exp {((E-Nx\\mu _t)/k_BT_s)}$ where $N$ is the total number of atoms.",
"Then we calculate the partition function $Z(T,\\mu )=\\sum _{E,x} W(E,x) e^{-(E-Nx\\mu )/k_BT}$ which is accurate over a range of temperatures and chemical potentials close to $T_s$ and $\\mu _t$ .",
"Free energy F, internal energy U, specific heat $c_v$ and other thermodynamic properties can be obtained directly from Z by differentiation.",
"Moreover, by combining histograms taken at temperatures and chemical potentials the density of states can be self-consistently reconstructed [17] so that the free energy becomes accurate over all intervening temperatures and chemical potentails provided the tails of the histograms overlap.",
"Fig REF shows the overlapping marginal distributions of energy histogram for a 6x6x6 supercell at different temperatures fixing $\\bar{\\mu }=\\mu _t/k_BT=1.0$ (left) and marginal distributions of polar carbon composition at different $\\bar{\\mu }^{\\prime }$ s fixing T$_s$ =720K (right).",
"The rapid evolution of energy histogram between 660K and 720K indicates a phase transition in this temperature region at this chemical potential.",
"Figure: Marginal distributions of H(E,x)H(E,x) for 6x6x6 supercell.",
"Left: energy histogram at various Ts fixing μ ¯=1.0\\bar{\\mu }=1.0.",
"Right: polar carbon histogram at various μ ¯ ' \\bar{\\mu }^{\\prime }s fixing T=720K.The 2d density of states $W(E,x_C)$ can be further broken down according to the order parameters $P_z$ and $P_{xy}$ , yielding $W(E,x_C,P_z)$ and $W(E,x_C,P_{xy})$ , which are joint distribution of energy, chemical potential and the corresponding order parameter.",
"Evaluating powers of these parameters $\\langle |P_z| \\rangle $ , $\\langle |P_z|^2 \\rangle $ , $\\langle |P_{xy}| \\rangle $ and $\\langle |P_{xy}|^2 \\rangle $ from the corresponding density of states, we introduce the longitudinal susceptibility $\\chi _z$ and in-plane susceptibility $\\chi _{xy}$ which are fluctuations of the related order parameters, e.g.",
"$\\chi _z(T,\\mu )=N\\frac{\\langle |P_z|^2\\rangle - \\langle |P_z|\\rangle ^2}{ k_BT}.$ The susceptibility $\\chi _{xy}(T)$ is obtained in a similar way."
],
[
"Order parameters",
"Plotting order parameters at different temperatures and chemical potentials helps to determine the phases and transitions.",
"An upper limit on chemical potential of $\\mu $ =0.575eV is determined by comparing the stable structures on the convex hull at zero temperature B$_4$ C and graphite.",
"In reality $\\mu $ is a function of temperature but we shall neglect this dependence.",
"Fig REF illustrates order parameters $P_z$ and $P_{xy}$ as a function of temperature, at the high $\\mu $ limit (solid) which favors high carbon composition or intermediate $\\mu $ (broken) with carbon composition between 0.13 and 0.2.",
"At intermediate $\\mu $ , both $|P_z|$ and $P_{xy}$ decrease as the system size grows, suggesting that both order parameters vanish at this chemical potential.",
"Then the rhombohedral phase of symmetry $R\\bar{3}m$ covers the whole temperature range from 450K to 1000K.",
"At the high $\\mu $ limit where carbon composition approaches 0.2, the average longitudinal polarization $\\langle |P_z|\\rangle $ vanishes for $T\\gtrsim 570$ K but approaches to finite values for $T\\lesssim 570$ K, while $\\langle P_{xy}\\rangle $ decreases with increasing supercell size for $T\\gtrsim 730$ K but approaches finite values for $T\\lesssim 730$ K. We judge there are three phases, separated by two phase transitions.",
"At high temperature both $P_z$ and $P_{xy}$ vanish and the phase has symmetry $R\\bar{3}m$ .",
"Below 730K the in-plane polarization $P_{xy}$ grows so one particular direction of $i=0,1,2$ is favored, however $m_i\\approx m_{i^{\\prime }}$ so $P_z$ remains small.",
"The phase becomes bipolar with space group $C2/m$ by losing 3-fold rotation symmetry.",
"As temperature decreases further, passing 570K, the longitudinal polarization $P_z$ grows continuously and the phase becomes monoclinic with space group $Cm$ , losing inversion symmetry.",
"Figure: NO_CAPTIONOrder parameters $\\langle |P_z| \\rangle $ (upper left), $\\langle P_{xy} \\rangle $ (upper right) and corresponding susceptibilities $\\chi _z$ (lower left), $\\chi _{xy}$ (lower right), for supercells from $L$ =3 to 8, marked with different colors.",
"Solid curves show order parameters or susceptibilities at high $\\mu =0.575$ eV, while the broken curves are the corresponding order parameters and susceptibilities at intermediate $\\mu =-0.013$ eV."
],
[
"Specific heat and susceptibility",
"The order parameters are first derivatives of the free energy with respect to applied fields, and the corresponding second derivatives are susceptibilities.",
"Specific heat is the second derivative of free energy with respect to temperature.",
"All are evaluated from Monte Carlo data via the fluctuations of energy or order parameters, such as Eq.",
"(REF ).",
"Fig REF shows the specific heat for a series of increasing supercell sizes at the high $\\mu $ limit and intermediate $\\mu $ .",
"A strong peak grows with system size around T=730K and another weak peak begins to appear for large system size L=7 around T=570K.",
"These two temperatures coincide to those where the above order parameters change rapidly.",
"Fig.",
"REF shows the longitudinal and perpendicular (i.e.",
"in-plane) susceptibilities, $\\chi _z$ and $\\chi _{xy}$ respectively.",
"Both grow with increasing system size.",
"The peak of $\\chi _{xy}$ coincides with the strong specific heat peak and big change of $P_{xy}$ , while $\\chi _{z}$ coincides with the weak specific heat peak and change of $P_z$ .",
"Figure: Specific heat for L=3 to 8 supercells, at high μ\\mu limit (solid) and intermediate μ\\mu (broken).At intermediate $\\mu =-0.013$ eV, both specific heat and susceptibility converge to nonzero analytic functions, indicating a single phase with no phase transition.",
"From the order parameters $P_z$ and $P_{xy}$ both of which vanish, we judge this region as a single rhombohedral phase."
],
[
"3-state Potts-like transition at high $\\mu $",
"As the high temperature transition from $R\\bar{3}m$ to $C2/m$ coincides with a breaking of 3-fold rotation symmetry, we expect a three-state Potts-like phase transition which is weakly first order in three dimensions [24].",
"Then the corresponding order parameter $P_{xy}$ should jump discontinuously, and the fluctuations per atom of energy and polarization should grow proportionally to the number of atoms, i.e.",
"as $L^3$ .",
"Our largest supercell size $L=8$ has not yet reached this limit, with divergence around $L^{0.93}$ and $L^{2.6}$ seen for $c_v$ and $\\chi _{xy}$ respectively.",
"A similar issue also occurs in our previous work [15] and is due to the limited size of $L$ .",
"The Lee-Kosterlitz criterion [25], [26] is an alternative method to confirm a first order transition.",
"Because the two coexisting phases exhibit finite differences in properties such as energy and polarization, probability distributions of such properties should be bimodal, with each peak sharpening as system size grows.",
"Fig.",
"REF illustrates this distribution for $P_{xy}$ .",
"This distribution is obtained by marginalizing the joint energy and polarization histogram $H_{T_s}(E,P_{xy})$ over energy, then reweighting with the factor $\\exp (E/k_BT_e-E/k_BT_s)$ , where the temperature $T_e$ is chosen so as to make the heights of the two peaks equal.",
"Clearly the distributions of polarization illustrate coexistence of a state with low $P_{xy}< 0.1$ and a state with $P_{xy}\\sim 0.5$ .",
"Thus we conclude the transition is first order, as expected for symmetry-breaking of the 3-state Potts type in three dimensions.",
"Figure: Validation of universality classes.",
"(left) Lee-Kosterlitz histograms of P xy P_{xy} for supercells from LL=3 to 8.; (right) Ising scaling function for χ z \\chi _z, for supercells from LL=5 to 8."
],
[
"Ising-like transition at high $\\mu $",
"As the low temperature transition from $C2/m$ to $Cm$ coincides with a breaking of inversion symmetry, we expect the transition to be in the universality class of the three-dimensional Ising model.",
"Some associated critical exponents are $\\alpha =0.110$ (specific heat), $\\gamma =1.2372$ (susceptibility) and $\\nu =0.6301$ (correlation length) [27].",
"Based on finite size scaling theory [25], for Ising-like phase transition in three dimensions, the specific heat peak should diverge as $L^{\\alpha /\\nu }$ ($\\alpha /\\nu =0.175$ ), and the susceptibility peak should diverge as $L^{\\gamma /\\nu }$ ($\\gamma /\\nu =1.963$ ).",
"When plotting the scaled susceptibility $\\chi _z/L^{\\gamma /\\nu }$ as a function of an expanded temperature scale $\\epsilon L^{1/\\nu }$ with reduced temperature $\\epsilon =(T-T_c)/T_c$ , Ising universality requires convergence to a common scaling function.",
"As shown in Fig REF (right), the scaling function curves begin to converge as the system size grows, although with $T_c=627$ K, which is higher than the temperature of the peaks of susceptilibity $\\chi _z$ , implying the systems under study are still too small to show well converged behavior of Ising-like phase transition."
],
[
"Phase diagram in $x_C-T$ plane",
"Our simulation on a $2d$ grid of $(\\mu ,T)$ , together with $2d$ multihistogram analysis, allows us to predict the phase diagram in the $x_{\\rm C}-T$ plane.",
"At the high $\\mu $ limit it displays three phases and two phases transitions while at the intermediate $\\mu $ it displays a single rhombohedral phase.",
"We do not consider the low $\\mu $ case because our model excludes chain variants like C-B-B or B-V-B which are important at low $\\mu $ , allowing the low $\\mu $ rhombohedral phase boundary of rhombohedral phase to extend below $x_{\\rm C}=2/15=0.133$ .",
"Thus we predict the phase diagram only for $0.133\\le x_{\\rm C}\\le 0.2$ .",
"Figure: Phase diagram prediction from supercell LL=6 (left) and 7 (right).",
"Thresholds for phases: rhombohedral (P z <P_z<0.5 and P xy <P_{xy}<0.4), bipolar (P z <P_z<0.5 and P xy >P_{xy}>0.6), and monoclinic (P z >P_z>0.5 and P xy >P_{xy}>0.6).",
"Dash curves indicate our qualitatively suggested phase boundaries.Fig REF shows our predicted phase diagrams based on system size $L$ =6 (left) and 7 (right).",
"Threshold values of $P_z$ and $P_{xy}$ define phases based on the corresponding order parameters as a function of temperature (Fig REF ) at the high $\\mu $ limit.",
"Black circles indicate the rhombohedral phase $R\\bar{3}m$ , red squares show bipolar phase $C2/m$ , while green diamonds show the monoclinic phase $Cm$ .",
"Phase coexistence regions separate the rhombohedral phase from bipolar and from monoclinic, indicating first order phase transitions across the phase boundaries.",
"Even though at high $\\mu $ the bipolar phase (red) changes to monoclinic (green) via a continuous Ising like phase transition, at lower $\\mu $ the phase diagram implies a gap showing a first order phase transtion, indicating the possible occurence of tricritical point [28].",
"Including variable carbon composition in an Ising-like free energy model could serve as a possible explanation for the change from continuous to first order phase transition as carbon composition decreases, similar to the behavior of the compressible ferromagnet [29]."
],
[
"Conclusion",
"We construct a poly-pair interaction model for boron carbide by placing zero, one, or two polar carbons in each icosahedral cluster, introducing B$_{12}$ , B$_{11}$ C$^p$ or B$_{10}$ C$_2^p$ icosahedra, respectively, while fixing the chain to be C-B-C. Our model, which includes pairwise interaction and a nonlinear function of composition, fits DFT total energies well and is amendable to computer simulation.",
"With this model we study the phase diagram and phase transitions over a range of carbon composition.",
"We focus on the range from intermediate to high $\\mu $ , since we neglect chain variants that are important at low $\\mu $ .",
"Monte Carlo simulations on a $2d$ grid of temperatures and chemical potentials, together with $2d$ replica exchange to attain better equilibrium sampling, reveal a wide range of the experimentally observed rhombohedral phase $R\\bar{3}m$ at intermediate $\\mu $ .",
"Three phases occur at the high $\\mu $ limit, where two phase transitions occur, one of which at higher temperature is 3-state Potts like, first order, breaking three fold rotation symmetry, and one of which at lower temperature is Ising like, continuous, breaking inversion symmetry.",
"The three phases are rhombohedral ($R\\bar{3}m$ ), bipolar ($C2/m$ ) and monoclinic ($Cm$ ), as temperature goes from high to low.",
"These low temperature ordered phases have not been observed experimentally, presumably because atomic diffusion is slow at low T, preventing equilibration.",
"Globally, we predict the phase diagram in the $x_C-T$ plane, showing regions of the three phases: rhombohedral, bipolar and monoclinic (Fig REF ).",
"Empty regions representing the coexistence between phases arise from the first order phase transitions.",
"At the high $\\mu $ limit the transition from bipolar to monoclinic phase is continuous and Ising-like.",
"It could be first order at lower chemical potential due to variable carbon composition, thus a tricritical point may exist."
],
[
"Acknowledgement",
"This work was supported by DOE grant DE-SC0014506."
]
] | 1612.05527 |
[
[
"Embedded surfaces with Anosov geodesic flows, approximating spherical\n billiards"
],
[
"Abstract We consider a billiard in the sphere S^2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic.",
"We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface in the ambiant space S^3.",
"As an application, we show that every orientable surface of genus at least 11 admits an isometric embedding into S^3 (equipped with the standard metric) such that its geodesic flow is Anosov.",
"Finally, we explain why this construction cannot provide examples of isometric embeddings of surfaces in the Euclidean R^3 with Anosov geodesic flows."
],
[
"Anosov geodesic flows for embedded surfaces",
"The geodesic flow of any Riemannian surface whose curvature is negative everywhere is Anosov: in particular, any orientable surface of genus at least 2 can be endowed with a hyperbolic metric, for which the geodesic flow is Anosov.",
"On the contrary, there is no Riemannian metric on the torus or the sphere with an Anosov geodesic flow.",
"If a closed surface admits an isometric embedding in $\\mathbb {R}^3$ , then it needs to have positive curvature somewhere.",
"However, it is still possible to obtain an Anosov geodesic flow for such a surface, as shown by Donnay and Pugh [4].",
"More precisely, they showed that there exists a genus $g_0$ such that for all $g \\ge g_0$ , the orientable surface of genus $g$ admits an isometric embedding in $\\mathbb {R}^3$ whose geodesic flow is Anosov (see [5]).",
"The value of $g_0$ is completely unknown: in particular, it is unknown whether it is possible to embed isometrically a surface of genus smaller than one million in $\\mathbb {R}^3$ so that its geodesic flow is Anosov.",
"The same question may be asked for embeddings in the sphere $\\mathbb {S}^3$ endowed with the standard metric.",
"Here, we have the following situation: Proposition 1.1 Any closed surface $M$ isometrically embedded in $\\mathbb {S}^3$ admits at least one point at which the Gauss curvature is at least 1 (except if $M$ is a torus or a Klein bottle).",
"The curvature of the surface is given at each point by $K = k_1 k_2 + 1$ , where $k_1$ and $k_2$ are the principal curvatures of the surface.",
"If $K < 1$ everywhere, then the principal curvatures are nonzero and have different signs at each point.",
"Thus the principal directions induce two nonsingular vector fields on $M$ , which implies that the Euler characteristic of $M$ must be zero.",
"We will show for the first time that it is possible to embed isometrically in $\\mathbb {S}^3$ a Riemannian surface with Anosov geodesic flow.",
"More precisely: Theorem 1.2 Every orientable surface of genus at least 11 admits an isometric embedding into $\\mathbb {S}^3$ such that its geodesic flow is Anosov."
],
[
"Approximating billiards by flattened surfaces",
"Birkhoff [2] seems to be the first to suggest that billiards could be approximated by geodesic flows on flattened surfaces: he took the example of an ellipsoid whose vertical axis tends to zero, which converges to the billiard in an ellipse.",
"Later, Arnold [1] writes that smooth Sinaï billiards could be approximated by surfaces with nonpositive curvature, with Anosov geodesic flows.",
"In [6], we proved this fact for a large class of flattened surfaces.",
"In this paper, we prove a similar result for another class of objects.",
"Consider a finite family of open disks $\\Delta _i$ , whose closures are disjoint, which have radii $r_i < \\pi /2$ , on the sphere $\\mathbb {S}^2$ : we will say that the billiard $D = \\mathbb {S}^2 \\setminus \\bigcup _{i} \\Delta _i$ is a spherical billiard with circular obstacles (Figure REF ).",
"Define the billiard flow in the following way: outside the obstacles, the particle follows the geodesics of the sphere with unit speed; when the particle hits an obstacle, it bounces, following the usual billiard reflection law.",
"Figure: A spherical billiard with 12 obstacles.The horizon $H$ of a spherical billiard is the length of the longest geodesic of $\\mathbb {S}^2$ contained in the billard $D$ .",
"The phase space $\\Omega $ of the billiard is defined as $\\Omega = T^1(\\mathrm {Int}~(D))$ (the unit tangent bundle of the interior of $D$ ).",
"To define uniform hyperbolicity, we need to consider the set $\\tilde{\\Omega }$ of all $(x, v) \\in T^1(\\mathrm {Int}(D))$ such that the trajectory starting at $(x, v)$ does not contain any grazing collision (that is, each obstacle is reached with a nonzero angle).",
"We will use a definition of “uniformly hyperbolic billiard” which can be found in [3]: Definition 1.3 The billiard flow $\\phi ^t$ is uniformly hyperbolic if at each point $x \\in \\tilde{\\Omega }$ , there exists a decomposition of $T_x\\Omega $ , stable under the flow, $ T_{x} \\Omega = E_x^0 \\oplus E_x^u \\oplus E_x^s $ where $E_x^0 = \\mathbb {R} \\left.",
"\\frac{d}{dt}\\right|_{t=0} \\phi ^t(x)$ , such that $ \\Vert D\\phi _x^t|_{E_x^s} \\Vert \\le a \\lambda ^t, \\ \\Vert D\\phi _x^{-t}|_{E_x^u} \\Vert \\le a \\lambda ^{t} $ (for some $a > 0$ and $\\lambda \\in (0,1)$ , which do not depend on $x$ ).",
"Smooth flat billiards with negative curvature and finite horizon are known to be uniformly hyperbolic [8], but it is not the case for spherical billiards, as the following example shows: Example 1.4 Consider six disjoint disks on the sphere, with the same radius $r$ , whose centers are the vertices of a regular octahedron which is inscribed in the sphere.",
"If the radius $r$ is large enough, the billiard has finite horizon.",
"However, there is a family of billiard trajectories which are parallel to the geodesic which is drawn on Figure REF , and thus, the billiard is not uniformly hyperbolic.",
"Figure: The trajectory which is shown on the right-hand side is not hyperbolic.However, we will show that there exists a spherical billiard with 12 circular obstacles which is uniformly hyperbolic (see Figure REF )."
],
[
"Approximation by a closed surface",
"In [6], we have shown that, under some conditions, uniformly hyperbolic billiards may be approximated by smooth surfaces whose geodesic flow is Anosov.",
"The main result of [6] only applies to flat billiards, but we will show in this paper that it is possible to approximate our spherical billiard by a surface in the ambiant space $\\mathbb {S}^3$ such that the geodesic flow is Anosov (Theorem REF ): see Figure REF .",
"Figure: An approximation of the spherical billiard by a surface of genus 11.",
"This surface is isometrically embedded in 𝕊 3 \\mathbb {S}^3 and is seen here in stereographic projection.",
"Its geodesic flow is Anosov.It is tempting to try the same construction in the ambiant space is $\\mathbb {R}^3$ : however, we will see that in this framework, the geodesic flow of a surface which approximates a spherical billiard is never Anosov (see Theorem REF ).",
"This result, which might seem surprising at first sight, is due to the accumulation of a high quantity of positive curvature beside the negative curvature which appears near the boundary of the billiard."
],
[
"Main results",
"Consider a billiard $D$ in the sphere $\\mathbb {S}^2$ with circular obstacles.",
"Consider the largest obstacle and the length of its radius $A$ (for the spherical metric), and the horizon $H$ of the billiard.",
"First we will prove: Theorem 2.1 If $H < \\pi /2$ and $2 \\tan (\\pi /2 - A) > \\tan (H)$ , then $D$ is uniformly hyperbolic.",
"In particular, it is the case if $A + H < \\pi /2$ .",
"We will now see how uniform hyperbolicity “transfers” to surfaces which approximate such a billiard.",
"Consider the stereographic projection of $\\mathbb {S}^3$ , and the surface $ \\mathbb {S}^2 = \\left\\lbrace (x, y, z) \\in \\mathbb {S}^3 ~ \\vert ~ x^2 + y^2 + z^2 = 1 \\right\\rbrace .",
"$ For $q \\in \\mathbb {R}^3$ denote by $\\rho (q)$ the radial unit vector at $q$ (the unit vector which is positively colinear to the vector joining the origin to $q$ ) and by $\\pi $ the natural projection of $\\mathbb {S}^3$ (minus the poles) onto $\\mathbb {S}^2$ : $ \\begin{aligned} \\pi : \\mathbb {R}^3 \\setminus \\lbrace 0\\rbrace & \\rightarrow \\mathbb {S}^2 \\\\ (x, y, z) & \\rightarrow (x / (x^2 + y^2 + z^2), y / (x^2 + y^2 + z^2), z / (x^2 + y^2 + z^2)).",
"\\end{aligned} $ We also consider the “flattening map” $f_\\epsilon $ for $\\epsilon \\in (0,1)$ : $ \\begin{aligned} f_\\epsilon : \\mathbb {R}^3 \\setminus \\lbrace 0\\rbrace & \\rightarrow \\mathbb {R}^3 \\setminus \\lbrace 0\\rbrace \\\\ q & \\rightarrow \\epsilon q + (1-\\epsilon ) \\pi (q).",
"\\end{aligned}.",
"$ The main theorem of this paper is the following: Theorem 2.2 Consider a spherical billiard $D$ with spherical obstacles $\\Delta _i$ , which satisfies $H < \\pi /2$ and $2 \\tan (\\pi /2 - A) > \\tan (H)$ (see the notations above), and a surface $\\Sigma $ in $\\mathbb {S}^3$ such that $\\pi (\\Sigma ) = D$ .",
"We assume that: (Transversality to the fibers of the projection.)",
"For all $x \\in \\Sigma $ , if $\\pi (x) \\notin \\partial D$ , then $\\rho \\notin T_x\\Sigma $ .",
"(Nonzero vertical principal curvature.)",
"For all $x \\in \\Sigma $ , if $\\pi (x) \\in \\partial D$ , then $\\mathrm {II}_x(\\rho ) \\ne 0$ (where $\\mathrm {II}$ is the second fundamental form).",
"(Symmetry.)",
"For all $i$ , $\\partial \\Delta _i \\subseteq \\Sigma $ ; moreover, there is a neighborhood of $\\partial \\Delta _i$ in $\\Sigma $ which is invariant by inversion with respect to $\\mathbb {S}^2$ , and by rotation in $\\mathbb {S}^3$ around the axis $(0, c_i)$ , where $c_i$ is the center of $\\Delta _i$ .",
"Then there exists $\\epsilon _0 \\in (0,1)$ such that for all $\\epsilon \\le \\epsilon _0$ , the geodesic flow on the flattened surface $\\Sigma _\\epsilon = f_\\epsilon (\\Sigma )$ is Anosov.",
"The assumptions of Theorem REF are very close to those which appear in the main theorem of [6] (which deals with the case of flat billiards), but the proof is made more difficult by the positive curvature of the sphere.",
"It is actually simple, from a given billiard, to construct a surface which satisfies these assumptions.",
"More precisely: Theorem 2.3 If $D$ is a spherical billiard with $n$ circular obstacles, then there exists a surface $\\Sigma $ of genus $n-1$ such that $\\pi (\\Sigma ) = D$ , which satisfies the assumptions 1, 2 and 3 of Theorem REF .",
"We assume that the circular obstacles have radii $r_1, r_2, \\ldots , r_n$ .",
"Choose $\\delta > 0$ such that the circles $\\tilde{\\Delta }_i$ of radii $r_i + \\delta $ , with the same centers as the obstacles, remain disjoint.",
"Consider the image $S_1$ of $\\mathbb {S}^2 \\setminus \\bigcup _i \\tilde{\\Delta }_i$ by a homothety (in $\\mathbb {R}^3$ ) of center $(0,0,0)$ and ratio $1-\\epsilon $ , with a small $\\epsilon > 0$ .",
"Consider the image $S_2$ of $S_1$ by inversion with respect to $\\mathbb {S}^2$ .",
"Finally, construct symmetric tubes which connect the pairs of “holes” on the surfaces $S_1$ and $S_2$ .",
"Theorem REF will allow us to prove Theorem REF in Section .",
"On the other hand, we prove the following: Theorem 2.4 Consider a spherical billiard $D$ with spherical obstacles $\\Delta _i$ , of radii $r_i < \\pi /2$ , and a surface $\\Sigma $ in $\\mathbb {R}^3$ such that $\\pi (\\Sigma ) = D$ and $\\partial \\Delta _i \\subseteq \\Sigma $ .",
"Write $r_{\\max } = \\max _i (r_i)$ and, for all $\\delta > 0$ , denote by $V_i^\\delta $ the open neighborhood of $\\partial \\Delta _i$ in $\\Sigma $ which consists of all points at distance less than $\\delta $ from $\\partial \\Delta _i$ .",
"We will say that the surface $\\Sigma $ is $\\epsilon $ -$C^1$ -close to the billiard $D$ if there exists locally a parametrization $f_1$ of $\\Sigma $ and a parametrization $f_2$ of $D$ such that $\\left\\Vert f_1 - f_2 \\right\\Vert _{C^2} \\le \\epsilon $ for the $C^2$ -norm in the Euclidean $\\mathbb {R}^3$ .",
"For all $\\delta _1 \\in (0, \\pi /2 - r_{\\max })$ , there exists $\\delta _2 > 0$ , such that the geodesic flow of any surface satisfying the following conditions has conjugate points: (Symmetry.)",
"The neighborhood $V_i^{\\delta _1}$ is invariant by rotation in $\\mathbb {R}^3$ around the axis $(0, c_i)$ , where $c_i$ is the center of $\\Delta _i$ .",
"(Approximation of the billiard.)",
"The surface $\\Sigma \\setminus \\bigcup _i V_i^{\\delta _2}$ is $\\delta _2$ -$C^2$ -close to $D$ .",
"In particular, for the surface $\\Sigma _\\epsilon $ in Theorem REF , endowed with the metric induced by $(\\mathbb {R}^3, g_\\mathrm {eucl})$ , the geodesic flow always has conjugate points, and thus, it is never Anosov."
],
[
"Structure of the paper.",
"The proof of Theorem REF relies on a theorem which is proved in [7]: in Section , we recall the statement of this theorem and finish the proof of Theorem REF .",
"In Section , we state and prove a lemma in the Euclidean ambiant space.",
"This lemma is transposed to the spherical ambiant space in Section .",
"The local dynamics near a circular obstacle are studied in Section .",
"Section ends the proof of Theorem REF by studying the global dynamics.",
"We prove Theorem REF in Section .",
"Finally, we show Theorem REF in Section ."
],
[
"The Riccati equation",
"The Riccati equation is an important tool for the proofs of this paper.",
"On a smooth Riemannian manifold $M$ , the Riccati equation is a differential equation along a geodesic $\\gamma : [a, b] \\rightarrow \\mathbb {M}$ given by: $ \\dot{u}(t) = -K(t) - u(t)^2 $ where $K$ is the Gaussian curvature of the surface.",
"The space of orthogonal Jacobi fields on the geodesic $\\gamma $ has dimension 1, thus it is possible to consider any orthogonal Jacobi field as a function $j : [a, b] \\rightarrow \\mathbb {R}$ (by choosing an orientation of the normal bundle of $\\gamma $ ).",
"In this case, it is well-known that $j$ satisfies the equation $j^{\\prime \\prime }(t) = - K(t) j(t)$ .",
"A short calculation then shows that $u = j^{\\prime } / j$ satisfies the Riccati equation whenever $j \\ne 0$ .",
"The following criterion is an improvement of a statement which appears in [4]; a complete proof may be found in [7].",
"Theorem 3.1 Let $M$ be a closed surface.",
"Assume that there exist $m > 0$ and $C > c > 0$ such that for any geodesic $\\gamma : \\mathbb {R} \\rightarrow M$ , there exists an increasing sequence of times $(t_k)_{k \\in \\mathbb {Z}} \\in \\mathbb {R}^{\\mathbb {Z}}$ with $c \\le t_{k+1} - t_k \\le C$ , such that the solution $u$ of the Riccati equation with initial condition $u(t_k) = 0$ is defined on the interval $[t_k, t_{k+1}]$ , and $u(t_{k+1}) > m$ .",
"Then the geodesic flow $\\phi _t : T^1 M \\rightarrow T^1 M$ is Anosov.",
"Now, consider a spherical billiard $D$ and a billiard trajectory $\\gamma $ .",
"It is possible to consider a small variation of $\\gamma $ , which is also called a Jacobi field, and consider $u = j^{\\prime }/j$ , as in the case of a geodesic flow.",
"Thus, there is a natural generalization of the Riccati equation.",
"We say that $u$ is a solution of this equation if: in the interval between two collisions, $\\dot{u}(t) = -K(t) - u(t)^2$ ; when the particle bounces against the boundary at a time $t$ , $u$ undergoes a discontinuity: we have $u(t^+) = u(t^-) - \\frac{2 \\kappa }{\\sin \\theta }$ , where $\\kappa $ is the geodesic curvature of the boundary of $D$ , and $\\theta $ is the angle of incidenceThe notation $u(t^+)$ stands for $\\lim _{h \\rightarrow 0, h > 0} u(t + h)$ , and likewise $u(t^-) = \\lim _{h \\rightarrow 0, h < 0} u(t + h)$ .. With this generalized Riccati equation, the following theorem holds (its proof may be found in [7]): Theorem 3.2 Consider a spherical billiard $D$ .",
"Assume that there exist positive constants $A, m, c$ and $C$ such that for any trajectory $\\gamma $ with $\\gamma (0) \\in \\tilde{\\Omega }$ , there exists an increasing sequence of times $(t_k)_{k \\in \\mathbb {Z}} \\in \\mathbb {R}^{\\mathbb {Z}}$ satisfying $c \\le t_{k+1} - t_k \\le C$ , such that for any $k \\in \\mathbb {Z}$ , the solution $u$ of the Riccati equation with initial condition $u(t_k^+) = 0$ satisfies $u(t^+) \\ge -A$ for all $t \\in [t_k, t_{k+1}]$ , and $u(t_{k+1}^+) > m$ .",
"Also assume that for each $k \\in \\mathbb {Z}$ , there is no collision in the interval $(t_k - c, t_k)$ , and at most one collision in the interval $(t_k, t_{k+1}]$ .",
"Then the billiard flow on $D$ is uniformly hyperbolic (see Definition REF ).",
"Knowing this theorem, we are ready to prove Theorem REF .",
"[Proof of Theorem REF ] Since the obstacles are circles of radius at most $A$ , their geodesic curvature is at least $\\tan (\\pi /2 - A)$ .",
"We define $m = 2 \\tan (\\pi /2 - A) - \\tan (H)$ (thus $m > 0$ ).",
"Consider a billiard trajectory $(q(t), p(t))_{t \\in [0, 2H]}$ in the billiard $D$ , with collision times $(t_k)_{k \\in \\mathbb {Z}}$ .",
"Fix $k$ and consider the solution $u$ of the Riccati equation along this trajectory with $u(t_k) = 0$ .",
"We want to show that $u(t_{k+1}) > \\delta $ .",
"On $(t_k, t_{k+1})$ , the solution $u$ satisfies $u^{\\prime }(t) = -1 - u(t)^2$ , so $u(t_{k+1}^-) = -\\tan (t_{k+1} - t_k) \\ge -\\tan (H)$ .",
"Knowing that $u(t_{k+1}^+) = u(t_{k+1}^-) - \\frac{2 \\kappa }{\\sin \\theta }$ , we obtain $u(t_{k+1}^+) \\ge -\\tan (H) + 2 \\tan (\\pi /2 - A) = m$ .",
"According to Theorem REF , this concludes the proof."
],
[
"Flattening a curve in the Euclidean plane",
"Lemma 4.1 Consider a smooth curve $c : (-a, a) \\rightarrow \\mathbb {R}^2$ with unit speed, and write its coordinates $c(t) = (r(t), z(t))$ .",
"Assume that $0 \\in (a, b)$ , $c(0) = R$ , $c^{\\prime }(0) = (0,1)$ , and the curvature of $c$ at 0 is nonzero.",
"For all $t \\in (a, b)$ , also assume that $c(t) \\ge R$ , $r(-t) = r(t)$ and $z(-t) = -z(t)$ .",
"Consider the flattened curve $c^\\epsilon (t) = (r(t), \\epsilon z(t))$ and its curvature $k^\\epsilon (t)$ .",
"The unit tangent vector to $c^\\epsilon (t)$ is $T^\\epsilon (t) = (T^\\epsilon _r(t), T^\\epsilon _z(t))$ , and the normal vector is $(T_z^\\epsilon (t), -T_r^\\epsilon (t))$ .",
"Then there exists $m_0 > 0$ such that for all $m \\le m_0$ , there exists $\\epsilon _0 > 0$ such that for all $\\epsilon \\le \\epsilon _0$ , there exists $t_c$ such that for all $t \\in (0, t_c)$ , $T^\\epsilon _z(t) \\ge m$ and $k^\\epsilon (t) \\ge m^4 / \\epsilon ^2$ , for all $t \\in (t_c, m)$ , $T^\\epsilon _z(t) \\le m$ and $k^\\epsilon (t) \\ge 0$ .",
"We will write $T^\\epsilon (t) = (\\cos \\alpha ^\\epsilon (t), \\sin \\alpha ^\\epsilon (t))$ and $s^\\epsilon $ a parametrization by arc length of $c^\\epsilon $ (such that $s^\\epsilon (0) = 0$ and $ds^\\epsilon /dt = \\left\\Vert dc^\\epsilon /dt \\right\\Vert $ ).",
"Since the curvature of $c$ at $t=0$ is nonzero, we may assume that the angle $t \\mapsto \\alpha (t)$ is decreasing on the interval $(-m, m)$ (reducing $m$ if necessary).",
"We may also assume that $\\alpha (t) \\in (0, \\pi )$ .",
"We have: $ T^\\epsilon (t) = \\frac{(\\cos (\\alpha ^1(t)), \\epsilon \\sin (\\alpha ^1(t)))}{\\sqrt{\\cos ^2 (\\alpha ^1(t)) + \\epsilon ^2 \\sin ^2 (\\alpha ^1(t))}}.",
"$ Thus, $ \\tan (\\alpha ^\\epsilon (t)) = \\epsilon \\tan \\alpha ^1(t) $ for $t \\in (0, m)$ .",
"In particular, $t \\mapsto \\alpha ^\\epsilon (t)$ is decreasing (thus $k^\\epsilon (t) \\ge 0$ ), so for $\\epsilon $ small enough, there exists $t_c \\in (0, m)$ such that $\\sin \\alpha ^\\epsilon (t_c) = m$ .",
"Thus for all $t \\in (t_c, m)$ , we have $T^\\epsilon _z(t) \\le m$ , whereas for all $t \\in (0, t_c)$ , we have $T^\\epsilon _z(t) \\ge m$ .",
"We will now show that $T^\\epsilon _z(t) \\ge m$ implies $k^\\epsilon (t) \\ge m^4/\\epsilon ^2$ , which will conclude the proof of the lemma.",
"After differentiation of (REF ), we obtain $ \\begin{aligned} \\frac{d\\alpha ^\\epsilon }{dt} (1 + \\tan ^2 (\\alpha ^\\epsilon (t))) & = \\epsilon \\frac{d\\alpha ^1}{dt} \\frac{1}{\\cos ^2 (\\alpha ^1(t))}\\\\ \\frac{d\\alpha ^\\epsilon }{dt} (1 + \\epsilon ^2 \\tan ^2 (\\alpha ^1(t))) & = \\epsilon \\frac{d\\alpha ^1}{dt} \\frac{1}{\\cos ^2 (\\alpha ^1(t))}\\\\ \\frac{d\\alpha ^\\epsilon }{dt} & = \\frac{d\\alpha ^1}{dt} \\frac{\\epsilon }{\\cos ^2 (\\alpha ^1(t)) + \\epsilon ^2 \\sin ^2 (\\alpha ^1(t))}\\\\ \\frac{d\\alpha ^\\epsilon }{dt} & = \\frac{d\\alpha ^1}{dt} \\frac{\\epsilon }{(ds^\\epsilon /dt)^2}\\end{aligned} $ Thus the curvature of $c_\\epsilon $ is $ k^\\epsilon (t) = \\frac{d\\alpha ^\\epsilon }{ds} = \\frac{d\\alpha ^\\epsilon }{dt} \\cdot \\frac{1}{dt/ds^\\epsilon } = k^1(t) \\cdot \\epsilon \\cdot \\frac{1}{(dt/ds^\\epsilon )^3}.",
"$ Assuming that $T^\\epsilon _z \\ge m$ , we obtain $ \\frac{\\epsilon \\sin \\alpha }{ds^\\epsilon /dt} = T^\\epsilon _z \\ge m $ and thus $ \\frac{ds^\\epsilon }{dt} \\le \\frac{\\epsilon }{m}.",
"$ Finally, $ k^\\epsilon (t) \\ge k^1(t) \\cdot \\frac{m^3}{\\epsilon ^2} \\ge \\frac{m^4}{\\epsilon ^2}.",
"$"
],
[
"Curvature in a flattened tube",
"In sections , and , we choose constants $\\nu $ , $m$ , $\\delta $ , and $\\epsilon $ , in the interval $(0,1)$ , such that: These constants are sufficiently small: how small they need to be depends on the choice of the billiard $D$ and the surface $\\Sigma $ .",
"These constants satisfy $\\nu \\gg m \\gg \\delta \\gg \\epsilon $ .",
"This means that the ratios $m / \\nu $ , $\\delta / m$ and $\\epsilon / \\delta $ are sufficiently small, again with respect to the choice of $D$ and $\\Sigma $ .",
"We even assume that the ratios $m / \\nu ^{1000}$ , $\\delta / m^{1000}$ and $\\epsilon / \\delta ^{1000}$ are small.",
"We consider these constants as fixed once and for all, to avoid adding in each statement a prefix such as “there exists $\\nu _0 > 0$ , such that for all $\\nu \\le \\nu _0$ , there exists $m_0 > 0$ , such that for all $m \\le m_0$ ...”.",
"In this section, we consider a circular obstacle $\\Delta _{i_0}$ , with center $q_0 \\in \\mathbb {S}^2$ , and use the stereographic projection of $\\mathbb {S}^3$ with $q_0$ as the south pole (that is, $q_0$ has coordinates $(0, 0, 0)$ ).",
"We will use cylindric coordinates $(r, \\theta , z)$ .",
"The circle $\\Delta _{i_0}$ is defined by the equation $r = R$ , $z = 0$ , where $R \\in (0,1)$ .",
"Consider $\\tilde{\\mathcal {T}} = \\left\\lbrace (r, \\theta , z) \\in \\Sigma _\\epsilon ~ \\vert ~ r \\le R + \\delta \\right\\rbrace $ , and $\\mathcal {T}$ the connected component of $\\tilde{\\mathcal {T}}$ which contains $\\partial \\Delta _{i_0}$ .",
"The “tube” $\\mathcal {T}$ is a surface of revolution (assumption 3 of Theorem REF ), obtained by rotation of a curve $\\mathcal {S}$ around the $z$ -axis.",
"More precisely, we define the curve $\\mathcal {S}$ as the intersection of $\\mathcal {T}$ with the half great sphere corresponding to the equation $\\theta = 0$ .",
"It has an upper part described by the equation $z = h(r)$ , and a lower part given by $z = -h(r)$ .",
"Here, the mapping $h$ is nonnegative, defined continuously on the interval $[R, R + \\delta ]$ , smooth on $(R, R + \\delta )$ , and such that $h(R) = 0$ (see Figure REF ).",
"Figure: The curve 𝒮\\mathcal {S} (on the right-hand side) seen in stereographic projection.",
"The two dotted lines correspond to two spheres in 𝕊 3 \\mathbb {S}^3 which are close to the great sphere 𝕊 2 \\mathbb {S}^2.We consider the Euclidean metric $g_\\mathrm {eucl}$ and the metric of the stereographic projection $ g_\\mathrm {stereo} = \\xi ^2 g_\\mathrm {eucl}, \\quad \\text{ where } \\quad \\xi = \\frac{2}{(1 + r^2 + z^2)}.",
"$ The Euclidean scalar product is denoted by $\\left\\langle \\cdot ~ \\vert ~ \\cdot \\right\\rangle = g_\\mathrm {eucl}(\\cdot , \\cdot )$ , and the Euclidean norm is $\\left\\Vert \\cdot \\right\\Vert = \\sqrt{\\left\\langle \\cdot ~ \\vert ~ \\cdot \\right\\rangle }$ .",
"The Levi-Civita connection of $g_\\mathrm {stereo}$ is written $\\nabla $ .",
"There are three unit vectors $e_r(q)$ , $e_\\theta (q)$ and $e_z(q)$ for each $q = (x, y, z) \\in \\mathbb {R}^3 \\setminus \\lbrace 0\\rbrace $ , where (in cartesian coordinates) $e_r(q) = (x / \\sqrt{x^2 + y^2}, y / \\sqrt{x^2 + y^2}, 0)$ , $e_z = (0,0,1)$ and $e_\\theta = e_z \\times e_r$ .",
"The Euclidean norm of these vectors is 1.",
"If $p$ is a vector in $\\mathbb {R}^3 \\setminus \\lbrace 0\\rbrace $ , we will write $p_r = \\left\\langle p ~ \\vert ~ e_r \\right\\rangle $ , $p_\\theta = \\left\\langle p ~ \\vert ~ e_\\theta \\right\\rangle $ and $p_z = \\left\\langle p ~ \\vert ~ e_z \\right\\rangle $ .",
"The second fundamental form of the surface $\\Sigma _\\epsilon $ is defined by $\\mathrm {II}_q(v) = \\xi ^2 \\left\\langle \\nabla _v v ~ \\vert ~ N(q) \\right\\rangle $ , where $N(q)$ is the unit normal vector to $\\Sigma _\\epsilon $ for the metric $g_\\mathrm {stereo}$ (thus $\\xi \\left\\Vert N \\right\\Vert = 1$ ).",
"Any geodesic ($q(t), p(t)$ ) on the tube satisfies the equation $ \\nabla _p p = \\mathrm {II}_q(p) N(q).",
"$ When studying a tube, we always assume that $N(q)$ points to the outside of the tube, and write $N_r$ , $N_\\theta $ and $N_z$ its spherical coordinates.",
"By symmetry, at each point $q$ , the two principal curvatures are $k_1 (q) = \\mathrm {II}_q(e_\\theta /\\xi )$ and $k_2 = \\mathrm {II}_q(e_s/\\xi )$ , where $e_s$ is a unit vector which is orthogonal to $e_\\theta $ and tangent to $\\Sigma _\\epsilon $ .",
"The curvature of $\\Sigma $ at $q$ is $k_1 k_2 + 1$ .",
"Lemma 5.1 Consider the waist $W$ of the tube $\\mathcal {T}$ (the smallest horizontal circle contained in the tube), and $\\kappa $ its curvature for the metric $g_\\mathrm {stereo}$ .",
"Then for all $q = (q_r, q_\\theta , q_z)$ in the tube: $\\left|k_1(q) - \\xi N_r(q) \\kappa \\right| \\le m^2$ ; $k_1(q) \\le 0$ .",
"We consider a parametrization $\\mathcal {C}(t)$ of the horizontal circle contained in $\\Sigma $ such that $\\mathcal {C}(0) = q$ , with unit speed (for $g_\\mathrm {stereo}$ ).",
"The principal curvature $k_1$ is $ k_1 = \\xi ^2 \\left\\langle \\nabla _{\\mathcal {C}^{\\prime }(0)} \\mathcal {C}^{\\prime }(0) ~ \\vert ~ N(q) \\right\\rangle .",
"$ Since the circle $\\mathcal {C}$ is close to the circle $W$ , we have $ \\left\\Vert \\nabla _{\\mathcal {C}^{\\prime }(0)} \\mathcal {C}^{\\prime }(0) - \\kappa \\frac{e_r}{\\xi } \\right\\Vert \\le m^3 $ and thus $ \\left|k_1(q) - N_r(q) \\kappa \\right| = \\left|\\xi ^2 \\left\\langle \\nabla _{\\mathcal {C}^{\\prime }(0)} \\mathcal {C}^{\\prime }(0) ~ \\vert ~ N(q) \\right\\rangle - \\xi \\kappa N_r(q) \\right| \\le m^3 \\xi ^2 \\le m^2 $ which proves the first statement.",
"We now prove the second statement: by symmetry, we may assume that $q_z \\ge 0$ .",
"Writing $\\nabla _{\\mathcal {C}^{\\prime }(0)} \\mathcal {C}^{\\prime }(0) = (r, \\theta , z)$ , we obtain $k_1 = \\xi ^2(r N_r(q) + z N_z(q))$ .",
"Notice that $r \\le 0$ and $z \\ge 0$ .",
"At the same time, $N_r(q) \\ge 0$ and $N_z(q) \\le 0$ .",
"Thus, $k_1 \\le 0$ .",
"Lemma 5.2 Consider a smooth curve $q(t)$ in $\\Sigma _\\epsilon $ and $p(t) = \\dot{q}(t)$ .",
"Then: $ \\left|\\nabla _{p(t)} p(t) - \\dot{p}(t) \\right| \\le \\left\\Vert p(t) \\right\\Vert ^2 / m^{1/10}.",
"$ $ \\nabla _{p(t)} p(t) = \\sum _{1 \\le i, j, k \\le 3} p_i(t) p_j(t) \\Gamma _{ij}^k e_k + \\dot{p}(t), $ where $p_1$ , $p_2$ , $p_3$ mean respectively $p_r$ , $p_\\theta $ and $p_z$ , and $e_1$ , $e_2$ , $e_3$ mean respectively $e_r$ , $e_\\theta $ and $e_z$ , and $\\Gamma _{ij}^k$ are the Christoffel symbols of $\\nabla $ .",
"Thus $ \\left|\\nabla _{p(t)} p(t) - \\dot{p}(t) \\right| \\le \\sum _{1 \\le i, j, k \\le 3} \\left\\Vert p(t) \\right\\Vert ^2 \\left|\\Gamma _{ij}^k \\right| \\le \\left\\Vert p(t) \\right\\Vert ^2 / m^{1/10}.",
"$ Lemma 5.3 Consider the curve $c: [a,b] \\rightarrow \\mathbb {R}^2$ , $c(t) = (r(t), z(t))$ , which is a unit-speed parametrization (for the metric $g_\\mathrm {stereo}$ ) of the curve $\\mathcal {S}$ .",
"Assume that $0 \\in (a, b)$ , $c(0) = R$ and $c^{\\prime }(0) = (0,1)$ .",
"Recall that the curvature of $c$ at 0 is nonzero.",
"For all $t \\in (a, b)$ , we have $r(t) \\ge R$ , $r(-t) = r(t)$ and $z(-t) = -z(t)$ .",
"The unit tangent vector to $c(t)$ is $T(t) = (T_r(t), T_z(t))$ (that is, $g_\\mathrm {eucl}(T, T) = 1$ ), and the normal vector is $N = (T_z(t), -T_r(t))$ .",
"The curvature of $c$ for the metric $g_\\mathrm {stereo}$ is written $k_\\mathrm {stereo}$ .",
"Then there exists $t_c$ such that for all $t \\in (0, t_c)$ , $T_z(t) \\ge m / 2$ and $k_\\mathrm {stereo}(t) \\ge m^5 / \\epsilon ^2$ , for all $t \\in (t_c, m)$ , $T_z(t) \\le 2 m$ and $k_\\mathrm {stereo}(t) \\ge - 1 / m^{1/4}$ .",
"$\\int _a^b k_\\mathrm {stereo} (t) dt \\le 1/m^{1/8}$ .",
"In the following, we will write $r_c = r(t_c)$ .",
"The strategy is to reduce the problem to a flat one, and then apply Lemma REF .",
"Consider the mapping $\\Phi : (0,1) \\times (- \\sqrt{\\epsilon }, \\sqrt{\\epsilon }) \\rightarrow \\mathbb {R}^2$ , whose restriction to the line $z = 0$ is the identity, which is a bijection onto its image, and such that $\\Phi ^{-1} \\circ f_\\epsilon \\circ \\Phi $ coincides with the affine map $A: (r, z) \\mapsto (r, \\epsilon z)$ on its domain.",
"Then $ \\sup _{\\mathbb {R} \\times (- \\sqrt{\\epsilon }, \\sqrt{\\epsilon })} \\left\\Vert \\Phi - \\mathrm {Id} \\right\\Vert < m^2 \\quad \\text{ and } \\quad \\sup _{\\mathbb {R} \\times (- \\sqrt{\\epsilon }, \\sqrt{\\epsilon })} \\left\\Vert D\\Phi - D\\mathrm {Id} \\right\\Vert < m^2 $ (“$\\Phi $ is $C^1$ -close to the identity”).",
"We will write $\\tilde{c}(t) = \\Phi ^{-1} \\circ c(t)$ .",
"Notice that $\\tilde{c}$ is a “flattened curve” in the Euclidean sense, and thus, Lemma REF applies to $\\tilde{c}$ .",
"We denote by $s$ the arc length of $c$ for the metric $g_\\mathrm {eucl}$ , and consider the curvature $k_\\mathrm {eucl}$ of $c$ for the metric $g_\\mathrm {eucl}$ .",
"Similarly, we denote by $\\tilde{s}$ the arc length of $\\tilde{c}$ for the metric $g_\\mathrm {eucl}$ , and consider the curvature $\\tilde{k}_\\mathrm {eucl}$ of $\\tilde{c}$ for the metric $g_\\mathrm {eucl}$ .",
"Also, $\\tilde{T}$ (resp.",
"$\\tilde{N}$ ) is the unit tangent (resp.",
"normal) vector to $\\tilde{c}$ for the metric $g_\\mathrm {eucl}$ .",
"Then: $ \\begin{aligned}k_\\mathrm {eucl} & = \\left\\langle \\frac{d \\left(D\\Phi (\\tilde{c}(t)) \\cdot \\frac{\\tilde{T}}{\\left\\Vert D\\Phi (\\tilde{c}(t)) \\cdot \\tilde{T} \\right\\Vert }\\right)}{d\\tilde{s}} \\cdot \\frac{d \\tilde{s}}{ds} ~ \\vert ~ N \\right\\rangle \\\\ & = \\left\\langle D^2 \\Phi (c(t)) \\cdot \\left(\\frac{\\tilde{T}}{\\left\\Vert D\\Phi (\\tilde{c}(t)) \\cdot \\tilde{T} \\right\\Vert }, \\tilde{T} \\right) \\cdot \\frac{d \\tilde{s}}{ds} ~ \\vert ~ N \\right\\rangle \\\\ & + \\left\\langle D\\Phi (\\tilde{c}(t)) \\cdot \\left( \\frac{d\\tilde{T}}{d\\tilde{s}} \\cdot \\frac{1}{\\left\\Vert D\\Phi (\\tilde{c}(t)) \\cdot \\tilde{T} \\right\\Vert } - \\tilde{T} \\frac{d}{ds}\\left(\\frac{1}{\\left\\Vert D\\Phi (\\tilde{c}(t)) \\cdot \\tilde{T} \\right\\Vert } \\right)\\right) \\cdot \\frac{d \\tilde{s}}{ds} ~ \\vert ~ N \\right\\rangle \\\\ & \\ge m^{1/10} \\tilde{k}_\\mathrm {eucl} (t) - \\frac{1}{m^{1/10}}.\\end{aligned}$ On the other hand, using Lemma REF : $ \\begin{aligned} \\left|k_\\mathrm {stereo} - \\frac{1}{\\xi } k_\\mathrm {eucl} \\right| & = \\left|\\xi ^2 \\left\\langle \\nabla _{T/\\xi } T/\\xi ~ \\vert ~ N/\\xi \\right\\rangle - \\frac{1}{\\xi } \\left\\langle \\frac{dT}{ds} ~ \\vert ~ N \\right\\rangle \\right|\\\\ & = \\left|\\frac{1}{\\xi } \\left\\langle \\nabla _T T - \\frac{dT}{ds} ~ \\vert ~ N \\right\\rangle \\right|\\\\ & \\le \\frac{1}{\\xi m^{1/10}}.",
"\\end{aligned} $ Hence $ k_\\mathrm {stereo}(t) \\ge m^{1/9} \\cdot k_\\mathrm {eucl}(t) - \\frac{1}{m^{1/9}}.",
"$ Finally, $ k_\\mathrm {stereo}(t) \\ge m^{1/4} \\cdot \\tilde{k}_\\mathrm {eucl}(t) - \\frac{1}{m^{1/4}} $ and thus Lemma REF applied to the curve $\\tilde{c}$ allows us to prove Statements 1 and 2.",
"We now prove Statement 3.",
"From the inequality $ \\left|k_\\mathrm {stereo} - \\frac{1}{\\xi } k_\\mathrm {eucl} \\right| \\le \\frac{1}{\\xi m^{1/10}}$ we also obtain $ k_\\mathrm {stereo} \\le \\frac{1}{\\xi } k_\\mathrm {eucl} + \\frac{1}{\\xi m^{1/10}} $ and thus $ \\begin{aligned} \\int _a^b k_\\mathrm {stereo} (t) dt & \\le \\int _a^b (k_\\mathrm {eucl} (t) + \\frac{1}{m^{1/10}}) \\frac{ds}{dt} dt \\\\ & \\le \\left(\\int _a^b 2\\frac{d\\alpha }{ds} ds + (b-a)\\frac{1}{m^{1/9}} \\right) \\\\ & \\le 4\\pi + \\frac{(b-a)}{m^{1/9}} \\le 1/m^{1/8}.",
"\\end{aligned} $"
],
[
"The dynamics in the tube",
"In this section, we consider a unit speed geodesic $(q(t), p(t))$ in $T^1 \\mathcal {T}$ , where $q$ is the position and $p$ is the speed.",
"We will write $(q_r, q_\\theta , q_z)$ and $(p_r, p_\\theta , p_z)$ the cylindric coordinates of $q$ and $p$ .",
"Also, we write $p_s = \\sqrt{p_r^2 + p_z^2}$ and define $e_s$ as the unit vector such that $p_s = \\left\\langle p ~ \\vert ~ e_s \\right\\rangle $ .",
"The field $r e_\\theta $ is a Killing field on $\\mathcal {T}$ .",
"Thus, the quantity $L = g(r e_\\theta , p) = \\xi ^2 r p_\\theta $ is constant on each geodesic (this is the Clairaut first integral).",
"The geodesic starts on the boundary of the tube at $t = t^\\mathrm {in}$ and exits at $t = t^\\mathrm {out}$ (if the geodesic does not exit the tube, we write $t^\\mathrm {out} = +\\infty $ ).",
"Lemma 6.1 For all $t \\in [t^\\mathrm {in}, t^\\mathrm {out}]$ , we have $ \\left|p_s(t) - p_s(t^\\mathrm {in}) \\right| \\le m^{10}.",
"$ For all $t$ , $ p_s^2 = 1/\\xi ^2 - p_\\theta ^2 = 1/\\xi ^2 - \\frac{L^2}{\\xi ^4 r^2}.",
"$ The coordinate $r$ varies between $R$ and $R + \\delta $ .",
"Moreover, knowing that $z \\le \\delta $ (with $\\epsilon $ sufficiently small), the quantity $\\xi $ varies between $\\frac{2}{1 + (R+\\delta )^2 + \\delta ^2}$ and $\\frac{2}{1 + R^2}$ .",
"Thus the variation of $p_s^2$ is less than $m^{10}$ .",
"Lemma 6.2 Assume that $\\left|p_s(t^\\mathrm {in}) \\right| \\ge m$ .",
"Then the time spent in the tube is smaller than $6 \\delta / m$ .",
"The length of the curve $c^\\epsilon $ is smaller than $3\\delta $ .",
"Moreover, Lemma REF implies that $\\left|p_s(t) \\right| \\ge m/2$ for all $t$ .",
"In particular, $ds/dt$ does not change sign.",
"Thus, the time spent in the tube is smaller than $3\\delta / (ds/dt) \\le 6 \\delta /m$ .",
"Lemma 6.3 Assume that $\\left|p_s(t^\\mathrm {in}) \\right| \\ge m$ .",
"Then $ \\left| \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} K(t) dt - \\frac{2 \\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right| \\le m^{1/3}.",
"$ Let us divide the problem into several steps by using the triangle inequality (the integrals are taken between the times $t^\\mathrm {in}$ and $t^\\mathrm {out}$ ).",
"$ \\begin{aligned} & \\left| \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} K(t) dt - \\frac{2 \\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right|\\\\ & \\le \\left|\\int K(t)dt - \\int k_1 k_2 dt \\right| + \\left|\\int k_1 k_2 dt - \\int k_2 \\xi N_r \\kappa dt \\right|\\\\ & + \\left|\\int k_2 \\xi N_r \\kappa dt - \\int k_2 \\xi \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right|\\\\ & + \\left|\\int k_2 \\xi \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\int \\frac{1}{\\xi (t) p_s(t)^2} \\mathrm {II}_q(p) \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right|\\\\ & + \\left|\\int \\frac{1}{\\xi (t) p_s(t)^2} \\mathrm {II}_q(p) \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\int \\frac{1}{\\xi (t) p_s(t)^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right|\\\\ & + \\left|\\int \\frac{1}{\\xi (t) p_s(t)^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\int \\frac{1}{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right|\\\\ & + \\left|\\int \\frac{1}{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\frac{2\\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right|.\\end{aligned} $ We will now show that each of the terms is smaller than $\\sqrt{m}$ .",
"With Lemma REF , $ \\left|\\int K(t)dt - \\int k_1 k_2 dt \\right| = \\left|\\int 1 dt \\right| \\le 6\\delta / m $ With Lemma REF , $ \\left|\\int k_1 k_2 dt - \\int k_2 \\xi N_r \\kappa dt \\right| \\le m\\int k_2 dt = m \\int \\frac{k_2}{p_s} ds \\le 2 \\int k_2 ds $ Moreover, with Lemma REF , $\\int k_2 ds \\le 1 / m^{1/8}$ , so $ \\left|\\int K(t) - \\int k_2 \\xi N_r \\kappa dt \\right| \\le \\sqrt{m}.",
"$ With Lemma REF , we know that $e_r(t) - e_r(t^\\mathrm {in}) \\le \\sqrt{\\delta }$ .",
"Thus: $ \\begin{aligned} \\left|\\int k_2 \\xi N_r \\kappa dt - \\int k_2 \\xi \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right| & = \\left|\\int k_2 \\xi \\left\\langle N ~ \\vert ~ e_r(t) - e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right| \\\\ & \\le \\delta ^{1/4} \\int k_2 ds \\\\ & \\le \\sqrt{m}.",
"\\end{aligned} $ Using the fact that $\\mathrm {II}_q(p) = \\xi ^2 (k_1 p_\\theta ^2 + k_2 p_s^2)$ , we obtain: $ \\begin{aligned} ~ & \\left|\\int k_2 \\xi \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\int \\frac{1}{\\xi (t) p_s(t)^2} \\mathrm {II}_q(p) \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right| \\\\ & = \\left|\\int k_1 \\frac{p_\\theta ^2}{p_s^2} \\xi \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right| \\\\ & \\le \\left|\\int \\xi \\kappa (1 + m) \\frac{1}{\\xi ^2(m/2)^2} \\xi \\kappa dt \\right| \\le \\sqrt{m} \\end{aligned} $ With Lemma REF , since $\\mathrm {II}_q(p)N = \\nabla _p p$ , we have: $ \\begin{aligned} & \\left|\\int \\frac{1}{\\xi (t) p_s(t)^2} \\mathrm {II}_q(p) \\left\\langle N ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\int \\frac{1}{\\xi (t) p_s(t)^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right|\\\\ & \\le \\int \\frac{\\kappa }{\\xi (m/2)^2 m}dt \\le \\sqrt{m} \\end{aligned} $ We use Lemma REF : $ \\begin{aligned} & \\left|\\int \\frac{1}{\\xi (t) p_s(t)^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\int \\frac{1}{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt \\right| \\\\ & \\le \\left|m^{2/3} \\int \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\right| \\le m^{2/3} \\left\\Vert p(t^\\mathrm {out}) - p(t^\\mathrm {in}) \\right\\Vert \\le \\sqrt{m} \\end{aligned} $ Since the tube is symmetric, we have $\\left\\langle p_r(t^\\mathrm {in}) ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle = - \\left\\langle p_r(t^\\mathrm {out}) ~ \\vert ~ e_r(t^\\mathrm {out}) \\right\\rangle $ .",
"Thus: $ \\begin{aligned} & \\left|\\int \\frac{1}{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})^2} \\left\\langle \\dot{p} ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle \\kappa dt - \\frac{2\\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right|\\\\ & = \\left|\\frac{1}{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})^2} \\kappa \\right| \\left| \\left\\langle p(t^\\mathrm {out}) - p(t^\\mathrm {in}) ~ \\vert ~ e_r(t^\\mathrm {in}) \\right\\rangle - 2 \\left\\langle p(t^\\mathrm {in}) ~ \\vert ~ e_s(t^\\mathrm {in}) \\right\\rangle \\right|\\\\ & \\le \\frac{2}{m} \\kappa m^2 \\le \\sqrt{m}.",
"\\end{aligned} $ In the following lemma, we consider the constant $r_c$ given by Lemma REF .",
"Lemma 6.4 In the tube, the time during which $r \\ge r_c$ is smaller than $\\delta ^{1/3}$ (in other words, $\\int _{r \\ge r_c} dt \\le \\delta ^{1/3}$ ).",
"First, we compute for $r \\ge r_c$ : $ \\begin{aligned} \\frac{dr}{dt} & = p_r\\\\ & = \\pm N_z \\xi p_s\\\\ & = \\pm N_z \\xi \\sqrt{\\frac{1}{\\xi ^2} - p_\\theta ^2}\\\\ & = \\pm N_z \\xi \\sqrt{\\frac{1}{\\xi ^2} - \\frac{L^2}{\\xi ^4 r^2}}\\\\ & = \\pm N_z \\sqrt{1 - L^2 f(r)} \\end{aligned} $ where $f(r) = \\frac{(1 + r^2 + h(r)^2)^2}{4r^2}$ .",
"We compute: $ \\begin{aligned} f^{\\prime }(r) & = 2(1 + r^2 + h(r)^2) (r^2 - 1 + h(r) (2h^{\\prime }(r) - h(r)))/r^3\\\\ & \\le - \\delta ^{1/10}.",
"\\end{aligned} $ Thus for all $r_0 \\ge r_c$ we may write: $ f(r) \\le f(r_0) - \\delta ^{1/10} (r-r_0).",
"$ Moreover, $f(r) \\le \\delta ^{-1/20}$ ."
],
[
"First case.",
"We assume that there exists $r_0 \\ge r_c$ such that $1 - L^2 f(r_0) = 0$ .",
"Then the geodesic has the following life: $r$ decreases from $R + \\delta $ to $r_0$ , reaches $r_0$ at some time $t_0$ , and then increases from $r_0$ to $R + \\delta $ .",
"In this case, the length of the geodesic is $ \\begin{aligned} 2 (t_0 - t^\\mathrm {in})& = 2 \\int _{t^\\mathrm {in}}^{t_0} dt\\\\ & = 2 \\int _{r_0}^{R + \\delta } \\frac{1}{- dr / dt} dr\\\\ & = 2\\int _{r_0}^{R + \\delta } \\frac{1}{\\left|N_z \\right| \\sqrt{1 - L^2 f(r)}} dr\\\\ & \\le 2 \\int _{r_0}^{R + \\delta } \\frac{2}{\\sqrt{1 - L^2 f(r_0) - L^2 \\delta ^{1/10} (r - r_0)}} dr\\\\ & \\le 4 \\int _{r_0}^{R + \\delta } \\frac{1}{\\sqrt{L^2 \\delta ^{1/10} (r - r_0)}} dr\\\\ & \\le \\frac{8 \\sqrt{R + \\delta - r_0}}{L \\delta ^{1/20}} \\end{aligned}$ Since $1 - L^2 f(r_0) = 0$ , we have $L = 1 / f(r_0)$ and thus $L \\ge \\delta ^{1/20}$ .",
"Hence $ 2 (t^\\mathrm {out} - t_0) \\le 8 \\frac{\\sqrt{\\delta }}{\\delta ^{1/10}} \\le \\delta ^{1/3}.",
"$"
],
[
"Second case.",
"We assume that $1 - L^2 f(r) > 0$ for all $r \\in (r_c, R + \\delta )$ .",
"Then the geodesic goes through the zone $r \\ge r_c$ and enters the zone $r \\le r_c$ at some time $t_c$ .",
"Then either it remains in this zone for all times, or it exits this zone and goes through the zone $r \\ge r_c$ once more.",
"Thus, the time spent in the zone $r \\ge r_c$ is at most $ \\begin{aligned} 2 \\int _{t^\\mathrm {in}}^{t} dt & = 2 \\int _{r_c}^{R + \\delta } \\frac{1}{- dr / dt} dr\\\\ & = 2\\int _{r_0}^{R + \\delta } \\frac{1}{\\left|N_z \\right| \\sqrt{1 - L^2 f(r)}} dr.\\end{aligned}$ If $L \\le \\delta ^{1/20}$ then $ 2 \\int _{t^\\mathrm {in}}^{t} dt \\le 4 \\int _{r_c}^{R + \\delta } \\frac{1}{\\sqrt{1 - \\delta ^{1/10} \\delta ^{-1/20}}} dr \\le \\delta ^{1/3} $ which concludes the proof.",
"Now assuming that $L \\ge \\delta ^{1/20}$ , we compute: $ \\begin{aligned}2 \\int _{t^\\mathrm {in}}^{t} dt & \\le 4 \\int _{r_c}^{R + \\delta } \\frac{1}{\\sqrt{1 - L^2 f(r_c) - L^2 \\delta ^{1/10} (r - r_c)}} dr\\\\ & \\le \\frac{8 \\sqrt{R + \\delta - r_c}}{L \\delta ^{1/20}}\\\\ & \\le \\delta ^{1/3}\\end{aligned}$ Lemma 6.5 The Gauss curvature $K$ of $\\Sigma _\\epsilon $ satisfies $K \\le \\frac{1}{m^{1/3}}$ .",
"Moreover, $K \\le -1/\\epsilon $ in the zone $r \\le r_c$ .",
"In the zone $r \\ge r_c$ , we have $k_1 \\le 0$ (Lemma REF ) and $k_2 \\ge -1/m^{1/4}$ (Lemma REF ), so that $K = k_1 k_2 + 1 \\le 1 / m^{1/3}$ .",
"In the zone $r \\le r_c$ , we have $k_1 \\le - \\kappa m / 2$ (Lemmas REF and REF ) and $k_2 \\ge m^5/\\epsilon ^2$ .",
"Thus, in this zone $K = k_1 k_2 + 1 \\le - \\kappa m^6 / (2\\epsilon ^2) + 1$ .",
"In particular, $K \\le -1/\\epsilon $ .",
"Lemma 6.6 In this lemma, we consider a geodesic $(q(t), p(t))_{t \\in [t^\\mathrm {in}, t^\\mathrm {out}]}$ in the tube $\\mathcal {T}$ , but we do not assume that $q(t^\\mathrm {in})$ or $q(t^\\mathrm {out})$ is on the boundary of $\\mathcal {T}$ .",
"Consider a solution $u$ of the Riccati equation $u^{\\prime }(t) = -K(t) - u(t)^2$ such that $\\left|u(t^\\mathrm {in}) \\right| \\le 1/m^2$ .",
"Then $u(t^\\mathrm {out}) \\ge u(t^\\mathrm {in}) - m$ ; if the time spent in the tube $\\mathcal {T}$ is at least $m$ , then $u(t^\\mathrm {out}) \\ge 1/m^2$ .",
"Let $t^1 = \\sup \\left\\lbrace t \\in [t^\\mathrm {in}, t^\\mathrm {out}] ~ \\vert ~ u(t) \\ge 2/m^2 \\right\\rbrace $ (if this set is empty, let $t^1 = t^\\mathrm {in}$ ), and $t^2 = \\inf \\left\\lbrace t \\in [t^1, t^\\mathrm {out}] ~ \\vert ~ u(t) \\le - 2/m^2 \\right\\rbrace $ (if this set is empty, let $t^2 = t^\\mathrm {out}$ ).",
"There is a (possibly empty) interval $(t^3, t^4) \\subseteq (t^1, t^\\mathrm {out})$ such that, for all $t \\in (t^1, t^\\mathrm {out})$ , $r(t) < r_c$ if and only if $t \\in (t^3, t^4)$ .",
"Assume that $t^2 \\le t^3$ .",
"Then using Lemmas REF and REF , we have: $ \\begin{aligned} u(t^2) & = u(t^1) + \\int _{t^1}^{t^2} - K(t) - u(t)^2 dt\\\\ & \\ge u(t^1) - \\frac{\\delta ^{1/3}}{m^{1/3}} - \\frac{4\\delta ^{1/3}}{m^4}\\\\ & \\ge u(t^1) - m/2 \\end{aligned} $ which contradicts the fact that $u(t^2) \\le -2/m^2$ .",
"Thus, $t^2 \\ge t^3$ and $u(t^3) \\ge u(t^1) - m/2$ .",
"For $t \\in (t^3, t^4)$ , we have: $ u^{\\prime }(t) = -K(t) - u(t)^2 \\ge \\frac{1}{\\epsilon } - 4 / m^4 \\ge 1/m^5 $ which implies that $t^2 \\ge t^4$ and $u(t^4) \\ge u(t^1) - m/2 + (t^4 - t^3)/m^5$ .",
"Moreover, $u(t^4) \\le 2 / m^2$ (because $t^1 \\le t^4$ ) so $t^4 - t^3 \\le m^2$ .",
"If $t^\\mathrm {out} - t^\\mathrm {in} \\ge m$ , this implies that $t^1 > t^\\mathrm {in}$ and thus $u(t^1) \\ge 2/m^2$ .",
"Finally, $ \\begin{aligned} u(t^2) & = u(t^4) + \\int _{t^4}^{t^2} - K(t) - u(t)^2 dt\\\\ & \\ge u(t^1) - m/2 - \\frac{\\delta ^{1/3}}{m^{1/3}} - \\frac{4\\delta ^{1/3}}{m^4}\\\\ & \\ge u(t^1) - m \\end{aligned} $ and thus $t^2 = t^\\mathrm {out}$ and $u(t^\\mathrm {out}) \\ge u(t^1) - m \\ge u(t^1) - m$ .",
"If the time spent in $\\mathcal {T}$ is at least $m$ , then $u(t^1) \\ge 2/m^2$ and thus $u(t^\\mathrm {out}) \\ge 1/m^2$ .",
"Lemma 6.7 Assume that $\\left|p_s(t^\\mathrm {in}) \\right| \\ge m$ .",
"Consider a solution $u$ of the Riccati equation $u^{\\prime }(t) = -K(t) - u(t)^2$ such that $\\left|u(t^\\mathrm {in}) \\right| \\le 2/m^2$ .",
"Then: $ \\left|u(t^\\mathrm {out}) - u(t^\\mathrm {in}) - \\frac{2 \\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right| \\le m^{1/4}.",
"$ Let $t^1 = \\inf \\left\\lbrace t \\in [t^\\mathrm {in}, t^\\mathrm {out}] ~ \\vert ~ \\left|u(t^\\mathrm {in}) \\right| \\ge 2/m^2 \\right\\rbrace $ (if this set is empty, let $t^1 = t^\\mathrm {out}$ ).",
"We write $K = K^+ - K^-$ , where $K^+ = \\max (K, 0)$ is the positive part of $K$ and $K^- = max (-K, 0)$ is the negative part.",
"Then, using Lemmas REF , REF and REF , $ \\begin{aligned} u(t^1) & = u(t^\\mathrm {in}) + \\int _{t^\\mathrm {in}}^{t^1} -K(t) - u(t)^2 dt\\\\ \\left|u(t^1) \\right| & \\le \\left|u(t^\\mathrm {in}) \\right| + \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} \\left|K(t) \\right| + \\left|u(t) \\right|^2 dt\\\\ \\left|u(t^1) \\right| & \\le \\left|u(t^\\mathrm {in}) \\right| + 2 \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} K^+(t) dt - \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} K(t) dt + \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} \\left|u(t) \\right|^2 dt\\\\ & \\le \\frac{1}{m^2} + 2 \\cdot \\frac{6\\delta }{m} \\cdot \\frac{1}{m^{1/3}} + \\frac{2 \\left|\\kappa \\right|}{m} + m^{1/3} + \\frac{6\\delta }{m} \\cdot \\frac{4}{m^4}\\\\ & < \\frac{2}{m^2}\\end{aligned} $ Thus $t^1 = t^\\mathrm {out}$ and $ \\small \\begin{aligned} \\left|u(t^\\mathrm {out}) - u(t^\\mathrm {in}) - \\frac{2 \\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right| & = \\left|u(t^\\mathrm {out}) - u(t^\\mathrm {in}) - \\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} K(t)dt \\right| + \\left|\\int _{t^\\mathrm {in}}^{t^\\mathrm {out}} K(t)dt - \\frac{2 \\kappa }{\\xi (t^\\mathrm {in}) p_s(t^\\mathrm {in})} \\right| \\\\ & \\le \\frac{6\\delta }{m} \\cdot \\frac{4}{m^4} + m^{1/3}\\\\ & \\le m^{1/4} \\end{aligned} $"
],
[
"End of the proof of Theorem ",
"Theorem 7.1 We say that a curve $\\varphi : [a, b] \\rightarrow D$ in the billiard $D$ is “$\\eta $ -almost a geodesic” if $\\left\\Vert \\phi ^{\\prime }(t) \\right\\Vert _g \\le 1$ for all $t \\in [a, b]$ , $d(\\varphi (b), \\varphi (a)) \\ge b - a - \\eta $ , where $d$ is the Riemannian distance and $g$ the Riemannian metric in the sphere $\\mathbb {S}^2$ .",
"Consider $H$ the horizon of $D$ .",
"Then there exists $\\eta > 0$ such that for all $\\eta $ -almost geodesic $\\varphi $ , $a-b < H + \\nu $ .",
"Assume that the conclusion is false.",
"Then there exists a sequence $\\varphi _n$ of $\\frac{1}{n}$ -almost geodesics such that $a = 0$ and $b = H + \\nu $ .",
"By the Arzelà-Ascoli theorem, the sequence $\\varphi _n$ converges in the $C^0$ -topology to a curve $\\varphi : [0,H+\\nu ] \\rightarrow D$ which is a real geodesic in $D$ .",
"This contradicts the definition of $H$ .",
"[End of the proof of Theorem REF ] Consider a geodesic $(q(t), p(t))_{t \\in [0, H + 2 \\nu ]}$ .",
"During its lifetime, the geodesic enters and exits the tubes.",
"We will say that the tube is almost avoided if the two following conditions are satisfied: $\\left|p_s(t^\\mathrm {in}) \\right| \\le m$ ; $t^\\mathrm {out} - t^\\mathrm {in} \\le m$ .",
"Lemma 7.2 If the geodesic almost avoids all the tubes in a time interval $(t^1, t^2)$ , then $t^2 - t^1 \\le H + \\nu $ .",
"The geodesic's projection $\\pi \\circ q$ is $\\nu $ -almost a geodesic in $\\mathbb {S}^2$ , so we may apply Lemma REF .",
"Now, consider an increasing sequence of times $(t_k)_{k \\in \\mathbb {Z}}$ such that: For each $k \\in \\mathbb {Z}$ , either $t_k$ is a time at which the geodesic exits a tube which is not almost avoided (“type A”), or $t_k = t_{k-1} + H + 2 \\nu $ (“type B”) ; If the geodesic exits a tube which is not almost avoided at a time $t^\\mathrm {out}$ , then there exists $k \\in \\mathbb {Z}$ such that $t_k = t^\\mathrm {out}$ .",
"For all $k \\in \\mathbb {Z}$ , $\\nu \\le t_{k+1} - t_k \\le H + 3 \\nu $ .",
"According to Theorem REF , we need to show that for any $k \\in \\mathbb {Z}$ and any $u$ solution of the Riccati equation along the geodesic $(p(t), q(t))$ with initial condition $u(t_k) = 0$ , the solution $u$ is well-defined on $[t_k, t_{k+1}]$ and $u(t_{k+1}) > m$ .",
"In the sphere, since the curvature is 1, the geodesics follow the Riccati equation $u^{\\prime }(t) = -1 - u(t)^2$ .",
"If $q(t)$ remains outside the tubes in the time interval $(t^1, t^2)$ , since the metric on $\\Sigma _\\epsilon $ is close to the metric of the Euclidean sphere, we have $u(t^2) \\ge \\tan (\\arctan (u(t^1)) - t^2) - \\nu $ .",
"This is also the case if one assumes that $q(t)$ almost avoids all the tubes in this time interval (by Lemma REF )."
],
[
"First case.",
"If $t_k$ is of type A, then consider the first time $t^\\mathrm {in}$ (with $t^\\mathrm {in} \\in [t_k, t_{k+1}]$ ) at which the geodesic enters a tube which is not almost avoided (such a time exists by Lemma REF ).",
"Then $u(t^\\mathrm {in}) \\ge - \\tan (H + \\nu ) - \\nu $ .",
"Then, by Lemmas REF and REF , $ u(t_{k+1}) \\ge u(t^\\mathrm {in}) + 2 \\tan (\\pi /2 - A) - \\nu \\ge - \\tan (H + \\nu ) + 2 \\tan (\\pi /2 - A) - 2\\nu \\ge m. $"
],
[
"Second case.",
"If $t_k$ is of type B, notice that $q(t_k)$ is inside a tube which is not almost avoided, because of Lemma REF .",
"Therefore, the geodesic remains in the tube during the interval $[t_k + H + \\nu , t_{k+1}]$ .",
"Since $t_{k+1} - (t_k + H + \\nu ) \\ge \\nu $ , we may apply Lemma REF and obtain: $u(t_{k+1}) \\ge 1/m^2$ .",
"Thus Theorem REF applies, and Theorem REF is proved."
],
[
"Embedding surfaces of genus at least 11",
"Consider a billiard $D_R$ obtained from 12 circles of equal radius $R$ whose centers are the vertices of a icosahedron which is inscribed in $\\mathbb {S}^2$ (Figure REF ).",
"The circles are disjoint if and only if $R < R_0 = \\arctan (2)/2$ .",
"The horizon $H_R$ of the billiard $D_R$ depends on $R$ .",
"Proposition 8.1 $ H_R \\underset{R \\rightarrow R_0}{\\rightarrow } \\pi - 2\\arctan (2).",
"$ Consider a sequence $R_n$ such that $R_n \\underset{n \\rightarrow \\infty }{\\rightarrow } R_0$ , and a sequence $\\gamma _n$ of portions of geodesics of $\\mathbb {S}^2$ of maximal length, which are contained in $D_{R_n}$ .",
"Then there is a subsequence of $\\gamma _n$ which converges uniformly to the portion of geodesic represented on Figure REF , whose length is $\\pi - 2\\arctan (2)$ .",
"Figure: The limit of a sequence of geodesics of maximum lengths in D R n D_{R_n}.Thus $ R + H_R \\underset{R \\rightarrow R_0}{\\rightarrow } \\pi - \\frac{3}{2} \\arctan (2) < \\pi /2.",
"$ Therefore, for $R$ sufficiently close to $R_0$ , we have $R + H_R < \\pi /2$ , and thus $2 \\tan (\\pi /2 - R) \\ge \\tan (\\pi /2 - R) > \\tan (H_R)$ .",
"By applying Theorem REF , we obtain Corollary 8.2 The billiard $D$ is uniformly hyperbolic.",
"To obtain a billiard with $n$ obstacles ($n \\ge 12$ ), one may add spherical obstacles with small radii, which are disjoint from the others.",
"Thus: Corollary 8.3 For any $n \\ge 12$ , there exists a spherical billiard with exactly $n$ circular obstacles, which is uniformly hyperbolic.",
"Finally, Theorem REF completes the proof of Theorem REF ."
],
[
"The Euclidean case: proof of Theorem ",
"We will study the geodesics in the “tube” $V_{i_0}^{\\delta _1}$ for some fixed $i_0$ .",
"We assume (after rotation) that the center $q_0$ of the disk $\\Delta _{i_0}$ is on the $z$ -axis.",
"The tube is a surface of revolution in $\\mathbb {R}^3$ , obtained by rotation of a curve $\\gamma $ along the $z$ -axis.",
"We will use the cylindric coordinates $(r, \\theta , z)$ .",
"An essential difference with the spherical case is that the curve $\\gamma $ is not invariant by a symmetry with respect to a horizontal plane.",
"We assume that $\\gamma (s)$ is parametrized by arc length, that $\\gamma $ is in the half-plane $\\lbrace \\theta = 0\\rbrace $ , and that $\\gamma (0) \\in \\partial \\Delta _{i_0}$ , with $\\gamma ^{\\prime }(0) = e_z$ .",
"Denote by $\\alpha (s)$ the angle of the tangent vector $\\gamma ^{\\prime }(s)$ with the unit vector $e_r$ , and consider the curvature $k(s) = \\frac{d\\alpha }{ds}$ .",
"Writing $\\gamma (s) = (\\gamma _r(s), \\gamma _z(s))$ , with the convention that the normal vector at $\\gamma (0)$ is $e_r$ , the principal curvatures of the surface $\\Sigma $ at a point $\\gamma (s)$ are $k_1 = -\\frac{\\cos (\\alpha (s) - \\pi /2)}{\\gamma _r(s)}$ and $k_2 = - k(s)$ ; thus the Gauss curvature is $ K(s) = \\frac{k(s)\\cos (\\alpha (s)-\\pi /2)}{\\gamma _r(s)} = \\frac{\\frac{d\\alpha }{ds} \\sin \\alpha (s)}{\\gamma _r(s)} = - \\frac{d(\\cos \\alpha )/ds}{\\gamma _r(s)}.",
"$ Assumption 2 of the theorem implies that $K(s) \\ge 0$ and $\\alpha (s) < - r_{i_0}/2$ for all $s \\in (\\delta _2, \\delta _1)$ .",
"Moreover, since $\\alpha (0) = \\pi /2$ , there exists $\\delta _3 \\in (0, \\delta _2)$ such that $\\alpha (\\delta _3) = 0$ .",
"Consider a geodesic $(p(t), q(t))$ in $\\Sigma $ , which enters the tube at a time $-t_1$ and exits at time $t_1$ .",
"The symmetry assumption implies that the quantity $L = \\left\\langle r(t) e_\\theta (t) ~ \\vert ~ p(t) \\right\\rangle = r(t) p_\\theta (t)$ is constant on each unit speed geodesic.",
"We assume that $L = \\gamma _r(\\delta _3)$ (the intermediate value theorem guarantees the existence of such a geodesic).",
"We will consider $s(t)$ such that $(r(t), z(t)) = \\gamma (s(t))$ , and write $K(s)$ the Gauss curvature at $q(s(t))$ .",
"We have $p_s = \\sqrt{1 - \\frac{L^2}{r^2}}$ ; moreover, $s(0) = \\delta _3$ and there is a unique time $t_2$ at which $s(t_2) = \\delta _2$ .",
"Lemma 9.1 The following estimate holds: $ \\int _0^{t_2} K(t) dt \\ge \\frac{1-\\cos (r_{i_0}/2)}{\\sqrt{\\gamma _r(\\delta _2)^2 - \\gamma _r(\\delta _3)^2}}.",
"$ We compute: $ \\begin{aligned} \\int _0^{t_2} K(t) dt & = \\int _{\\delta _3}^{\\delta _2} \\frac{K(s(t))}{ds/dt} ds\\\\ & = - \\int _{\\delta _3}^{\\delta _2} \\frac{d(\\cos \\alpha )/ds}{\\sqrt{\\gamma _r(s)^2 - \\gamma _r(\\delta _3)^2}} ds\\\\ & = \\int _{\\delta _3}^{\\delta _2} g(s) \\frac{d(\\cos \\alpha )}{ds} ds\\end{aligned} $ where $ g(s) = - \\frac{1}{\\sqrt{\\gamma _r(s)^2 - \\gamma _r(\\delta _3)^2}}.",
"$ Notice that $g$ is differentiable on $(\\delta _3, \\delta _2)$ with $g^{\\prime }(s) > 0$ , and that there exists $a > 0$ such that when $s$ tends to $\\delta _3$ , $ g(s) = - \\frac{a}{\\sqrt{s - \\delta _3}} + o\\left(\\frac{1}{\\sqrt{s - \\delta _3}}\\right).",
"$ Now integrating by parts, $ \\small \\begin{aligned} \\int _{\\delta _3}^{\\delta _2} g(s) \\frac{d(\\cos \\alpha )}{ds} ds = g(\\delta _2) (\\cos \\alpha (\\delta _2)-1) - \\lim _{s \\rightarrow \\delta _3} g(s) (\\cos \\alpha (s)-1) - \\int _{\\delta _3}^{\\delta _2} g^{\\prime }(s) (\\cos \\alpha (s)-1) ds.",
"\\end{aligned} $ Since $\\lim _{s \\rightarrow \\delta _3} g(s) (\\cos \\alpha (s)-1) = 0$ , and $g^{\\prime }(s) (\\cos \\alpha (s)-1) \\le 0$ , we obtain: $ \\begin{aligned} \\int _0^{t_2} K(t) dt & \\ge g(\\delta _2) (\\cos \\alpha (\\delta _2)-1)\\\\ & \\ge \\frac{1-\\cos (r_{i_0}/2)}{\\sqrt{\\gamma _r(\\delta _2)^2 - \\gamma _r(\\delta _3)^2}}.\\end{aligned} $ The existence of conjugate points in the geodesic flow is given by the following lemma: Lemma 9.2 Consider the Riccati equation along the geodesic $(p(t), q(t))$ : $ u(0) = 0, \\quad \\frac{du}{dt} = -K(t) - u(t)^2.",
"$ The solution of this equation blows up to $-\\infty $ in finite positive time, and to $+\\infty $ in finite negative time.",
"First, notice that $ u(t_2) \\le - \\int _{0}^{t_2} K(t) dt \\le - \\frac{1-\\cos (r_{i_0}/2)}{\\sqrt{\\gamma _r(\\delta _2)^2 - \\gamma _r(\\delta _3)^2}}.",
"$ Moreover, for $t \\in (t_2, t_1)$ , since $K(t) \\ge 0$ , we have $ \\frac{du}{dt} \\le -u(t)^2 $ and so, for $t \\in (t_2, t_1)$ , $ u(t) \\le \\frac{1}{t-t_2+1/u(t_2)}.",
"$ Thus, the solution of the Riccati equation blows up to $- \\infty $ before the time $ t_2 - \\frac{1}{u(t_2)} \\le t_2 + \\frac{\\sqrt{\\gamma _r(\\delta _2)^2 - \\gamma _r(\\delta _3)^2}}{1-\\cos (r_{i_0}/2)}, $ thus before the time $t_1$ , provided that $\\delta _2$ is sufficiently small with respect to $\\delta _1$ .",
"By symmetry, the solution blows up to $+ \\infty $ in negative times, after the time $-t_1$ .",
"This ends the proof of Theorem REF ."
],
[
"Acknowledgements",
"This work was partially supported by the European Research Council."
]
] | 1612.05430 |
[
[
"Quantum phases of disordered three-dimensional Majorana-Weyl fermions"
],
[
"Abstract The gapless Bogoliubov-de Gennes (BdG) quasiparticles of a clean three dimensional spinless $p_x+ip_y$ superconductor provide an intriguing example of a thermal Hall semimetal (ThSM) phase of Majorana-Weyl fermions in class D of the Altland-Zirnbauer symmetry classification; such a phase can support a large anomalous thermal Hall conductivity and protected surface Majorana-Fermi arcs at zero energy.",
"We study the effect of quenched disorder on such a topological phase with both numerical and analytical methods.",
"Using the kernel polynomial method, we compute the average and typical density of states for the BdG quasiparticles; based on this, we construct the disordered phase diagram.",
"We show for infinitesimal disorder, the ThSM is converted into a diffusive thermal Hall metal (ThDM) due to rare statistical fluctuations.",
"Consequently, the phase diagram of the disordered model only consists of ThDM and thermal insulating phases.",
"Nonetheless, there is a cross-over at finite energies from a ThSM regime to a ThDM regime, and we establish the scaling properties of the avoided quantum critical point which marks this cross-over.",
"Additionally, we show the existence of two types of thermal insulators: (i) a trivial thermal band insulator (ThBI) [or BEC phase] supporting only exponentially localized Lifshitz states (at low energy), and (ii) a thermal Anderson insulator (AI) at large disorder strengths.",
"We determine the nature of the two distinct localization transitions between these two types of insulators and ThDM.We also discuss the experimental relevance of our results for three dimensional, time reversal symmetry breaking, triplet superconducting states."
],
[
"Introduction",
"Three dimensional Weyl semimetals (WSM) are one the most prominent examples of a gapless topological phase; two non-degenerate bands touch at isolated points in the Brillouin zone causing the low energy quasiparticles to exhibit a linear dispersion in the vicinity of these diabolic points [1].",
"Such band touching points act as the (anti)monopoles of Abelian Berry curvature, giving rise to protected surface Fermi arcs, and exotic transport and electrodynamic properties [2], [3], [4], [5], [6], [7], [8], [9], [10].",
"Since the momentum separation vector $\\delta \\mathbf {K}$ between the right and left handed Weyl points selects out a preferred inertial frame, a WSM phase breaks Lorentz invariance despite possessing a linear dispersion [2], [3], [4].",
"Interestingly, the low energy quasiparticles of several time-reversal ($\\mathcal {T}$ ) and inversion ($\\mathcal {I}$ ) symmetry breaking ground states of strongly correlated materials can be described by Weyl fermions, which may or may not possess a conserved electric charge.",
"The gapless quasiparticles of various magnetically ordered states of 227 pyrochlore iridates [11], [12], [13], and time reversal symmetry breaking $p_x+ip_y$ [14], [15], [16], [17] and $d_{xz}+id_{yz}$ [18], [19] superconductors are some examples of proposed Weyl excitations in correlated materials.",
"According to the Altland-Zirnbauer classification scheme [20] of noninteracting fermions, a $\\mathcal {T}$ breaking WSM can be a member of the following three symmetry classes: unitary/A (a magnetic phase with conserved electric charge), C (chiral, spin singlet pairing like $d_{xz}+id_{yz}$ ), and D (spin rotational symmetry breaking chiral paired states like $p_x+ip_y$ ).",
"While class A WSMs can exhibit both anomalous charge and thermal Hall effects, a class D WSM only exhibits an anomalous thermal Hall effect.",
"By contrast, a class C WSM can support both anomalous spin (in response to a spin electric field caused by a spatially varying Zeeman coupling) and thermal Hall effects [18].",
"We also note that $\\mathcal {T}$ preserving but $\\mathcal {I}$ breaking WSMs , which possess an even number of left and right handed nodal points can be realized for symmetry classes AII (spin rotation breaking with conserved electric charge) and DIII (spin rotation breaking superconductors).",
"The experimentally observed WSMs in spin orbit coupled, noncentrosymmetric materials [21], [22], [23], [24], [25] are members of the symmetry class AII.",
"Such WSMs do not support any anomalous Hall effect, but they can exhibit natural optical activity [10].",
"Since impurities are ubiquitous in solid state materials, it is extremely important to study disorder effects on the stability of gapless topological states.",
"In all realistic situations, the fate of a disordered WSM is crucial for understanding experimental behavior since disorder is invariably present in all material systems.",
"The unconventional behavior of disordered Weyl fermions was first discussed by Fradkin [26], [27], long before any topological properties of WSM were understood.",
"Fradkin's conclusion was that the WSM is stable to weak finite disorder.",
"Over the past few years, disorder effects on WSMs belonging to class A and class AII have been extensively studied using both analytical [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] and numerical methods [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48].",
"Even though some theoretical works have discussed the effects of quenched disorder on three dimensional, gapped topological superconductors [49], [50], the problem of dirty Weyl superconductors [19] has remained relatively unexplored.",
"In addition, class D superconductors have been studied in detail in both one and two dimensions [51], [52], , , [55], [56], [57], [58], [59], whereas a systematic study in three dimensions remains (to the best of our knowledge) largely unexplored.",
"One aim of the present manuscript is to fill this gap.",
"In the present work, we study the effects of quenched disorder on a three dimensional $p_x+ip_y$ superconductor.",
"Interestingly, the Bogoliubov-de Gennes (BdG) Weyl excitations of this phase are real fermions and therefore are in fact Majorana-Weyl (MW) fermions, which can be realized in $^3$ He-A [14] and in time reversal symmetry breaking, triplet paired states of ferromagnetic superconductors [17].",
"Since a $p_x+ip_y$ superconductor can support a large anomalous thermal Hall effect, we will refer to the MW semimetal as a thermal Hall semimetal (ThSM).",
"For a comprehensive analysis of dirty superconductors one has to account for the complicated interplay between interaction and disorder.",
"Since unconventional superconductors with finite angular momentum pairing are not protected by Anderson's theorem (which applies only to $s$ -wave pairing), non-magnetic disorder can suppress the superconducting transition temperature ($T_c$ ) of $p$ -wave superconductors.",
"This type of disorder induced competition between the normal state (a diffusive Fermi liquid) and an unconventional paired state has been studied for a long time [60], and is not the focus of the present work.",
"Rather, we will develop a qualitative understanding of disorder effects on the gapless BdG quasiparticles.",
"Therefore, instead of performing any self-consistent calculation of the pairing gap, we will assume a constant pairing amplitude and study the effects of randomness on a class D quadratic Hamiltonian of a spinless $p_x+ip_y$ three-dimensional superconductor.",
"This should be a reasonable approximation deep inside the superconducting phase, and our interest is understanding the quantum phases of the system rather than the temperature-induced `classical' phase transition.",
"We begin by describing and summarizing our main findings for the disordered quantum phase diagram of three-dimensional class D MW fermions in Sec.",
", which provides a helpful guide for the detailed results derived in the rest of the paper.",
"The remainder of the paper is organized as follows: In Section we introduce the model of interest including the phases within the clean limit and the numerical method used.",
"In Section we discuss the non-perturbative effects of weak disorder, notably the effect of rare regions.",
"In Section and we discuss, respectively, the phase diagram and the properties of the avoided quantum critical point.",
"In Section we look into the quantum critical line separating the insulating phase from the thermally metallic phase, in Section the localization physics is considered up to large disorder, and we conclude in Section .",
"In Appendix , we present perturbative analytical results for the nature of the quantum phase transitions, in Appendix we study the anti-localization peak in the DOS, and in Appendix we discuss the details of determining the phase boundaries."
],
[
"Main Results",
"Our numerical calculations are performed on a lattice model for a spinless $p_x+ip_y$ three-dimensional superconductor defined on a simple cubic lattice [see Eq. ()].",
"In the absence of disorder, the lattice model can support (as a function of chemical potential) three distinct ThSMs (with different numbers of Weyl nodes or diabolic points) and two topologically trivial gapped insulating phases, as shown in Fig.",
"REF .",
"These `quantum phases' arise simply from the `band structure' properties of the BdG quasiparticles as described in depth in Sec. .",
"The ThSMs with two and four Weyl points are respectively denoted as ThSM2 and ThSM4, which both carry a non-quantized thermal Hall conductivity.",
"For the clean problem, the density of states (DOS) for MW fermions vanishes as $\\rho (E) \\sim |E|^2$ , while the DOS for a thermal band insulator (ThBI) or BEC phase has a hard spectral gap.",
"These phases are separated by clean, quantum critical points (QCPs) located at $\\mu /t= \\pm 2, \\; \\pm 6$ (for a chemical potential $\\mu $ and hopping $t$ ), with critical excitations that display anisotropic dispersions $E_{AW}(\\mathbf {k})=\\pm \\sqrt{t^2 k^4_3+ \\Delta ^2_p k^2_\\perp }$ [with $k_\\perp =\\sqrt{k^2_1+k^2_2}$ , and $\\Delta _p$ being the pairing amplitude].",
"Therefore the DOS for such critical excitations vary as $\\rho (E) \\sim |E|^{3/2}$ , and we can associate an effective dynamic scaling exponent from $\\rho (E)\\sim |E|^{d/z_{aw}-1}$ , with $z_{aw}=6/5$ .",
"Thus, at the clean anisotropic points the DOS is a non-analytic function of energy (or chemical potential).",
"The DOS in each region of the phase diagram in Fig.",
"REF (a) vanishes faster than $|E|$ and as a result disorder acts as an irrelevant perturbation, which suggests that the clean phase diagram is robust to a weak amount of disorder.",
"(This perturbative argument is similar to what happens in a normal non-superconducting WSM for weak disorder with the tentative conclusion that the WSM phase is stable to weak disorder.)",
"One interesting aspect of the present class D model is the possibility for the density of states to become non-analytic (as a predicted consequence of the perturbative RG) since rigorous bounds [62] do not apply.",
"The perturbative analysis also predicts that at finite disorder the semimetal and the band insulator have to undergo quantum phase transitions to a diffusive thermal Hall metal (ThDM), possessing a finite density of states at zero energy.",
"However, for very weak disorder we find that non-perturbative effects give rise to quasilocalized rare states that convert the ThSM into a ThDM, with an exponentially small density of states (DOS) at zero energy [32], [46], [47] independent of how weak the disorder is.",
"By contrast, we find that the gapped ThBI is robust to a weak amount of disorder with a clear average gap in the DOS, and the low energy spectrum is composed of Anderson localized mid-gap states (i.e.",
"Lifshitz states).",
"Thus, non-perturbative rare region physics destroys the ThSM phase (converting it generically into a ThDM phase even for weak disorder), but does not destroy the ThBI phase (despite Lifshitz states contributing a non-zero DOS inside the average band gap).",
"Our numerical results are consistent with $\\rho (0) \\ne 0$ across the entire disordered phase diagram and the DOS is always analytic at low-$|E|$ .",
"Therefore, the numerically determined phase diagram of the disordered model illustrated in Fig.",
"REF (a), only consists of a delocalized ThDM phase, a ThBI (or a trivial BEC) phase supporting exponentially localized Lifshitz states at $E=0$ , and a class D Anderson insulator (AI) for very large disorder.",
"The weak disorder controlled transitions and crossovers can be tracked by the average DOS, while an Anderson localization of the ThDM can only be tracked with non-self averaging quantities such as the typical DOS, which we compute with the kernel polynomial method [61] (KPM).",
"We stress that all of the disorder controlled phases described here are superconductors and therefore exhibit the Meissner effect.",
"However, the ThDM and an ordinary Fermi liquid both show similar linear-in-temperature specific heat and longitudinal thermal conductivity.",
"Figure: (a) The zero energy (or temperature) schematic phase diagram of Hamiltonian Eq.",
"() with μ\\mu as the chemical potential for the electrons, WW is the strength of disorder, and tt is the hopping strength.",
"ThDM is a diffusive thermal Hall metal with finite density of states at zero energy, ThBI is the thermal band insulator supporting exponentially localized Lifshitz states, and AI denotes an Anderson insulator phase of class D, BdG quasiparticles.The thin solid orange line indicates the crossover from thermal Hall semimetals (ThSMs) and ThDM which occur at finite energies, and are governed by an avoided QCP [as shown in (b)].",
"The dashed line at large W/tW/t and μ/t\\mu /t represents the possible separation between the two Anderson insulating phases.",
"We expect that a ThDM regime (of a width on the order of the hopping strength) could exist along this line due to the different nature of the two localized phases and their respective transitions.",
"(b) A sketch of what each regime means [taken as a cut in (a) for either fixed μ\\mu or WW such that one passes through both the ThBI to ThDM localization transition and the avoided transition], which we verify in each relevant section below.Due to the exponentially small DOS at E=0E=0 there is only a ThDM at low energies; despite this the DOS still resembles that of a ThSM above a cross over energy scale.",
"The perturbatively predicted critical point between ThSM and ThDM becomes avoided, but its quantum critical fan can be probed numerically over a wide range of energies.The ThBI phase is insulating and an actual critical point separates it from the ThDM phase with its own critical fan.",
"Nonetheless, ρ(0)>0\\rho (0)>0 in the ThBI phase purely from localized rare states (represented by the grayed region in the ThBI phase).Λ\\Lambda is some characteristic energy scale, above which short distance or lattice effects are important.This figure does not depict the Anderson localization transition occurring at a very large disorder strength (or energy).For weak to moderate disorder strengths, the phase diagram exhibits crossover between ThSMs and ThDM over a wide range of energies, governed by an avoided QCP [see Fig.",
"REF (b)].",
"Our analytical one loop renormalization group (RG) calculations predict that the universality class of the avoided QCP is controlled by the repulsive fixed point of “axial chemical potential\" type disorder for MW fermions, with a dynamic scaling exponent $z=3/2$ and a correlation length exponent $\\nu =1$ , which agrees well with our numerical calculations.",
"We emphasize that the axial chemical potential disorder is not present in our bare microscopic model, and we are only making a statement about the fixed point Hamiltonian that determines the crossover properties at finite energy between the ThSM and the ThDM.",
"The “axial chemical potential\" universality class provides a possible explanation for why the nonperturbative rare states of the current particle-hole symmetric problem and the rare states of a Dirac (or Weyl) semimetal in the presence of particle-hole asymmetric scalar random potential show similar properties [32], [46], [47].",
"We find a genuine localization-delocalization quantum phase transition between the ThBI and ThDM phases.",
"Physically, this is due to the average band gap being “filled in” by random mid-gap states that have wave functions which overlap significantly with states on neighboring sites resulting in a delocalized phase.",
"For infinitesimally weak disorder, perturbing about the anisotropic critical point gives rise to a phase boundary separating the ThDM and ThBI phases.",
"Along this line, perturbatively we find that the DOS still obeys $\\rho (E) \\sim |E|^{3/2}$ , and the effective dynamic scaling exponent along this critical line is $z_{aw}=6/5$ .",
"We find good agreement between the power law scaling of the DOS in the numerics and the perturbative RG expectations along the ThBI to ThDM phase boundary with a quantum critical fan at finite energy anchored by this localization-delocalization transition as shown in Fig.",
"REF (b).",
"Since both the exponentially localized Lifshitz states in the ThBI and the power law quasi-localized rare states in the ThDM contribute to a non-zero DOS at $E=0$ , our numerical results are consistent with $\\rho (0)\\ne 0$ at the ThBI to ThDM transition.",
"Moreover, we directly show that the non-analytic behavior in the DOS in the clean limit (at $\\mu = \\pm 6t$ ) is rounded out due to non-perturbative effects.",
"Thus, the scaling of the DOS in the quantum critical fan anchored by the ThBI to ThDM QCP only holds at finite energy and inevitably crosses over at low energy to an analytic DOS.",
"Thus, by tuning disorder strength $W$ or chemical potential $\\mu $ we can observe rich crossover behavior at finite energies, which is illustrated in Fig.",
"REF .",
"We find that the diffusive phase is well described by a class D nonlinear sigma model only when the disorder strength is larger than the avoided critical strength.",
"As a hallmark of class D diffusive model we observe antilocalization effects on the density of states.",
"The perturbative beta function for the nonlinear sigma model has been known up to four loop orders [63], [64].",
"However, the beta function consists of an alternating series and it is difficult to analytically predict the Anderson localization transition for class D. Thus, our exact numerical calculations are essential for unveiling Anderson localization transition of the class D diffusive metal.",
"At very large disorder strength we find an Anderson localization transition of the ThDM to an AI, which results from suppressing neighboring hopping with large fluctuations of the onsite potential.",
"Due to anti-localization effects, we find this occurs for a disorder strength much larger than the clean bandwidth as well as the existence of a sharp anti-localization peak in both typical and average DOS in the ThDM phase.",
"We determine the shape of the mobility edge and the power law governing how the typical DOS goes to zero at the AI transition.",
"Since the ThBI and AI arise from distinct physical mechanisms and are governed by different quantum phase transitions, it is possible that a ThDM phase exists to very large $W/t$ and $\\mu /t$ separating the ThBI and the AI [shown schematically as the dashed line in Fig.",
"REF (a)]."
],
[
"Model and Method",
"We study the following quadratic Hamiltonian of a three dimensional spinless $p_x+ip_y$ superconductor H = r [t cr+ cr + i cr+ cr+ h.c. ] + r (V(r) -) cr cr, where $c_{\\mathbf {r}}$ is the fermion annihilation operator at site $\\mathbf {r} $ on a cubic lattice, $\\hat{\\nu }=\\pm \\hat{\\mathbf {x}}, \\; \\pm \\hat{\\mathbf {y}}, \\; \\pm \\hat{\\mathbf {z}}$ are coordination vectors for nearest neighbors, $t$ is the nearest neighbor hopping strength, and $\\Delta _\\nu $ are nearest neighbor pairing amplitudes.",
"For describing $p_x+ip_y$ pairing, we choose $\\Delta _{\\hat{x}}= \\Delta , \\; \\Delta _{\\hat{y}}=-i \\Delta , \\; \\Delta _{\\hat{z}}=0$ , where $2\\Delta $ is the superconducting gap for a clean model.",
"Additionally, $\\mu $ and $V(\\mathbf {r})$ respectively denote the uniform and randomly varying chemical potentials for normal quasiparticles.",
"For our calculations the disorder potential will follow either (i) a Gaussian probability distribution with zero mean and standard deviation $W$ or (ii) a box distribution in the interval $[-W/2,W/2]$ (we will specify which probability distribution we are using in each relevant section).",
"The boundary conditions are taken to be periodic unless otherwise specified with a linear size $L$ , and volume $V=L^{3}$ .",
"Both disorder distributions give the same qualitative behavior, but differ in some quantitative details."
],
[
"Clean phase diagram",
"In the absence of disorder, the mean-field Hamiltonian in the momentum space can be written as $H_0=\\frac{1}{2}\\; \\sum _\\mathbf {k} \\; \\psi ^\\dagger _\\mathbf {k} h(\\mathbf {k}) \\psi _\\mathbf {k}$ where $\\psi ^\\dagger _\\mathbf {k}=(c^\\dagger _{\\mathbf {k}}, c_{-\\mathbf {k}})$ is the two component Nambu spinor.",
"The Hamiltonian operator is given by $h(\\mathbf {k})=[2t\\sum _{j=1}^{3}\\cos k_j -\\mu ]\\tau _3+ \\Delta (\\sin k_1 \\tau _1 +\\sin k_2 \\tau _2), \\nonumber \\\\$ where $\\tau _j$ 's are Pauli matrices operating in particle-hole space, and $h(\\mathbf {k})$ satisfies the following particle-hole symmetry condition $\\tau _1 \\; h^T(-\\mathbf {k}) \\; \\tau _1=-h(\\mathbf {k})$ of Altland-Zirnbauer symmetry class D. The quasiparticle spectra for this model are determined by $E_{\\pm }(\\mathbf {k})=\\pm \\; [ (2t\\sum _{j=1}^{3}\\cos k_j -\\mu )^2+\\Delta ^2 (\\sin ^2 k_1+\\sin ^2 k_2)]^{1/2}.$ Depending on the ratio $\\mu /t$ , the clean model can support three gapless and two gapped states, as shown in Fig.",
"REF , and the corresponding anomalous thermal Hall conductivities are displayed in Fig.",
"REF .",
"When $2t< \\mu <6t$ , the paired state has two Weyl nodes, and the left and right handed Weyl points are respectively located at $\\mathbf {k}=(0,0,\\pm K_1)$ , where $K_1=\\arccos \\left(\\frac{\\mu }{2t}-2\\right)$ .",
"Consequently, the Berry flux through the $xy$ plane will point along the $+\\hat{z}$ direction.",
"At low temperatures, the MW fermions display a longitudinal thermal conductivity $\\kappa _{xx} \\sim T^2$ (arising from residual inelastic scattering effects) and an anomalous thermal Hall conductivity $\\kappa _{xy}=\\frac{k^2_BT}{6\\hbar \\pi a} \\; \\arccos \\left(\\frac{\\mu }{2t}-2\\right),$ where $a$ is the lattice spacing.",
"We denote this phase as ThSM2.",
"Notice that $\\kappa _{xy}$ vanishes as $\\mu \\rightarrow 6t$ and acquires its maximum value $\\frac{k^2_BT}{6\\hbar \\pi a}$ when $\\mu \\rightarrow 2t$ .",
"For $\\mu >6t$ , there is no underlying Fermi surface, and the system is in the BEC regime, where the quasiparticle spectrum is fully gapped, and the paired state acts as a thermal band insulator (both $\\kappa _{xx}/T$ and $\\kappa _{xy}/T$ vanish in the $T \\rightarrow 0$ limit).",
"We denote this phase as ThBI.",
"In the vicinity of the QCP between ThSM2 and ThBI located at $\\mu =6t$ , the low energy excitations are described by $h(\\mathbf {k}) \\approx -\\Delta (k_1 \\tau _1 +k_2 \\tau _2)+ (6t-\\mu - t k^2_3)\\tau _3.$ The strongly anisotropic dispersion at the QCP is captured by $E_{AW}(\\mathbf {k})=\\pm \\sqrt{t^2 k^4_3+ \\Delta ^2_p k^2_\\perp }$ .",
"At any finite energy (E) we can define two distinct energy dependent correlation lengths: $\\xi _3(E) \\sim 1/\\sqrt{|E|}$ along the $z$ direction and $\\xi _\\perp \\sim 1/|E|$ in the $xy$ plane.",
"Similarly, at finite temperatures we can define two different de Broglie wavelengths or temperature dependent correlation lengths, by replacing $|E|$ with temperature $T$ .",
"All the critical properties can be understood in terms of these energy or temperature dependent correlation lengths.",
"As a consequence of such anisotropic scaling, the critical density of states behaves as $\\rho (E,\\mu =6t,W=0) \\sim \\xi ^{-1}_3(E) \\xi ^{-2}_\\perp (E) |E|^{-1} \\sim |E|^{3/2}.$ If we introduce an effective dynamic scaling exponent $z_{\\mathrm {aw}}$ such that $\\rho (E,\\mu =6t,W=0) \\sim |E|^{d/z_{\\mathrm {aw}}-1}$ , we obtain $z_{\\mathrm {aw}}=6/5$ .",
"The quantum critical fan for this QCP is described by the condition $|E|$ (or $T) > |6t-\\mu |$ .",
"Outside the critical fan $E$ (or $T) < |6t-\\mu |$ , the low energy physics of ThSM2 and ThBI is governed by the correlation lengths $\\xi _3 \\sim 1/\\sqrt{|6t-\\mu |}$ and $\\xi _\\perp \\sim 1/|6t-\\mu |$ .",
"Inside the ThSM2 the density of states is reduced and follows the power law $\\rho (E) \\sim E^2$ reflecting that the dynamic scaling exponent $z=1$ for MW excitations.",
"By contrast, the DOS for the ThBI phase exhibits a sharp spectral gap.",
"We can summarize such critical and off-critical behaviors of the DOS with the following results $\\rho (E,\\mu ^*+\\delta \\mu ,W=0) \\\\ \\sim {\\left\\lbrace \\begin{array}{ll} E\\sqrt{E - \\tfrac{\\delta \\mu }{2}}\\Theta (E - \\tfrac{\\delta \\mu }{2}), & \\delta \\mu >0, \\\\E\\left[ \\sqrt{E - \\tfrac{\\delta \\mu }{2}} - \\sqrt{\\tfrac{|\\delta \\mu |}{2}-E}\\Theta (\\tfrac{|\\delta \\mu |}{2}-E) \\right], & \\delta \\mu <0, \\end{array}\\right.",
"}$ where $\\Theta (x)$ is the Heaviside step function and $\\delta \\mu =6t-\\mu $ .",
"The nodal separation inside the ThSM2 is governed by $\\xi ^{-1}_3 \\sim \\sqrt{\\delta \\mu }$ , which also controls how $\\kappa _{xy}/T$ vanishes when the QCP is approached from the ThSM2 side.",
"It is important to note that this gives rise to a non-analytic DOS, e.g.",
"the second derivative of the DOS with respect to energy diverges as $\\rho ^{\\prime \\prime }(0) \\sim \\delta \\mu ^{-1/2}$ .",
"When we approach the QCP at $\\mu =2t$ from the ThSM2 side, the Weyl points move to the Brillouin zone boundaries as $K_1 \\rightarrow \\pi $ .",
"Precisely at $\\mu =2t$ , three flavors of anisotropic critical excitations (similar to the one described above) emerge at $\\mathbf {k}=(0,0,\\pi )$ , $\\mathbf {k}=(\\pi ,0,0)$ , and $\\mathbf {k}=(0,\\pi ,0)$ .",
"For $-2t<\\mu <2t$ we find a new ThSM phase with four MW fermions.",
"The Weyl points are located at $\\mathbf {k}=(0,\\pi ,\\pm K_2)$ and $\\mathbf {k}=(\\pi ,0,\\pm K_2)$ , with $K_2=\\arccos \\left(\\frac{\\mu }{2t}\\right)$ , and we denote this phase as ThSM4.",
"The right and left handed Weyl points are now respectively placed at $k_3=+K_2$ and $k_3=-K_2$ .",
"Consequently, the Berry flux due to these four Weyl points through the $xy$ plane is directed along the negative $z$ axis, leading to $\\kappa _{xy}=\\frac{k^2_BT}{6\\hbar \\pi a}\\left[\\pi -2\\arccos \\left(\\frac{\\mu }{2t}\\right)\\right].$ In the range $0<\\mu <2t$ , $\\kappa _{xy}$ decreases from its maximum positive value toward zero.",
"For $\\mu <0$ , $\\kappa _{xy}$ changes its sign and attains the maximum negative value $-\\frac{k^2_BT}{6\\hbar a}$ at $\\mu =-2t$ .",
"For the QCP located at $\\mu =-2t$ , the left and right handed Weyl points of ThSM4 merge at $(0,\\pi , \\pi )$ and $(\\pi ,0,\\pi )$ , giving rise to two flavors of anisotropic critical excitations.",
"In addition, a third flavor of critical excitation appears at $(\\pi ,\\pi ,0)$ .",
"When $-6t<\\mu <-2t$ , a different ThSM2 phase is realized, where the left and right handed Weyl points are respectively located at $(\\pi ,\\pi ,\\pm K_3)$ , with $K_3=\\arccos \\left(2+\\frac{\\mu }{2t}\\right)$ .",
"The Berry flux through the $xy$ plane due to these Weyl points is directed along the positive $z$ axis and the thermal Hall conductivity for $-6t<\\mu <-2t$ becomes $\\kappa _{xy}=\\frac{k^2_BT}{6\\hbar \\pi a}\\left[\\arccos \\left(\\frac{\\mu }{2t}+2\\right) -\\pi \\right].$ As we approach $\\mu =-6t$ , the new set of Weyl points move toward the zone boundaries and $\\kappa _{xy}$ gradually goes to zero.",
"For $\\mu <-6t$ , again there is no underlying Fermi surface and we enter a BEC or ThBI phase.",
"The anisotropic critical excitations at $\\mu =-6t$ occur at $\\mathbf {k}=\\mathbf {(}\\pi ,\\pi ,\\pi )$ .",
"In Appendix , we present some perturbative analysis of the disordered problem, which will guide our qualitative understanding of the numerically exact results to be presented later in the paper."
],
[
"Typical and average DOS",
"In the remainder of the paper, we study the evolution of the clean phase diagram shown in Fig.",
"REF as a function of the strength of disorder.",
"To study the various regimes in this problem, we compute the average DOS using the KPM technique, which is particularly well-suited here.",
"For each disorder realization it is defined as (E) = 12 Vi (E -Ei) where $E_i$ are the eigenstates of the disordered Hamiltonian.",
"We then average this quantity over a number of disorder realizations (see Table REF ).",
"Additionally, in order to study localization phenomena, we are also interested in the possibility of a thermal insulating phase (i.e.",
"the AI phase) driven by very large disorder (i.e.",
"a much larger disorder strength than the putative weak-disorder avoided QCPs); this localization of the Majorana BdG quasiparticles can be tracked by computing the typical density of states [61] t(E) = [i(E) ] where $\\langle \\langle \\cdots \\rangle \\rangle $ denotes a disorder average and the local density of states at site $i$ is defined as i(E) = 12V n |i| En|2 (E - En).",
"To reach sufficiently large system sizes, we avoid a direct diagonalization of $H$ by using the KPM.",
"This method allows us to compute the average and typical density of states for sufficiently large system sizes due to the sparsity of the tight binding Hamiltonian (for technical details see Ref. [61]).",
"Essentially, the KPM expands the DOS (and also the local DOS) in terms of Chebyshev polynomials with an appropriate Kernel (here we use the Jackson Kernel).",
"The KPM then just needs to use matrix multiplication techniques to find the Chebyshev expansion coefficients.",
"There are a couple of numerical inputs into this method: We must specify the number of Chebyshev coefficients we are keeping $N_C$ , the number of random vectors we use to evaluate the stochastic trace $N_R$ (see Sec.",
"REF for details about this), the type of disorder distribution used [here we use either Gaussian or box disorder], and how many disorder realizations we take $N_{\\rm dis}$ .",
"Unless otherwise stated, we generally use the values indicated in Table REF for the average-DOS, and for the typical-DOS we use $L=30$ , $N_C=2^{11}$ to $2^{13}$ , $N_R=10$ and $N_{\\mathrm {dis}}=800$ .",
"Table: The numerical input for most calculations.The size of the system is L 3 L^3, N C N_C is the number of Chebyshev coefficients taken, N R N_R is the number of random vectors in the stochastic trace.For box disorder the random vectors are unnormalized while for Gaussian disorder the random vectors are normalized, see Sec.",
".Lastly, N dis N_{\\rm dis} is the number of disorder realizations.The system is self-averaging, so not very many are needed.However, for Gaussian disorder, we use many more in order to see rare-region effects."
],
[
"Modification of stochastic trace",
"In order to probe the effects of rare regions on the low energy DOS we need to be able to detect an exponentially small DOS on the order of $10^{-5}$ .",
"However, as shown in [46] the KPM method has an artificial background DOS that is flat and sets a lower bound on what size DOS can be accurately detected.",
"Here, we show what the origin of this fake KPM background is and establish how to remove it alltogether.",
"As a result the lower bound of the DOS is set by the intersection of the broadening of the Dirac delta-functions in the definition of the DOS.",
"Figure: The stochastic trace evolution of the density of states produces an artificial background as seen by the use of “Unnormalized” vectors.However, if one just uses a small number of “Normalized vectors” much more accurate results at low values of ρ(E)\\rho (E) can be found.The calculation in this plot was done with 100 disorder realizations and Gaussian disorder at μ=0\\mu =0.This effect persists at lower system sizes where it is confirmed by computing the exact trace.Evaluating the KPM requires the trace of a matrix, and the most numerically efficient way to evaluate the trace of a large matrix is with stochastic vectors [61].",
"Usually, one takes Gaussian sampled vectors $\\mathinner {|{r}\\rangle } = \\sum _i \\xi _{ri}\\mathinner {|{i}\\rangle }$ such that ri*r'i' = rr'ii' where $r$ denotes the random vector and $i$ its component.",
"However, this means that the vectors we are sampling over are only normalized on average, and the fluctuations away from average are what lead to the artificial KPM background that masks the real value of the DOS, as shown in Fig.",
"REF .",
"As has been pointed out [65], [66], normalization of the stochastic vectors used in the evaluation of the trace can reduce the statistical noise in the data.",
"To implement this formally, we insist that each individual vector is normalized.",
"Thus after sampling the vectors, we impose the normalization condition i|ri|2 = 2V.",
"We demonstrate how this works in practice in Fig.",
"REF : for $\\rho (E) \\gtrsim 10^{-4}$ all vectors work (normalized or unnormalized), but the unnormalized vectors hit the artificial KPM background between $10^{-4}$ to $10^{-5}$ that is just marginally improved by taking more random vectors.",
"On the other hand, the normalized vectors work down to very low numbers dictated by, in this case, the tail of the Gaussian-broadened states (due to the Jackson Kernel).",
"Even if we use many more normalized random vectors, the results shown in Fig.",
"REF do not change substantially.",
"Furthermore, at smaller sizes where an exact trace is computationally reasonable, the agreement is much better.",
"This technical advancement in our implementation of KPM is substantial as it now opens the door for using the KPM to detect the existence of low energy rare region effects, which prior to our work, was only possible at moderate disorder strengths above the fake KPM background.",
"Our modification essentially gets rid of the constraint arising from the KPM-induced artificial DOS background problem."
],
[
"Non-perturbative effects at weak disorder",
"Since the DOS for both the MW fermions and the critical excitations at the anisotropic QCP, vanish faster then $|E|$ , weak disorder is expected to be an irrelevant perturbation (in the RG sense) to the clean phase diagram in Fig.",
"REF .",
"The self-consistent Born approximation (SCBA) calculation also suggests that the ballistic MW fermions remain stable (i.e., $\\rho (E=0)=0$ ) up to a critical strength of disorder $W_c(\\mu )$ .",
"For $W>W_c(\\mu )$ , disorder induces a finite density of states at zero energy, giving rise to a diffusive thermal Hall metal (i.e.",
"the ThDM phase).",
"The perturbative irrelevance of disorder at the anisotropic QCP (i.e.",
"$\\mu =\\pm 6t$ ) also suggests the existence of a perturbatively accessible disorder driven (metal to insulator) transition $W_I(\\mu )$ , consistent with the SCBA.",
"However, this picture is drastically modified by non-perturbative effects of disorder as we show in this section.",
"We find numerically that the DOS is analytic across the entire $W-\\mu $ phase diagram and our results are consistent with the DOS always being non-zero in the thermodynamic limit.",
"Despite its shortcomings, the perturbative RG does correctly capture the shape of the phase boundary separating the ThDM and ThBI phases and provides a quantitative description of the power law scaling of the DOS and the average band gap.",
"Due to its technical nature and to avoid confusion with full numerical solution (that incorporates all effects) we present the perturbative RG in Appendix ."
],
[
"Thermal band insulator $|\\mu |>\\mu _I(W)$",
"We begin by discussing the low energy eigenstates inside the band gap of the thermal band insulator in the phase diagram of Fig.",
"REF .",
"For weak disorder, mid-gap states fall randomly inside the band gap that induce Lifshitz states, which are fairly well understood [67], [68].",
"These eigenstates are exponentially localized around the sites (or cluster of sites) with a very large disorder strength.",
"These eigenstates round out the gap in the DOS and give rise to an exponentially decaying energy dependent DOS that goes like $\\rho (E)\\sim A(E) e^{-B(E)}$ [69] (e.g.",
"$\\rho (E)\\sim A e^{-B |E-E_0|^{-3/2}}$ for box disorder and $\\rho (E)\\sim A e^{-B|E-E_0|^{1/2}}$ for Gaussian disorder [67], [68] in three dimensions).",
"We demonstrate this in the ThBI phase in Fig.",
"REF where we fit the DOS to the Lifshitz form near the band edge.",
"In the ThBI phase the zero energy eigenstates are Anderson localized insulating states and their contribution to the zero energy DOS is negligible, but strictly speaking non-zero.",
"Thus, despite the inability to probe such zero energy states numerically, our data satisfying the Lifshitz form is consistent with $\\rho (0) >0$ in the disordered ThBI phase.",
"For sufficiently large disorder strength, these mid-gap states become sufficiently dense to completely fill in the average bulk band gap.",
"In this limit these Lifshitz states at $E=0$ develop sufficient overlap eventually driving a quantum phase transition from the ThBI to a ThDM.",
"We explore this transition non-perturbatively in Section REF and its scaling properties in Section .",
"Figure: Deep within the insulating phase, we fit the Lifshitz Tail due to rare region effects: ρ(E)-ρ(0)≈Ae b|E-E 0 | -3/2 \\rho (E) - \\rho (0) \\approx A e^{b|E - E_0|^{-3/2}} for box disorder (with A≈5.31×10 -12 A\\approx 5.31\\times 10^{-12}, b≈0.209b\\approx 0.209, and E 0 ≈0.233tE_0\\approx 0.233 t).",
"The thin individual lines are averages over 1000 realizations while the thick (red) line is an average over 28,000 realizations—all calculations are done with N c =2 13 N_c=2^{13}.",
"The thin lines clearly show rare contributions to the DOS deep in the average band gap.E 0 E_0 is a good approximation for the average band edge, drawn as a vertical line in both plot and inset.",
"(Inset) Same data on a linear-linear scale."
],
[
"Thermal semi-metal regime $|\\mu |<\\mu _c(W)$",
"In this section, we provide numerical evidence at weak disorder that non-perturbative rare-regions exist in the current model (despite the strict particle hole symmetry) and contribute a non-zero DOS at $E=0$ .",
"We focus on a gaussian distribution of disorder because the unbounded tails of the distribution lead to large local fluctuations of the potential enhancing the probability of generating a rare event [32], [47].",
"Moreover, as these events are rare we use $10,000$ disorder realizations to find a statistically significant result.",
"Figure: The finite size DOS at weak disorder.",
"(a) Depending on the physics we are looking at, it is useful to sometimes keep one Weyl peak at zero energy as shown here.",
"(b) The DOS for various LL with Gaussian disorder as a function of LELE.",
"At weak disorder the peaks are spaced like 1/L1/L and therefore are composed of perturbatively dressed Weyl states that are smoothly connected to their W=0W=0 counterparts.There are no states at zero energy, the finite value of the DOS at E=0E=0 is solely due to the overlap of the two Weyl peaks at EL/t≈5EL/t \\approx 5 broadened by the KPM.",
"(c) The DOS versus EE computed with a gaussian disorder distribution displaying the cross over regimes at low EE.",
"At very low energies the density of states is non-zero and essentially EE-independent displaying the ThDM regime.",
"At finite EE, the ThSM regime is intact with characteristic ρ(E)∼E 2 \\rho (E)\\sim E^2 dispersion, and at large EE, we start to see the critical fan ρ(E)∼|E|\\rho (E)\\sim |E|.As shown in Ref.",
"[46], in order to diagnose rare region effects and eliminate the finite size effects on the DOS in the SM regime, we need to move the zero energy states away from $E=0$ so that for $W=0$ , $\\rho (0)=0$ at all $L$ .",
"In short, by modifying boundary conditions alone we can move from a situation with a state at $E=0$ as shown in Fig.",
"REF (a) and move the lowest energy Weyl state maximally from zero as shown in Fig.",
"REF (b).",
"To accomplish this in Nambu space we either consider $L$ that is not a multiple of four with periodic boundary conditions or we introduce antiperiodic boundary conditions, which both move the clean (i.e.",
"disorder-free) states away from zero energy.",
"In particular, for anti-periodic boundary conditions $\\psi (0) = - \\psi (L \\mathbf {e}_j)$ for the directions $j=x,y,z$ when $L$ is a multiple of four The location of disorder-free states in $k$ -space depends heavily on the size of the system due to the location of the nodes going as $k_z^* = (2n+1)\\pi /2 = 2\\pi m/L$ for integers $n$ and $m$ .",
"Therefore, we have zero-energy states appear as nodes when $L$ is a multiple of 4 and do not appear when $L$ is even but not a multiple of 4.",
"Therefore, in the former case we use antiperiodic boundary conditions while in the latter we use periodic boundary conditions in the $z$ -direction only..",
"In the disorder-free limit with anti-periodic boundary conditions the eigenstates labeled by ${\\bf k}$ are shifted to ${\\bf k}\\rightarrow {\\bf k} + \\pi /L(1,1,1)$ .",
"After performing this shift, the lowest lying Weyl state is maximally moved away from zero energy.",
"Also, we want to minimize the broadening of each energy eigenvalue by disorder, and we do this by enforcing $\\sum _r V(r) = 0$ for each disorder realization.",
"Here, we are seeking an exponentially small $\\rho (0)$ and therefore the modified stochastic trace (see Sec.",
"REF ) is essential to be able to access a DOS of such a small magnitude.",
"For very weak disorder the low energy DOS is well described by Weyl peaks that are broadened and move in energy due to disorder, see Figs.",
"REF (a) and (b).",
"At slightly larger disorder, (where we can find a statistically significant amount of rare states) the low energy DOS is an essentially flat background (between the Weyl peaks) that extends to $E=0$ , see Fig.",
"REF (c).",
"These Weyl peaks are well described by perturbation theory in disorder and are essentially perturbatively renormalized Weyl states [46], which are spaced like $~1/L$ (see Fig.",
"REF ).",
"The flat background DOS on the other hand originate from quasi-localized rare eigenstates.",
"Focusing on a disorder sample that produces a low energy state contributing to the flat part of the DOS, we compute the two component spinor wave function $\\psi (x,y,z)$ of the this state using Lanczos on $H^2$ for $\\mu =0$ , $L=18$ and $W=0.8t$ .",
"Note that for $L=18$ and periodic boundary conditions this places the first low energy state (in the clean limit) to be at $E_0\\approx 0.3t$ and thus satisfies $\\rho (0)=0$ for this value of $L$ .",
"We project the probability amplitude into two dimensions via $\\sum _z|\\psi (x,y,z)|^2$ for plotting purposes.",
"We find that the wavefunction is quasi-localized in real space [see Fig.",
"REF (a)] about two sites one with a value of $V_i\\sim 4W$ and the other $V_i\\sim 2W$ , with a probability $\\sim \\exp (-10)$ which is indeed a rare eigenstate relative to $V_i \\sim W$ .",
"Figure: (a) Projected probability density ∑ z |ψ(x,y,z)| 2 \\sum _z|\\psi (x,y,z)|^2 in the xyxy-plane of a rare wavefunction that corresponds to a state in the low energy tail of the DOS with W/t=0.8W/t=0.8, L=18L=18, and periodic boundary conditions.",
"(b) A scatter plot of the wave function decay away from its maximum value on a log-log plot, showing a clear power-law trend.",
"(c) The scatter plot data is binned and is fit to a/r x a/r^x showing a clear power law decay at small rr with x=1.6x=1.6 and the rare wavefunction is indeed quasi-localized.We now compute the decay of the wavefunction from its maximal value.",
"To do this we first compute the distance to the site where the wave function has its maximum ${\\bf r}_{\\mathrm {max}}$ and then compute the distance from it $\\psi (r)\\equiv \\psi (|{\\bf x} - {\\bf r}_{\\mathrm {max}}|)$ (respecting the periodic boundary conditions $|x^{\\mu }- r_{\\mathrm {max}}^{\\mu }| <L/2$ ), the scatter plot of this is shown in Fig.",
"REF (b).",
"We then bin the wavefunction along $r$ and compute its power law decay, which leads to one of our main results, namely for this particular rare state we find (r) 1r1.6.",
"This power law decay can vary from one disorder sample to the next but we do find good agreement with the expected analytic prediction of $1/r^2$ (Ref. [32]).",
"Thus, we conclude that these quasi-localized rare eigenstates are unaffected by the the presence of particle-hole symmetry.",
"It is important to contrast these quasi-localized eigenstates in the ThSM regime with the exponentially localized Lifshitz states in the Anderson localized ThBI phase.",
"These quasi-localized eigenstates have level repulsion [46] and are not Anderson localized.",
"Therefore focusing on weak disorder and tuning $\\mu $ across $\\mu _I(W)$ (i.e.",
"the ThDM to THBI transition) is a true metal to insulator quantum phase transition where the power law quasi-localized rare states are converted into exponentially localized Anderson insulating zero energy states.",
"Figure: (a) The DOS at zero energy ρ(0)\\rho (0) as a function of disorder strength WW.",
"A Weyl state at zero energy produces a very large finite size effect in ρ(0)\\rho (0), whereas the DOS without a Weyl state becomes LL independent at weak disorder (W≳0.75tW\\gtrsim 0.75t).",
"With a Weyl state at zero energy, we can pinpoint the crossover due to the avoided transition to occur roughly around W/t≈1.0W/t\\approx 1.0 (for Gaussian disorder by finite size scaling).",
"(b) Without a Weyl state at E=0E=0, rare states begin populating the low-|E||E| DOS and they appear as an LL independent DOS.",
"The data is well fit to the rare region form ρ(0)∼exp(-a/W 2 )\\rho (0)\\sim \\exp (-a/W^2) and our results are consistent with only the ThDM phase persisting at zero energy for weak disorder.Even without an E=0E=0 state, we hit an artificial background DOS due to the Gaussian broadening of the Dirac-delta function in the DOS of nearby states around W/t≈0.7W/t\\approx 0.7.Having identified the eigenstates that make up the low energy relatively flat $L$ independent background DOS extending to $E=0$ we are in a good position to determine the evolution of $\\rho (0)$ versus W as seen in Fig.",
"REF .",
"In Fig.",
"REF (b) we plot the $L$ independent background DOS with an excellent fit to the rare region form [32] $\\log \\rho (0) \\sim {(t/W)}^{2}.$ The data is well fit to this form over 4 orders of magnitude of $\\rho (0)$ ranging from $W=0.75t$ to $1.2t$ .",
"For disorder strength less then $W\\approx 0.7$ the rare states are generated with such a low probability that we cannot accurately estimate their contribution to $\\rho (0)$ on these size samples for this number of disorder realizations.",
"The non-zero DOS at $E=0$ has converted the ThSM into a ThDM at $E=0$ .",
"In Fig.",
"REF (a) we compare the data with anti-periodic boundary conditions to the case of using periodic boundary conditions, which for this $L$ give rise to a Weyl peak centered at $E=0$ , inducing a large finite size effect and obscuring the DOS at weak disorder.",
"The data with periodic boundary conditions does help however in providing an estimate of the avoided QCP.",
"It is natural to expect that the non-zero $\\rho (0)$ rounds out the QCP into an avoided transition.",
"To show this explicitly we study the strength of the avoidance by assuming that the DOS is always analytic.",
"This implies that $\\rho (E) = \\rho (0) + \\frac{1}{2!",
"}\\rho ^{\\prime \\prime }(0)E^2 + \\frac{1}{4!",
"}\\rho ^{(4)}(0)E^4 +\\dots .,$ and if the DOS becomes non-analytic $\\rho ^{\\prime \\prime }(0)$ and $\\rho ^{(4)}(0) \\rightarrow \\infty $ .",
"Therefore, we use the size of the second derivative (with respect to energy) of the zero energy DOS $\\rho ^{\\prime \\prime }(0)$ to measure the strength of avoidance [47].",
"We compute $\\rho ^{\\prime \\prime }(0)$ directly using the KPM [47].",
"As shown in Fig.",
"REF we find that for box disorder $\\rho ^{\\prime \\prime }(0)$ is saturated in both system size and KPM expansion order.",
"We conclude that the DOS is analytic and the ThSM to ThDM QCP is avoided.",
"Figure: (a) The second derivative of the DOS at zero energy at μ=0\\mu =0 and as a function of disorder strength W/tW/t for box disorder.",
"The peak describes the avoided QCP and remains finite as we we saturate the value of LL at each particular N C N_C.",
"(b) Focusing at W/t=3.4W/t=3.4, and increasing with N C N_C, we eventually see the value of the peak saturating suggesting the critical point is indeed avoided.",
"This data has been averaged over the twisted boundary conditions.Similar to what has been discussed in Refs.",
"[46], [47], these rare states lead to a destruction of the ThSM at low energy, but as we will explore later there is still a regime at finite energy where the effective semimetallic scaling in the DOS can be seen.",
"We sketch this for the QCPs and avoided QCPs relevant in this work in Fig.",
"REF (b).",
"Rare regions turn the critical point to an avoided critical point, but we can still probe higher energy cross over features of the DOS that are dictated by the hidden QCP.",
"Thus, although strictly from a theoretical viewpoint any disorder destroys the ThSM phase creating the ThDM phase, for all practical purposes an effective ThSM regime can still be observed in the crossover behavior at higher energy and lower disorder."
],
[
"Thermal diffusive metal to band insulator transition $\\mu _I(W)$",
"We now focus on the evolution of the clean QCP separating the ThSM2 and ThBI in the presence of disorder.",
"As we have discussed in Section REF , the anisotropic QCP at $\\mu _I(W=0) = \\pm 6 t$ has a non-analytic DOS $\\rho (E) \\sim |E|^{3/2}$ , which gives $\\rho ^{\\prime \\prime }(0) \\sim | \\mu -\\mu _I(0)|^{-1/2}$ .",
"We now study the evolution of this point in the presence of disorder.",
"In Appendix we treat the effect of disorder on the anisotropic QCP within a perturbative RG approach.",
"We find that (perturbatively) the QCP survives in the presence of disorder with a renormalized phase boundary, and the non-analytic behavior in the DOS remains with the same critical exponents as in the clean limit.",
"As we show in Sec.",
", the RG predictions provide an accurate estimate of the power law scaling in both the DOS and the average band gap.",
"Here, however, we are concerned with the asymptotic low energy behavior of the DOS and whether or not the non-analytic behavior that the power law implies holds all the way down to $E=0$ .",
"To address this numerically, we focus on weak disorder and vary the chemical potential passing from the ThDM to ThBI phase.",
"This transition is an Anderson localization transition and will be characterized by non-self averaging quantities (such as the typical DOS) developing single parameter scaling.",
"However, we are concerned with following the clean quantum critical properties in the presence of disorder and therefore focus on the average DOS.",
"Near the ThDM to ThBI transition, it is natural to expect that there will be some non-trivial interplay between rare regions that are either exponentially localized Lifshitz states or quasi-localized power law states.",
"Since both of these effects are inherently non-perturbative, we study the strength of the non-analyticity in the DOS at weak disorder by computing $\\rho ^{\\prime \\prime }(0)$ within the KPM.",
"As shown in Figs.",
"REF (a) and (b), we find that there is a sharp peak in $\\rho ^{\\prime \\prime }(0)$ which provides an accurate estimate of $\\mu _I(W)$ .",
"We find that the peak is saturated in both $L$ and $N_C$ .",
"Thus, by removing all of the extrinsic rounding due to finite size effects, we conclude that non-perturbative effects of disorder give rise to an intrinsic rounding that suppresses the divergence of $\\rho ^{\\prime \\prime }(0)$ and the DOS remains analytic at the ThDM to ThBI transition.",
"There is a regime at moderate disorder strengths, where the avoided quantum critical line $\\mu _c(W)$ approaches the phase boundary $\\mu _I(W)$ .",
"As shown in Fig.",
"REF (a) and (b), in this regime our data does reveal the existence of two peaks, suggestive that the two lines never intersect.",
"For very large $W$ , we find these peaks become very broad and are smeared out, Fig.",
"REF (c).",
"Figure: (a) The second derivative of the DOS across the transition from ThSM to ThBI.",
"At W=0W=0, this peak is non-analytic, but for finite disorder, we find that it saturates as we increase N C N_C.",
"We see this saturating clearly in (b) where the peak's height is independent of the size for N C =2 10 N_C=2^{10}.",
"(c) The second derivative of the DOS as a function of μ/t\\mu /t across the transition from ThDM to ThBI.",
"We saturate the second derivative of the DOS across the transition.",
"The results are suggestive that the DOS remains analytic across the transition, but are inconclusive as to whether an intermediate ThSM regime remains between the ThBI phase and ThDM regime.Each of these was computed with 1,000 disorder realizations."
],
[
"Phase Diagram",
"Now that we have determined the physics of this model at weak disorder we now move onto establishing the full thermal phase diagram as a function of the chemical potential ($\\mu $ ) and disorder ($W$ ) in Fig.",
"REF .",
"We will then move onto study the avoided criticality between ThSM and ThDM regimes as well as the quantum phase transitions separating the zero energy ThDM and ThBI phases in detail in Sections and respectively.",
"In order to determine the regimes and phases in Fig.",
"REF using the density of states alone, we use slightly different techniques to find the various transitions and crossovers.",
"Generally, we try to keep the Weyl state near zero-energy as in Fig.",
"REF (a) to study the avoided critical phenomena at finite energy in the quantum critical fan [as illustrated in Fig.",
"REF (b)], and we use a box potential with $V(\\mathbf {r})\\in [-W/2,W/2]$ .",
"We find that each ThSM regime is only present at non-zero energy, which then crosses over to a diffusive metal upon lowering $E$ or increasing $W$ .",
"For fixed $E$ , tuning $W$ allows us to pass through a quantum critical crossover regime that is anchored by the avoided QCP line $W_c(\\mu )$ , (see the thin solid orange line in Fig.",
"REF and the dotted orange line in Fig.",
"REF ).",
"In the ThSM regime, we find the DOS has the form $\\rho (E) \\sim E^2$ for $E>E^*$ [where $E^*$ is the crossover energy to the ThDM regime with an $L$ -independent zero energy DOS that is roughly $E$ independent, see Fig.",
"REF (a) and (b)], with $\\rho (0)$ decreasing with increasing $L$ due to the Weyl peak centered about $E=0$ .",
"Near the avoided QCP the low energy DOS goes like $\\rho (E) \\sim |E|$ for $E>E^*$ [see Fig.",
"REF (b)], which defines the quantum critical regime.",
"At finite energy, upon crossing the avoided critical line $W_c(\\mu )$ the model crosses over into the ThDM with a finite DOS at zero energy that becomes $L$ -independent.",
"Even though this shares similarities with its non-superconducting counterparts [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [37], the superconducting case is quite different due to particle-hole symmetry.",
"For example, unlike the metallic case, the beta function from the non-linear sigma model analysis does not possess a zero [52], , , which is suggestive that disorder cannot easily localize the zero energy state.",
"Within our calculations this gives rise to a peak in the low energy average DOS that goes like $-\\sqrt{E}$ at sufficiently low energy deep in the ThDM regime, as described in detail in Appendix .",
"We find that this peak does not occur until crossing a boundary that is always slightly larger then $W_c(\\mu )$ (see the dashed line in Fig REF ).",
"This is very suggestive that the non-linear sigma model description of the problem does not apply until the disorder is strong enough to induce a relatively large zero energy DOS.",
"In order to distinguish the two ThSM regimes at finite energy, we follow the chemical potential at which the anistropic Weyl nodes occur at finite disorder (see Section REF below), and thus identify the crossover between the two ThSM regimes $W_{\\mathrm {SM}}(\\mu )$ .",
"This is shown clearly in Fig.",
"REF .",
"For fixed disorder strength below the ThDM regime (i.e.",
"$W<W_c(\\mu )$ ), upon tuning $\\mu $ the BdG quasiparticles become gapped out entering a ThBI phase, see Fig.",
"REF .",
"Even considering rare regions, $\\rho (0)$ is technically not a good order parameter on either side of this transition as $L\\rightarrow \\infty $ , but its behavior at finite size allows us to nonetheless characterize the transition.",
"In addition, the low energy power law of the DOS upon entering the ThBI phase clearly indicates the existence of an average band gap, as shown in Fig.",
"REF .",
"As discussed previously in section , we expect Lifshitz tails should fill in a small $\\rho (0)$ , making $\\rho (0)>0$ on both sides of the transition; however, they are Anderson localized states in the band insulator.",
"Separating these two phases is a QCP between a delocalized and localized phase, where in the clean limit the band structure has an anistropic Weyl point.",
"This transition line clearly evolves under disorder and chemical potential defining the ThBI critical line $W_I(\\mu )$ , and we connect this transition to its clean counterpart in Section .",
"We discuss how each of these crossovers and transitions is determined from the data in Appendix for the average density of states.",
"We discuss the thermal Anderson localization physics at much larger disorder strength separately in Section .",
"Figure: (a) The crossover from ThSM4 (left of peak) to ThSM2 (right of peak) by observing the finite-size induced peak for ρ(0)\\rho (0) as a function of μ\\mu .Notice that away from the peak the data becomes noisy.This is the expected behavior for ρ(0)\\rho (0) in the semi-metallic regime at finite size [see Fig (c)].This data is taken at L=60L=60.",
"An estimate for the full-width at half-maximum is used for the error.",
"(b) The avoided multi-critical point can be visually captured here by considering ρ(0)\\rho (0) as a function of μ/t\\mu /t.The peak that began at μ=2t\\mu =2t for W=0W=0 smoothly transitions to the peak seen here around μ∼2.4t\\mu \\sim 2.4t.On the right side, we have either ThDM or ThSM4 depending on the value of W/tW/t.On the right side, it quickly becomes ThSM2Figure: The transition from ThDM to ThBI.",
"(a) ρ(0)\\rho (0) as it tends to zero from the ThDM to the ThBI (left) and the average gap that forms in the insulator (right).The dashed line represents a power-law fit of the gap for L=120L=120.",
"(b) The density of states across the transition.Notice that roughly when the gap closes, ρ(0)>0\\rho (0)>0 indicating the ThDM phase.",
"(c) The transition from the ThSM2 regime (left of peak) to the ThBI phase (right of peak) up to disorder W/t=4W/t = 4.As with the ThSM4 to ThSM2 crossover in Fig.",
", the peak at the anisotropic Weyl node is expected and the subsequent fall to ρ(0)=0\\rho (0) = 0 even at this finite size of L=60L=60 is expected in the ThBI phase.The transition and error bars are read off the plot by the position of the peak and resolution of the data.Figure: When fitting ρ(E)∼E d/z * -1 \\rho (E) \\sim E^{d/z^*-1} across the phase transition from ThSM2 to ThBI, we can observe a sudden increase in the power d/z * -1d/z^*-1 due to entering the ThBI phase.",
"This can be used to numerically give an upperbound to the transition μ I (W)\\mu _I(W)."
],
[
"ThSM to ThDM crossover",
"For $\\mu =0$ we find a crossover from ThSM4 to ThDM, which is captured in Fig.",
"REF (a) as $\\rho (0)$ becomes finite.",
"As discussed in Section , we find that the avoided quantum critical point is at $W_c(\\mu =0)/t = 3.525 \\pm 0.075$ , in the inset to Fig.",
"REF (a).",
"We can also see how the density of states $\\rho (E)$ itself changes across the transition, as seen in Fig.",
"REF (b).",
"Near $W=W_c$ we find the DOS for $E>E^*$ varies like (E) |E|d/z-1 One can qualitatively see the that change from $\\rho (E) \\sim E^2$ to $\\rho (E) \\sim |E|$ occurs as a function of $W$ on a linear scale in Fig.",
"REF (b).",
"with a critical exponent $z = 1.50\\pm 0.09$ [as shown in Fig.",
"REF (c)].",
"This result is in excellent agreement with the field theoretic one loop calculation for a random axial chemical potential [29].",
"As we show in Appendix , various disorder couplings are generated in the RG process that are not present in the bare model.",
"When these are all taken into account, we find that the avoided QCP in this particle-hole symmetric model is in fact dictated by the universality class of random axial chemical potential [29].",
"Thus, we find that the RG provides an accurate prediction for the finite energy power law scaling of the DOS.",
"The agreement between the non-trivial power law scaling in the data and the one loop RG estimates of the critical exponents lead us to identify this as the quantum critical regime.",
"Figure: In this figure, we determine the critical exponents for the quantum critical region between the ThSM4 to ThDM regimes.The exponent ν\\nu is determined by two methods:(a) by ρ(0)∝δ ν(d-z) \\rho (0) \\propto \\delta ^{\\nu (d-z)} where δ=(W-W c )/W c \\delta = (W - W_c)/W_c and (b) by data collapse minimizing the collapse function S(ν)S(\\nu ).This is illustrated here for W c =3.525W_c = 3.525.The exponent zz is found simply in (c) by fitting ρ(E)∝(E-b) 3 z-1 \\rho (E) \\propto (E - b)^{\\frac{3}{z} - 1}.The dashed lines are numerical fits.The computed values for ν\\nu and zz are quoted in Table .Table: Avoided quantum critical points and critical exponents for the crossover from ThSM to ThDM.",
"* ^*This point was found just by fitting a power law of ρ(0)∼|W-W c | b \\rho (0)\\sim |W-W_c|^b on the ThDM side; it is less reliable as a result.We now come to numerically determining the correlation length exponent $\\nu $ .",
"We have pointed out in Ref.",
"[42] that computing $\\nu $ from the KPM calculation of $\\rho $ suffers from large fluctuations due to the accuracy problem in $W_c$ and the size of the critical region (which is hard to determine as it depends sensitively on the strength of the avoidance).",
"Despite this we can still provide a reasonable estimate of $\\nu $ from the data via the power law dependence and finite size scaling similar to the method described in detail in Refs.",
"[40], [42].",
"Due to the diverging correlation length at the transition $\\xi \\sim |\\delta |^{-\\nu }$ (where we have defined the distance to the avoided critical point by $\\delta = (W-W_c)/W_c$ ) we expect the scaling hypothesis to hold, which implies $\\rho (0) \\propto \\delta ^{\\nu (d-z)}$ with $\\delta >0$ for sufficiently large $L$ .",
"Here, it is important to note that this power law dependence exists on top of the exponentially small rare region contribution, and strictly speaking $\\rho (0)\\ne 0$ for $W=W_c$ in the thermodynamic limit.",
"Fitting $\\rho (0) \\propto \\delta ^{\\nu (d-z)}$ as shown in Fig.",
"REF (a) (using the value of $W_c$ and $z$ already determined) we find $\\nu _{\\mathrm {Fit}} = 1.38\\pm 0.38$ .",
"The scaling hypothesis implies the finite size scaling form $\\rho (0) = L^{z-d} f(\\delta L^{1/\\nu })$ , which we use to perform data collapse yielding $\\nu _{\\mathrm {DC}} = 1.24\\pm 0.20$ , as shown in Fig.",
"REF (b).",
"The two values of $\\nu $ are in relatively good agreement.",
"Similarly, we consider the crossover from ThSM2 to ThDM for various different values of $\\mu $ (not shown) with the results for the avoided critical points and their corresponding critical exponents in the quantum critical crossover regime summarized in Table REF .",
"Lastly, we have also estimated $z$ and $\\nu $ for a larger number of points focusing on $L=120$ along the avoided quantum critical line $W_c(\\mu )$ as depicted in Fig.",
"REF .",
"Interestingly, we find that $z \\approx 1.5$ holds along the entire line.",
"Focusing on the avoided multi-critical point where the avoided critical line $W_c(\\mu )$ [separating the ThSM regime and the ThDM regime] intersects the crossover line $W_{\\mathrm {SM}}(\\mu )$ [separating the ThSM4 and ThSM2 regimes], despite the uncertainty in $\\nu $ , we find a systematic increase in the value of $\\nu $ near the multi-critical point.",
"From a one-loop RG calculation at this multi critical point there are two relevant scaling variables and as a result $\\nu _{\\mathrm {MC}}=2$ , which is in reasonable agreement with the numerics.",
"Figure: The critical exponents zz (a) and ν\\nu (b) along the critical line connecting the ThSM phases to the ThDM.",
"It becomes more difficult to pin down a value of zz the closer we get to the ThBI phase.",
"In fact, the uncertainty in zz is related to our inability to pin down the phase transition to a good accuracy in those regions.Similarly, it becomes difficult to pin down ν\\nu (when μ>8t\\mu >8t, ν\\nu cannot even be determined reliably due to proximity to the transition to ThBI).",
"The ν\\nu data was found by finding the inflection point of logρ(0)\\log \\rho (0) vs. logδ\\log \\delta by cubic interpolation and determining the tangent at that point.Most of this data was found with ρ(E)\\rho (E) computed with L=120L=120 (for μ=0\\mu =0, 2.322.32, 2.422.42, 2.522.52, and 5, L=140L=140 data is used)."
],
[
"ThSM4 to ThSM2 crossover",
"We now turn to the crossover from one thermally semimetallic regime (ThSM4) to the other (ThSM2).",
"The starting point for this transition is clearly given by the peak $\\mu /t=2$ in Fig.",
"REF (c).",
"This peak can be followed to finite disorder and finally through the ThDM crossover as illustrated in Fig.",
"REF (a) and (b).",
"Table: The values of the crossover point μ SM (W)\\mu _{\\mathrm {SM}}(W) for various WW values for the crossover ThSM4 to ThSM2.",
"Additionally, the exponent z SM * z_{\\mathrm {SM}}^* that describes ρ(E)∼E d/z SM * -1 \\rho (E)\\sim E^{d/z_{\\mathrm {SM}}^*-1} is shown.",
"The value at W=0W=0 is the clean value that is analytically known; the rest are calculated numerically.In the clean limit, there are 3 anisotropic Weyl points (linear in $k_x$ and $k_y$ and parabolic in $k_z$ ), and we know that just as for the ThSM2 to ThBI transition, $\\rho (E)\\sim E^{3/2}$ or $z_{\\mathrm {SM}}^* = 1.2$ by considering again (E)Ed/zSM* -1.",
"When we follow this to higher energies we get the values enumerated in Table REF .",
"We see that the value $z_\\mathrm {SM}^* \\approx 1.2$ is valid until we start getting close to the transition to the ThDM at which point, we obtain results consistent with the value of $z = 1.5$ .",
"The DOS for small disorder ($W/t=1$ and $W/t=2$ in particular) have a large finite size effect from the Weyl peaks.",
"Thus, to obtain error bars (for fixed $L$ ) we perform a fit on the systematically-noisy data that we obtain by performing a moving average."
],
[
"ThDM to ThBI quantum phase transtion",
"We now turn to the diffusive metal to ThBI transition—which can be accessed at both weak and strong disorder.",
"In this section, we focus on the evolution of the clean anisotropic QCP (at $W=0$ and $\\mu =\\pm 6t$ ) in the presence of disorder.",
"We consider the DOS and average band gap to see how effects of the clean QCP survive in the presence of disorder.",
"Despite, focusing on self-averaging quantities we are able to study the ThSM to ThBI transition (see Fig.",
"REF ) from both sides of $\\mu _I(W)$ using $\\rho (E)$ and the average gap $\\Delta _g$ .",
"Note that we will use the notation for the transition line $\\mu _I(W)$ and its functional inverse $W_I(\\mu )$ interchangeably.",
"We are able to connect this QCP to the clean limit by studying the energy dependence of the density of states.",
"Along the critical line $W_I(\\mu )$ we compute the dynamic exponent $z^*$ from (E) Ed/z* -1.",
"We show the corresponding numerical results for $z^*$ in Fig.",
"REF (a), obtained from the fitting shown in Fig.",
"REF .",
"We find good agreement with the dynamic exponent in the clean limit ($=1.2$ ) along $W_I(\\mu )$ until $W_I(\\mu )\\gtrapprox 7.0t$ , which is in good agreement with the expectation that disorder acts like an irrelevant perturbation to the anisotropic QCP.",
"As we have shown in Section REF , non-perturbative effects of disorder round out this non-analyticity in the DOS at the lowest energy (or longest length scale), and the quantum critical scaling only holds at finite energy above a non-universal cross over scale.",
"Similar to the avoided QCP, due to the good agreement between the numerics and the analytic estimate of critical exponents from the one loop RG, we associate this with the quantum critical regime.",
"When $W_I(\\mu )\\gtrapprox 7.0t$ , the quantum critical fan from the avoided quantum critical point begins to contaminate the calculation for $z^*$ , leaving us with a wide range of exponents and large error bars.",
"Our KPM numerics are no longer reliable for determining critical exponents in this regime.",
"Upon entering the ThBI phase an average band gap opens near zero energy, which is captured in $\\rho (E)$ depicted in Fig.",
"REF (b).",
"This allows us to determine the average band gap in the ThBI phase.",
"In both cases, we also use the vanishing of gap as an order parameter for the transition.",
"Approaching $\\mu _I(W)$ from the ThBI side we find that the average gap vanishes in a power law fashion g |-I(W)|, where $\\gamma = \\nu z$ since $\\Delta _g\\sim \\xi ^{-z}$ with a correlation length $\\xi \\sim |\\mu - \\mu _I(W)|^{-\\nu } $ .",
"The values of $\\gamma $ are given in Fig.",
"REF (b) which vary between 1 and 1.3 upon entering the ThBI and jump up to 1.7 for $W_I(\\mu )=9$ , closer to where the disorder is so large that there is no remnant of the SM regime any longer.",
"Note that, since the non-analytic behavior in the DOS is rounded out, this implies that the scaling of the average band gap is also rounded on the largest length scales.",
"Figure: The critical exponents on the critical line μ I (W)\\mu _I(W).",
"(a) The critical exponent z * z^* extracted from fitting ρ(E)∼E d/z * -1 \\rho (E) \\sim E^{d/z^* -1}.In the ThBI phase z * →0 + z^* \\rightarrow 0^+.For W=0W=0 (the blue data point), this value is known exactly from analytics.For 1≤W≤41\\le W\\le 4 (the orange data points), ρ(E)\\rho (E) is noisy, so we find this exponent for larger bounds that could see higher band effects.For W≥5W\\ge 5, we can fit ρ(E)\\rho (E) with E/t≪1E/t\\ll 1 and these are the exponents we find.",
"(b) The critical exponent γ=νz\\gamma =\\nu z, that describes how the band insulator the gap increases in the ThBI phase: Δ g ∼(μ-μ c ) γ \\Delta _g \\sim (\\mu -\\mu _c)^\\gamma .Systematics that could lead to appreciable error in this quantity cannot be reliably estimated.Our data is consistent with $\\nu \\approx 1$ for most of the critical line if $z^*\\approx 1.2$ , until it becomes closer in proximity to the avoided QCP and $z\\approx 1.5$ , while $\\nu $ remains roughly the same.",
"The crossing of quantum critical fans can be seen in Fig.",
"REF where the different regimes can be seen explicitly in the DOS.",
"This is a verification of our schematic in Fig.",
"REF (b) near the QCP.",
"For larger disorder, the ThDM regime gets so close to the ThBI phase that we can no longer properly discern a ThSM regime.",
"Our results in Fig.",
"REF (c) are inconclusive in regards to whether the avoided QCP and the true transition merge; instead we merely see a broad and ill-defined peak in $\\rho ^{\\prime \\prime }(0)$ as a function of $\\mu $ .",
"The critical exponents ($\\nu $ and $z^*$ ) that we have computed in this section describe the power law scaling in the DOS and the average band gap.",
"Our numerical estimates are in good agreement with the one loop RG calculations in Appendix .",
"Approaching this transition from the ThBI side, the localization transition will be described by a diverging localization length $\\xi _l\\sim |W-W_I|^{-\\nu _l}$ and will give rise to single parameter scaling in quantities that are not self averaging (such as the inverse participation ratio or the typical DOS), with robust non-analytic behavior at $E=0$ .",
"It will be interesting to study this localization transition in the future using observables that explicitly track the localization length and see if there is any relation between these two sets of exponents (e.g.",
"$\\nu $ and $\\nu _l$ ).",
"Figure: Plots of the mobility edge defined by M edge (E)=ρ t (E)/ρ(E)M_{\\mathrm {edge}}(E) = \\rho _t(E)/\\rho (E) tuning disorder WW while keeping μ=0\\mu =0 constant.We see a large anti-localization peak in the DOS develop in addition to the standard behavior that states far from E=0E=0 localize first.The function ρ(E)\\rho (E) is calculated at L=60L=60 and ρ t (E)\\rho _t(E) is calculated at L=30L=30 and N C =2 13 N_C = 2^{13}.In (a) we have the whole range given by a colorplot, and in (b) we see some cuts illustrating how drastic the peak is."
],
[
"Thermal Anderson Insulator and localization at large disorder",
"We now turn to the thermal Anderson insulator properties at large disorder.",
"It is important to note that we are now considering a much larger disorder strength than we have considered so far.",
"In order to study localization phenomenon, we consider both the typical $[\\rho _t(E)]$ and the average $[\\rho (E)]$ density of states just as in Sec. .",
"Since the typical DOS goes to zero in the thermal AI phase we can define the mobility edge, as $M_{\\mathrm {edge}}(E) = \\rho _t(E)/\\rho (E)$ .",
"As shown in Fig.",
"REF , we find a thermal mobility edge at finite energy, separating states that are a ThDM and thermal Anderson insulator.",
"The peak from Sec.",
"shows up here before the transition.",
"In addition, as shown in Fig.",
"REF (a), the thermal mobility edge squeezes in towards zero energy where states far away from zero energy localize first.",
"Figure: Typical DOS displaying the Anderson localization transition at large disorder for μ=0\\mu =0 and L=30L=30.",
"(a) The N C N_C dependence of the typical DOS.",
"In the localized phase, ρ t (0)\\rho _t(0) decreases as N C N_C is increased whereas in the ThDM phase ρ t (0)\\rho _t(0) is relatively insensitive to N C N_C.While this is plotted for μ=0\\mu =0, the plot is nearly identical for any μ\\mu [see Fig.",
"(a)].",
"(b) The typical DOS goes to zero while the average DOS remains finite.",
"The average DOS is computed with L=60L=60.",
"(c) The peak in the typical-DOS before the transition to the AI.",
"One sees little or even positive N C N_C dependence at E=0E=0.",
"(d) The peak splits and now ρ t (E)\\rho _t(E) has a strong N C N_C dependence.With the theory for the ThDM breaking down and other measures, we conclude this is the localized phase.However, in order to see localization, we need to check the $N_C$ -dependence of the typical DOS $\\rho _t(0)$ .",
"The finite KPM expansion order ($N_C$ ) controls the broadening of the Dirac-delta function in the local DOS.",
"This introduces an artificial length scale that can make $\\rho _t(0)$ “look” more delocalized due to the convolution of the single particle wave functions and the broadened Dirac-delta functions.",
"For increasing $N_C$ this length scale in $\\rho _t(0)$ should vanish as $N_C\\rightarrow \\infty $ , and thus in the localized phase $\\rho _t(0)$ should go to zero with $N_C\\rightarrow \\infty $ .",
"As shown in Fig.",
"REF (a) we find the transition point by fitting the data at each $N_C$ to a power law $\\rho _t(0)\\sim (W_l - W)^\\beta $ and then fitting $W_l$ to a polynomial in $N_C$ , $W_l = a/N_C^2 + b/N_C + c$ .",
"We then extrapolate $W_l$ to $N_c\\rightarrow \\infty $ to obtain the true transition point.",
"This is represented in Fig.",
"REF (b).",
"However, one might wonder what happens to the peak in the average and typical DOS in the AI phase.",
"The peak in the average gets broadened [see Fig.",
"REF (b)], but the peak in the typical DOS behaves very differently.",
"Before the transition, one can see that the peak is more or less intact in Fig.",
"REF (c), but after the transition to the AI, the peak splits and gets a strong $N_C$ -dependence as seen in Fig.",
"REF (d).",
"Since there is a localization phase transition, we can also characterize the critical exponent by $\\rho _t(0)\\sim \\delta _l^\\beta $ where $\\delta _l = |W_l-W|/W_l$ .",
"However, despite the uniformity of the data for all $\\mu $ , the slowly encroaching phase transition for large $\\mu $ modifies this behavior dramatically and leads to what appears to be a lowering of $\\beta $ for larger $\\mu $ .",
"We cannot confidently say that this is a real affect on the critical exponent $\\beta $ , so we merely state its value at $\\mu =0$ which is $\\beta = 2.9\\pm 0.1$ and the fit can be seen in Fig.",
"REF (b).",
"However, $\\rho _t(0)$ and $\\rho (0)$ seem rather $\\mu $ -independent at around $W\\sim 22t$ where the Anderson transition occurs.",
"Figure: (a) The Anderson insulator transition line at large disorder extracted from the N C N_C dependence of the typical DOS.",
"(b) The fit of the power law for the typical DOS across the Anderson localization transition in the regime that is N C N_C independent.",
"We find β=2.9±0.1\\beta = 2.9\\pm 0.1 and the fit is the red line.",
"All data here is taken at L=30L=30."
],
[
"Discussion and Conclusion",
"In this work we have shown that three-dimensional disordered spinless $p_x+ip_y$ superconductors have a rich quantum phase diagram with various types of crossovers, non-perturbative effects of disorder, quantum phase transitions between different superconducting phases, and Anderson localization.",
"This opens the door for the study of disordered three-dimensional nodal superconductors in general.",
"Our work can also be thought of as the general theory for the consideration of quantum phases in Majorana-Weyl fermions in the presence of quenched disorder.",
"We have established the existence of non-perturbative quasi-localized rare states in the presence of particle-hole symmetry.",
"(It is important to emphasize here that the earlier extensive work [46], [47] in this context on nonperturbative rare region effects by two of the coauthors were restricted to systems without (in italics) particle-hole symmetery.)",
"This finding is highly non-trivial as prior to our work it was natural to assume that particle hole symmetry could somehow “protect” the zero energy eigenstates, but we have found that this is not true.",
"As a result of disorder-induced rare regions all three-dimensional nodal superconductors will always have a non-vanishing DOS at the Fermi energy.",
"However, this effect is exponentially small in disorder, and may be therefore difficult to detect experimentally (but perhaps no more difficult than in the corresponding non-superconducting systems [46], [47]).",
"Nonetheless, we do expect that the avoided quantum critical fan can be probed in thermodynamic quantities such as the specific heat or even the thermal conductivity.",
"We therefore expect that our results will be particularly relevant to situations involving doping of various heavy fermion superconductors.",
"Our main theoretical accomplishments are: (1) showing that nonperturbative rare region effects convert various disorder-driven `semimetal' to `diffusive metal' transitions to crossovers with avoided criticality (with the system being always a diffusive metal even at very weak disorder in contrast to the perturbative RG theory); (2) the underlying avoided critical physics can be well-described by a one-loop RG calculation with reasonable agreement between the RG theory and exact numerical calculations; (3) non-perturbative effects of disorder can round out non-analytic behavior in the clean DOS; (4) at strong disorder, the three-dimensional class D diffusive metal phase undergoes an Anderson localization transition to an Anderson insulator.",
"We emphasize that our work now establishes these conclusions to be definitive for systems with particle-hole symmetry with earlier work [46], [47] establishing it without this symmetry.",
"Particle-hole symmetry allows additional phases and phase transitions in the system which were not considered before.",
"A new feature of the model we have considered here, that is distinct from our previous work on Dirac and Weyl semimetals (Refs.",
"[40], [42], [46], [47]), is the presence of a (thermal) semimetal to band insulator QCP in the band structure.",
"At this transition, the single particle dispersion still has nodal points but the scaling is anisotropic.",
"This leads to a non-analytic DOS that vanishes like $\\rho (E)\\sim |E|^{3/2}$ , and represents a distinct phase from that of the semimetal.",
"Our RG results predict that disorder acts as an irrelevant perturbation at this clean fixed point, and the non-analytic behavior should hold along the renormalized phase boundary (in disorder and chemical potential).",
"As we have shown, however, non-perturbative effects of disorder round this out and the DOS becomes analytic across the entire phase diagram.",
"The explicit theory of these anisotropic rare states is unknown at present and is an interesting problem for future work.",
"Nonetheless, our results point to the generic scenario that non-perturbative effects of disorder dominate the generic behavior of three-dimensional Fermi points (at the longest length scales) independent of symmetry classifications.",
"Disorder acts like an irrelevant perturbation making this class of problems distinct from those with a DOS that does not vanish faster than $|E|$ (e.g.",
"graphene or two-dimensional $d$ -wave superconductors), which is an important new distinction arising from our work.",
"It will be important in the future to treat the superconducting phase in a self consistent manner.",
"Here we described phases, crossover regimes, avoided criticality, and other properties of the a nodal class-D Hamiltonian, but a real system will have a fluctuating order parameter affected by disorder.",
"The existence of these rare states are likely not very sensitive to a spatially fluctuating superconducting gap.",
"As a result, our work opens the prospect of finding rare region mediated superconductivity, where the quasi-localized large probability amplitude is likely to produce tunneling of the Bogoliubov-de Gennes quasiparticles between the rare regions leading to large puddles of superconductivity that will eventually become phase coherent [32] at sufficiently low temperatures.",
"A detailed study of this physics is well outside the scope of the current work focusing on the quantum criticality (or lack thereof), but should be an important future extension of our work.",
"Such a rare region mediated superconductivity will essentially be a novel and exotic phase of matter.",
"The phase diagrams in Fig.",
"REF and Fig.",
"REF summarize all of the physics presented in this manuscript.",
"Our simple model captures much of the essential physics of gapless Weyl nodes in the presence of particle-hole symmetry.",
"We explored much of this physics in the current work: the role of rare-regions, the nontrivial density of states in the diffusive metal, and the localization transition at higher values of disorder.",
"Despite rare-regions, we were still able to probe field theoretic quantites like critical exponents near the avoided critical point, and characterize the physics near true transitions such as the thermal insulator to thermal metal transition.",
"Our analytical and numerical works agree well with each other where ever they both apply.",
"The current work along with the earlier works presented in Refs.",
"[40], [42], [46], [47] essentially complete the basic theoretical study of disorder-driven quantum criticality and nonperturbative rare region physics in three dimensional Weyl systems, bringing to an end a quest that started thirty years ago with Refs.",
"[26], [27].",
"We thank Olexei Motrunich, Gil Refael, Matthew Foster, Sarang Gopalakrishnan, and Rahul Nandkishore for useful discussions.",
"We also thank David A. Huse for various discussions and collaborations on earlier work.",
"This work is partially supported by JQI-NSF-PFC and LPS-MPO-CMTC (JHP, PG, and SDS), and the Airforce Office for Scientific Research (JHW).",
"The authors acknowledge the University of Maryland supercomputing resources (http://www.it.umd.edu/hpcc) made available for conducting the research reported in this paper.",
"Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293."
],
[
"Perturbative renormalization group analysis",
"In this appendix we determine the perturbative effects of disorder on the model defined in Eq. ().",
"Our goal is to establish the universality class, within one loop RG, that governs the perturbative QCPs.",
"Despite this method missing the non-perturbative effects of disorder, the RG does provide an analytic understanding of various features we have observed in the numerics."
],
[
"Anisotropic QCP between ThSM2 and ThBI",
"We first consider the disorder effects on the single flavor of critical excitations governing the QCP between ThSM2 and ThBI.",
"In the presence of a random chemical potential $V(\\mathbf {r})$ (for normal quasiparticles) following Gaussian white noise distribution, the replicated Euclidean effective action around the critical point becomes $&&\\bar{S}_c=\\int d^3x d\\tau \\psi ^\\dagger _a[\\partial _\\tau - i v_p \\nabla _\\perp \\cdot \\tau + (c_1 \\partial ^2_3 + \\delta \\mu )\\tau _3]\\psi _a \\nonumber \\\\&-&\\frac{\\Delta _0}{2} \\int d^3x d\\tau d\\tau ^\\prime [\\psi ^\\dagger _a \\tau _3 \\psi _a](\\mathbf {x},\\tau )[\\psi ^\\dagger _b \\tau _3 \\psi _b](\\mathbf {x},\\tau ^\\prime ),$ where $a$ and $b$ are replica indices, and $\\delta \\mu =6t-\\mu $ is the deviation from the QCP, $v_p=\\Delta _p$ is the quasiparticle velocity in the $xy$ plane, $c_1=t$ is the inverse effective mass along the $z$ direction, and $\\Delta _0$ is the disorder strength.",
"The quadratic part of $\\bar{S}_c$ remains invariant under the scale transformations $\\mathbf {x}_\\perp \\rightarrow \\mathbf {x}_\\perp e^l$ , $\\tau \\rightarrow \\tau e^{l}$ and $x_3 \\rightarrow x_3 e^{l/2}$ , and $\\psi \\rightarrow \\psi e^{-5l/4}$ , while the disorder coupling changes according to $\\Delta _0(l)=\\Delta _0(0) e^{-l/2}$ .",
"Therefore, weak disorder is an irrelevant perturbation, which can only modify the location of the QCP or the phase boundary between ThSM2 and TBI.",
"At one loop level the RG flow equation for $\\Delta _0$ is given by $\\frac{d\\Delta _0}{dl}=-\\frac{\\Delta _0}{2}-\\mathcal {B}_1 \\Delta ^2_0,$ where $\\mathcal {B}_1$ is a nonuniversal constant that depends on the precise method of mode elimination.",
"Therefore, the leading order quantum corrections make disorder a more irrelevant perturbation at the QCP.",
"This is reminiscent of mass disorder effects on a two dimensional, two component Majorana fermion that separates a thermal quantum Hall and a thermal band insulator.",
"In fact, by setting $c_1=0$ , one accesses this particular case of a two dimensional quantum phase transition.",
"To summarize, the QCP separating ThSM2 and ThBI remains stable against disorder, even after accounting for quantum corrections at one loop level, in contrast to the predictions of SCBA.",
"The RG flow equation for $\\delta \\mu $ is given by $\\frac{d\\delta \\mu }{dl}=(1-\\mathcal {B}_2 \\Delta _0)\\delta \\mu + \\mathcal {B}_3 c_1 \\Delta _0$ where $\\mathcal {B}_{2}$ and $\\mathcal {B}_{3}$ are two additional regulator dependent, positive constants.",
"After solving the two flow equations simultaneously, we find $[\\delta \\mu (l) + \\frac{2}{3} \\mathcal {B}_3 c_1 \\Delta _0(l)] \\approx [\\delta \\mu (0) + \\frac{2}{3} \\mathcal {B}_3 c_1 \\Delta _0(0)] e^l,$ for weak disorder.",
"Therefore, $\\delta \\tilde{\\mu }=[\\delta \\mu (0) + \\frac{2}{3} \\mathcal {B}_3 c_1 \\Delta _0(0)]=0$ defines the renormalized phase boundary between ThSM2 and ThBI.",
"After noting that $\\delta \\mu (0)=6t-\\mu $ , we find that disorder shifts the phase boundary to a larger value of chemical potential $\\mu =t(6 + \\frac{2}{3} \\mathcal {B}_3 \\Delta _0 )$ , thus expanding the ThSM2 region.",
"If $\\delta \\tilde{\\mu } <0$ , we need to integrate the flow equations up to $l_\\ast =\\log \\left(1/|\\delta \\tilde{\\mu }| \\right)$ , and work with the low energy theory of disordered MW fermions.",
"Hence, in the vicinity of the ThSM2 to ThBI transition, $\\Delta _0(l_\\ast )<\\Delta _0(0)$ acts as the bare disorder coupling for the MW fermions.",
"Consequently, we expect $W_c(\\mu )$ to increase when $\\mu $ approaches the ThSM2 to ThBI phase boundary.",
"Figure: (Color Online) Renormalized disorder couplings vs. RG flow time l=log(Λ 0 /Λ)l=\\log (\\Lambda _0/\\Lambda ).",
"The disorder couplings Δ 1 \\Delta _1, Δ 2 \\Delta _2, Δ 3 \\Delta _3 and Δ 4 \\Delta _4 are respectively showed as the green, orange, blue and purple lines.",
"(a) for the initial values Δ 1 (0)=Δ 2 (0)=0.8701\\Delta _1(0)=\\Delta _2(0)=0.8701, Δ 3 (0)=Δ 4 (0)=0\\Delta _3(0)=\\Delta _4(0)=0 the renormalized disorder couplings flow to zero at long wavelength limit, signifying a perturbatively stable thermal semimetal phase.",
"(b) for the initial values Δ 1 (0)=Δ 2 (0)=0.8702\\Delta _1(0)=\\Delta _2(0)=0.8702, Δ 3 (0)=Δ 4 (0)=0\\Delta _3(0)=\\Delta _4(0)=0, the renormalied axial chemical potential coupling Δ 4 (l)\\Delta _4(l) diverges at long wavelength limit, indicating a diffusive metal phase.",
"Therefore the universality class of quantum phase transition between semimetal and diffusive metal phases is described by the axial chemical potential disorder controlled quantum critical point, even though we are explicitly tuning the strength of random chemical potential for normal quasiparticles."
],
[
"Majorana-Weyl fermions: ThSM to ThDM perturbative QCP",
"Inside the ThSM2 phase, the Nambu spinor $\\psi (x)$ can be written in terms of the two component right and left handed MW fermion fields as $\\psi (\\mathbf {x}) = R(\\mathbf {x}) e^{i K_1 x_3} +L(\\mathbf {x}) e^{-i K_1 x_3}$ .",
"After combining the right and left handed fields $R$ and $L$ into a four component spinor $\\Psi ^T=(R^T, L^T\\tau _3)$ , the effective action for the MW fermions can be written in the following form $S_{MW}&=&\\int d^3x d\\tau \\Psi ^\\dagger [\\partial _\\tau - i v \\partial _j \\Gamma _j]\\Psi ,$ where $\\Gamma _j=\\sigma _3 \\otimes \\tau _j$ are three anticommuting gamma matrices.",
"We have rescaled the spatial coordinates according to $\\mathbf {x}_\\perp \\rightarrow (v_F/v_p)^{1/3} \\mathbf {x}_\\perp $ , $x_3 \\rightarrow (v_p/v_F)^{2/3} x_3$ , where $v_F=2t\\sin (K_1)$ is the velocity of Weyl fermions along the $z$ or nodal direction.",
"Consequently, we have obtained an isotropic quasiparticle velocity $v=v^{1/3}_{F} v^{2/3}_{p}$ .",
"Since the other two anticommuting gamma matrices $\\Gamma _4=\\eta _1 \\otimes \\tau _0$ and $\\Gamma _5=\\eta _2 \\otimes \\tau _0$ are absent from the effective action, the system has a continuous chiral symmetry under the operation $\\Psi \\rightarrow e^{i \\theta \\Gamma _{45}}$ where $\\Gamma _{45}=-\\Gamma _4 \\Gamma _5=\\eta _3 \\otimes \\tau _0$ .",
"Physically the chiral symmetry originates from the underlying translational symmetry.",
"The random chemical potential for normal quasiparticles gives rise to (i) intranode scattering term of the form $V_{3,45}(\\mathbf {x}) \\Psi ^\\dagger \\Gamma _3 \\Gamma _{45} \\Psi $ , and (ii) two internode scattering terms $V_{b,1}(\\mathbf {x})\\Psi ^\\dagger \\Gamma _4 \\Psi $ and $V_{b,2}(\\mathbf {x})\\Psi ^\\dagger \\Gamma _5 \\Psi $ .",
"The intranode scattering term acts as the third component of the random axial vector potential, causing a random variation of the nodal separation along the $z$ direction.",
"By contrast, the internode scattering terms act as random Dirac masses.",
"We will consider these intranode and internode scattering terms as independent random variables, following Gaussian white noise distributions, and respectively assign the coupling constants $\\Delta _1$ , and $\\Delta _{b,1}=\\Delta _{b,2}=\\Delta _2$ .",
"For a short range random chemical potential, the bare intranode and internode coupling constants are almost equal.",
"When we coarse grain the replicated, disorder averaged effective action obtained from the above random potentials, additional intranode scattering terms are generated.",
"They are described by (i) $\\sum _{j=1}^{2} V_{j,45}(\\mathbf {x})\\Psi ^\\dagger \\Gamma _j \\Gamma _{45}\\Psi $ and (ii) $V_{45}(\\mathbf {x}) \\Psi ^\\dagger \\Gamma _{45} \\Psi $ .",
"At the microscopic level, the first type of scattering arises from random triplet pairing along the $z$ direction (i.e., with a form factor $\\sin k_3$ ).",
"For the low energy problem, they serve as other two components of random axial vector potential, causing a shift of Weyl nodes in the $xy$ plane.",
"The second intranode term describes a random axial chemical potential, and at the microscopic level it originates from the random Doppler shift.",
"Therefore, the RG analysis has to be carried out by including these additional scattering processes.",
"We will assume the $V_{j,45}$ 's and $V_{45}$ to be independent random variables following Gaussian white noise distributions, respectively possessing the coupling constants $\\Delta _{1,45}=\\Delta _{2,45}=\\Delta _3$ and $\\Delta _4$ .",
"Therefore, we perform the RG analysis of the following replicated effective action $&&\\bar{S}_{MW}=\\int d^3x d\\tau \\Psi ^\\dagger _a [\\partial _\\tau - i v \\partial _j \\Gamma _j]\\Psi _a-\\frac{\\Delta _1}{2} \\int d^3x d\\tau d\\tau ^\\prime [\\Psi ^\\dagger _a \\Gamma _3\\Gamma _{45} \\Psi _a](\\mathbf {x},\\tau )[\\Psi ^\\dagger _b \\Gamma _3 \\Gamma _{45}\\Psi _b](\\mathbf {x},\\tau ^\\prime )-\\frac{\\Delta _2}{2} \\int d^3x d\\tau d\\tau ^\\prime \\nonumber \\\\ && \\times \\lbrace [\\Psi ^\\dagger _a \\Gamma _4 \\Psi _a](\\mathbf {x},\\tau )[\\Psi ^\\dagger _b \\Gamma _4 \\Psi _b](\\mathbf {x},\\tau ^\\prime )+[\\Psi ^\\dagger _a \\Gamma _5 \\Psi _a](\\mathbf {x},\\tau )[\\Psi ^\\dagger _b \\Gamma _5 \\Psi _b](\\mathbf {x},\\tau ^\\prime )\\rbrace -\\frac{\\Delta _3}{2} \\sum _{j=1}^{2} \\int d^3x d\\tau d\\tau ^\\prime [\\Psi ^\\dagger _a \\Gamma _j\\Gamma _{45} \\Psi _a](\\mathbf {x},\\tau ) \\times \\nonumber \\\\ && [\\Psi ^\\dagger _b \\Gamma _j \\Gamma _{45} \\Psi _b](\\mathbf {x},\\tau ^\\prime )-\\frac{\\Delta _4}{2} \\int d^3x d\\tau d\\tau ^\\prime [\\Psi ^\\dagger _a \\Gamma _{45} \\Psi _a](\\mathbf {x},\\tau )[\\Psi ^\\dagger _b \\Gamma _{45}\\Psi _b](\\mathbf {x},\\tau ^\\prime ).$ Under the scale transformation $\\mathbf {x} \\rightarrow \\mathbf {x} e^l$ , $\\tau \\rightarrow \\tau e^{l}$ , $\\Psi \\rightarrow \\Psi e^{-3l/2}$ the quadratic part of the action remains invariant.",
"But, the disorder couplings change according to $\\Delta _j (l)=\\Delta _j(0) e^{-l}$ .",
"Therefore, weak disorder is an irrelevant perturbation.",
"After carrying out a one loop RG calculation (this is controlled by a $d=2+\\epsilon $ continuation) we find the following RG flow equations $\\frac{d\\Delta _1}{dl}&=&\\Delta _1\\left[-1-\\frac{2\\Delta _1}{3}-\\frac{4\\Delta _2}{3}+\\frac{4\\Delta _3}{3}-\\frac{2\\Delta _4}{3}\\right]+\\frac{4\\Delta ^2_2}{3}\\nonumber \\\\ &&+\\frac{8\\Delta _3\\Delta _4}{3}, \\\\\\frac{d\\Delta _2}{dl}&=&\\Delta _2\\left[-1-\\frac{2\\Delta _1}{3}-\\frac{4\\Delta _3}{3}+2\\Delta _4\\right],\\\\\\frac{d\\Delta _3}{dl}&=&\\Delta _3\\left[-1+\\frac{2\\Delta _1}{3}-\\frac{4\\Delta _2}{3}+\\frac{2\\Delta _4}{3}\\right] +\\frac{4\\Delta ^2_2}{3}\\nonumber \\\\ &&+\\frac{4\\Delta _1\\Delta _4}{3},\\\\\\frac{d\\Delta _4}{dl}&=&\\Delta _4\\left[-1+2\\Delta _1-4\\Delta _2+4\\Delta _3+2\\Delta _4\\right]+\\frac{4\\Delta ^2_3}{3} \\nonumber \\\\ &&+\\frac{8\\Delta _1\\Delta _3}{3},$ and a scale dependent dynamic scaling exponent $z(l)=1+\\Delta _1(l)+2\\Delta _2(l)+2\\Delta _3(l)+\\Delta _{4}(l)$ .",
"Apart from the attractive clean fixed point, these flow equations support a repulsive fixed point at $\\Delta _1=\\Delta _2=\\Delta _3=0$ and $\\Delta _4=1/2$ , with a dynamic scaling exponent $z=3/2$ .",
"Therefore, the universality class of the semimetal to diffusive metal transition of MW fermions is governed by the random axial chemical potential.",
"This claim can be further substantiated by numerically solving the coupled RG flow equations, with the initial condition $\\Delta _1(0)=\\Delta _2(0) \\ne 0$ and $\\Delta _3(0)=\\Delta _4(0)=0$ .",
"Notice that the intranode part of the random chemical potential disorder by itself is an irrelevant perturbation, and the interplay of intranode and internode scattering processes is essential for driving the phase transition.",
"In Fig.",
"REF (a) we plot the disorder couplings $\\Delta _j(l)$ when $\\Delta _1(0)=\\Delta _2(0)=0.8701$ .",
"At a large length scale, all the renormalized disorder couplings flow to zero, thus indicating a perturbatively stable ThSM2 phase.",
"By contrast, for $\\Delta _1(0)=\\Delta _2(0)=0.8702$ , renormalized axial chemical potential coupling $\\Delta _4$ flows to strong coupling, as shown in Fig.",
"REF (b).",
"This signals a disorder driven diffusive metal phase.",
"Notice that the renormalized strengths of the other three disorder couplings are negligible in comparison to $\\Delta _4$ , which helps us to verify that the universality class of disorder driven semimetal to metal transition is indeed controlled by random axial chemical potential.",
"For a single Weyl cone with a random axial chemical potential, one can directly apply the arguments for non-perturbative effects of rare regions as in Ref.",
"[32] to show the existence of low energy quasi-localized eigenstates that produce a non-zero but exponentially small DOS at weak disorder.",
"The fact that the universality class of the transition for MW fermions reduces to that of the random axial chemical potential provides a plausible explanation for why the non-perturbative effects of disorder (described in detail in the next section) in this particle hole symmetric model are well described by rare regions in the presence of potential disorder, which breaks this symmetry.",
"Essentially, nonperturbative rare region effects are outside the realm of the perturbative RG theory developed in this section (and the situation does not change in higher loops in RG either) with the very basic ingredient of the RG argument, namely, that weak disorder is irrelevant (which also remains true in the self-consistent Born approximation) breaks down in the presence of disorder-induced rare region effects leading to a destabilization of the ThSM phase to the ThDM phase already at infinitesimal disorder.",
"We study this nonperturbative physics numerically in Sec.",
"."
],
[
"Thermal diffusive metal",
"In this appendix we briefly study the properties of the low energy DOS deep in the ThDM phase.",
"In order to make sure we are not in an insulating phase, we consider both the average and typical DOS.",
"Both of these quantities are nonzero, and the DOS follows closely the non-linear sigma model analysis [52], , , [55].",
"Figure: The peak in both the normal-DOS (a) and typical-DOS (b) around E=0E=0 as predicted by perturbation theory for a ThDM.The peak becomes well-defined at finite disorder and persists up to all disorders considered here.The DOS is calculated at L=60L=60 and the typical-DOS is calculated at L=30L=30.",
"(c) We show here two example fits to the expected analytical form for the anti-localizaiton peak in ρ(E)\\rho (E) to the numerical data defined with the integral in Eq.",
"() for small EE.",
"(d)The fit (the dashed red line) works surprisingly well to even lower energies for ρ t (E)\\rho _t(E)As predicted, we see a characteristic peak in the average and typical DOS within this regime as seen in Fig.",
"REF (a) and (b).",
"In particular, from perturbation theory quantum interference phenomena induces a peak in the DOS that can be calculated from (E) = + 1d3 k(2)3 1D k2 - 2 i E. where $\\bar{\\rho }$ is some constant, $\\Re $ is specifies the real part, and $D$ is the diffusion constant (we fit our peak to $\\bar{\\rho }$ , $D$ , and a cutoff scale $\\Lambda $ ).",
"In particular, near $E=0$ , we have (E)- (0) - |E|, with $\\rho (0) >0$ .",
"The data is well fit by the analytic form as indicated in Fig.",
"REF (c).",
"It is important to note that this peak does not manifest itself until well after the avoided QCP and its onset is indicated by a dashed green line in Fig.",
"REF .",
"In fact, the fit is even better for the typical DOS as seen in Fig.",
"REF (d).",
"However, higher energy states begin to localize as indicated by the typical DOS and we eventually find a transition into an Anderson insulator."
],
[
"The avoided QCP between ThSM to ThDM",
"For the ThSM$n$ to ThDM crossover, we have studied $\\mu /t = 0,\\, 2.32,\\, 2.42,\\, 2.52,\\, 5.0$ at various system sizes ranging from $L=60-140$ in steps of 20.",
"As a function of $W$ , we analyze $\\rho (0)$ for various system sizes to obtain an estimate of the avoided critical line $W_c(\\mu )$ between the ThSM$n$ and ThDM crossovers.",
"As $1/L \\rightarrow 0$ , $\\rho (0)$ tends to its rare-region value $\\rho (0)\\ll 1$ in the ThSM$n$ regime, but in the ThDM regime, it saturates to a larger value $\\rho (0)\\sim O(1)$ , and for $\\mu /t = 0,\\, 2.32,\\, 2.42,\\, 2.52$ we use this to locate $W_c(\\mu )$ , see Fig.",
"REF .",
"However, in general tracking the size dependence is complicated by the renormalization of $\\mu $ : The number of states at $E=0$ can fluctuate at finite disorder, and with the numerics presented in this section we cannot distinguish between rare-region effects and numerical background effects when the zero-energy state is not present.",
"We therefore cannot use this approach in general.",
"For the other values of $\\mu $ along this transition line ($\\mu = 0-10$ in increments of $0.5$ ) we focus on $L=120$ and fit a power law well into the ThDM regime to find the transition point, i.e.",
"from $\\rho (0) \\propto {(W - W_c)}^b$ on the ThDM side of the transition for $L=120$ .",
"From this we obtain the approximate location of the avoided quantum critical point between the ThSM and ThDM regimes as shown in Fig.",
"REF .",
"We have checked that these two procedures give consistent estimates of $W_c(\\mu )$ ."
],
[
"ThSM4 to ThSM2",
"For the crossover between the ThSM4 to ThSM2 regimes we utilize the fact that for a fixed $L$ , $\\rho (0)$ has a maximum at the transition for $W=0$ (see Fig REF (c)).",
"We can understand this maximum by an increase in the number of zero energy states due to the three anisotropic points, and we can easily track it as seen in Fig.",
"REF .",
"Due to the perturbative irrelevance of disorder to the anisotropic Weyl nodes and an analysis of $\\rho (E)$ (see Section REF ) we conclude that this is a signature of the crossover.",
"This maximum broadens and eventually intersects the avoided quantum critical line."
],
[
"ThSM2/ThDM to ThBI",
"To determine the transition from the ThDM to the ThBI phase, the value of $\\rho (0)$ drops abruptly (due to an average gap there are very few if not zero states at or near $E=0$ , even at finite size) which is easily discernible up to $W\\sim 5 t$ .",
"Since $\\rho (0)$ isn't an order parameter for the transition we use two complementary methods to find the critical line $\\mu _I(W)$ : (A) By fitting $\\rho (E) \\sim E^{d/z^*-1}$ to the DOS inside the ThSM2 regime and observing where $z^*\\ll 1$ (numerically $d/z^*\\sim O(10^2)$ suddenly, see Fig.",
"REF ) as a result of an average gap in the spectrum.",
"(B) Approaching the transition from the ThBI, we find the average band gap in the ThBI phase and extrapolate it to zero via a power law fit, see Fig.",
"REF (a)."
]
] | 1612.05648 |
[
[
"Growth over time-correlated disorder: a spectral approach to Mean-field"
],
[
"Abstract We generalize a model of growth over a disordered environment, to a large class of It\\=o processes.",
"In particular, we study how the microscopic properties of the noise influence the macroscopic growth rate.",
"The present model can account for growth processes in large dimensions, and provides a bed to understand better the trade-off between exploration and exploitation.",
"An additional mapping to the Schr\\\"ordinger equation readily provides a set of disorders for which this model can be solved exactly.",
"This mean-field approach exhibits interesting features, such as a freezing transition and an optimal point of growth, that can be studied in details, and gives yet another explanation for the occurrence of the $\\textit{Zipf law}$ in complex, well-connected systems."
],
[
"Conventions for the disorder",
"We will first introduce the details of the disorder, making the assumption that the resources $\\eta (x,t)$ obey an Itō equation with time independent drift and diffusion: d (t) = D1() dt + 2 D2() dWt where $W_t$ is a Brownian process.",
"The probability distribution $P(\\eta ,t)$ obeys the Fokker-Planck equation (with the Itō prescription): P(,t)t = - (D1 () P(,t)) + 22 (D2() P(,t)) = L0 P(,t) $\\mathcal {L}_0$ being the Fokker-Planck operator of the disorder is written.",
"As commonly stated [33], the $\\eta $ dependence of the diffusion $D_2(\\eta )$ can be absorbed by a change of variable and we assume it constant in the following.",
"We also assume $\\eta $ to have a stationary distribution: Q() = N-1 e- () with N = de-() () = D2 - D1(u)D2 du with natural boundary conditions (or reflecting in case of bounded support).",
"We comment on this hypothesis later in Section .",
"As $\\Phi $ is defined up to a constant, it can be written as $f$ : () = f()/D2 f() = - D1 (u) du with $f$ the potential of the process.",
"Finally, a non zero mean $\\mu =\\int \\eta Q(\\eta ) d\\eta $ simply adds a constant contribution to the growth rate.",
"We set such mean to 0 and focus on the contribution stemming from the fluctuations of the disorder.",
"We will especially examine the interplay between exploration and time correlations.",
"To quantify those correlations, it is natural to introduce the -normalized- integrated time correlation function [33]: T=0 K(t)K(0) dt K(t) = (t) (0) Q where $\\langle \\cdots \\rangle _Q$ denotes in the following the average with respect to $\\eta $ .",
"$T$ can be expressed in terms of the terms in Eq.REF as [34]: T=1K(0)- dxD2(x) Q(x) (- x s Q(s) ds)2 In the following, we will start all our stochastic processes at stationarity, so $K_{\\eta }(0)$ reduces to the variance $\\langle \\eta ^2 \\rangle _Q$ of $Q(\\eta )$ ."
],
[
"The evolution equation of the growth process",
"We consider a large number $N$ of sites, each populated by $Z_i(t)$ elements and resources $\\eta _i(t)$ , $i=1,... ,N$ .",
"According to the rules presented in the introduction, each $Z_i(t)$ evolves as: Zi(t+ dt) = {ll Zi(t) [i(t) dt ] prob.",
"1- dt 12 (Zi(t) + Zj(t)) prob.",
"dt .",
"where $j \\ne j $ labels a site chosen at random amongst the rest.",
"There is considerable freedom in choosing the branching process.",
"We stick to the most common Poisson branching, with a fixed rate $\\lambda $ , but the derivation below can be easily generalized (see [16] for some examples).",
"Owing to its wide fluctuations, the magnitude of $Z_i(t)$ can be estimated in two ways: picking one realization of the disorder and considering its almost sure behaviour (the quenched setting), or averaging $Z_i$ over all possible realizations of the disorder (the annealed setting).",
"Therefore central quantities are the typical growth rate $c_q$ and the more common average growth rate $c_a$ : cq =1t N j Zj(t) ca = 1t N j Zj(t) The first quantity, although harder to calculate, is more representative of the typical, most likely, growth, and we focus on it, following the approach of Derrida and coworkers [17], [35] and [16] and defining the generating functions, for any $i$ : Gt(x,) := [-e-xZi(t)][i(t)-] Gt(x) := -d Gt(x,) = [-e-xZi(t)] Gt=0(x,) = (-e-x) Q() We assume the disorder initialized at stationarity and $Z_i(t=0)=1$ .",
"Due to the temporal persistance of the disorder $\\eta $ , we need to keep track of its value through $G_t(x,\\eta )$ .",
"For long times, the behaviour of $\\hat{G}$ is akin to that of a wave front, traveling with constant velocity in the $x$ direction, and reaching 1 exponentially for large $x$ : Gt(x) 1- e-(x-c t) for x .",
"Moreover, for $x \\rightarrow +\\infty $ , the generating function goes rather flatly to 0 and so: Gt(+ ,) [i(t) - ] = Q() This suggests to look at the following Ansatz for $G_t(x,\\eta )$ , where dependence in $x$ and $\\eta $ are factorized [16]: Gt(x,) Q() - R()e-(x-ct) under the constraints: Q() d= R() d= 1 Combining the definition Eq.REF with the evolutation equation Eq.REF and averaging over the disorder, one obtains the following evolution equation for $G_t$ : Gt+dt(x,) = (1- dt) [-e-x+i(t)dtZi(t)][G i(t) -] + dt[-e-x-(2)Zi(t)][i(t)-] [-e-x-(2)Zj(t)]with $\\mathcal {G}$ the infinitesimal propagator of $\\eta _t$ over an increment of time $dt$ .",
"Expanding the arguments of $G_t$ for small $\\mathrm {d}t$ : t Gt(x,) = L0 G - x G + ( Gt(x - (2),) Gt(x-(2)) - Gt(x,) ) with $\\mathcal {L}_0$ the Fokker Planck operator given Eq.REF .",
"The above equation obeyed by $\\hat{G}_t$ is, under disguise, a wave propagation equation, in the $x$ -direction -although it lacks the diffusion term in $x$ , such as in [17]-.",
"This partial differential equation seems rather difficult to solve, but our analysis only requires the asymptotic speed of the wave, obtained from the exponential decay of the front $\\gamma $ .",
"Plugging the Ansatz Eq.REF and identifying terms of order 1 and terms of order $e^{-\\gamma (x-ct)}$ , Eq.REF reduces to the following system: 0 = L0 Q() R c = L0 R + R + (1/2)Q dR() + R() ((1/2)-1) Eq.REF states that $Q(\\eta )$ is the stationary distribution of the process $\\eta (t)$ , consistently with Eq.REF .",
"The scaling degrees of freedom of both equations are fixed by the normalization contraints Eq.REF .",
"Once $Q$ is known, Eq.REF is simply an inhomogeneous Sturm Liouville problem, whose operator is very similar to the Fokker-Planck operator of $\\eta $ , aside from the bias $\\gamma \\, \\eta $ , intimately related to the decay of the front.",
"It is convenient to introduce $Q_H(\\eta ) = \\sqrt{Q(\\eta )}= \\mathcal {N}^{-1/2} \\exp (-\\Phi (\\eta )/2)$ .",
"Transforming further $\\mathcal {L}_0$ into an hermitian operator is accomplished by the change $R(\\eta ) = \\lambda 2^{-\\gamma } Q_H (\\eta ) S(\\eta )$ , and multiplying the whole equation by $ \\sqrt{\\mathcal {N}} \\exp \\left(\\Phi (\\eta )/2 \\right)$ leaves us with the new system: LH S + S - S ()= - QH = (1-(1/2)) + c LH = e/2 L0 e-/2 Solving this system is standard, and we adopt the Green Function (or resolvent) formalism [36].",
"We first consider the homogeneous version of Eq.REF : LH S + S + S = 0 for any real $\\alpha $ .",
"As we assume a constant diffusion coefficient $D_2$ , it is instructive to cast Eq.REF into a Schrödinger form: 2 S 2 = V()S - D2 S V() = D1()24 D22 - D1()' + 2 2 D2 This analogy is particularly useful for exploring some exactly solvable models, as we can draw from the wisdom in Quantum Mechanics, and we will illustrate it through some examples below.",
"The regular Sturm-Liouville theory asserts that solutions of Eq.REF can be decomposed over the eigenset $\\lbrace \\alpha _n \\rbrace $ and $\\lbrace \\phi _n \\rbrace $ , $n \\in \\mathbb {N}^{+}$ .",
"We will assume from now on that Eq.REF has at least one bound state solution (in other words, at least $\\alpha _0$ is isolated, at the bottom of the spectrum), the significance of such hypothesis will become clearer later on.",
"We also use the common convention to write the decomposition as a discrete sum: (LH + ) n = - n n S() = n sn n () Plugging it into Eq.REF : n (- n - ) sn n = -QH and because $\\lbrace \\phi _n \\rbrace $ is a complete orthonormal basis: sn (n + ) = QH | n We can therefore write $S(\\eta )$ , and $R(\\eta )$ decomposed as: S() = n QH | n n + n () R() = 2- n QH | n n + QH() n () Given proper boundary conditions, one can finally recover $c(\\gamma )$ as an implicit equation by enforcing the self consistent condition $ \\int d \\eta \\, R(\\eta ) =1$ , leading to: 2 = n QH | n n + dQH() n () = n QH | n 2n + The quantity: G(z) = n n | n n - z is known as the resolvent operator.",
"$G_0(z)$ corresponds to the Green function of the system with no bias $\\gamma $ , and its lowest eigenvector is precisely $Q_H(\\eta )$ , of eigenvalue $\\alpha _0 =0$ .",
"We can compactly rewrite the above system as: QH | G ( (2--1) - c ) | QH = 2 The above formula allows to extract $c$ as a function of the front decay $\\gamma $ .",
"But it requires a rather suble analysis of the behaviour of the frond decay in travelling wave equations [17].",
"We recall this analysis in our setup in the following.",
"Figure: (Sketch) The growth rate cc as a function of γ\\gamma .",
"Typically, c(γ)c(\\gamma ) decreases at small γ\\gamma and increases at large γ\\gamma .",
"The increasing branch is not realized and any front initially prepared with a decay on this branch γ t=0 >γ min \\gamma _{t=0}> \\gamma _{min} will relax towards the minimum, asymptotically propagating at a speed c min c_{min} (the frozen regime)."
],
[
"The front relaxation",
"As written, Eq.REF is an implicit relation between $c$ and the decay of the front $\\gamma $ , with the parameters of the noise $D_1$ , $D_2$ and the diffusion rate $\\lambda $ as parameters.",
"Generically, the curve $c(\\gamma )$ exhibits a minimum at $\\gamma _{min}$ : for a range of $c$ , the variable $\\gamma $ is double valued (see Fig.REF ).",
"In the wave propagation literature [17], [37], it is known that, in fact, the increasing branch $\\gamma > \\gamma _{min}$ is never realized and the following mechanism takes place: when a front is prepared with a sharper decay than $\\gamma _{min}$ , this decay relaxes over time towards $\\gamma _{min}$ .",
"Given Eq.REF , the front is initially prepared with a decay $\\gamma _0 = 1$ .",
"Two cases are possible: If $\\gamma _{min} >1$ , the propagation of the wave with $\\gamma _{t=0}=1$ is possible: such situation corresponds to the annealed regime, and occurs, for example, at large diffusion $\\lambda $ .",
"Plugging $\\gamma =1$ in Eq.REF leaves us with $c$ as a implicit function of the noise and $\\lambda $ .",
"We refer to this portion of the curve as the annealed branch.",
"Instead, if $\\gamma _{min} <1$ , the front broadens towards $\\gamma _{min}$ , the frozen regime (see Fig.REF ).",
"Then $\\gamma $ is asymptotically fixed to $\\gamma _{min}$ (itself a function of the parameters) and plugging its value in Eq.REF gives back $c$ as a function of $\\lambda $ .",
"We refer to this portion of the curve as the quenched branch.",
"When $\\gamma _{min}=1$ , the parameters are tuned right on the freezing transition, and we denote $\\lambda _c$ the critical diffusion rate.",
"The typical shape of the whole curve is shown on the sketch Fig.REF .",
"The junction of both branches is therefore at $\\lambda _c$ and they also have a global maximum at $\\lambda _m$ .",
"The remaining part of the paper will be dedicated to the analysis of such curve for general processes, and illustrated on particular examples.",
"Figure: (Sketch) The growth rate cc as a function of λ\\lambda , in log-log plot.",
"Once the critical diffusion rate λ c \\lambda _c is fixed by γ min =1\\gamma _{min}=1, it separates two regimes corresponding to quenched (dashed-dot) and annealed (dahsed) branch solutions of Eq., depending on the value of γ\\gamma .",
"Both curves touch at λ c \\lambda _c.",
"The correct curve is depicted in large, red, dashes for noises with a non-zero correlation time.",
"The white-noise case is also depicted, with black squares, and reaches a plateau equal to |α 0 (γ=1)||\\alpha _0 (\\gamma =1)| at λ c \\lambda _c.",
"Typical growth rates exhibit a maximum at a value λ m \\lambda _m, in the quenched phase λ m <λ c \\lambda _m < \\lambda _c.As described by Eq.REF , to obtain $c(\\lambda )$ , one merely needs to obtain the resolvent of Eq.REF , the same as the resolvent of the operator $\\mathcal {L_H}$ with an additional linear bias of amplitude $\\gamma \\eta $ .",
"Green functions are usually difficult to compute and such task is not easy.",
"This very problem is nonetheless not new and has triggered a large activity in the somewhat unrelated field of Quantum Mechanics (QM), under the name of Stark effect [38]: how is a bounded electron perturbed when an electric field is switched on?",
"Of course, the mapping from Itō process to Schrödinger potential may sometimes lead to complicate expressions of $f(\\eta )$ , but it also provides a way to leverage the computational means developed to tackle the Stark effect.",
"Let us illustrate the similarity of both problems.",
"Note that $\\hat{\\lambda }>0$ for any $\\gamma $ , and consider first the case $\\gamma $ very small.",
"In the right side of Eq.REF , all the terms in the sum, except for $n=0$ , are close to 0, due to the vanishing overlaps.",
"Hence, Eq.REF reduces to good approximation to: 2 QH |0 20() + from which we will extract the asymptotics for $\\gamma \\rightarrow 0$ .",
"At $\\gamma =0$ , excited states all have a higher positive energy, and $\\hat{\\lambda }>0$ .",
"Once $\\gamma $ differs from 0, $\\alpha _0(\\gamma )$ necessarily becomes negative, a well-known result in QM [38].",
"As $\\gamma $ goes to 1, the behaviour of the series $\\lbrace \\alpha _n \\rbrace _n$ strongly depends on the details of the disorder, and some eigenvalues may cross the $y$ -axis, also becoming negative.",
"Therefore, many branches of solutions of Eq.REF appear, but because we expect $c(\\gamma )$ continuous, the physical solution remains close to the pole at $\\alpha _0$ , and so $c(\\gamma ) < |\\alpha _0(\\gamma )|$ for any $\\gamma $ .",
"An important quantity, usually coined the polarisability $\\epsilon $ , is defined as: 0 () = - 2 + O(3) The vanishing of the first order term stems from the fact that the mean of $\\eta $ is set to 0.",
"The value of $\\epsilon $ is obtained either using the Rayleigh-Schrodinger theory, or simply expanding the stationary probability distribution $Q(\\eta )=|Q_H(\\eta )|^2$ in small $\\gamma $ , obtaining for the energy at second order, after some manipulations: = 1D2 - dxQ(x)(-x s Q(s) ds )2 = 2 Q T We have also used the general expression for the correlation time defined in Eq.REF .",
"This is an example of Green-Kubo identity: $T$ is obtained by integrating the temporal two point function $\\langle \\eta (0) \\eta (t) \\rangle $ , whereas $\\epsilon $ describes the response to a linear forcing proportional to $\\gamma $ .",
"In our context, $\\epsilon $ quantifies the propensity of $\\eta $ to “yield” under the effect of the bias $\\gamma $ .",
"The characteristics of a soft -or very sensitive to the biais- process become rather clear from Eq.REF : it should widely fluctuate or be long time-correlated.",
"Note that Eq.REF is, in many cases, a suitable approximation also for large $\\gamma $ .",
"Indeed, as $\\gamma $ grows, the left side of Eq.REF blows up exponentially, forcing $\\hat{\\lambda }$ to concentrate around $\\alpha _0$ .",
"This ultimately depends on the asymptotic behaviour of $\\lbrace \\alpha _n \\rbrace $ .",
"We therefore turn onto a more detailed study of the asymptotics of both annealed and quenched branches."
],
[
"The annealed branch",
"We first consider the behaviour of $c_a(\\lambda )$ with $\\gamma =1$ in the limit of weak diffusion $\\lambda \\rightarrow 0$ .",
"It provides a useful upper bound for $c_q(\\lambda )$ : ca() =|0(=1)| - (1- QH | 0 2=1)2 + O(2) Knowing that $\\alpha _0(\\gamma = 0) = 0$ , $|\\alpha _0(\\gamma =1)|$ again measures the ability of the noise to \"polarize\" under the field $\\gamma = 1$ .",
"The extreme case of a \"stiff\" process is the constant Langevin noise, for which a variation of $\\lambda $ has no effect at all.",
"Interestingly, Eq.REF also provides a compact way to compute the Laplace transform of integrated Markov processes $\\langle \\exp (\\int _0 ^t \\eta (t) dt) \\rangle $ , an important endeavour in finance [39].",
"The large $\\lambda $ limit requires to expand the right hand side of Eq.REF in inverse powers of $\\lambda $ (to lighten the notations, all the overlaps and eigenvalues in the remaining of this subsection are evaluated at $\\gamma = 1$ ), assuming the overlaps decay exponentially fast at large $i$ : 1 =i QH | i 22 i/+ 1 + 2 c()/ 1 = i QH | i 2 (1 -2 c() + i +) Using together the normalisation of $Q_H$ , and the fact that $Q(\\eta )$ has zero mean, hence $\\langle E \\rangle =\\sum _i \\alpha _i \\langle Q_H | \\phi _i \\rangle ^2= \\langle Q_H |\\hat{x} | Q_H \\rangle =0$ , we obtain: c() = 2 i i 2 QH | i 2 - 4 i i 3 QH | i 22+ O(-3) The coefficient of the dominant decay can be rewritten using: i i 2 QH | i 2 = QH() (LH +)2 QH() = 2 QH()2 = 2 Q Hence the dominant decay is given by the variance of $Q$ .",
"Higher order terms include higher moments of $Q$ and can be systematically computed."
],
[
"The quenched branch",
"The quenched branch is more difficult to investigate, as one has also to obtain the location of the minimum $\\gamma _{min}$ of $c(\\gamma )$ .",
"We extract the small $\\lambda $ expansion, obtained by considering Eq.REF , under the assumption that both $\\gamma _{min}$ and $c(\\gamma _{min})$ go to 0 as $\\lambda \\rightarrow 0$ .",
"Plugging Eq.REF into Eq.REF and balancing all the terms, we obtain: min = + O() c(min) = 2 + O() By expanding to higher order the lowest eigenvalues and overlaps of the resolvent, the approximation can be systematically improved, but the computation quickly becomes tedious.",
"Note that both asymptotics, large and small $\\lambda $ , depend on the variance and the time correlation of the noise only, a manifestation of universality.",
"One can shed light on those scalings REF and Eq.REF using more hand-waving arguments and the Feynman-Kac representation [40]: Z(x,t) = (0 X(s)(t-s) ds ) X where $X$ is a Poisson process over the space of sites, of rate $\\lambda $ and distribution $\\pi _X$ .",
"We consider first the small diffusion $\\lambda $ case.",
"Over a total time $t$ , $\\lambda t$ jumps occur, breaking $\\int _0 \\eta _X(s)(t-s) ds$ into $\\lambda t$ pieces.",
"Each of those pieces is the integral, over a time $1/\\lambda $ , of a time $T$ -correlated noise, and so has a typical amplitude of $\\sqrt{\\langle \\eta ^2 \\rangle _Q T t/\\lambda }$ .",
"Deep in the quenched phase, the measure is dominated by the maximum over $X$ , being roughly estimated by: (Z) t 2 Q T/ c() (Z)/t 2 Q T The high-$\\lambda $ limit goes along similar same lines and has been presented in [16] in a different form: first recall that, for the white noise model, the free energy in the annealed phase is fixed to $\\langle \\eta ^2 \\rangle _Q$ .",
"At finite $T$ and large $\\lambda $ , the random walk is so fast, compared with $T$ , that it only sees a frozen disorder on each site, before jumping onto another.",
"Again $\\int _0 \\eta _X(s)(t-s) ds$ breaks into $\\lambda t$ pieces, but each is now simply the integration, over a time $1/\\lambda $ , of a frozen random variable $\\eta $ , independently drawn from $Q(\\eta )$ .",
"Therefore in this case: (Z) t 2 Q/2 c() 2 Q/ In this section, we illustrate the computational aspect of the approach, first solving the case of the Ornstein Ulhenbeck by an alternative, but equivalent, route to the one presented in [16].",
"We then go onto processes of bounded support, or with varying tails in their stationary distributions.",
"Other solvable examples could be inspired by the literature on Stark effect [41], [42], [43].",
"Figure: c(λ)c(\\lambda ) as a function of λ\\lambda for the bounded process, with the set of parameters (from top to bottom): D 2 D_2 and aa fitted from Eq.",
"; D 2 =0.5D_2=0.5, a=1a=1; D 2 =1D_2=1, a=1a=1.",
"The dotted line is a fit obtained from the OU process, matching the asymptotic behaviour.",
"The dashed line is the expansion Eq..",
"The numerics are performed on a system of size N=10 6 N=10^6 sites, up to a time t tot =500t_{tot}=500, with the discretization time step dt=0.001dt=0.001."
],
[
"The Ornstein-Ulhenbeck process",
"The Ornstein-Ulhenbeck (OU) process was the first colored generalization made [16], to our knowledge.",
"This corresponds in QM, to the harmonic oscillator, whose solution is completely known.",
"The Fokker-Planck operator and stationary solution are: L0 = D2 22 + k f() = k 22 Q()= k2 D2 e-k 22 D2 QH () = (k2 D2 )1/4 e-k 24 D2 The problem is equivalent to solving the Schrodinger equation in a potential given by Eq.REF : V() = k2 24 D22-k + 2 2 D2 a tilted harmonic potential.",
"Using $\\tilde{\\eta }=\\sqrt{\\frac{k}{2 D_2}} \\eta - \\frac{\\sqrt{2 D_2}}{k^{3/2}} \\gamma $ , we reduce it to: 2 S 2 = (2 - ) S = 1+ 2 k + 2 D2 2k3 The propagator of the Harmonic oscillator goes by the name of the $\\textit {Mehler formula}$ .",
"In the $(\\tilde{\\eta },\\epsilon )$ set of variables: K(1,2,t) = 12 (2t) ( (2 t) (12 +22)/2 + cosech(2 t) 1 2 ) The resolvent is simply the Laplace transform of the propagator $K(\\tilde{\\eta }_1,\\tilde{\\eta }_2,t)$ with respect to $t$ : 2 =1,2 d1 d2 t=0 dt e- t K(1, 2,t) Performing both gaussian integrals in $\\tilde{\\eta }_1$ and $\\tilde{\\eta }_2$ , we are left with: 2 = 0 dt (D2 2k3 (e-t k-1) + t (2 D2k2 - ) ) Another route (detailed in Appendix ) is to fully diagonalize $\\mathcal {L_H}$ and write down the resolvent as an infinite sum.",
"A numerical confirmation of the above result is plotted Fig.REF .",
"The upper bound is $|\\alpha _0(\\gamma =1)| = D_2/k^2$ , and the set of parameters in Fig.REF has been chosen so that this upper bound is fixed to $1/2$ .",
"As $T = 1/k$ tends to 0 (the white noise limit), $c(\\lambda )$ saturates at the plateau $c(\\lambda ) = |\\alpha _0(\\gamma =1)| $ in the annealed phase.",
"This limit is singular however, as for any small $T>0$ , $c(\\lambda )$ decays as $\\lambda ^{-1}$ ."
],
[
"The bounded noise",
"Another case of common interest, especially in condensed matter, is the noise of bounded support.",
"It corresponds to a particle in an infinite well, submitted to a uniform electric field, and is again solvable [44], although we end up with a set of transcendental equations.",
"To simplify slightly the analysis, we set $V(\\eta )$ to be a square infinite well, which translates into a bounded but rather contrived Itō process.",
"At $\\gamma = 0$ , we have: V0() = - 24 a2 for || < a f()= - 2 D2 ((2 a) ) Q() = a-1 2 ( 2 a ) QH() = a-1/2 ( 2 a )The eigenset is simply made of Airy functions.",
"The potential with bias is $V(\\eta ) = -\\pi ^2/(4 a^2) - \\gamma \\eta /D_2$ and after the change of variables: = -( D2 ) 1/3 (+ +2 D24 a2 ) we obtain the following eigenbasis, with their according boundary conditions: n() = an Ai () + bn Bi () b= -( D2 ) 1/3 (a + +2 D24 a2 ) Ai(b+)Bi(b-)=Ai(b-)Bi(b+) The discrete eigenvalues $\\lbrace \\alpha _n \\rbrace _n$ are solutions of the above transcendental equation.",
"Once this discrete set of eigenvalues is determined, $a_n$ and $b_n$ can be fixed so that the set $\\phi _n$ is normalized and obeys the boundary conditions.",
"We compute the first $N=10$ terms of the resolvent as an estimate.",
"$c$ as a function of $\\gamma $ is plotted Fig.REF and compared with numerical simulations.",
"Once again, the agreement is excellent.",
"On Fig.REF , we also have compared this bounded process with an Ornstein-Ulhenbeck one, matching both $T$ and $\\langle \\eta ^2 \\rangle $ , which read: 2 Q = a2(1 -6/2)3 = 12 T = a2(15/2-1)D2(2-6) = 1 Both curves are quite similar, the largest deviation occurs around the freezing transition.",
"It emphasizes the difficulty of choosing a faithful modelling of systems that sit around $\\lambda _c$ ."
],
[
"The role of the tails",
"Growth processes can be seen as extremal in some sense: their statistics are dominated by those space-time paths that manage to collect the largest amount of resources.",
"Inspired by the theory of extreme statistics, one would expect the tails of $Q(\\eta )$ to play a prevalent role.",
"The asymptotics mentioned in Section only depend on $\\langle \\eta ^2 \\rangle _Q$ and $T$ .",
"To analyse the effects of the tail of $D_1(\\eta )$ on $T$ , for example, we define the famility distribution $Q_{\\mu }$ obtained from $D_1(\\eta ) \\sim sign(\\eta ) \\, |\\eta |^{\\mu }$ , such that $D_2$ and $\\langle \\eta ^2 \\rangle _{Q_{\\mu }}$ are normalized to 1.",
"This family smoothly interpolates from the harmonic potential $\\mu =1$ to the infinite well $\\mu =\\infty $ .",
"We then compute $\\epsilon (\\mu )$ from Eq.REF using the expression of $Q_{\\mu }$ .",
"It turns out that $\\epsilon (\\mu )$ has a minimum at $\\mu =1$ (the case of the OU process $\\epsilon (1) = 1$ ) and tends to $6/5$ at infinity (the process with a uniform stationary distribution and unit variance).",
"The conclusion is that, at fixed variance, thinner tails yield an enhanced growth.",
"Although somewhat conterintuitive, it can be traced to the flatter nature of the potential $\\Phi (\\eta )$ at large $\\mu $ , when $\\eta $ is close to 0, increasing the polarizability of $\\eta $ .",
"The perturbative results in the range $\\mu \\in (0,1)$ have to be taken with a grain of salt, as the perturbation becomes singular and requires a more elaborate treatment [45].",
"For example, at $\\mu =0$ , the process has a Laplace stationary distribution, for which an exact solution exists and one can show that the energy gap between $\\alpha _0$ and the rest of the spectrum vanishes even at small $\\gamma $ .",
"We return to it later, when examining the limitations of this spectral approach."
],
[
"Discussion ",
"The previous exactly solvable cases and the expansions make all the more obvious the existence -and robustness- of both a freezing transition point $\\lambda _c$ , and a maximum at $\\lambda _m$ in the growth rate.",
"At diffusion low enough, the total population is not a self-averaging quantity, and so $c_q < c_{a} $ .",
"The gap between $c_q$ and $c_{a}$ is due to heavy tails and strong correlations between the $Z_i(t)$ .",
"Those factors grow as diffusion decreases, favorizing condensation onto few sites.",
"At $\\lambda =0$ , $Z_i$ merely reduces to the exponential of $\\int _t dt \\eta (t)$ , the integrated Itō process: $\\log Z$ is essentially a Gaussian of zero average and growing variance $\\langle \\eta ^2 \\rangle T t$ , and $Z$ , a log-normal, heavy-tailed distribution.",
"There seems to be no close formula neither for the value $\\lambda _m$ at which the freezing transition occurs, nor for the point of optimal growth $\\lambda _c$ .",
"Nonetheless, for any process $\\eta $ , the annealed branch $c_a(\\lambda )$ is monotonically decreasing with $\\lambda $ (see Appendix for a proof), and $c_a(0)=|\\alpha _0| \\simeq \\epsilon $ .",
"Assuming the quenched branch is differentiable, we deduce that necessarily $\\lambda _m \\le \\lambda _c$ , with equality in the limiting case of white noise.",
"The fact that the optimum always lays in the quenched phase is intriguing, and reminiscent of the Zipf law, a very general attempt to explain the predominance of power-laws in natural systems.",
"The present case falls in the category of highly optimized tolerance [46]: when optimized, complex systems have a tendency to develop algebraic tails and experimental studies have shown that they are found close from the optimal point [27].",
"Given that $\\lambda _m$ and $\\lambda _c$ are not far, it also shows how systems poised at optimality could deceiptively look critical [47], [48].",
"Figure: Scaling of λ c \\lambda _c for the OU process (Left) Scaling of λ c \\lambda _c with TT, and 〈η 2 〉=1/2\\langle \\eta ^2 \\rangle = 1/\\sqrt{2}.",
"(Right) Scaling of λ c \\lambda _c with 〈η 2 〉\\langle \\eta ^2 \\rangle , and T=1.0T = 1.0.",
"Both dashed lines are guidelines of unit slopes.Figure: Scaling of the difference λ c -λ m \\lambda _c - \\lambda _m for the OU process.",
"We have fixed ϵ=D 2 k 2 =1 2\\epsilon = \\frac{D_2}{k^2} = \\frac{1}{2}.",
"The blue dots are the result of the numerical solution of Eq.",
"for specific values of D 2 D_2 and k=1/Tk=1/T, while the black dashed line is a small TT expansion: λ c -λ m \\lambda _c - \\lambda _m widens linearly with TT, close to T=0T=0.A rough estimation of the position of $\\lambda _c$ (or $\\lambda _m$ ) is obtained by balancing the asymptotics with the upper bound $c_a(\\lambda =0)$ , leading to: c m 2 T In Fig.REF , we tested its validity by numerically solving Eq.REF for the specific OU process, fixing either $\\langle \\eta ^2 \\rangle $ or $T$ .",
"It is also possible to investigate the behaviour of $\\lambda _m$ and $\\lambda _c$ close to the white noise limit $T \\rightarrow 0$ , at fixed $\\epsilon = 1/2$ (the case presented in Fig.REF ).",
"A tedious expansion at small $T$ from Eq.REF yields both $\\lambda _c(T) \\simeq \\frac{2 \\epsilon }{\\log 4 + T \\epsilon (1-\\log 2)}$ ($\\lambda _c(0)=0.7213...$ ) and the fact that the gap $\\lambda _c - \\lambda _m$ grows linearly with $T$ , with a complicated prefactor that we do not report.",
"This confirms the scaling presented in Eq.REF .",
"In the regime of widely fluctuating noises $T \\ll \\langle \\eta ^2 \\rangle $ , $\\lambda _m \\simeq \\lambda _c$ and a large plateau in $c(\\lambda )$ develops around those transition points: the diffusion is still small enough for the noise to be seen as a quasi white noise.",
"This suggests that optimal growth is more robust in a widly varying environment, a rather surprising finding.",
"On a more practical side, it is often difficult to characterize the properties of the miscrocopic noise $\\eta (t)$ , and only macroscopic observables are measured.",
"Such situations are common occurences in biology for example, where concentrations of proteins or bacteries are much easier to obtain than levels of mRNA or nutrients they harvest.",
"Within the present class of growth models, $\\langle \\eta ^2 \\rangle _Q$ and $T$ can be extracted from both small and large $\\lambda $ (assuming $\\lambda $ is a control parameter of the experiment).",
"Those two values, the most salient features of $\\eta $ , are enough to fit one of the solvable models onto the experience at hand and estimate $\\lambda _m$ and $\\lambda _c$ .",
"We believe the mechanims presented above to be of more general scope and investigating both the interplay between $\\lambda _m$ and $\\lambda _c$ , as well as their presence in other, non mean-field growth models, would be a worthy subject of investigation.",
"To conclude, we would like to comment on some limitations.",
"The original case made in [17], and most of the subsequent literature, concerns the pure white noise (also called branching Brownian Motion).",
"Its evolution cannot be cast into a well-defined Itō equation, but may be obtained as a rather singular limit with $T \\rightarrow 0$ .",
"On the other hand, Itō processes with no stationary distribution $Q(\\eta )$ -such as the Brownian Motion-, fall out of the present analysis.",
"Yet we expect them to have no freezing transition: the wandering of those processes is so important that few branches of the tree, if not a single one, should always dominate the statistics, but a more precise study would also be welcomed.",
"The requirement of at least one isolated state at the bottom of the spectrum is a more subtle issue.",
"In principle, such restriction is not necessary, although one would have to tackle the continuous part of the spectrum describing the extended states.",
"The process with the stationary Laplace distribution $Q(\\eta ) \\sim \\exp (-|x|)$ is an enlightening example.",
"It translates as a Dirac potential $V(\\eta ) \\sim \\delta (\\eta )$ in Eq.REF .",
"It is known that a particle in such a narrow potential, and also submitted to an electric field, has no bound state, even for $\\gamma $ infinitesimally small (see [49] and Appendix for more details).",
"Therefore the resolvent has no simple pole, and the expansions presented in Section are not valid anymore.",
"One has to integrate over the branch cut of $G_{\\gamma }$ , which extends over the whole real axis, and regularize it with an $\\epsilon $ -prescription.",
"While one can write down such equation (see Appendix ), the resulting integrand involves complex, oscillating, terms, that are very difficult to tackle numerically.",
"In models of the same flavour (such as the Random Energy Model [50], [51] or the Parabolic Anderson Model [52], [53]), distributions with such exponential decay lay at the boundary between two different universality classes, and we surmise that the disappearance of the lowest bound state might have a deeper, statistical, meaning.",
"Enlarging the present derivation to disorders with stretched exponential or even power-law tails, would however require a different approach."
],
[
"Conclusion",
"In the present work, we have developed a mean-field approach to growth models with temporally correlated disorder.",
"We extended the scope of the well-known travelling wave equation approach, building on work done in [16].",
"This method allows for a detailed analysis for a general Itō processes, and even leads to exact formulas of growth rates for a variety of disorders.",
"We gave three examples, with gaussian, uniform or Laplace stationary distributions.",
"It unveils universal features in growth from microscopic details, in particular in the small and large diffusion regimes.",
"This suggests a methodology to fit such models on experimental data.",
"The mean-field computation presents both an optimal growth point and a distinct freezing transition, features that have been also observed in many finite dimension models.",
"In the present case, the optimal growth always lays in the quenched phase but a more detailed study of the statistics of $Z_i$ is dearly needed, and should be possible along the lines of [17].",
"To match the numerous directions more phenomenological approaches of growth have taken, we suggest possible extensions of the present study.",
"We wonder how to extend the analysis to heavy-tailed disorders, as they are now recognized as crucial ingredients of the large sensitivity of growth to environmental, financial or economic shocks [46].",
"On the same side, the effect of non-stationary environments, adding a temporal dependence to the Itō equation itself, would further our understanding of delayed effects also commonly observed, such as population “momentum” [54].",
"Finally we return to the primary motivations of the \"polymers on tree\", a spin glass toy model, and surmise our analysis could be made as rigorous as the original, white noise case [20], an important step towards a theory of such processes.",
"Nonetheless, those models are often treated with the replica tool, a very different and general approach, up to now limited to white-noise disorder.",
"A better understanding of the above derivation in the language of replicas might open many other disordered systems to colored disorder.",
"Acknowledgements.",
"We thank Alexander Dobrinevsky and Jean-Philippe Bouchaud for starting this line of thoughs, and many stimulating discussions."
],
[
"An alternative solution of the OU process ",
"Another possibility [16] to solve The OU noise model is to write down the eigenvectors, and eigenvalues of the harmonic oscillator, and leave the implicit equation as a sum.",
"The normalized eigenbasis is built over the Hermite functions, and given by: n = k n-2 D2k2 , n0 n = ( 2n n!",
"2 D2k )-1/2 e-2/2 Hn() with $H_n$ the Hermite polynomials.",
"Remains to compute the projection of the eigenvectors over $Q_H$ : QH | n 2 = 1n!",
"( D2 2k3)n e-D2 2 / k3 Plugging this expression into Eq.REF finally leads to an implicit expression for the curve $c(\\gamma )$ : 2 = e-D2 2 / k3 n=0 ( D2 2 / k3 )nn !",
"(k n-2 D2/k2 - ) It gives back the result from [16] with the convention $D_2 = \\sigma ^2 / 2 \\tau ^2$ and $k = 1/ \\tau $ ."
],
[
"Monotonous decay of the annealed branch ",
"Here we show that the annealed branch $c_a(\\lambda )$ , according to Eq.REF , is necessarily a decreasing function of $\\lambda $ .",
"Let us first recall Eq.REF in the annealed regime $\\gamma =1$ : 2 = n 0 QH | n 2n +/2 + c() and derivate it w.r.t to $\\lambda $ : 22 = (12 + c ) n 0 QH | n 2(n +/2 + c())2 Substituting the left hand-side with Eq., we are left with: 2 c n 0 QH | n 2(n +/2 + c())2 = (n 0 QH | n 2n +/2 + c() )2 - n 0 QH | n 2(n +/2 + c())2 But using the Cauchy-Schwart inequality over the first term of the right hand side: (n 0 QH | n 2n +/2 + c() )2 = (n 0QH | n QH | n n +/2 + c() )2 ( n QH | n 2 ) (n QH | n 2(n +/2 + c())2 ) n QH | n 2(n +/2 + c())2 using the fact that: n QH | n 2 = ||QH||2 2 =1N d(-())=1 and so: c 0"
],
[
"The exponential model ",
"The process with a Laplace stationary distribution represents a singular case in this class of models.",
"It follows: V() = k24 D23 - kD2 () = A - B () f()= k |x| Q() = k2 D2 (-kD2 |x|) QH() = k2 D2 (-k2 D2 |x|) For $\\delta $ potentials, the Dyson equation can be solved exactly in coordinate representation, and gives the Green function $G_{\\gamma }$ as a function of the well-known Green function, noted $G_0$ , for the free particle under an electric field [49]: G(x,y;z) = G0(x,y;z) + B G0(x,0;z)G0(0,y;z)1-B G0(0,0;z) G0(x,y;z) = - i (2 i)-1/2 0 t-1/2 [i (z t + (x+y)t2 D2 - 2 t324 D22) ] dt Eq.",
"readily shows that no bound state survives to the electric field in a Dirac potential.",
"Eq.REF , regularized by the addition of a small imaginary part $\\hat{\\lambda } \\rightarrow \\hat{\\lambda } + i \\epsilon $ , reduces to (setting $D_2=1$ for simplicity): 0 dt 8 ei t k3 (k2+t 2)2 ( ei t5/2 2/24 - ei tk)-1 = 2 Although the above equation should lead to the growth rate, the appearance of oscillating terms makes it unsuitable for numerical estimations, and we have been unable to confirm its validity."
]
] | 1612.05463 |
[
[
"Darboux transformations and global explicit solutions for nonlocal\n Davey-Stewartson I equation"
],
[
"Abstract For the nonlocal Davey-Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained.",
"Like the nonlocal equations in 1+1 dimensions, many solutions may have singularities.",
"However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of $t$ respectively, by a Darboux transformation of degree $n$ are global solutions of the nonlocal Davey-Stewartson I equation.",
"The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have $n$ peaks, and \"line dark soliton\" solutions when the seed solution is an exponential function of $t$, in the sense that they are bounded and their norms change fast along some straight lines."
],
[
"Introduction",
"In [1], Ablowitz and Musslimani introduced the nonlocal nonlinear Schrödinger equation and got its explicit solutions by inverse scattering.",
"Quite a lot of work were done after that for this equation and other equations.",
"[2], [3], [4], [5], [6], [7], [8], [9], [10], [11] In [12], Fokas studied high dimensional equations and introduced the nonlocal Davey-Stewartson I equation $\\begin{array}{l}u_t=u_{xx}+u_{yy}+2\\sigma u^2\\bar{u}^*+2uw_y,\\\\w̥_{xx}-w_{yy}=2\\sigma (u\\bar{u}^*)_{y},\\\\\\end{array} $ where $w$ satisfies $\\bar{w}^*=w$ .",
"Here $\\bar{f}(x,y,t)=f(-x,y,t)$ for a function $f$ , $*$ refers to complex conjugation.",
"The solution of (REF ) is PT symmetric in the sense that if $(u(x,y,t),w(x,y,t))$ is a solution of (REF ), then so is $(u^*(-x,y,-t), w^*(-x,y,-t))$ .",
"This leads to a conserved density $u\\bar{u}^*$ , which is invariant under $x\\rightarrow -x$ together with complex conjugation.",
"As is known, the usual Davey-Stewartson I equation does not possess a Darboux transformation in differential form.",
"Instead, it has a binary Darboux transformation in integral form.",
"[13], [14] However, for the nonlocal Davey-Stewartson I equation (REF ), we can construct a Darboux transformation in differential form.",
"Like the nonlocal equations in 1+1 dimensions, the solutions may have singularities.",
"Starting from the seed solutions which are zero and an exponential function of $t$ , we prove that the derived solutions can be globally defined and bounded for all $(x,y,t)\\in \\hbox{\\bf R}^3$ if the parameters are suitably chosen.",
"Unlike the usual Davey-Stewartson I equation where localized solutions are dromion solutions if the seed solution is zero,[15], [16], [17] the derived solutions here are soliton solutions in the sense that there are $n$ peaks in the solutions obtained from a Darboux transformation of degree $n$ .",
"If the seed solution is an exponential function of $t$ , the norms of the derived solutions change a lot along some straight lines.",
"We call them “line dark soliton” solutions.",
"In Section of this paper, the Lax pair for the nonlocal Davey-Stewartson I equation is reviewed and its symmetries are considered.",
"Then the Darboux transformation is constructed and the explicit expressions of the new solutions are derived.",
"In Section and Section , the soliton solutions and “line dark soliton” solutions are constructed respectively.",
"The globalness, boundedness and the asymptotic behaviors of those solutions are discussed."
],
[
"Lax pair and Darboux transformation",
"Consider the $2\\times 2$ linear system $\\begin{array}{l}_x=\\tau J\\Phi _y+\\tau P\\Phi =\\tau \\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\Phi _y+\\tau \\left(\\begin{array}{cc}0&-u\\\\v&0\\end{array}\\right)\\Phi ,\\\\_t=-2I\\tau ^2J\\Phi _{yy}-2I\\tau ^2P\\Phi _y+IQ\\Phi \\\\=-2I\\tau ^2\\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\Phi _{yy}-2I\\tau ^2\\left(\\begin{array}{cc}0&-u\\\\v&0\\end{array}\\right)\\Phi _y\\\\+I\\tau \\left(\\begin{array}{cc}-\\tau uv-\\tau w_y-w_x&u_x+\\tau u_y\\\\v_x-\\tau v_y&\\tau uv+\\tau w_y-w_x\\end{array}\\right)\\Phi \\end{array}$ where $\\tau =\\pm 1$ , $u,v,w$ are functions of $(x,y,t)$ .",
"The compatibility condition $\\Phi _{xt}=\\Phi _{tx}$ gives the evolution equation $\\begin{array}{l}u_t=u_{xx}+\\tau ^2u_{yy}+2\\tau ^2u^2v+2\\tau ^2uw_y,\\\\Iv_t=v_{xx}+\\tau ^2v_{yy}+2\\tau ^2uv^2+2\\tau ^2vw_y,\\\\w̥_{xx}-\\tau ^2w_{yy}=2\\tau ^2(uv)_{y}.\\\\\\end{array}$ When $\\tau =1$ , $v=\\sigma \\bar{u}^*$ $(\\sigma =\\pm 1)$ , (REF ) becomes the nonlocal Davey-Stewartson I equation (REF ).",
"The Lax pair (REF ) becomes $\\begin{array}{l}_x=U(\\partial )\\Phi \\overset{\\triangle }{=}J\\Phi _y+P\\Phi =\\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\Phi _y+\\left(\\begin{array}{cc}0&-u\\\\\\sigma \\bar{u}^*&0\\end{array}\\right)\\Phi ,\\\\_t=V(\\partial )\\Phi \\overset{\\triangle }{=}-2IJ\\Phi _{yy}-2IP\\Phi _y+IQ\\Phi \\\\=-2I\\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\Phi _{yy}-2I\\left(\\begin{array}{cc}0&-u\\\\\\sigma \\bar{u}^*&0\\end{array}\\right)\\Phi _y\\\\+I\\left(\\begin{array}{cc}-\\sigma u\\bar{u}^*-w_y-w_x&u_x+u_y\\\\\\sigma (\\bar{u}^*)_x-\\sigma (\\bar{u}^*)_y&\\sigma u\\bar{u}^*+w_y-w_x\\end{array}\\right)\\Phi \\end{array}$ where $=\\frac{\\partial }{\\partial y}$ .",
"Here $U(\\partial )$ implies that $U$ is a differential operator with respect to $y$ .",
"The coefficients in the Lax pair (REF ) satisfies $\\bar{J}^*=-KJK^{-1},\\quad \\bar{P}^*=-KPK^{-1},\\quad \\bar{Q}^*=-KQK^{-1}$ where $K̥=\\left(\\begin{array}{cc}0&\\sigma \\\\1&0\\end{array}\\right)$ .",
"Here $M^*$ refers to the complex conjugation (without transpose) of a matrix $M$ .",
"(REF ) implies $\\overline{U(\\partial )^*}=-KU(\\partial )K^{-1},\\quad \\overline{V(\\partial )^*}=KV(\\partial )K^{-1}.$ Hence we have Lemma 1 If $=\\left(\\begin{array}{c}\\xi \\\\\\eta \\end{array}\\right)$ is a solution of (REF ), then so is $K̥\\bar{\\Phi }^*=\\left(\\begin{array}{c}\\sigma \\bar{\\eta }^*\\\\\\bar{\\xi }^*\\end{array}\\right)$ .",
"By Lemma REF , take a solution $(\\begin{array}{c}\\xi \\\\\\eta \\end{array}$ of (REF ) and let $H̥=\\left(\\begin{array}{cc}\\xi &\\sigma \\bar{\\eta }^*\\\\\\eta &\\bar{\\xi }^*\\end{array}\\right)$ , then $G(\\partial )=\\partial -S$ with $S=H_yH^{-1}$ gives a Darboux transformation.",
"[18], [19] Written explicitly, $S=\\frac{1}{\\xi \\bar{\\xi }^*-\\sigma \\eta \\bar{\\eta }^*}\\left(\\begin{array}{cc}\\bar{\\xi }^*\\xi _y-\\sigma \\eta (\\bar{\\eta }^*)_y&\\sigma \\xi (\\bar{\\eta }^*)_y-\\sigma \\bar{\\eta }^*\\xi _y\\\\\\bar{\\xi }^*\\eta _y-\\eta (\\bar{\\xi }^*)_y&\\xi (\\bar{\\xi }^*)_y-\\sigma \\bar{\\eta }^*\\eta _y.\\end{array}\\right).$ $G(\\partial )$ also keeps the symmetries (REF ) invariant.",
"After the action of $G(\\partial )$ , $(u,w)$ is transformed to $(\\widetilde{u},\\widetilde{w})$ by $\\begin{array}{l}G(\\partial )U(\\partial )+G_x(\\partial )=\\widetilde{U}(\\partial )G(\\partial ),\\quad G(\\partial )V(\\partial )+G_t(\\partial )=\\widetilde{V}(\\partial )G(\\partial ).\\end{array} $ That is, $\\begin{array}{l}u=u+2\\sigma \\frac{\\bar{\\eta }^*\\xi _y-\\xi (\\bar{\\eta }^*)_y}{\\bar{\\xi }^*\\xi -\\sigma \\bar{\\eta }^*\\eta },\\\\w=w+2\\frac{(\\bar{\\xi }^*\\xi -\\sigma \\bar{\\eta }^*\\eta )_y}{\\bar{\\xi }^*\\xi -\\sigma \\bar{\\eta }^*\\eta }.\\end{array}$ The Darboux transformation of degree $n$ is given by a matrix-valued differential operator $G(\\partial )=\\partial ^n+G_1\\partial ^{n-1}+\\cdots +G_n$ of degree $n$ which is determined by $G(\\partial )H_j=0\\quad (j=1,\\cdots ,n)$ for $n$ matrix solutions $H_j$ $(j=1,\\cdots ,n)$ of (REF ).",
"By comparing the coefficients of $\\partial ^j$ in (REF ), the transformation of $(P,Q)$ is $\\widetilde{P}=P-[J,G_1],\\quad \\widetilde{Q}=Q+2[J,G_2]-2[JG_1-P,G_1]+4JG_{1,y}-2nP_y.$ Rewrite (REF ) as $\\partial ^nH_j+G_1\\partial ^{n-1}H_j+\\cdots +G_nH_j=0\\quad (j=1,\\cdots ,n),$ then $\\begin{array}{l}(\\begin{array}{cccc}G_1&G_2&\\cdots &G_n\\end{array}\\left(\\begin{array}{cccc}\\partial ^{n-1}H_1&\\partial ^{n-1}H_2&\\cdots &\\partial ^{n-1}H_n\\\\\\partial ^{n-2}H_1&\\partial ^{n-2}H_2&\\cdots &\\partial ^{n-2}H_n\\\\\\vdots &\\vdots &&\\vdots \\\\H_1&H_2&\\cdots &H_n\\end{array}\\right)\\\\\\left(\\begin{array}{cccc}-\\partial ^nH_1&-\\partial ^nH_2&\\cdots &-\\partial ^nH_n\\end{array}\\right).\\end{array}$ Write $H̥_j=\\left(\\begin{array}{cc}h_{11}^{(j)}&h_{12}^{(j)}\\\\h_{21}^{(j)}&h_{22}^{(j)}\\end{array}\\right)$ .",
"By reordering the rows and columns, we have $\\begin{array}{l}(\\begin{array}{cccccc}(G_1)_{11}&\\cdots &(G_n)_{11}&(G_1)_{12}&\\cdots &(G_n)_{12}\\\\(G_1)_{21}&\\cdots &(G_n)_{21}&(G_1)_{22}&\\cdots &(G_n)_{22}\\end{array}W=-R\\\\\\end{array}$ where $W̥=(W_{jk})_{1\\le j,k\\le 2}$ , $R̥=(R_{jk})_{1\\le j,k\\le 2}$ , $W_{jk}=\\left(\\begin{array}{cccc}\\partial ^{n-1}h_{jk}^{(1)}&\\partial ^{n-1}h_{jk}^{(2)}&\\cdots &\\partial ^{n-1}h_{jk}^{(n)}\\\\\\partial ^{n-2}h_{jk}^{(1)}&\\partial ^{n-2}h_{jk}^{(2)}&\\cdots &\\partial ^{n-2}h_{jk}^{(n)}\\\\\\vdots &\\vdots &&\\vdots \\\\h_{jk}^{(1)}&h_{jk}^{(2)}&\\cdots &h_{jk}^{(n)}\\\\\\end{array}\\right),$ $R_{jk}=\\left(\\begin{array}{cccc}\\partial ^nh_{jk}^{(1)}&\\partial ^nh_{jk}^{(2)}&\\cdots &\\partial ^nh_{jk}^{(n)}\\end{array}\\right)\\quad (j,k=1,2).$ Solving $G$ from (REF ), we get the new solution of the equation (REF ) from (REF ).",
"Especially, $\\widetilde{u}=u+2(G_1)_{12}.$"
],
[
"Single soliton solutions",
"Let $u=0$ , then $=\\left(\\begin{array}{c}\\xi \\\\\\eta \\end{array}\\right)$ satisfies $\\begin{array}{l}_x=\\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\Phi _y,\\quad _t=-2I\\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right)\\Phi _{yy}.\\end{array} $ Take a special solution $\\begin{array}{l}=\\hbox{e}^{\\lambda x+\\lambda y-2I\\lambda ^2t} +\\hbox{e}^{-\\lambda ^*x-\\lambda ^*y-2I\\lambda ^{*2}t} ,\\\\=a\\hbox{e}^{\\lambda x-\\lambda y+2I\\lambda ^2t} +b\\hbox{e}^{-\\lambda ^*x+\\lambda ^*y+2I\\lambda ^{*2}t} ,\\end{array}$ where $\\lambda ,a,b$ are complex constants.",
"(REF ) gives the explicit solution $\\begin{array}{l}u\\frac{2\\sigma \\lambda _R(a^*-b^*)}{D}\\hbox{e}^{2I\\lambda _I y-4I(\\lambda _R^2-\\lambda _I^2)t} \\end{array}$ of (REF ) where $\\begin{array}{l}D=(2-\\sigma |a|^2-\\sigma |b|^2)\\cosh (2\\lambda _R y+8\\lambda _R\\lambda _It)\\\\\\qquad +\\sigma (|a|^2-|b|^2)\\sinh (2\\lambda _R y+8\\lambda _R\\lambda _It)\\\\\\qquad +2(1-\\sigma \\hbox{\\hspace{1.0pt}\\rm Re\\hspace{1.0pt}}(ab^*))\\cosh (2\\lambda _R x)-2I\\sigma \\hbox{\\hspace{1.0pt}\\rm Im\\hspace{1.0pt}}(ab^*)\\sinh (2\\lambda _Rx).\\end{array}$ Here $z_R=\\hbox{\\hspace{1.0pt}\\rm Re\\hspace{1.0pt}}z$ and $z_I=\\hbox{\\hspace{1.0pt}\\rm Im\\hspace{1.0pt}}z$ for a complex number $z$ .",
"Since $\\widetilde{w}$ is looked as an auxiliary function in (REF ), hereafter we will study mainly the behavior of $\\widetilde{u}$ .",
"Note that $\\widetilde{u}$ is global if $|a|<1$ and $|b|<1$ since $\\hbox{\\hspace{1.0pt}\\rm Re\\hspace{1.0pt}}D>0$ in this case.",
"Moreover, the peak moves in the velocity $(v_x,v_y)=(0,-4\\lambda _I)$ .",
"Remark 1 The solution (REF ) may have singularities when the parameters are not chosen suitably, say, when $\\sigma =-1$ , $a=1$ , $b=-2$ , or $\\sigma =1$ , $a=2$ , $b=1/4$ .",
"Figure 1 shows a 1 soliton solution with parameters $\\sigma =-1$ , $t=20$ , $\\lambda =0.07-1.5I$ , $a=0.2$ , $b=0.1I$ .",
"Figure: |u ˜||\\widetilde{u}| of a 1 soliton solution."
],
[
"Multiple soliton solutions",
"For an $n\\times n$ matrix $M$ , define $|M||=\\sup _{x\\in \\hbox{C}^n,||x||=1}||Mx||$ where $||\\cdot ||$ is the standard Hermitian norm in $\\hbox{\\bf C}^n$ .",
"The following facts hold obviously.",
"(i) $||MN||\\le ||M||\\,||N||$ .",
"(ii) Each entry $M_{jk}$ of $M$ satisfies $|M_{jk}|\\le ||M||$ .",
"(iii) $|\\det M|\\le ||M||^n$ .",
"(iv) If $||M||<1$ , then $|(I+M)^{-1}||\\le (1-||M||)^{-1}$ .",
"(v) If $||M||<1$ , then $\\det (I+M)|\\ge (1-||M||)^n$ .",
"Now we construct explicit solutions according to (REF ).",
"As in (REF ), take $\\begin{array}{l}_k=\\hbox{e}^{\\lambda _k(x+y)-2I\\lambda _k^2t} +\\hbox{e}^{-\\lambda _k^*(x+y)-2I\\lambda _k^{*2}t} ,\\\\_k=a_k\\hbox{e}^{\\lambda _k(x-y)+2I\\lambda _k^2t} +b_k\\hbox{e}^{-\\lambda _k^*(x-y)+2I\\lambda _k^{*2}t} ,\\end{array}$ then $W_{jk}$ 's in (REF ) and $R_{jk}$ 's in (REF ) are $\\begin{array}{l}(W_{11})_{jk}=\\lambda _k^{n-j}e_{k+}+(-\\lambda _k^*)^{n-j}e_{k+}^{*-1},\\quad (W_{12})_{jk}=\\sigma a_k^*(-\\lambda _k^*)^{n-j}e_{k+}^{*-1}+\\sigma b_k^*\\lambda _k^{n-j}e_{k+},\\\\(W_{21})_{jk}=a_k(-\\lambda _k)^{n-j}e_{k-}+b_k(\\lambda _k^*)^{n-j}e_{k-}^{*-1},\\quad (W_{22})_{jk}=(\\lambda _k^*)^{n-j}e_{k-}^{*-1}+(-\\lambda _k)^{n-j}e_{k-},\\\\(R_{11})_{1k}=\\lambda _k^ne_{k+}+(-\\lambda _k^*)^ne_{k+}^{*-1},\\quad (R_{12})_{1k}=\\sigma a_k^*(-\\lambda _k^*)^ne_{k+}^{*-1}+\\sigma b_k^*\\lambda _k^ne_{k+},\\\\(R_{21})_{1k}=a_k(-\\lambda _k)^ne_{k-}+b_k(\\lambda _k^*)^ne_{k-}^{*-1},\\quad (R_{22})_{1k}=(\\lambda _k^*)^ne_{k-}^{*-1}+(-\\lambda _k)^ne_{k-}\\\\(j,k=1,\\cdots ,n)\\end{array}$ where $e_{k\\pm }=\\hbox{e}^{\\lambda _k(x\\pm y)\\mp 2I\\lambda _k^2t} .$ However, temporarily, we assume $e_{k+},e_{k-}$ $(k=1,\\cdots ,n)$ are arbitrary complex numbers rather that (REF ) holds.",
"Denote $L̥=\\,\\hbox{\\rm diag}((-1)^{n-1},(-1)^{n-2},\\cdots ,-1,1)$ , $F=(\\lambda _k^{n-j})_{1\\le j,k\\le n},\\quad f=(\\lambda _1^n,\\cdots ,\\lambda _n^n),$ $\\begin{array}{l}A=\\,\\hbox{\\rm diag}(a_1,\\cdots ,a_n),\\quad B=\\,\\hbox{\\rm diag}(b_1,\\cdots ,b_n),\\end{array}$ $\\begin{array}{l}E̥_\\pm =\\,\\hbox{\\rm diag}(e_{1\\pm },\\cdots ,e_{n\\pm }).\\end{array}$ Then $W=\\left(\\begin{array}{cc}FE_++LF^*E^{*-1}_{+}&\\sigma FB^*E_++\\sigma LF^*A^*E^{*-1}_{+}\\\\LFAE_-+F^*BE^{*-1}_{-}&LFE_-+F^*E^{*-1}_{-}\\end{array}\\right),$ $R=\\left(\\begin{array}{cc}fE_++(-1)^nf^*E^{*-1}_{+}&\\sigma fB^*E_++\\sigma (-1)^nf^*A^*E^{*-1}_{+}\\\\(-1)^nfAE_-+f^*BE^{*-1}_{-}&(-1)^nfE_-+f^*E^{*-1}_{-}\\end{array}\\right).$ By using the identity $\\left(\\begin{array}{cc}A&B\\\\C&D\\end{array}\\right)^{-1}=\\left(\\begin{array}{cc}A^{-1}+A^{-1}B\\Delta ^{-1}CA^{-1}&-A^{-1}B\\Delta ^{-1}\\\\-\\Delta ^{-1}CA^{-1}&\\Delta ^{-1}\\end{array}\\right)$ for a block matrix where $\\Delta =D-CA^{-1}B$ , (REF ) gives $\\Big ((G_1)_{12},\\cdots ,(G_n)_{12}\\Big )=-(RW^{-1})_{12}=-(R_{12}-R_{11}W_{11}^{-1}W_{12}){\\overset{\\circ }{W}}^{-1} $ where $\\begin{array}{l}{W}=W_{22}-W_{21}W_{11}^{-1} W_{12}\\\\=L\\Big (FE_-+LF^*E^{*-1}_{-}-\\sigma (FAE_-+LF^*BE^{*-1}_{-})\\cdot \\\\\\quad \\cdot (FE_++LF^*E^{*-1}_{+})^{-1}(FB^*E_++LF^*A^*E^{*-1}_{+})\\Big ).\\end{array}$ Lemma 2 Suppose $a_j$ and $b_j$ are nonzero complex constants with $|a_j|<1$ , $|b_j|<1$ $(j=1,\\cdots ,n)$ , $\\kappa _1,\\cdots ,\\kappa _n$ are nonzero real constants with $|\\kappa _j|\\ne |\\kappa _k|$ $(j,k=1,\\cdots ,n;\\,j\\ne k)$ , then there exist positive constants $\\delta $ , $C_1$ and $C_2$ , which depend on $a_j$ 's, $b_j$ 's and $\\kappa _j$ 's, such that $|\\det W|\\ge C_1$ and $\\begin{array}{l}(G_1)_{12}|\\le C_2\\max _{1\\le k\\le n}\\frac{|e_{k+}|}{1+|e_{k+}|^2}\\max _{1\\le k\\le n}\\frac{|e_{k-}|}{1+|e_{k-}|^2}\\end{array}$ hold whenever $|\\lambda _j-I\\kappa _j|<\\delta $ and $e_{j\\pm }\\in \\hbox{\\bf C}$ $(j=1,\\cdots ,n)$ .",
"Proof.",
"Denote $F^{-1}LF^*=I+Z$ , then $Z=0$ if $\\lambda _1,\\cdots ,\\lambda _n$ are all purely imaginary.",
"From (REF ) and (REF ), $\\det W=\\det (FE_++LF^*E^{*-1}_{+})\\det {\\overset{\\circ }{W}},$ ${\\overset{\\circ }{W}}=L(FE_-+LF^*E^{*-1}_{-})(I-\\sigma \\chi _-\\chi _+)$ where $\\begin{array}{l}\\chi _+=(FE_++LF^*E^{*-1}_{+})^{-1}(FB^*E_++LF^*A^*E^{*-1}_{+})\\\\\\qquad =\\Xi _{1+}\\Xi _{0+}^{-1}+\\Xi _{0+}^{-1}(I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})^{-1}ZE^{*-1}_{+}(A^*-\\Xi _{1+}\\Xi _{0+}^{-1}),\\\\\\chi _-=(FE_-+LF^*E^{*-1}_{-})^{-1}(FAE_-+LF^*BE^{*-1}_{-})\\\\\\qquad =\\Xi _{1-}\\Xi _{0-}^{-1}+\\Xi _{0-}^{-1}(I+ZE^{*-1}_{-}\\Xi _{0-}^{-1})^{-1}ZE^{*-1}_{-}(B-\\Xi _{1-}\\Xi _{0-}^{-1}),\\\\\\end{array}$ $\\Xi _{0\\pm }=E_\\pm +E^{*-1}_{\\pm },\\quad \\Xi _{1-}=AE_-+BE^{*-1}_{-},\\quad \\Xi _{1+}=B^*E_++A^*E^{*-1}_{+}.$ Let $c̥_0=\\max _{1\\le k\\le n}\\lbrace |a_k|,|b_k|\\rbrace <1$ .",
"Suppose $|Z||<\\frac{1-c_0}{2}$ , then we have the following estimates.",
"$||A||\\le c_0<1,\\quad ||B||\\le c_0<1,$ $||E_{\\pm }\\Xi _{0\\pm }^{-1}||\\le 1,\\quad ||E^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1}||\\le 1,$ $||\\Xi _{0\\pm }||\\ge 2,\\quad ||\\Xi _{0\\pm }^{-1}||\\le \\frac{1}{2},\\quad ||\\Xi _{1\\pm }\\Xi _{0\\pm }^{-1}||\\le c_0<1,$ $\\begin{array}{l}|E^{*-1}_{+}(\\Xi _{1+}\\Xi _{0+}^{-1}-A^*)||=\\max _{1\\le k\\le n}\\frac{|a_k-b_k|\\,|e_{k+}|}{1+|e_{k+}|^2}\\le 1,\\\\|E^{*-1}_{-}(\\Xi _{1-}\\Xi _{0-}^{-1}-B)||=\\max _{1\\le k\\le n}\\frac{|a_k-b_k|\\,|e_{k-}|}{1+|e_{k-}|^2}\\le 1,\\end{array}$ $||(I+Z E^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1})^{-1}||\\le ||(1-||Z||\\,||E^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1}||)^{-1}||\\le (1-||Z||)^{-1}\\le 2.$ Hence $||\\chi _\\pm -\\Xi _{1\\pm }\\Xi _{0\\pm }^{-1}||\\le ||Z||$ , $||\\chi _\\pm ||\\le c_0+||Z||\\le \\frac{1+c_0}{2}<1.$ Denote $\\pi _0=|\\det F|\\Big |_{\\lambda _j=\\hbox{i}\\kappa _j\\atop {j=1,\\cdots ,n}},\\quad \\pi _1=||F^{-1}||\\Big |_{\\lambda _j=\\hbox{i}\\kappa _j\\atop {j=1,\\cdots ,n}},\\quad \\pi _2=||f||\\Big |_{\\lambda _j=\\hbox{i}\\kappa _j\\atop {j=1,\\cdots ,n}}.$ Clearly, $\\pi _0$ , $\\pi _1$ , $\\pi _2$ are all positive since $F|_{\\lambda _j=\\hbox{i}\\kappa _j\\atop {j=1,\\cdots ,n}}$ is a Vandermonde determinant.",
"By the continuity, there exists $\\delta >0$ such that ${\\pi _0}2\\le |\\det F|\\le 2\\pi _0$ , $||F^{-1}||\\le 2\\pi _1$ , $||f||\\le 2\\pi _2$ , and $|F^{-1}LF^*-I||=||Z||<\\frac{1-c_0}{2}$ whenever $|\\lambda _j-I\\kappa _j|<\\delta $ .",
"(REF ) and (REF ) lead to $\\begin{array}{l}|\\det W|\\!=\\!|\\det F|^2\\,|\\det \\Xi _{0+}|\\,|\\det (I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})|\\cdot \\\\\\qquad \\cdot |\\det \\Xi _{0-}|\\,|\\det (I+ZE^{*-1}_{-}\\Xi _{0-}^{-1})|\\,|\\det (1-\\sigma \\chi _-\\chi _+)|\\\\\\ge \\pi _0^2(1-||Z||)^{2n}(1-||\\chi _+||\\,||\\chi _-||)^n\\ge \\pi _0^2\\Big (\\frac{1+c_0}{2}\\Big )^{2n}\\Big (1-\\Big (\\frac{1+c_0}{2}\\Big )^2\\Big )^{n},\\end{array}$ which is a uniform positive lower bound for any $e_{j\\pm }\\in \\hbox{\\bf C}$ $(j=1,\\cdots ,n)$ when $|\\lambda _j-I\\kappa _j|<\\delta $ .",
"By (REF ), (REF ), (REF ) and (REF ), $\\begin{array}{l}(G_1)_{12},(G_2)_{12},\\cdots ,(G_n)_{12})=-(R_{12}-R_{11}W_{11}^{-1}W_{12}){\\overset{\\circ }{W}}^{-1}\\\\=-\\sigma fE_+\\Xi _{0+}^{-1}(I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})^{-1}(I+Z)(B^*-A^*)E^{*-1}_{+}{\\overset{\\circ }{W}}^{-1}\\\\\\qquad \\quad -\\sigma (-1)^nf^*E^{*-1}_{+}\\Xi _{0+}^{-1}(I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})^{-1}(A^*-B^*)E_+{\\overset{\\circ }{W}}^{-1}\\\\=-\\sigma \\Big (f-(-1)^nf^*+(fE_+\\Xi _{0+}^{-1}+(-1)^nf^*E^{*-1}_{+}\\Xi _{0+}^{-1})(I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})^{-1}Z\\Big )\\cdot \\\\\\qquad \\quad \\cdot (B^*-A^*)E_+E^{*-1}_{+}\\Xi _{0+}^{-1}(I-\\sigma \\chi _-\\chi _+)^{-1}\\Xi _{0-}^{-1}(I+ZE^{*-1}_{-}\\Xi _{0-}^{-1})^{-1}F^{-1}L^{-1}.\\end{array}$ Here we have used $I+Z=(I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})+ZE_+\\Xi _{0+}^{-1}$ .",
"Hence, by using (REF )–(REF ), $\\begin{array}{l}(G_1)_{12}|\\le 8(||f||+||f^*||)||F^{-1}||\\,||(I-\\sigma \\chi _-\\chi _+)^{-1}||\\,||\\Xi _{0+}^{-1}||\\,||\\Xi _{0-}^{-1}||\\\\\\le 64\\pi _1\\pi _2\\Big (1-\\Big (\\frac{1+c_0}{2}\\Big )^2\\Big )^{-1}\\max _{1\\le k\\le n}\\frac{|e_{k+}|}{1+|e_{k+}|^2}\\max _{1\\le k\\le n}\\frac{|e_{k-}|}{1+|e_{k-}|^2}.\\end{array}$ The lemma is proved.",
"Now we consider the solutions of the nonlocal Davey-Stewartson I equation.",
"That is, we consider the case where $e_{j\\pm }$ 's are taken as (REF ).",
"Theorem 1 Suppose $a_j$ and $b_j$ are nonzero complex constants with $|a_j|<1$ , $|b_j|<1$ $(j=1,\\cdots ,n)$ , $\\kappa _1,\\cdots ,\\kappa _n$ are nonzero real constants with $|\\kappa _j|\\ne |\\kappa _k|$ $(j,k=1,\\cdots ,n;\\,j\\ne k)$ , then there exists a positive constant $\\delta $ such that the following results hold for the derived solution $\\widetilde{u}=2(G_1)_{12}$ of the nonlocal Davey-Stewartson I equation when $\\hbox{\\hspace{1.0pt}\\rm Re\\hspace{1.0pt}}\\lambda _j\\ne 0$ and $|\\lambda _j-I\\kappa _j|<\\delta $ $(j=1,\\cdots ,n)$ .",
"(i) $\\widetilde{u}$ is defined globally for $(x,y,t)\\in \\hbox{\\bf R}^3$ .",
"(ii) For fixed $t$ , $\\widetilde{u}$ tends to zero exponentially as $(x,y)\\rightarrow 0$ .",
"(iii) Let $y=\\widetilde{y}+vt$ and keep $(x,\\widetilde{y})$ bounded, then $\\widetilde{u}\\rightarrow 0$ as $t\\rightarrow \\infty $ if $v\\ne -4\\lambda _{kI}$ for all $k$ .",
"Proof.",
"We have known that $\\widetilde{u}$ is a solution of the nonlocal Davey-Stewartson equation in Section .",
"(i) According to Lemma REF , $|\\det W|$ has a uniform positive lower bound.",
"Hence $\\widetilde{u}$ is defined globally.",
"(ii) When $x\\ge 0$ and $y\\ge 0$ , $\\begin{array}{l}|e_{k+}|\\ge \\hbox{e}^{\\lambda _{kR}\\sqrt{x^2+y^2}+4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }\\lambda _{kR}>0,\\\\|e_{k+}|\\le \\hbox{e}^{-|\\lambda _{kR}|\\sqrt{x^2+y^2}+4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }\\lambda _{kR}<0.\\end{array}$ Hence $_{1\\le k\\le n}\\frac{|e_{k+}|}{1+|e_{k+}|^2}$ tends to zero exponentially when $x\\ge 0$ , $y\\ge 0$ and $(x,y)\\rightarrow \\infty $ .",
"Likewise, we have $\\begin{array}{l}|e_{k-}|\\le \\hbox{e}^{-\\lambda _{kR}\\sqrt{x^2+y^2}-4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }x\\le 0,\\;y\\ge 0,\\;\\lambda _{kR}>0,\\\\|e_{k-}|\\ge \\hbox{e}^{|\\lambda _{kR}|\\sqrt{x^2+y^2}-4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }x\\le 0,\\;y\\ge 0,\\;\\lambda _{kR}<0,\\\\|e_{k+}|\\le \\hbox{e}^{-\\lambda _{kR}\\sqrt{x^2+y^2}+4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }x\\le 0,\\;y\\le 0,\\;\\lambda _{kR}>0,\\\\|e_{k+}|\\ge \\hbox{e}^{|\\lambda _{kR}|\\sqrt{x^2+y^2}+4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }x\\le 0,\\;y\\le 0,\\;\\lambda _{kR}<0,\\\\|e_{k-}|\\ge \\hbox{e}^{\\lambda _{kR}\\sqrt{x^2+y^2}-4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }x\\ge 0,\\;y\\le 0,\\;\\lambda _{kR}>0,\\\\|e_{k-}|\\le \\hbox{e}^{-|\\lambda _{kR}|\\sqrt{x^2+y^2}-4\\lambda _{kR}\\lambda _{kI}t} \\hbox{ if }x\\ge 0,\\;y\\le 0,\\;\\lambda _{kR}<0.\\end{array}$ Lemma REF implies that $\\widetilde{u}\\rightarrow 0$ exponentially as $(x,y)\\rightarrow \\infty $ .",
"(iii) $|e_{k\\pm }|=\\hbox{e}^{\\lambda _{kR}(x\\pm \\widetilde{y})\\pm \\lambda _{kR}(v+4\\lambda _{kI})t} .$ If $v\\ne -4\\lambda _{kI}$ for all $k=1,\\cdots ,n$ , then either $e_{k+}\\rightarrow 0$ or $e_{k+}\\rightarrow \\infty $ for all $k=1,\\cdots ,n$ when $t\\rightarrow \\infty $ .",
"Lemma REF implies that $\\widetilde{u}\\rightarrow 0$ when $t\\rightarrow \\infty $ .",
"The theorem is proved.",
"A 3 soliton solution is shown in Figure 2 where the parameters are $\\sigma =-1$ , $t=20$ , $\\lambda _1=0.07-1.5I$ , $\\lambda _2=0.05+2I$ , $\\lambda _3=0.1+I$ , $a_1=0.2$ , $a_2=0.1I$ , $a_3=0.1$ , $b_1=0.1I$ , $b_2=-0.2$ , $b_3=-0.2$ .",
"The figure of the solution appears similarly if $\\sigma $ is changed to $+1$ , although it is not shown here.",
"Figure: |u ˜||\\widetilde{u}| of a 3 soliton solution."
],
[
"Single “line dark soliton” solutions",
"Now we take $u=\\rho \\hbox{e}^{-2I\\sigma |\\rho |^2t} ,\\quad w=0$ as a solution of (REF ) where $\\rho $ is a complex constant.",
"The Lax pair (REF ) has a solution $\\begin{array}{l}(\\begin{array}{c}\\hbox{e}^{\\alpha (\\lambda )x+\\beta (\\lambda )y+\\gamma (\\lambda )t} \\\\{\\lambda }{\\rho }\\hbox{e}^{\\alpha (\\lambda )x+\\beta (\\lambda )y+(\\gamma (\\lambda )+2I\\sigma |\\rho |^2)t} \\end{array},\\end{array}$ where $\\begin{array}{l}(\\lambda )=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda }-\\lambda \\Big ),\\quad \\beta (\\lambda )=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda }+\\lambda \\Big ),\\\\(\\lambda )=I(\\alpha (\\lambda )^2-2\\alpha (\\lambda )\\beta (\\lambda )-\\beta (\\lambda )^2)=I\\lambda ^2-\\frac{I}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda }+\\lambda \\Big )^2,\\end{array}$ $\\lambda $ is a complex constant.",
"Note that $\\alpha (-\\lambda ^*)=-(\\alpha (\\lambda ))^*$ , $\\beta (-\\lambda ^*)=-(\\beta (\\lambda ))^*$ , $\\gamma (-\\lambda ^*)=-(\\gamma (\\lambda ))^*$ .",
"Now take $=\\left(\\begin{array}{c}\\xi \\\\\\eta \\end{array}\\right)$ where $\\begin{array}{l}=\\hbox{e}^{\\alpha x+\\beta y+\\gamma t} +\\hbox{e}^{-\\alpha ^*x-\\beta ^*y-\\gamma ^*t} ,\\\\=\\frac{\\lambda }{\\rho }\\hbox{e}^{\\alpha x+\\beta y+(\\gamma +2I\\sigma |\\rho |^2)t} -\\frac{\\lambda ^*}{\\rho }\\hbox{e}^{-\\alpha ^*x-\\beta ^*y-(\\gamma ^*-2I\\sigma |\\rho |^2)t} .\\end{array}$ Here $\\alpha =\\alpha (\\lambda )$ , $\\beta =\\beta (\\lambda )$ , $\\gamma =\\gamma (\\lambda )$ .",
"This $\\Phi $ is a linear combination of the solutions of form (REF ).",
"Then (REF ) gives the new solution $\\widetilde{u}=\\rho \\hbox{e}^{-2I\\sigma |\\rho |^2t} \\frac{\\frac{\\lambda ^*}{\\lambda }c_1\\hbox{e}^{2\\beta _Ry+2\\gamma _Rt} +\\frac{\\lambda }{\\lambda ^*}c_1\\hbox{e}^{-2\\beta _Ry-2\\gamma _Rt} -c_2\\hbox{e}^{2\\alpha _Rx} -c_2^*\\hbox{e}^{-2\\alpha _Rx} }{c̥_1(\\hbox{e}^{2\\beta _Ry+2\\gamma _Rt} +\\hbox{e}^{-2\\beta _Ry-2\\gamma _Rt} )+c_2\\hbox{e}^{2\\alpha _Rx} +c_2^*\\hbox{e}^{-2\\alpha _Rx} }$ of the nonlocal Davey-Stewartson I equation where $\\begin{array}{l}c̥_1=1-\\sigma \\frac{|\\lambda |^2}{|\\rho |^2},\\quad c_2=1+\\sigma \\frac{\\lambda ^2}{|\\rho |^2}\\end{array}$ This solution is smooth for all $(x,y,t)\\in \\hbox{\\bf R}^3$ if $|\\lambda |<|\\rho |$ .",
"Especially, if $\\lambda $ is real, then $\\gamma _R=0$ , so we get a standing wave solution.",
"Figure 3 shows a 1 “line dark soliton” solution with parameters $\\sigma =-1$ , $t=10$ , $\\rho =1$ , $\\lambda =0.3+0.1I$ .",
"The figure on the right describes the same solution but is upside down.",
"Figure: |u ˜||\\widetilde{u}| of a 1 “line dark soliton” solution."
],
[
"Multiple “line dark soliton” solutions",
"Now we take $n$ solutions $\\begin{array}{l}_k=\\hbox{e}^{\\alpha _k x+\\beta _k y+\\gamma _k t} +\\hbox{e}^{-\\alpha _k^*x-\\beta _k^*y-\\gamma _k^*t} ,\\\\_k=\\frac{\\lambda _k}{\\rho }\\hbox{e}^{\\alpha _k x+\\beta _k y+(\\gamma _k+2I\\sigma |\\rho |^2)t} -\\frac{\\lambda _k^*}{\\rho }\\hbox{e}^{-\\alpha _k^*x-\\beta _k^*y-(\\gamma _k^*-2I\\sigma |\\rho |^2)t} \\end{array}$ of form (REF ) to get multiple “line dark soliton” solutions where $\\begin{array}{l}_k=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda _k}-\\lambda _k\\Big ),\\quad \\beta _k=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda _k}+\\lambda _k\\Big ),\\\\\\gamma _k=I(\\alpha _k^2-2\\alpha _k\\beta _k-\\beta _k^2).\\end{array}$ Similar to (REF ) and (REF ), $W$ and $R$ in (REF ) are $W=\\left(\\begin{array}{cc}FE_++LF^*E^{*-1}_{+}&-\\sigma \\rho ^{*-1}\\hbox{e}^{-I\\phi } (LF\\Lambda E_--F^*\\Lambda ^*E^{*-1}_{-})\\\\\\rho ^{-1}\\hbox{e}^{I\\phi } (F\\Lambda E_+-LF^*\\Lambda ^*E^{*-1}_{+})&LFE_-+F^*E^{*-1}_{-}\\end{array}\\right),$ $R=\\left(\\begin{array}{cc}fE_++(-1)^nf^*E^{*-1}_{+}&-\\sigma \\rho ^{*-1}\\hbox{e}^{-I\\phi } ((-1)^nf\\Lambda E_--f^*\\Lambda ^*E^{*-1}_{-})\\\\\\rho ^{-1}\\hbox{e}^{I\\phi } (f\\Lambda E_+-(-1)^nf^*\\Lambda ^*E^{*-1}_{+})&(-1)^nfE_-+f^*E^{*-1}_{-}\\end{array}\\right),$ where $\\begin{array}{l}E_\\pm =\\,\\hbox{\\rm diag}(e_{k\\pm })_{k=1,\\cdots ,n},\\quad F=(\\beta _k^{n-j})_{1\\le j,k\\le n},\\quad f=(\\beta _1^n,\\cdots ,\\beta _n^n),\\\\_k=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda _k}-\\lambda _k\\Big ),\\quad \\beta _k=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\lambda _k}+\\lambda _k\\Big ),\\\\\\gamma _k=I(\\alpha _k^2-2\\alpha _k\\beta _k-\\beta _k^2),\\end{array}$ $=\\,\\hbox{\\rm diag}(\\lambda _k)_{1\\le k\\le n}$ , $\\phi =2\\sigma |\\rho |^2t$ .",
"Moreover, $e_{k\\pm }=\\hbox{e}^{\\alpha _kx\\pm \\beta _ky\\pm \\gamma _kt} $ .",
"However, as in the soliton case, we suppose temporarily that $e_{k\\pm }$ 's are arbitrary complex numbers.",
"Lemma 3 Suppose $\\kappa _1,\\cdots ,\\kappa _n$ are distinct nonzero real numbers, then there exist positive constants $\\rho _0$ , $\\delta $ , $C_1$ and $C_2$ , which depend on $\\kappa _j$ 's, such that $|\\det W|\\ge C_1$ and $|(G_1)_{12}|\\le C_2$ hold whenever $|\\rho |>\\rho _0$ , $|\\lambda _j-I\\kappa _j|<\\delta $ and $e_{j\\pm }\\in \\hbox{\\bf C}$ $(j=1,\\cdots ,n)$ .",
"Proof.",
"Denote $F^{-1}LF^*=I+Z$ , then $Z=0$ if $\\lambda _1,\\cdots ,\\lambda _n$ are all purely imaginary.",
"Hence $||Z||$ is small enough if $|\\lambda _1-I\\kappa _1|,\\cdots ,|\\lambda _n-I\\kappa _n|$ are all small enough.",
"Let $c̥_0=\\max _{1\\le k\\le n}|\\kappa _k|$ , $\\pi _3=|\\det F|\\Big |_{\\lambda _k=\\hbox{i}\\kappa _k\\atop k=1,\\cdots ,n}$ , $\\pi _4=||F^{-1} LF||\\Big |_{\\lambda _k=\\hbox{i}\\kappa _k\\atop k=1,\\cdots ,n}$ .",
"Then there exists $\\delta $ with $0<\\delta <c_0$ such that $|Z||\\le \\frac{1}{2}$ , $\\det F|\\ge \\frac{\\pi _3}{2}$ , $||F^{-1}LF||\\le 2\\pi _4$ if $|\\lambda _k-I\\kappa _k|<\\delta $ $(k=1,\\cdots ,n)$ .",
"In this case, $|\\lambda _k|<c_0+\\delta <2c_0$ .",
"From (REF ), $\\det W=\\det (FE_++LF^*E^{*-1}_{+})\\det {\\overset{\\circ }{W}},$ where $\\begin{array}{l}{\\overset{\\circ }{W}}=LFE_-+F^*E^{*-1}_{-}+\\sigma |\\rho |^{-2}(F\\Lambda E_+-LF^*\\Lambda ^*E^{*-1}_{+})\\cdot \\\\\\qquad \\cdot (FE_++LF^*E^{*-1}_{+})^{-1}L(F\\Lambda E_--LF^*\\Lambda ^*E^{*-1}_{-})\\\\\\qquad =(1+\\sigma |\\rho |^{-2}F\\chi _+F^{-1} LF\\chi _-F^{-1} L^{-1})LF(I+ZE^{*-1}_{-}\\Xi _{0-}^{-1})\\Xi _{0-}\\end{array}$ $\\begin{array}{l}\\chi _\\pm =F^{-1}(F\\Lambda E_\\pm -LF^*\\Lambda ^*E^{*-1}_{\\pm })(FE_\\pm +LF^*E^{*-1}_{\\pm })^{-1}F\\\\\\qquad =\\Xi _{1\\pm }\\Xi _{0\\pm }^{-1}-(Z\\Lambda ^*+\\Xi _{1\\pm }\\Xi _{0\\pm }^{-1} Z)E^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1}(I+ZE^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1})^{-1},\\end{array}$ $\\Xi _{0\\pm }=E_\\pm +E^{*-1}_{\\pm },\\quad \\Xi _{1\\pm }=\\Lambda E_\\pm -\\Lambda ^*E^{*-1}_{\\pm }.$ We have the following estimates: $||E_{\\pm }\\Xi _{0\\pm }^{-1}||\\le 1,\\quad ||E^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1}||\\le 1,\\quad ||\\Xi _{1\\pm }\\Xi _{0\\pm }^{-1}||\\le 2c_0,$ $||\\Xi _{0\\pm }||\\ge 2,\\quad ||\\Xi _{0\\pm }^{-1}||\\le \\frac{1}{2},\\quad |\\det \\Xi _{0\\pm }|\\ge 2^n,$ $||(I+Z E^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1})^{-1}||\\le (1-||Z||)^{-1}\\le 2,\\quad |\\det (I+ZE^{*-1}_{\\pm }\\Xi _{0\\pm }^{-1})|\\ge (1-||Z||)^n\\ge \\frac{1}{2^n}.$ Hence (REF ) implies $\\begin{array}{l}|\\chi _\\pm ||\\le 2c_0+8c_0||Z||\\le 6c_0,\\end{array}$ $\\begin{array}{l}|\\chi _+F^{-1} LF\\chi _-F^{-1} L^{-1}F||\\le ||F^{-1} LF||^2\\,||\\chi _+||\\,||\\chi _-||\\le 144c_0^2\\pi _4^2.\\end{array}$ By (REF ) and (REF ), $\\begin{array}{l}|\\det W|=|\\det F|^2\\,|\\det \\Xi _{0+}|\\,|\\det \\Xi _{0-}|\\,|\\det (I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})|\\cdot \\\\\\qquad \\cdot |\\det (I+ZE^{*-1}_{-}\\Xi _{0-}^{-1})|\\,|\\det (I+\\sigma |\\rho |^{-2}\\chi _+F^{-1} LF\\chi _-F^{-1} L^{-1} F)|\\\\\\frac{\\pi _3^2}{4}(1-144c_0^2\\pi _4^2|\\rho |^{-2})>0\\end{array}$ if $\\rho |>12c_0\\pi _4$ .",
"Therefore, $|\\det W|$ has a uniform positive lower bound if $\\rho |>12c_0\\pi _4$ and $|\\lambda _k-I\\kappa _k|<\\delta $ $(k=1,\\cdots ,n)$ .",
"By (REF ), (REF ), (REF ), (REF ) and (REF ), $\\begin{array}{l}(G_1)_{12},(G_2)_{12},\\cdots ,(G_n)_{12})=-(R_{12}-R_{11}W_{11}^{-1}W_{12}){\\overset{\\circ }{W}}^{-1}\\\\\\sigma \\rho ^{*-1}\\hbox{e}^{-I\\phi } (-1)^n\\Big (f\\Lambda E_--(-1)^nf^*\\Lambda ^*E^{*-1}_{-}-(fE_++(-1)^nf^*E^{*-1}_{+})\\cdot \\\\\\cdot (-1)^n(FE_++LF^*E^{*-1}_{+})^{-1}L(F\\Lambda E_--LF^*\\Lambda ^*E^{*-1}_{-})\\Big ){\\overset{\\circ }{W}}^{-1}\\\\\\sigma \\rho ^{*-1}\\hbox{e}^{-I\\phi } (-1)^n\\Big (f\\Lambda E_-\\Xi _{0-}^{-1}-(-1)^nf^*\\Lambda ^*E^{*-1}_{-}\\Xi _{0-}^{-1}-\\\\-(-1)^n(fE_+\\Xi _{0+}^{-1}+(-1)^nf^*E^{*-1}_{+}\\Xi _{0+}^{-1})(I+ZE^{*-1}_{+}\\Xi _{0+}^{-1})^{-1} F^{-1}LF\\cdot \\\\\\cdot (\\Xi _{1-}\\Xi _{0-}^{-1}-Z\\Lambda ^*E^{*-1}_{-}\\Xi _{0-}^{-1})\\Big )(I+ZE^{*-1}_{-}\\Xi _{0-}^{-1})^{-1}F^{-1}L^{-1}\\cdot \\\\\\cdot (1+\\sigma |\\rho |^{-2}F\\chi _+F^{-1} LF\\chi _-F^{-1} L^{-1})^{-1}.\\end{array}$ $(G_1)_{12}$ is bounded when $\\rho |>12c_0\\pi _4$ and $|\\lambda _k-I\\kappa _k|<\\delta $ $(k=1,\\cdots ,n)$ because of the estimates (REF )–(REF ).",
"The lemma is proved.",
"Now we have the following theorem for the multiply “line dark soliton” solution.",
"Theorem 2 Suppose $\\kappa _1,\\cdots ,\\kappa _n$ are distinct nonzero real numbers, then there exist positive constants $\\rho _0$ and $\\delta $ such that the following results hold for the derived solution $\\widetilde{u}=u+2(G_1)_{12}$ of the nonlocal Davey-Stewartson I equation when $|\\rho |>\\rho _0$ and $|\\lambda _j-I\\kappa _j|<\\delta $ $(j=1,\\cdots ,n)$ .",
"(i) $\\widetilde{u}$ is globally defined and bounded for $(x,y,t)\\in \\hbox{\\bf R}^3$ .",
"(ii) Suppose the real numbers $v_x,v_y$ satisfy $\\alpha _{kR}v_x\\pm \\beta _{kR}v_y\\ne 0$ for all $k=1,\\cdots ,n$ where $\\alpha _k$ 's and $\\beta _k$ 's are given by (REF ), then $_{s\\rightarrow +\\infty }|\\widetilde{u}|=|\\rho |$ along the straight line $x=x_0+v_xs$ , $y=y_0+v_ys$ for arbitrary $x_0,y_0\\in \\hbox{\\bf R}$ .",
"Proof.",
"(i) follows directly from Lemma REF .",
"Now we prove (ii).",
"Since $|e_{k\\pm }|=\\hbox{e}^{(\\alpha _{kR}v_x\\pm \\beta _{kR}v_y)s+(\\alpha _{kR}x_0\\pm \\beta _{kR}y_0\\pm \\gamma _{kR}t)} $ along the straight line $x=x_0+v_xs$ , $y=y_0+v_ys$ , $\\alpha _{kR}v_x\\pm \\beta _{kR}v_y\\ne 0$ implies that for each $k$ , $e_{k+}\\rightarrow 0$ or $e_{k+}\\rightarrow \\infty $ , and $e_{k-}\\rightarrow 0$ or $e_{k-}\\rightarrow \\infty $ as $s\\rightarrow +\\infty $ .",
"Let $\\begin{array}{l}_k=\\left\\lbrace \\begin{array}{ll}\\lambda _k&\\hbox{ if }\\alpha _{kR}v_x+\\beta _{kR}v_y>0,\\\\-\\lambda _k^*&\\hbox{ if }\\alpha _{kR}v_x+\\beta _{kR}v_y<0,\\end{array}\\right.\\\\_k=\\left\\lbrace \\begin{array}{ll}-\\lambda _k&\\hbox{ if }\\alpha _{kR}v_x-\\beta _{kR}v_y>0,\\\\\\lambda _k^*&\\hbox{ if }\\alpha _{kR}v_x-\\beta _{kR}v_y<0,\\end{array}\\right.\\\\ḁ_k=\\left\\lbrace \\begin{array}{ll}\\beta _k&\\hbox{ if }\\alpha _{kR}v_x+\\beta _{kR}v_y>0,\\\\-\\beta _k^*&\\hbox{ if }\\alpha _{kR}v_x+\\beta _{kR}v_y<0,\\end{array}\\right.\\\\b̥_k=\\left\\lbrace \\begin{array}{ll}-\\beta _k&\\hbox{ if }\\alpha _{kR}v_x-\\beta _{kR}v_y>0,\\\\\\beta _k^*&\\hbox{ if }\\alpha _{kR}v_x-\\beta _{kR}v_y<0,\\end{array}\\right.\\\\\\end{array}$ then $a_k=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\mu _k}+\\mu _k\\Big ),\\quad b_k=\\frac{1}{2}\\Big (\\frac{\\sigma |\\rho |^2}{\\nu _k}+\\nu _k\\Big ).$ Rewrite (REF ) as $\\begin{array}{l}(\\begin{array}{cccccc}(G_1)_{11}&\\cdots &(G_n)_{11}&\\rho ^{-1}\\hbox{e}^{I\\phi } (G_1)_{12}&\\cdots &\\rho ^{-1}\\hbox{e}^{I\\phi } (G_n)_{12}\\\\(G_1)_{21}&\\cdots &(G_n)_{21}&\\rho ^{-1}\\hbox{e}^{I\\phi } (G_1)_{22}&\\cdots &\\rho ^{-1}\\hbox{e}^{I\\phi } (G_n)_{22}\\end{array}SWS^{-1}=-RS^{-1}\\\\\\end{array}$ where $S̥=\\left(\\begin{array}{cc}I_n\\\\&\\rho \\hbox{e}^{-I\\phi } I_n\\end{array}\\right)$ , $I_n$ is the $n\\times n$ identity matrix.",
"Applying Cramer's rule to (REF ) and using (REF ), we have $\\lim _{s\\rightarrow +\\infty }\\widetilde{u}=\\lim _{s\\rightarrow +\\infty }(\\rho \\hbox{e}^{-I\\phi } +2(G_1)_{12})=\\rho \\hbox{e}^{-I\\phi } \\Big (1-2\\frac{\\det W_1}{\\det W_0}\\Big )=\\rho \\hbox{e}^{-I\\phi } \\frac{\\det W_2}{\\det W_0}$ where $W_0=\\left(\\begin{array}{cccccc}a_1^{n-1}&\\cdots &a_n^{n-1}&\\sigma |\\rho |^{-2}b_1^{n-1}\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n^{n-1}\\nu _n\\\\a_1^{n-2}&\\cdots &a_n^{n-2}&\\sigma |\\rho |^{-2}b_1^{n-2}\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n^{n-2}\\nu _n\\\\\\vdots &&\\vdots &\\vdots &&\\vdots \\\\a_1&\\cdots &a_n&\\sigma |\\rho |^{-2}b_1\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n\\nu _n\\\\1&\\cdots &1&\\sigma |\\rho |^{-2}\\nu _1&\\cdots &\\sigma |\\rho |^{-2}\\nu _n\\\\a_1^{n-1}\\mu _1&\\cdots &a_n^{n-1}\\mu _n&b_1^{n-1}&\\cdots &b_n^{n-1}\\\\a_1^{n-2}\\mu _1&\\cdots &a_n^{n-2}\\mu _n&b_1^{n-2}&\\cdots &b_n^{n-2}\\\\\\vdots &&\\vdots &\\vdots &&\\vdots \\\\a_1\\mu _1&\\cdots &a_n\\mu _n&b_1&\\cdots &b_n\\\\\\mu _1&\\cdots &\\mu _n&1&\\cdots &1\\\\\\end{array}\\right),$ $W_1$ is obtained from $W_0$ by replacing the $(n+1)$ -th row with $\\left(\\begin{array}{cccccc}a_1^n&\\cdots &a_n^n&\\sigma |\\rho |^{-2}b_1^n\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n^n\\nu _n\\end{array}\\right),$ and $W_2=\\left(\\begin{array}{cccccc}a_1^{n-1}&\\cdots &a_n^{n-1}&\\sigma |\\rho |^{-2}b_1^{n-1}\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n^{n-1}\\nu _n\\\\a_1^{n-2}&\\cdots &a_n^{n-2}&\\sigma |\\rho |^{-2}b_1^{n-2}\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n^{n-2}\\nu _n\\\\\\vdots &&\\vdots &\\vdots &&\\vdots \\\\a_1&\\cdots &a_n&\\sigma |\\rho |^{-2}b_1\\nu _1&\\cdots &\\sigma |\\rho |^{-2}b_n\\nu _n\\\\1&\\cdots &1&\\sigma |\\rho |^{-2}\\nu _1&\\cdots &\\sigma |\\rho |^{-2}\\nu _n\\\\-\\sigma |\\rho |^2a_1^{n-1}\\mu _1^{-1}&\\cdots &-\\sigma |\\rho |^2a_n^{n-1}\\mu _n^{-1}&-\\sigma |\\rho |^{-2}b_1^{n-1}\\nu _1^2&\\cdots &-\\sigma |\\rho |^{-2}b_n^{n-1}\\nu _n^2\\\\a_1^{n-2}\\mu _1&\\cdots &a_n^{n-2}\\mu _n&b_1^{n-2}&\\cdots &b_n^{n-2}\\\\\\vdots &&\\vdots &\\vdots &&\\vdots \\\\a_1\\mu _1&\\cdots &a_n\\mu _n&b_1&\\cdots &b_n\\\\\\mu _1&\\cdots &\\mu _n&1&\\cdots &1\\\\\\end{array}\\right).$ Denote $\\hbox{ROW}_k$ and $\\hbox{COL}_k$ to be the $k$ -th row and $k$ -th column of $W_2$ respectively.",
"The elementary transformations $\\begin{array}{l}\\hbox{ROW}_{n+k+1}-2\\cdot \\hbox{ROW}_k\\rightarrow \\hbox{ROW}_{n+k+1}\\quad (k=1,\\cdots ,n-1),\\\\\\mu _k\\cdot \\hbox{COL}_k\\rightarrow \\hbox{COL}_k\\quad (k=1,\\cdots ,n),\\\\\\sigma |\\rho |^2\\nu _k^{-1}\\cdot \\hbox{COL}_{n+k}\\rightarrow \\hbox{COL}_{n+k}\\quad (k=1,\\cdots ,n),\\\\-\\sigma |\\rho |^{-2}\\cdot \\hbox{ROW}_{n+k}\\rightarrow \\hbox{ROW}_{n+k}\\quad (k=1,\\cdots ,n),\\\\\\hbox{ROW}_k\\leftrightarrow \\hbox{ROW}_{n+k}\\quad (k=1,\\cdots ,n)\\end{array}$ transform $W_2$ to $W_0$ .",
"Hence $W_2=\\prod _{k=1}^n\\frac{\\nu _k}{\\mu _k}\\det W_0$ .",
"This leads to $_{s\\rightarrow +\\infty }|\\widetilde{u}|=|\\rho |$ since $_{k=1}^n|\\mu _k|=\\prod _{k=1}^n|\\nu _k|=\\prod _{k=1}^n|\\lambda _k|$ .",
"The theorem is proved.",
"A 2 “line dark soliton” solution is shown in Figure 4 where the parameters are $\\sigma =-1$ , $t=10$ , $\\rho =1$ , $\\lambda _1=0.8+0.1I$ , $\\lambda _2=-0.6-0.3I$ .",
"The figure on the right describes the same solution but is upside down.",
"Figure: |u ˜||\\widetilde{u}| of a 2 “line dark soliton” solution."
],
[
"Acknowledgements",
"This work was supported by the Natural Science Foundation of Shanghai (No.",
"16ZR1402600) and the Key Laboratory of Mathematics for Nonlinear Sciences of Ministry of Education of China.",
"The author is grateful to Prof. S.Y.Lou for helpful discussion."
]
] | 1612.05689 |
[
[
"Durfee rectangles and pseudo-Wronskian equivalences for Hermite\n polynomials"
],
[
"Abstract We study an equivalence class of iterated rational Darboux transformations applied on the harmonic oscillator, showing that many choices of state adding and state deleting transformations lead to the same transformed potential.",
"As a by-product, we derive new identities between determinants whose entries are Hermite polynomials.",
"These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize the equivalence class of rational Darboux transformations.",
"Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, or equivalently, the minimum number of Darboux transformations.",
"The solution to this problem has an elegan graphical interpretation.",
"The results are applied to provide alternative and more efficient representations for exceptional Hermite polynomials and rational solutions of the Painlev\\'e IV equation."
],
[
"Introduction",
"In this paper we introduce an infinite number of identities between determinants whose entries are Hermite polynomials.",
"The remarkable fact is that the identities involve determinants of different size.",
"An example of such an identity would be given by $\\begin{vmatrix}{1} & {1}^{\\prime } & {1}^{\\prime \\prime } & {1}^{\\prime \\prime \\prime }\\\\{2} & {2}^{\\prime } & {2}^{\\prime \\prime } & {2}^{\\prime \\prime \\prime }\\\\{3} & {3}^{\\prime } & {3}^{\\prime \\prime } & {3}^{\\prime \\prime \\prime }\\\\{6} & {6}^{\\prime } & {6}^{\\prime \\prime } & {6}^{\\prime \\prime \\prime }\\end{vmatrix}=\\frac{1}{48} \\begin{vmatrix}\\tilde{H}_{1} & \\tilde{H}_{1}^{\\prime } & \\tilde{H}_{1}^{\\prime \\prime } \\\\\\tilde{H}_{2} & \\tilde{H}_{2}^{\\prime } & \\tilde{H}_{2}^{\\prime \\prime } \\\\\\tilde{H}_{6} & \\tilde{H}_{6}^{\\prime } & \\tilde{H}_{6}^{\\prime \\prime } \\\\\\end{vmatrix}=\\frac{1}{7680}\\begin{vmatrix}{2} & {2}^{\\prime } \\\\\\tilde{H}_{3} & \\tilde{H}_{4} \\\\\\end{vmatrix}$ where $H_n(x)$ denotes the $n$ -th Hermite polynomial and $\\tilde{H}_{n}(x)= {\\rm i}^{-n} H_n ({\\rm i} x)$ .",
"The first two determinants are Wronskian determinants, and this type of identities had already been shown in [14], and possibly known before [42].",
"As we shall explain below, they are a particular case of the whole equivalence class described in this paper that involves a given partition and its conjugate partition.",
"The third determinant in (REF ) is not a Wronskian, but can be constructed in a similar fashion and thus we have coined the name pseudo-Wronskian for it.",
"A pseudo-Wronskian involves two sequences, one of ordinary Hermite polynomials ${n}$ and one of conjugate Hermite polynomials $\\tilde{H}_{n}$ .",
"The determinant is built by taking derivatives of the ordinary Hermite polynomials as in the usual Wronskian, but shifting upwards the degree (which is almost an integration) for the conjugate Hermite polynomials.",
"Wronskian determinants are a key object in the theory of linear differential equations, but they also arise naturally when iterating Darboux transformations in Schrödinger's equation, as it was shown by Crum [10].",
"This factorization method has found many applications [11], ranging from the theory of integrable systems [17], [55], soliton theory [18], [39] and bispectral problems [12], [6], [30], among others.",
"Darboux transformations are also fundamentally related to exceptional orthogonal polynomials, [23], [20], and it was precisely this research that motivated the results reported in this paper.",
"The determinantal identities discussed here express an equivalence class of rational Darboux transformations that lead to the same transformed potential up to a spectral shift, and the two families of entries (${n}$ and $\\tilde{H}_{n}$ ) correspond to the two families of seed functions for state-adding and state-deleting rational Darboux transformations in the harmonic oscillator.",
"At the Schrödinger picture, these equivalences had already been noticed by in the Laguerre and Jacobi cases by Odake in [43], and by the authors in [36].",
"The most convenient way to visualize the equivalence is by shifting the origin in a Maya diagram, a type of diagram that was originally introduced by Sato in integrable systems theory [47].",
"They were applied to the context of Darboux transformations and exceptional polynomials by Takemura [53].",
"Wronskian determinants of Hermite polynomials appear already in the work of Karlin and Szegő [37], who provide expressions for the number of real zeros of the Wronskian of a sequence of consecutive Hermite polynomials.",
"Those sequences for which the Wronskian determinant has no real zeros were characterized by Adler in [1], and the result for arbitrary sequences has been recently derived by García-Ferrero et al.",
"[19], using oscillatory type arguments.",
"The complex zeros of these Wronskians of Hermite polynomials display very intriguing symmetric patterns on the complex plane, [14], [8], [15].",
"Beyond their obvious interest in Sturm-Liouville theory, Wronskian determinants of classical polynomials play a role in the construction of rational solutions to nonlinear integrable equations of Painlevé type.",
"The symmetry approach based on Bäcklund transformations of Painlevé equations developed by Noumi and his collaborators (see [41] and references therein) shows how to construct rational solutions by applying the symmetry group to a number of seed solutions.",
"Maya diagrams and partitions are also an essential part of their description.",
"In particular, two families of rational solutions to $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ can be constructed using generalized Hermite and Okamoto polynomials, which can both be expressed as Wronskians of specific sequences of Hermite polynomials [9].",
"We show how to apply our results to provide an alternative pseudo-Wronskian representation of Okamoto and generalized Hermite polynomials, which happens to be more efficient in the former case.",
"It is also possible to view the pseudo-Wronskian determinants introduced in this paper as an extension of Jacobi-Trudi formulas in the theory of symmetric functions, [16].",
"The original Jacobi-Trudi formulas express the Schur polynomial $s_\\lambda $ associated to a given partition $\\lambda $ as a determinant whose entries are complete homogeneous symmetric polynomials, or as a determinant whose entries are elementary symmetric polynomials associated to the conjugate partition $\\lambda ^{\\prime }$ .",
"Jacobi-Trudi formulas were already extended to include classical orthogonal polynomials in [51] and lead to exceptional orthogonal polynomials in [29].",
"The pseudo-Wronskian determinants in this paper can be regarded as a generalization of Jacobi-Trudi formulas that involve not just the partition $\\lambda $ or its conjugate partition $\\lambda ^{\\prime }$ , but also mixed representations described by the Durfee symbols introduced by Andrews in [3].",
"Analogous results for the Laguerre and Jacobi classes, including appropriately defined pseudo-Wronskians, will be presented in [28].",
"The paper is organized as follows: in Section we introduce the basic definitions and we review the connection between partitions and Maya diagrams.",
"In Section we show how to associate a pseudo-Wronskian determinant to each labelled Maya diagram and we prove the main theorem stating the proportionality of pseudo-Wronskian determinants for all labelled Maya diagrams in the same equivalence class.",
"In Section we address the problem of finding the minimal order pseudo-Wronskian determinant in each equivalence class.",
"Finally, in the last two sections we apply our results to derive alternative and more efficient pseudo-Wronskian representations of exceptional Hermite polynomials and special polynomials related to rational solutions of Painlevé type equations."
],
[
"Definitions and preliminaries",
"Definition 2.1 We define a Maya diagram to be a set of integers $M\\subset \\mathbb {Z}$ that contains a finite number of positive integers, and excludes a finite number of negative integers.",
"Definition 2.2 It is clear that if $M$ is a Maya diagram, then for $k\\in \\mathbb {Z}$ so is $M+k = \\lbrace m+k \\colon m\\in M \\rbrace $ .",
"We say that $M$ and $M+k$ are equivalent Maya diagrams, and we define an unlabelled Maya diagram to be the equivalence class of Maya diagrams related by such shifts.",
"We visualize a Maya diagram as a horizontally extended sequence of filled and empty boxes, with an origin placed between the box in position $-1$ and box in position 0, and with filled boxes indicating membership in $M$ .",
"By contrast, an unlabelled Maya diagram should be visualized as a sequence of filled and unfilled boxes, without a choice of origin.",
"In this formulation, the defining assumption of a Maya diagram is that all boxes sufficiently far to the left are filled, and that all boxes sufficiently far to the right are empty.",
"From a physical point of view, a Maya diagram depicts the empty and filled energy levels corresponding to the spectrum of a Hamiltonian with a pure point spectrum.",
"At the filled energy levels the Hamitonian would have a bound state, leading to a state-deleting rational Darboux transformations while at the empty levels there would be a quasi-rational formal eigenfunction, leading to a rational state-adding transformation.For simplicity we restrict in this paper to the case of the harmonic potential, for which only state-adding and state-deleting rational Darboux transformations exist.",
"We postpone the analysis of hamiltonians with rational isospectral Darboux transformations to a forthcoming publication.",
"Definition 2.3 We say that a Maya diagram $M\\subset \\mathbb {Z}$ is in standard form if $0\\notin M$ and $k\\in M$ for every $k<0$ .",
"Visually, a standard diagram has a gap just to the right of the origin, and no gaps to the left of the origin.",
"Proposition 2.1 Let $M\\subset \\mathbb {Z}$ be a Maya diagram.",
"Then there exists a unique $k\\in \\mathbb {Z}$ such that $M-k$ is in standard form.",
"The desired shift is given by $k=\\min \\, \\mathbb {Z}\\setminus M$ .",
"Figure: Three equivalent Maya diagrams corresponding to the partitionλ=(4,4,3,1,1)\\lambda =(4,4,3,1,1), together with their Frobenius representation.",
"The third diagram is in standard form.Definition 2.4 A partition $\\lambda $ is a non-increasing sequence of integers $\\lambda _1\\ge \\lambda _2\\ge \\cdots $ such that $\\lambda _i = 0$ for sufficiently large $i$ .",
"Let $\\ell $ be the largest index such that $\\lambda _\\ell >0$ .",
"We call $\\ell $ the length of the partition.",
"We call $|\\lambda | = \\lambda _1+\\cdots + \\lambda _\\ell $ the size of the partition.",
"Let $M\\subset \\mathbb {Z}$ be a Maya diagram and let $m_1>m_2>\\cdots $ be its elements ordered in decreasing order.",
"Consider the partition defined by $ \\lambda _i = \\#\\lbrace m\\notin M \\colon m < m_i\\rbrace ,\\quad i=1,2\\ldots .$ Proposition 2.2 The correspondence $M\\mapsto \\lambda $ given by (REF ) defines a bijection between the set of unlabelled Maya diagrams and the set of partitions.",
"Observe that $M+k$ defines the same partition as $M$ .",
"By Proposition REF every equivalence class of Maya diagrams contains a unique representative in standard form.",
"Thus, it suffices to establish a 1-1 correspondence between partitions and standard diagrams.",
"The desired correspondence is given by the relation $ m_i -\\lambda _i = \\ell -i,\\quad i=1,2,\\ldots $ If the above relation holds, then $\\lambda $ is a partition of length $\\ell $ if and only if $M$ is a standard diagram with $\\ell $ positive elements.",
"It is convenient to represent a partition by means of a Ferrer's diagram, a finite collection of points arranged in left-justified rows, with the row lengths in non-increasing order.",
"Definition 2.5 Let $\\curlyeqprec $ denote the box partial order on $\\mathbb {N}\\times \\mathbb {N}$ .",
"Formally, $(i_1, j_1) \\curlyeqprec (i_2,j_2)$ if and only if $i_1\\le i_2$ and $j_1\\le j_2$ .",
"We now define a Ferrer's diagram to be a finite subset of $\\mathbb {N}\\times \\mathbb {N}$ Throughout the paper we will use the following notation: $\\mathbb {N}= \\lbrace 1,2,\\ldots \\rbrace $ and ${\\mathbb {N}_0}=\\lbrace 0,1,2,\\ldots \\rbrace $ .",
"which is down-closed with respect to the box order.",
"Formally, the correspondence between Ferrer's diagrams and partitions is as follows.",
"Given a Ferrer's diagram $F\\subset \\mathbb {N}\\times \\mathbb {N}$ , let $ \\lambda _j = \\# \\lbrace i\\in \\mathbb {N}\\colon (i,j) \\in F\\rbrace ,\\quad j=1,2,\\ldots ;$ i.e., $\\lambda _j$ is the number of points in row $j$ .",
"Conversely, if $\\lambda $ is a partition, then the corresponding Ferrer's diagram is given by $ F = \\lbrace (i,j) \\in \\mathbb {N}\\times \\mathbb {N}\\colon i \\le \\lambda _j \\rbrace .$ Definition 2.6 For a partition $\\lambda =(\\lambda _1,\\dots ,\\lambda _\\ell ) $ of length $\\ell $ , set $ \\lambda ^{\\prime }_j = \\# \\lbrace i\\in \\mathbb {N}\\colon \\lambda _i \\ge j\\rbrace ,\\quad j=1,2,\\ldots $ The resulting sequence $\\lambda ^{\\prime }$ is called the conjugate partition of $\\lambda $ .",
"The following is well known.",
"Proposition 2.3 Let $\\lambda , \\lambda ^{\\prime }$ be as above, and let $F,F^{\\prime }$ be the corresponding Ferrer's diagrams.",
"Then $F^{\\prime }$ is the transpose of $F$ , meaning that $ F^{\\prime } = \\lbrace (j,i) \\in \\mathbb {N}\\times \\mathbb {N}\\colon (i,j) \\in F.\\rbrace .$ There is another way to represent Maya diagrams, one that makes the relation to Ferrer's diagrams more explicit, and which will be useful when we consider the minimal order problem.",
"Definition 2.7 We define a bent Maya diagram to be a doubly infinite sequence $B=\\lbrace (i_n,j_n)\\in {\\mathbb {N}_0}\\times {\\mathbb {N}_0}\\colon n\\in \\mathbb {Z}\\rbrace $ such that $ (i_{n+1},j_{n+1})-(i_n, j_n ) \\in \\lbrace (1,0), (0,-1) \\rbrace ,\\quad n \\in \\mathbb {Z}$ and such that $i_n j_n= 0$ for all but finitely many $n$ .",
"Note that since the displacement $(i_n,j_n)\\mapsto (i_{n+1},j_{n+1})$ is either down or to the right, the above definition implies that $i_n=0$ for all $n$ sufficiently small and that $j_n=0$ for all $n$ sufficiently large.",
"Proposition 2.4 For a Maya diagram $M\\subset \\mathbb {Z}$ set $i_n = \\# \\lbrace m\\notin M : m< n \\rbrace ,\\qquad j_n = \\# \\lbrace m\\in M : m\\ge n \\rbrace ,\\quad n\\in \\mathbb {Z}.$ Then, the doubly infinite sequence $B=\\lbrace (i_n,j_n)\\rbrace _{n\\in \\mathbb {Z}}$ is a bent Maya diagram.",
"By assumption, there exists an $N>0$ such that $n\\notin M$ for all $n\\ge N$ and such that $n\\in M$ for all $n\\le -N$ .",
"Hence, $j_n=0$ for all $n\\ge N$ and $i_n=0$ for all $n\\le -N$ .",
"If $n\\in M$ , then $(i_{n+1},j_{n+1})=(i_n,j_n-1)$ .",
"If $n\\notin M$ , then $(i_{n+1},j_{n+1})=(i_n+1,j_{n+1})$ .",
"Therefore, in both cases the defining condition of a bent diagram is satisfied.",
"Informally, a bent Maya diagram is a 2-dimensional representation of a Maya diagram, with a filled box at position $n$ corresponding to a unit downward displacement $(0,-1)$ and an empty box corresponding to a unit rightward displacement $(1,0)$ , as depicted in Figure REF .",
"A translation $M^{\\prime }= M-k$ corresponds to an index shift in the bent diagram: $(i^{\\prime }_n,j^{\\prime }_n) = (i_{n+k},j_{n+k}).$ There is a connection between bent diagrams and Ferrer's diagrams.",
"Definition 2.8 Let $F\\subset \\mathbb {N}\\times \\mathbb {N}$ be a Ferrer's diagram.",
"We define the rim of $F$ to be subset $F^{\\prime }=\\lbrace (i,j)\\in F \\colon (i+1,j+1)\\notin F\\rbrace .$ Proposition 2.5 Let $M\\subset \\mathbb {Z}$ be a Maya diagram, $B$ the corresponding bent diagram defined by (REF ), $\\lambda $ the corresponding partition defined by (REF ), and $F$ the corresponding Ferrer's diagram defined by (REF ).",
"Then, $F^{\\prime } = B\\cap (\\mathbb {N}\\times \\mathbb {N})$ .",
"Thus, the rim is that finite subset of the bent diagram whose points have non-zero coordinates.",
"We first show that the non-zero part of $B$ lies in $F^{\\prime }$ .",
"Suppose that $i_n,j_n>0$ .",
"Let $m_1>m_2>\\cdots $ be the elements of $M$ in decreasing order.",
"By (REF ), $m_{j_n+1}< n \\le m_{j_n}$ .",
"Since $ \\lbrace m\\notin M \\colon m< m_{j_n+1}\\rbrace \\subset \\lbrace m\\notin M \\colon m<n\\rbrace \\subset \\lbrace m\\notin M \\colon m< m_{j_n}\\rbrace $ it follows that $\\lambda _{j_n+1}\\le i_n \\le \\lambda _{j_n}$ .",
"Therefore by (REF ), $(i_n,j_n)\\in F$ but $(i_{n}+1,j_{n}+1)\\notin F$ .",
"We now prove the converse.",
"Arguing by contradiction, suppose that there is an $(i,j)\\in F^{\\prime }$ which does not belong to $B$ .",
"In $F^{\\prime }\\setminus B$ choose the points with $j$ as small as possible, and of those choose the point that has $i$ as large as possible.",
"By assumption, $i\\le \\lambda _j$ .",
"We consider two cases.",
"Case 1: assume that $i<\\lambda _j$ .",
"Then $i+1\\le \\lambda _j$ , which means that $(i+1,j)\\in F$ .",
"Since $(i+1,j+1) \\notin F$ , the same is true for $(i+2,j+1)$ .",
"Hence, $(i+1,j)\\in F^{\\prime }$ also.",
"Because of the assumed maximality of $i$ , we must have $(i+1,j)\\in B$ ; i.e.",
"$(i+1,j)=(i_n,j_n)$ for some $n\\in \\mathbb {Z}$ .",
"By the definition of a bent diagram, $(i_{n-1},j_{n-1})$ is either $(i_n,j_n+1)=(i+1,j+1)$ or $(i_n-1,j_n)= (i,j)$ .",
"The second possibility is excluded because we have assumed that $(i,j)\\notin B$ .",
"By the first part of the proof, $(i_{n-1},j_{n-1})\\in F^{\\prime }$ .",
"This means that $(i+1,j+1)\\in F$ , which contradicts the assumption that $(i,j)\\in F^{\\prime }$ .",
"Case 2: assume that $i=\\lambda _j$ .",
"If $j>1$ , then $i\\le \\lambda _{j-1}$ which means that $(i,j-1)\\in F$ .",
"By assumption, $(i+1,j)\\notin F$ .",
"Hence $(i,j-1)\\in F^{\\prime }$ and hence $(i,j-1)\\in B$ by the minimality of $j$ .",
"We now repeat the above argument to conclude that $(i,j)\\in B$ also — a contradiction.",
"Hence $j=1$ and $i=\\lambda _1$ .",
"Set $n=\\max M$ .",
"By (REF ), $i_n=\\lambda _1$ and $j_n=1$ .",
"This contradicts the assumption that $(i,j)\\notin B$ .",
"Definition 2.9 For a given Maya diagram $M\\subset \\mathbb {Z}$ , define $ M_+ = \\lbrace m\\in M \\colon m\\ge 0\\rbrace ,\\qquad M_-=\\lbrace -m-1 \\colon m<0,m\\notin M\\rbrace .$ In other words, $M_+$ gives positions of the filled boxes to the right of the origin, and $M_-$ the positions of the holes to the left of the origin.",
"The numbers in $M_+$ and $M_-$ indicate distance to the origin, with 0 indicating a position adjacent to the origin.",
"Since $M_+$ and $M_-$ fully define $M$ , the defining assumptions of a Maya diagram are equivalent to the condition that $M_+$ and $M_-$ should be finite subsets of ${\\mathbb {N}_0}$ .",
"Definition 2.10 Let $M\\subset \\mathbb {Z}$ be a Maya diagram.",
"Let $\\lbrace s_1,\\dots ,s_p\\rbrace ,\\, p=i_0,$ be the elements of $M_-$ and $\\lbrace t_1,\\dots ,t_q\\rbrace ,\\, q=j_0,$ the elements of $M_+$ , arranged in decreasing order.",
"The double list $(s_1,\\ldots , s_p \\mid t_1,\\ldots , t_q)$ is called the Frobenius symbol of $M$ [44].",
"The classical Frobenius symbol [4], [44], [5], [3] corresponds to the case where $i_0=j_0$ ; i.e.",
"the case where $M_-$ and $M_+$ have the same cardinality.",
"Such a choice of origin can be visualized as the unique intersection of the rim and the main diagonal in $\\mathbb {N}\\times \\mathbb {N}$ .",
"Figure: Correspondence between an unlabelled Maya diagram and apartition λ=(4,4,3,1,1)\\lambda =(4,4,3,1,1).",
"The squares are points that belong tothe Ferrer diagram FF, while black squares belong to the rim of FF.Let us also note the following connection between the Frobenius symbol and bent diagrams.",
"Proposition 2.6 Let $M\\subset \\mathbb {Z}$ be a Maya diagram and $B=\\lbrace (i_n,j_n)\\rbrace _{n\\in \\mathbb {Z}}$ the corresponding bent diagram.",
"Then, $i_n$ and $j_n$ are the cardinalities of $(M-n)_-$ and $(M-n)_+$ , respectively.",
"By (REF ), $i_0$ is the cardinality of $M_-$ and $j_0$ the cardinality of $M_+$ .",
"The general relation follows by (REF )."
],
[
"Hermite pseudo-Wronskians",
"In this section we will associate to each labelled Maya diagram a certain determinant whose entries are Hermite polynomials.",
"We then prove that determinants associated to equivalent Maya diagrams are proportional to each other.",
"This is our main result in this Section.",
"For $n\\ge 0$ , let $n(x) = (-1)^n e^{x^2} D_x^n e^{-x^2},\\quad D_x=\\frac{d}{dx}$ denote the degree $n$ Hermite polynomial, and $\\tilde{H}_n(x)={\\rm i}^{-n} {n}({\\rm i}x)$ the conjugate Hermite polynomial.",
"Recall that $y=n$ is a solution of the Hermite differential equation $y^{\\prime \\prime }-2x y^{\\prime } + 2n y=0$ for $n\\ge 0$ , and that $y=e^{x^2} \\tilde{H}_{-n-1}$ is a solution of (REF ) for $n<0$ .",
"Definition 3.1 Let $M\\subset \\mathbb {Z}$ be a Maya diagram.",
"Let $\\lbrace s_1,\\dots ,s_p\\rbrace $ be the elements of $M_-$ and $\\lbrace t_1,\\dots ,t_q\\rbrace $ the elements of $M_+$ , both arranged in descending order.",
"We define the Hermite pseudo-Wronskian associated to $M$ to be $H_M= e^{-px^2}\\operatorname{Wr}[ e^{x^2} \\tilde{H}_{s_1},\\ldots , e^{x^2}\\tilde{H}_{s_p}, {t_q},\\ldots {t_1} ],$ where $\\operatorname{Wr}$ denotes the Wronskian determinant of the indicated functions.",
"The polynomial nature of $H_M$ becomes evident once we represent it using a slightly different determinant.",
"Proposition 3.1 A Hermite pseudo-Wronskian admits the following alternative determinantal representation $H_M=\\begin{vmatrix}\\tilde{H}_{s_1} & \\tilde{H}_{s_1+1} & \\ldots & \\tilde{H}_{s_1+p+q-1}\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\tilde{H}_{s_p} & \\tilde{H}_{s_p+1} & \\ldots & \\tilde{H}_{s_p+p+q-1}\\\\{t_q} & D_x {t_q} & \\ldots & D_x^{p+q-1}{t_q}\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\{t_1} & D_x {t_1} & \\ldots & D_x^{p+q-1}{t_1}\\end{vmatrix}$ The desired conclusion follows by the fundamental identities $\\begin{aligned} &D_x n(x) = 2n {n-1}(x),\\quad n\\ge 0,\\\\ &D_x \\tilde{H}_n(x) = 2n \\tilde{H}_{n-1}(x),\\quad n\\ge 0,\\\\ &2x n(x) = {n+1}(x) + 2n {n-1}(x),\\\\ &2x \\tilde{H}_n(x) = \\tilde{H}_{n+1}(x) - 2n \\tilde{H}_{n-1}(x),\\\\&D_x (e^{x^2} \\tilde{H}_n(x))= e^{x^2}\\tilde{H}_{n+1}(x),\\\\&D_x (e^{-x^2} h_n(x)) = -e^{-x^2} h_{n+1}(x).\\end{aligned}$ and the Wronskian identity $\\operatorname{Wr}[g f_1,\\ldots , g f_s] = g^s \\operatorname{Wr}[f_1,\\ldots , f_s],$ We refer to $H_M$ as a pseudo-Wronskian because it is constructed by means of a modified Wronskian operator that replaces $D_x$ with an indicial shift for the rows with the conjugate Hermites.",
"More specifically, if $j\\in M_+$ , then the $(j,k)$ entry is $D_x^{(k-1)}H_j$ , as in the ordinary Wronskian.",
"If $i\\in M_-$ , then the $(i,k)$ entry is the conjugate Hermite polynomial $\\tilde{H}_{i+k-1}$ .",
"Example 3.1 Consider the first Maya diagram in Figure REF .",
"The Frobenius symbol is $(5,2,1\\mid 2,1)$ .",
"The Hermite pseudo-Wronskian $H_M$ associated to $M$ is given by $H_M= {\\rm e}^{-3x^2} \\operatorname{Wr}[{\\rm e}^{x^2} \\tilde{H}_{5},{\\rm e}^{x^2} \\tilde{H}_{2},{\\rm e}^{x^2} \\tilde{H}_{1},H_1,H_2 ]= \\begin{vmatrix}\\tilde{H}_{5} & \\tilde{H}_{6} & \\tilde{H}_{7} &\\tilde{H}_{8} & \\tilde{H}_{9}\\\\\\tilde{H}_{2} & \\tilde{H}_{3} & \\tilde{H}_{4} &\\tilde{H}_{5} & \\tilde{H}_{6}\\\\\\tilde{H}_{1} & \\tilde{H}_{2} & \\tilde{H}_{3} &\\tilde{H}_{4} & \\tilde{H}_{5}\\\\{1} & {1}^{\\prime } & {1}^{\\prime \\prime } & {1}^{\\prime \\prime \\prime } & {1}^{(4)}\\\\{2} & {2}^{\\prime } & {2}^{\\prime \\prime } & {2}^{\\prime \\prime \\prime } & {2}^{(4)}\\end{vmatrix}$ The main result of this section is the following class of fundamental determinantal identities enjoyed by these polynomials Theorem 3.1 Let $M$ and $M^{\\prime }=M-k,\\; k>0$ be two equivalent Maya diagrams.",
"Set $E_k&= \\lbrace m\\in M \\colon 0\\le m < k \\rbrace & G_k&= \\lbrace m\\notin M \\colon 0\\le m < k \\rbrace \\\\\\epsilon _i &= (-1)^{\\# \\lbrace m\\notin M \\colon m<i \\rbrace } \\prod _{m\\in M\\atop m>i} (2m-2i)& \\gamma _i &= (-1)^{\\# \\lbrace m\\in M \\colon m>i \\rbrace } \\prod _{m\\notin M\\atop m<i} (2m-2i)$ Then, $\\left(\\prod _{i\\in G_k} \\gamma _i\\right) H_{M^{\\prime }} = \\left(\\prod _{i\\in E_k} \\epsilon _i \\right) H_M$ Once the following two Lemmas are established, it suffices to verify that the factors shown in (REF ) and (REF ) below are equal to the above-defined $\\gamma _i$ and $\\epsilon _i$ symbols, respectively.",
"Throughout, $\\lbrace s_1,\\dots ,s_p\\rbrace $ and $\\lbrace t_1,\\dots ,t_q\\rbrace $ are, respectively, the elements of $M_-$ and $M_+$ arranged in descending order.",
"Lemma 3.1 Suppose that $M^{\\prime }=M-1$ and that $0\\in M$ .",
"Then, $H_M = (-1)^{p}\\,2^{q-1}\\left( \\prod _{b=1}^{q-1} t_b\\right) H_{M^{\\prime }}.$ By assumption, $t_q=0$ and $ M^{\\prime }_- = \\lbrace s_1+1,\\ldots , s_p+1\\rbrace ,\\quad M^{\\prime }_+ = \\lbrace t_1-1,\\ldots ,t_{q-1}-1\\rbrace .$ The identity $\\operatorname{Wr}[1, f_1,\\ldots , f_s] = \\operatorname{Wr}[Df_1,\\ldots , Df_s],$ together with (REF ) implies that $H_M&= e^{-p x^2} \\operatorname{Wr}[e^{x^2} \\tilde{H}_{s_1},\\ldots , e^{x^2}\\tilde{H}_{s_p},1,H_{t_{q-1}},\\ldots , {t_q}],\\\\&=(-1)^p e^{-px^2} \\operatorname{Wr}[ D(e^{x^2}\\tilde{H}_{s_1}), \\ldots , D(e^{x^2} \\tilde{H}_{s_p}), D{t_{q-1}},\\ldots , D {t_1}]\\\\&=(-1)^{p} 2^{q} \\left(\\prod _{b=1}^{q-1} t_b\\right) e^{-p x^2}\\operatorname{Wr}[ e^{x^2}\\tilde{H}_{s_1+1}, \\ldots , e^{x^2} \\tilde{H}_{s_p+1},{t_{q-1}-1},\\ldots , {t_1-1}]$ Lemma 3.2 Suppose that $M^{\\prime }=M+1$ and that $-1\\notin M$ .",
"Then, $H_M = (-1)^{p+q-1}2^{p-1}\\left( \\prod _{a=1}^{p-1} s_a\\right) H_{M^{\\prime }}$ By assumption, $s_p = 0$ and $ M^{\\prime }_-=\\lbrace s_1-1,\\ldots , s_{p-1}-1\\rbrace ,\\quad M^{\\prime }_+=\\lbrace t_1+1,\\dots ,t_q+1\\rbrace .$ The identity (REF ) together with (REF ) implies $H_M&= e^{-p x^2} \\operatorname{Wr}[e^{x^2}\\tilde{H}_{s_1},\\ldots ,e^{x^2} \\tilde{H}_{s_{p-1}},e^{x^2},h_{t_q},\\ldots {t_1}]\\\\&= e^{qx^2}\\operatorname{Wr}[\\tilde{H}_{s_1},\\ldots ,\\tilde{H}_{s_{p-1}},1,e^{-x^2} h_{t_q},\\ldots , e^{-x^2}{t_1}]\\\\&= (-1)^{p-1} e^{q x^2} \\operatorname{Wr}[D\\tilde{H}_{s_1},\\ldots ,D\\tilde{H}_{s_{p-1}},De^{-x^2} h_{t_q},\\ldots , De^{-x^2}{t_1}]\\\\&= (-1)^{p+q-1} 2^{p-1} \\left(\\prod _{a=1}^{p-1} s_a\\right) e^{q x^2}\\operatorname{Wr}[\\tilde{H}_{s_1-1},\\ldots ,\\tilde{H}_{s_{p-1}-1},e^{-x^2} h_{t_q+1},\\ldots , e^{-x^2}{t_1+1}]\\\\&= (-1)^{p+q-1} 2^{p-1} \\left(\\prod _{a=1}^{p-1} s_a\\right)e^{-(p-1) x^2}\\operatorname{Wr}[e^{x^2} \\tilde{H}_{s_1-1},\\ldots ,e^{x^2}\\tilde{H}_{s_{p-1}-1},H_{t_q+1},\\ldots , H_{t_1+1}]$ Proposition REF immediately gives the following.",
"Corollary 3.1 Every Hermite pseudo-Wronskian is a scalar multiple of a Hermite Wronskian $ \\operatorname{Wr}[H_{m_\\ell }, \\ldots , H_{m_1}] $ for some unique choice of positive integers $m_1>\\cdots > m_\\ell >0$ .",
"Example 3.2 Consider the three equivalent Maya diagrams in Figure REF .",
"We have $M^{\\prime }=M+3 :\\quad &M^{\\prime }_-= \\lbrace 2\\rbrace \\quad &M^{\\prime }_+=\\lbrace 2,4,5\\rbrace \\\\M^{\\prime \\prime }=M+6:\\quad &M^{\\prime \\prime }_-=\\emptyset \\quad &M^{\\prime \\prime }_+=\\lbrace 1,2,5,7,8\\rbrace $ Hence, $H_{M^{\\prime }}= \\begin{vmatrix}\\tilde{H}_{2} & \\tilde{H}_{3} & \\tilde{H}_{4} &\\tilde{H}_{5}\\\\{2} & {2}^{\\prime } & {2}^{\\prime \\prime } & {2}^{\\prime \\prime \\prime }\\\\{4} & {4}^{\\prime } & {4}^{\\prime \\prime } & {4}^{\\prime \\prime \\prime }\\\\{5} & {5}^{\\prime } & {5}^{\\prime \\prime } & {5}^{\\prime \\prime \\prime }\\end{vmatrix},$ $H_{M^{\\prime \\prime }}=\\operatorname{Wr}[H_1,H_2,H_5,H_7,H_8]$ By (REF ), $ -483840 H_{M^{\\prime \\prime }}=-1935360 H_{M^{\\prime }}= H_{M} .$ Before continuing, we mention that a “pure” pseudo-Wronskian corresponding to a Maya diagram without any positive elements can also be expressed as a Wronskian determinant of conjugate Hermite polynomials.",
"Proposition 3.2 Let $M\\subset \\mathbb {Z}$ be a Maya diagram consisting entirely of negative integers, that is $M_+=\\emptyset $ .",
"Let $\\lbrace s_1, \\ldots , s_p \\rbrace $ be the elements of $M_-$ arranged in descending order.",
"Then, $ H_M = \\operatorname{Wr}[\\tilde{H}_{s_1},\\ldots , \\tilde{H}_{s_p}].$ Recall that $ H_M = e^{-px^2} \\operatorname{Wr}[e^{x^2} \\tilde{H}_{s_1},\\ldots , e^{x^2} \\tilde{H}_{s_p}]$ The desired identity now follows using (REF ).",
"As a special case of Theorem REF we obtain the following identity; see [42] [14] and the references therein.",
"Corollary 3.2 Let $\\lambda $ be a partition of length $\\ell $ , $\\lambda ^{\\prime }$ the conjugate partitions of length $\\ell ^{\\prime }=\\lambda _1$ .",
"Let $m_1>\\ldots > m_\\ell $ the positive integers defined by (REF ) and $m^{\\prime }_1> \\ldots > m^{\\prime }_{\\ell ^{\\prime }}$ the analogous integers for $\\lambda ^{\\prime }$ .",
"Then, $ 2^{\\ell ^{\\prime }} V(m^{\\prime }_1,\\ldots , m^{\\prime }_{\\ell ^{\\prime }}) \\operatorname{Wr}[H_{m_1},\\ldots ,H_{m_\\ell }] = 2^\\ell V(m_1,\\ldots , m_\\ell ) \\operatorname{Wr}[\\tilde{H}_{m^{\\prime }_{1}},\\ldots , \\tilde{H}_{m^{\\prime }_{\\ell ^{\\prime }}}],$ where $ V(a_1,\\ldots , a_k) = \\prod _{1\\le i<j<k} (a_j-a_i) $ is the usual Vandermonde determinant."
],
[
"The minimal order problem",
"It is quite remarkable to have an infinite family of identities among determinants of different order.",
"In this section we pose and solve the following question: given an unlabelled Maya diagram, which of the corresponding equivalent pseudo-Wronskian determinants has the smallest order?",
"This question will have some applications in the following sections to derive simpler, alternative representations of certain class of special functions.",
"The precise formulation of the question requires the following.",
"Definition 4.1 We define $|M|$ , the girth of a Maya diagram $M$ , to be the length of its Frobenius symbol, that is the sum of the cardinalities of $M_-$ and $M_+$ .",
"Since the girth of $M$ is just the order of the corresponding pseudo-Wronskian determinant $H_M$ , our aim is to determine $ \\min \\lbrace |M-k| \\colon k \\in \\mathbb {Z}\\rbrace ,$ a quantity that we will call the minimal girth of $M$ .",
"A $k\\in \\mathbb {Z}$ such that $|M-k|$ is minimal will be called a minimal girth origin for $M$ .",
"As we now show, it suffices to check for minimal girth at a finite number of $k$ -values.",
"Proposition 4.1 If $k\\in \\mathbb {Z}$ is a minimal girth origin for a Maya diagram $M$ , then necessarily $k-1\\in M$ and $k\\notin M$ .",
"More plainly, the minimal girth origins of a Maya diagram must occur at locations where a full box is succeeded by an empty box.",
"Let $(s_1,\\ldots ,s_p\\mid t_1,\\ldots , t_q)$ be the Frobenius symbol of $M-k$ .",
"If $k\\in M$ , then $t_q=0$ .",
"Observe that the Frobenius symbol of $M^{\\prime }=M-k-1$ is $(s_1+1,\\ldots , s_p+1\\mid t_1-1 , \\cdots , t_{q-1}-1)$ .",
"If $k-1\\notin M$ , then $s_p=0$ .",
"In this case, observe that the Frobenius symbol of $M^{\\prime }=M-k+1$ is $(s_1-1,\\ldots , s_{p-1}-1\\mid t_1+1,\\ldots , t_q+1)$ .",
"In both cases, $|M^{\\prime }| = |M-k|-1$ .",
"Therefore $k\\notin M$ and $k-1\\in M$ .",
"The solution of the minimal order problem is closely related to the geometry of Ferrer's diagrams.",
"Definition 4.2 Let $F\\subset \\mathbb {N}\\times \\mathbb {N}$ be a Ferrer's diagram corresponding to a partition $\\lambda $ .",
"We call $(i,j)\\in F$ an inside corner if $(i,j+1), (i+1,j)\\in F$ but $(i+1,j+1)\\notin F$ .",
"Proposition 4.2 Let $B=\\lbrace (i_n,j_n)\\rbrace _{n\\in \\mathbb {Z}}$ be the bent diagram corresponding to a Maya diagram $M\\subset \\mathbb {Z}$ .",
"Then $|M-n| = i_n+j_n.$ Recall that $i_n$ is the cardinality of $(M-n)_-$ while $j_n$ is the cardinality of $(M-n)_+$ .",
"Proposition 4.3 Let $n\\in Z$ be a minimal girth origin for a Maya diagram $M\\subset \\mathbb {Z}$ corresponding to a partition $\\lambda =(\\lambda _1,\\ldots , \\lambda _\\ell )$ .",
"Then, one of the following three possibilities holds: (a) $(i_n,j_n)$ is an inside corner of the corresponding Ferrer's diagram, (b) $(i_n,j_n)=(0,\\ell )$ (c) $(i_n,j_n) = (\\lambda _1,0)$ .",
"Without loss of generality, assume that $M$ is in standard form.",
"By Proposition REF , $n-1\\in M$ and $n\\notin M$ .",
"Hence, by (REF ), $(i_{n-1},j_{n-1}) = (i_n,j_n-1)$ and $(i_{n+1},j_{n+1}) = (i_n+1,j_n)$ .",
"If $0<n<\\max M$ , then $i_n,j_n>0$ and hence by Proposition REF , $(i_n,j_n)$ is an inside corner of $F$ .",
"The only other possibilities are $n=0$ and $n=m_1+1$ , where $m_1=\\max M$ .",
"In the former case, $(i_0,j_0)=(0,\\ell )$ .",
"In the second case, $(i_{m_1+1},j_{m_1+1}) = (\\lambda _1,0)$ .",
"Corollary 4.1 Let $M\\subset \\mathbb {Z}$ be a Maya diagram and $\\lambda $ the corresponding partition.",
"Then, the minimal girth is given by $\\min \\lbrace \\lambda _{j+1} + j \\colon 0\\le j\\le \\ell \\rbrace $ .",
"By (REF ) the inside corners of the corresponding Ferrer's diagram occur at $(\\lambda _{j+1},j)$ for $j=1,\\ldots , \\ell -1$ .",
"If $j=0$ , then $(\\lambda _{j+1},j) = (\\lambda _1,0)$ .",
"If $j=\\ell $ , then $(\\lambda _{j+1},j) = (0,\\ell )$ .",
"The desired conclusion now follows by Proposition REF Definition 4.3 Let $F$ be a Ferrer's diagram.",
"We define a Durfee rectangle [3] of $F$ to be a rectangle with vertices $(0,0), (i,0), (i,j), (0,j)$ where $(i,j)$ is an inside corner of $F$ .",
"If $M$ is the corresponding Maya diagram with an origin located at an inside corner, then the corresponding Frobenius symbol $(s_1,\\ldots , s_p \\mid t_1, \\ldots , t_q)$ satisfies $s_p,t_q>0$ .",
"We are thus able to define the partitions $\\mu &= (s_1-p+1, \\ldots , s_{p-1}+1, s_p) \\\\\\nu &= (t_q-q+1,\\ldots , t_{q-1}+1, t_q)$ of length $p$ and $q$ , respectively.",
"We call $[\\mu \\mid \\nu ]_{p\\times q}$ the Durfee symbol of $M$ .",
"Visually, the partitions $\\mu $ and $\\nu $ describe the complement of the Durfee rectangle in $F$ (see Figure REF ).",
"Partition $\\mu $ is the transpose of the remnant above the rectangle, and $\\nu $ is the remnant to the right of the rectangle.",
"The girth of the corresponding Maya diagram is simply the distance of the inside corner to the origin relative to the taxi-cab metric.",
"To determine the minimal girth, the corresponding girths have to be compared to the height $\\ell $ of the Ferrer's diagram, and the width $\\lambda _1$ , which can be considered as Durfee rectangles of width zero and height zero, respectively (black circle dots in Figure REF ).",
"Figure: Durfee rectangles and Durfee symbols for the threeequivalent Maya diagrams depicted in Figure .",
"Notethat the shortest girth corresponds to the left diagram, which is anon-unique solution to the minimal order problem for thispartition.",
"The right diagram where the Durfee symbolcoincides with the partition, corresponds to the Maya diagram instandard form."
],
[
"Application to Exceptional Hermite polynomials",
"One possible application of Theorem REF and the minimal order problem is to provide a more efficient computation for exceptional Hermite polynomials.",
"Exceptional Hermite polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem on the real line, [21], [22].",
"The degree sequence of each family does not range over all positive integers, i.e.",
"there is a finite number of gaps or missing degrees.",
"Let $\\lambda =(\\lambda _1,\\dots ,\\lambda _\\ell )$ be a partition and $M\\subset \\mathbb {Z}$ the corresponding standard Maya diagram with $M_+=\\lbrace m_1,\\dots ,m_\\ell \\rbrace $ its positive elements as determined by (REF ).",
"Following [25], [26] we define an infinite number of polynomials in the following manner: $H^{(\\lambda )}_n=\\operatorname{Wr}[H_{m_\\ell },\\dots ,H_{m_1},H_{\\ell -|\\lambda |+n}],\\quad n\\notin M+|\\lambda |-\\ell $ By construction, $H^{(\\lambda )}_n$ is a polynomial with $\\deg H^{(\\lambda )}_n = \\sum _{i=1}^\\ell \\left(m_i-i+1\\right) +\\ell -|\\lambda |+n - \\ell = n$ The degree sequence for the exceptional Hermite family indexed by partition $\\lambda $ is $\\mathbb {Z}\\setminus (M+|\\lambda |-\\ell )$ .",
"Thus, degrees $0,1,\\ldots ,|\\lambda | -\\ell -1$ and the degrees $m_1+|\\lambda |-\\ell ,\\ldots , m_\\ell + |\\lambda |-\\ell $ are missing, so that the polynomial sequence $\\lbrace H^{(\\lambda )}_n\\rbrace _n$ is missing a total of $|\\lambda |$ degrees, the codimension of the sequence.",
"Exceptional Hermite polynomials are a generalization the classical Hermite family because they satisfy a Hermite-like differential equation $T_\\lambda \\left[H^{(\\lambda )}_n\\right] = 2(N-n) H^{(\\lambda )}_n,\\qquad n\\notin M+|\\lambda |-\\ell ,$ where $T_\\lambda [y]:=y^{\\prime \\prime }-2\\left(x+\\frac{H_M^{\\prime }}{H_M} \\right)y^{\\prime }+ \\left(\\frac{H_M^{\\prime \\prime }}{H_M}+ 2 x\\frac{H_M^{\\prime }}{H_M} \\right) y.$ We say that $\\lambda $ is an even partition if $\\ell $ is even and $\\lambda _{2i-1} = \\lambda _{2i}$ for every $i$ .",
"If $\\lambda $ is even, then the exceptional Hermite polynomials $H^{(\\lambda )}_n$ satisfy the orthogonality relations $ \\int _\\mathbb {R}H^{(\\lambda )}_n H^{(\\lambda )}_m W_\\lambda (x) dx =\\delta _{m,n} \\sqrt{\\pi }\\, 2^{j+\\ell } j!\\prod _{i=1}^{\\ell }(j-m_i),\\qquad n,m\\notin M+|\\lambda |-\\ell $ where $j = n+\\ell -N$ , and the orthogonality weight is defined as $ W_\\lambda (x) = \\frac{{e^{-x^2}}}{{H_M(x)^2}}.$ Moreover, if $\\lambda $ is an even partition then $\\text{span}\\lbrace H^{(\\lambda )}_n:n\\notin M+|\\lambda |-\\ell \\rbrace $ is dense in $\\mathrm {L}^ 2(\\mathbb {R},W_\\lambda )$ .",
"Exceptional polynomials appear in a number of applications in mathematical physics, mostly as solutions to exactly solvable quantum mechanical problems describing bound states [25], [45], [49].",
"They appear also in connection with super-integrable systems [48], [38], exact solutions to Dirac's equation [50], diffusion equations and random processes [32], finite-gap potentials [31] and point vortex models [34].",
"From a mathematical point of view, the main results are concerned with the full classification of exceptional polynomials [23], [20], properties of their zeros [24], [33], [35], and recurrence relations [40], [13], [46], [26].",
"Here we are concerned with the most economical presentation of a given exceptional Hermite polynomial family.",
"The definition (REF ) involves the computation of a Wronskian determinant of order $\\ell +1$ .",
"In light of the preceding results, this order can be potentially reduced by replacing the Wronskian with an appropriate pseudo-Wronskian.",
"Let $M\\subset \\mathbb {Z}$ be a Maya diagram, $m_1>m_2> \\cdots $ the elements of $M$ arranged in descending order, and $\\lambda $ the corresponding partition.",
"Set $ O_r = \\lbrace m_j +1 \\colon \\lambda _{j+1} + j = r,\\; 0\\le j \\le \\ell \\rbrace \\qquad r\\ge 0.$ Let $k_r = \\max O_r$ , or $-\\infty $ if the latter is the empty set.",
"Visually, the elements of $O_r$ are the labels of the inside corners that lie on the anti-diagonal $i+j=r$ , with $k_r$ the largest such.",
"By Corollary REF , if $r$ is the minimal girth, then $k_r$ is the largest minimal girth origin.",
"Proposition 5.1 Let $M\\subset \\mathbb {Z}$ be a Maya diagram, $m\\notin M$ and $M^{\\prime }=M\\cup \\lbrace m \\rbrace $ .",
"Let $r$ be the minimal girth of $M$ .",
"Let $r^{\\prime }$ denote the minimal girth of $M^{\\prime }$ and $O^{\\prime }_{r^{\\prime }}$ the minimal girth origins of $M^{\\prime }$ .",
"Then one of the following mutually exclusive possibilities holds.",
"If $m< k_r$ then $r^{\\prime } = r-1$ and $O^{\\prime }_{r^{\\prime }} = \\lbrace k\\in O_r \\colon k>m \\rbrace $ .",
"If $m=k_r$ then $r^{\\prime }=r$ and $O^{\\prime }_{r^{\\prime }} = \\lbrace m+ 1\\rbrace \\cup \\lbrace k\\in O_{r+1} \\colon k>m \\rbrace $ .",
"If $k_r<m<k_{r+1}$ , then $r^{\\prime } =r$ and $O^{\\prime }_{r^{\\prime }} = \\lbrace k\\in O_{r+1} \\colon k>m \\rbrace $ .",
"If $m\\ge \\max \\lbrace k_r,k_{r+1}\\rbrace $ then $r^{\\prime }=r+1$ and $O^{\\prime }_{r^{\\prime }} = O_r\\cup \\lbrace k\\in O_{r+2} \\colon k>m \\rbrace $ .",
"Let $B=\\lbrace (i_k,j_k) \\rbrace _{k\\in \\mathbb {Z}}$ and $B^{\\prime }=\\lbrace (i^{\\prime }_k,j^{\\prime }_k)\\rbrace _{k\\in \\mathbb {Z}}$ be the bent diagrams for $M$ and $M^{\\prime }$ , respectively.",
"The proof is based on the following key observation: $ (i^{\\prime }_k,j^{\\prime }_k) ={\\left\\lbrace \\begin{array}{ll}(i_k, j_k+1) & \\text{ if } k\\le m \\\\(i_k-1,j_k) & \\text{ if } k> m.\\end{array}\\right.",
"}$ We argue each of the above cases in turn.",
"Suppose that (a) holds.",
"If $k\\in O_r$ and $k>m$ , then $i^{\\prime }_k+j^{\\prime }_k=r-1$ .",
"For all other $k$ , we have $i^{\\prime }_k+j_k \\ge r$ .",
"Suppose that (b) holds.",
"Hence $(i_{m+1},j_{m+1}) = (i_m+1,j_m)$ and hence $i^{\\prime }_{m+1}+j^{\\prime }_{m+1} = i_m+j_m=r$ .",
"If $k\\le m$ , then $i^{\\prime }_k+j^{\\prime }_k \\ge r+1$ .",
"If $k>m$ , then $i^{\\prime }_k + j^{\\prime }_k = i_k + j_k-1$ is equal to $r$ if $k\\in O_{r+1}$ and is $>r$ , otherwise.",
"Suppose that (c) holds.",
"If $k\\le m$ , then $i^{\\prime }_k+j^{\\prime }_k \\ge r+1$ .",
"If $k>m$ then $i^{\\prime }_k+j^{\\prime }_k = r$ if $k\\in O_{r+1}$ and is $>r$ otherwise.",
"Suppose that (d) holds.",
"If $k\\le m$ then $i^{\\prime }_k+j^{\\prime }_k= r+1$ if $k\\in O_r$ and is $>r$ otherwise.",
"If $k>m$ then $i^{\\prime }_k+j^{\\prime }_k =r+1$ if $k\\in O_{r+2}$ and is $>r$ otherwise.",
"Let $M\\subset \\mathbb {Z}$ be a Maya diagram and $\\lambda $ the corresponding partition.",
"For $n\\notin M+|\\lambda |-\\ell $ set $M^{(\\lambda )}_n = M\\cup \\lbrace n+\\ell -|\\lambda |\\rbrace $ .",
"Observe that $H^{(\\lambda )}_n = (-1)^\\ell H_{M^{(\\lambda )}_n}$ .",
"We therefore seek the origin of minimal girth for $M^{(\\lambda )}_n$ .",
"Let $r^{(\\lambda )}_n = \\min \\lbrace |M^{(\\lambda )}_n-k|\\colon k\\in \\mathbb {Z}\\rbrace $ be the minimal order for $H^{(\\lambda )}_n$ .",
"We call any such $k$ that realizes this minimum an origin of minimal order.",
"Corollary 5.1 Let $r=r_M$ and $n\\notin M+|\\lambda |-\\ell $ .",
"If $k_{r+1}>k_r$ , then $r^{(\\lambda )}_n ={\\left\\lbrace \\begin{array}{ll}r-1 & \\text{ if } n<k_r+|\\lambda |-\\ell ,\\\\r & \\text{ if } k_r+|\\lambda |-\\ell \\le n < k_{r+1}+|\\lambda |-\\ell \\\\r+1 & \\text{ if } n> k_{r+1}+|\\lambda |-\\ell \\end{array}\\right.",
"}$ An origin of minimal order for each of the above cases is, respectively, $k_r, k_{r+1}, k_r$ .",
"If $k_{r+1}<k_r$ , then $r^{(\\lambda )}_n ={\\left\\lbrace \\begin{array}{ll}r-1 & \\text{ if } n<k_r+|\\lambda |-\\ell ,\\\\r & \\text{ if } n =k_r+|\\lambda |-\\ell \\\\r+1 & \\text{ if } n> k_r+|\\lambda |-\\ell \\end{array}\\right.",
"}$ An origin of minimal order for each of the above case is, respectively, $k_r, k_r+1, k_r$ .",
"Example 5.1 Consider the case of $\\lambda =(2,2,1,1)$ The corresponding Frobenius symbol is $(\\emptyset | 1,2,4,5)$ .",
"The minimal girth is $r=2$ with $k_2=6$ the unique minimal girth origin.",
"The Frobenius symbol of $M-6$ is $(5,2\\mid \\emptyset )$ .",
"Hence, pseudo-Wronskian of smallest order is $ H_{M-6} = \\begin{vmatrix}\\tilde{H}_{5} & \\tilde{H}_{6} \\\\\\tilde{H}_{2} & \\tilde{H}_{3}\\end{vmatrix}$ with $ H_M=\\operatorname{Wr}[H_1,H_2,H_4,H_5] = -2^5\\times 24 H_{M-6}.",
"$ The exceptional Hermite polynomials for this partition are given by $ H^{(\\lambda )}_n=\\operatorname{Wr}[H_1,H_2,H_4,H_5,H_{n-2}],\\qquad n\\in \\lbrace 2,5,8,9,10,\\dots \\rbrace .$ We can replace the above $5\\times 5$ determinant with a pseudo-Wronskian of smaller order.",
"Here $k_3 = 3<6$ and hence we must apply (REF ).",
"An origin of minimal order is 7 if $n=7$ , and 6 otherwise.",
"Explicitly, $H^{(\\lambda )}_2 &= 2^{12}\\times 120 \\; \\tilde{H}_2 \\\\H^{(\\lambda )}_5 &= 2^{12} \\times 72\\;\\tilde{H}_5 \\\\H^{(\\lambda )}_8 &= K_8 \\begin{vmatrix}\\tilde{H}_6 & \\tilde{H}_7\\\\\\tilde{H}_3 & \\tilde{H}_4\\end{vmatrix} \\\\H^{(\\lambda )}_n &=K_n\\begin{vmatrix}\\tilde{H}_{5} & \\tilde{H}_{6} & \\tilde{H}_{7}\\\\\\tilde{H}_{2} & \\tilde{H}_{3} & \\tilde{H}_{4}\\\\{n-8} & {n-8}^{\\prime } & {n-8}^{\\prime \\prime }\\end{vmatrix},\\qquad n>8,$ where $ K_n = -2^9 \\times 24\\times (n-3)(n-4)(n-6)(n-7) $ Example 5.2 Consider the case of $\\lambda =(4,4,1,1)$ The corresponding Frobenius symbol is $(\\emptyset \\mid 7,6,2,1)$ .",
"The minimal girth is $r=3$ with $k_3=3$ the unique minimal girth origin.",
"The Frobenius symbol of $M-3$ is $(2\\mid 4,3)$ .",
"Hence, pseudo-Wronskian of smallest order is $ H_{M-3} = \\begin{vmatrix}\\tilde{H}_{2} & \\tilde{H}_{3} & \\tilde{H}_4 \\\\H_{3} & H^{\\prime }_{3} & H^{\\prime \\prime }_3\\\\H_{4} & H^{\\prime }_{4} & H^{\\prime \\prime }_4\\end{vmatrix}$ with $ H_M=\\operatorname{Wr}[H_1,H_2,H_6,H_7] = 2^5\\times 600\\, H_{M-3}.",
"$ The exceptional Hermite polynomials for this partition are given by $ H^{(\\lambda )}_n=\\operatorname{Wr}[H_1,H_2,H_6,H_7,H_{n-6}],\\qquad n\\in \\lbrace 6,9,10,11,14,15,\\dots \\rbrace .$ Here $k_4 = 8>3$ and hence we must apply (REF ).",
"An origin of minimal order is 8 if $9\\le n\\le 14$ and 3 otherwise.",
"Explicitly, $H^{(\\lambda )}_6 &= 2^{14}\\times 9\\times 7\\times 25 \\; \\operatorname{Wr}[H_3,H_4] \\\\H^{(\\lambda )}_9 &= 2^{11} \\times 9\\times 5\\begin{vmatrix}\\tilde{H}_2 & \\tilde{H}_3 & \\tilde{H}_4\\\\\\tilde{H}_{3} & \\tilde{H}_{4} & \\tilde{H}_{5}\\\\\\tilde{H}_7 & \\tilde{H}_8 & \\tilde{H}_9\\end{vmatrix},\\\\H^{(\\lambda )}_{10} &= 2^{11} \\times 9\\times 5\\begin{vmatrix}\\tilde{H}_2 & \\tilde{H}_3 & \\tilde{H}_4\\\\\\tilde{H}_4 & \\tilde{H}_5 & \\tilde{H}_6\\\\\\tilde{H}_7 & \\tilde{H}_8 & \\tilde{H}_9\\end{vmatrix},\\\\H^{(\\lambda )}_{11} &= -2^{11} \\times 3\\times 25\\begin{vmatrix}\\tilde{H}_3 & \\tilde{H}_4 & \\tilde{H}_5\\\\\\tilde{H}_4 & \\tilde{H}_5 & \\tilde{H}_6\\\\\\tilde{H}_7 & \\tilde{H}_8 & \\tilde{H}_9\\end{vmatrix},\\\\H^{(\\lambda )}_n &=K_n\\begin{vmatrix}\\tilde{H}_{2} & \\tilde{H}_{3} & \\tilde{H}_{4} & \\tilde{H}_5\\\\2 & 2^{\\prime } & ^{\\prime }_2 & ^{\\prime \\prime }_2\\\\4 & 4^{\\prime } & ^{\\prime }_4 & ^{\\prime \\prime }_4\\\\{n-9} & {n-9}^{\\prime } & {n-9}^{\\prime \\prime } & ^{\\prime \\prime }_{n-9}\\end{vmatrix},\\qquad n\\ge 14$ where $ K_n = -2^{10} \\times 45\\times (n-7)(n-8)$"
],
[
"Application to rational solutions of Painleve IV",
"In this section we apply our minimal order results to the description of rational solutions of $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ , the fourth Painlevé equation: $ y^{\\prime \\prime } = \\frac{(y^{\\prime })^2}{2 y} + \\frac{3}{2} y^3 + 4t y^2+ 2(t^2-a)y +\\frac{b}{y},\\quad y = y(t).$ In order to connect $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ to pseudo-Wronskians we need to recall the notion of a factorization chain, and to describe the class of rational solvable extensions of the harmonic oscillator.",
"A factorization chain is a sequence of Schrödinger operators $ L_i = -D_x^2 + U_i ,\\quad D_x= \\frac{d}{dx},\\; U_i=U_i(x)$ that is related by Darboux transformations, $\\begin{aligned}L_i &= (D_x + f_i)(-D_x + f_i)+\\lambda _i, \\quad f_i = f_i(x),\\\\L_{i+1} &= (-D_x + f_i)(D_x + f_i)+\\lambda _i.\\end{aligned}$ It follows that $f_i$ is the solution of the Riccati equations $ f_i^{\\prime } + f_i^2 = U_i - \\lambda _i,\\quad -f_i^{\\prime } +f_i^2 = U_{i+1}- \\lambda _i.$ Equivalently, $L_i\\psi _i = \\lambda _i\\psi _i,\\qquad \\text{where } f_i = \\frac{\\psi _i^{\\prime }}{\\psi _i}.$ It also follows that the potentials of the chain are related by $ U_{i+n} =U_i - 2 \\left( f^{\\prime }_i+ \\cdots + f^{\\prime }_{i+n-1}\\right).$ Eliminating the potentials, we obtain a chain of coupled equations $ (f_i + f_{i+1})^{\\prime } + f_{i+1}^2 - f_i^2 = \\alpha _i,\\quad \\alpha _i =\\lambda _{i} - \\lambda _{i+1}.$ We speak of an $n$ -step cyclic factorization chain if $U_{n+1} = U_1+\\Delta $ for some $n\\in \\mathbb {N}$ and constant $ \\Delta = -(\\alpha _1 + \\cdots + \\alpha _n).$ It is well known [2], [54] that $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ is equivalent to the 3-step cyclic factorization chain $\\begin{aligned}(f_1 + f_2)^{\\prime } + f_2^2 - f_1 ^2 &= \\alpha _1,\\quad f_i= f_i(z),\\; i=1,2,3\\\\(f_2 + f_3)^{\\prime } + f_3^2 - f_2 ^2 &=\\alpha _2 \\\\(f_3 + f_1)^{\\prime } + f_1^2 - f_3 ^2 &=\\alpha _3\\end{aligned}$ The reduction of (REF ) to $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ is via the following relations $f_1 = -W - \\frac{\\Delta }{2} z,\\quad f_{2,3} = \\frac{W}{2} \\pm \\frac{W^{\\prime }-\\alpha _2}{2W},\\quad \\Delta = -(\\alpha _1+\\alpha _2+\\alpha _3),\\; W=W(z),\\\\ \\nonumber W(z) = 2^{-\\frac{1}{2}} \\Delta ^{\\frac{1}{2}}\\, y(t),\\; z=2^{\\frac{1}{2}}\\Delta ^{-\\frac{1}{2}}\\, t,\\qquad a = \\Delta ^{-1}(\\alpha _3-\\alpha _1),\\; b= -2\\Delta ^{-2}\\alpha _2^2.$ Next, we recall some relevant definitions from [25].",
"A rational extension of the harmonic oscillator is a potential of the form $ U(x) = x^2 + \\frac{a(x)}{b(x)},$ where $a(x), b(x)$ are polynomials with $\\deg a< \\deg b$ .",
"We say that the corresponding Schrödinger operator $L = -D^2 + U$ is exactly solvable by polynomials if there exists functions $\\mu (x),\\zeta (x)$ such that for all but finitely many $k\\in {\\mathbb {N}_0}$ there exists a degree $n$ polynomial $y_n(z)$ such that $ \\psi _n = \\mu (x) y_n(\\zeta (x)) $ is a (formal) eigenfunction of $L$ — that is $ -\\psi _n^{\\prime \\prime } + U \\psi _n = \\lambda _n \\psi _n,$ for some constant $\\lambda _n$ .",
"The following result was proved in [25].",
"Theorem 6.1 Every rational extension of the harmonic oscillator that is exactly solvable by polynomials has the form $ U(x) = x^2 - 2 D_x^2 \\log \\operatorname{Wr}[H_{m_1},\\ldots , H_{m_\\ell }]+C,$ where $\\lbrace m_1,\\dots ,m_\\ell \\rbrace $ are positive integers and $C$ is an additive constant.",
"For a Maya diagram $M\\subset \\mathbb {Z}$ , define $U_M = x^2 - 2 D_x^2 \\log H_M + 2 |M_+|- 2 |M_-| .$ Proposition 6.1 Let $M\\subset \\mathbb {Z}$ and $M^{\\prime }=M+k,\\; k\\in \\mathbb {Z}$ be equivalent Maya diagrams.",
"Then $U_{M^{\\prime }} = U_M+2k,\\quad k\\in \\mathbb {Z}.$ By Theorem REF , $ D_x^2 \\log H_M = D_x^2 \\log H_{M^{\\prime }}$ To establish the value of the shift in (REF ), it suffices to consider the case of $k=1$ .",
"The general case then follows by induction.",
"Now there are two subcases.",
"If $-1\\in M$ then $M^{\\prime }_- = M_-$ and $M^{\\prime }_+ = M_++1$ .",
"If $-1\\notin M$ , then $M^{\\prime }_- = M_--1$ and $M^{\\prime }_+ = M_+$ .",
"Therefore, in both cases, $ M^{\\prime }_+ - M^{\\prime }_- = M_+ - M_- + 1.$ Thus, the set of rational extensions of the harmonic oscillator modulo additive constants is in bijective correspondence with the set of unlabelled Maya diagrams.",
"Definition 6.1 We define a Maya diagram chain to be a sequence of Maya diagrams $M_1, M_2,\\ldots , M_\\ell $ such that there exist $m_1,\\ldots , m_\\ell \\in \\mathbb {Z}$ satisfying $ M_{i+1} ={\\left\\lbrace \\begin{array}{ll}M_{i} \\cup \\lbrace m_i \\rbrace & \\text{ if } m_i \\notin M_i,\\\\M_{i} \\setminus \\lbrace m_i \\rbrace & \\text{ if } m_i \\in M_i,\\end{array}\\right.",
"},\\quad i=1,\\ldots , \\ell -1.$ The following are proved in [27].",
"Also see [8] and the references therein.",
"Proposition 6.2 Let $M_i,\\; i=1,\\ldots , \\ell $ be a Maya diagram chain.",
"Then, $L_i = -D^2 + U_{M_i},\\; i=1,\\ldots , \\ell $ is a factorization chain.",
"Conversely, every factorization chain of rational extensions of the harmonic oscillator that are solvable by polynomials arises in precisely this fashion.",
"Theorem 6.2 Every rational solution of $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ corresponds to a 3-cyclic chain of rational extensions of the harmonic oscillator.",
"In light of the above, we wish to classify 3-cyclic chains of Maya diagrams $M_1, M_2, M_3, M_4$ such that $M_4 = M_1+k$ for some $k\\in \\mathbb {Z}$ .",
"To accomplish this, we define the following classes of Maya diagrams: $\\operatorname{GH}(m,\\ell ) = \\mathbb {Z}_{-} \\cup \\lbrace j\\in \\mathbb {Z}\\colon m\\le j\\le m+\\ell -1 \\rbrace ,\\quad m,\\ell \\in {\\mathbb {N}_0},\\\\\\operatorname{O}(\\ell _1,\\ell _2) = \\mathbb {Z}_{-} \\cup \\lbrace 3j+1 \\colon 0\\le j < \\ell _1 \\rbrace \\cup \\lbrace 3j+2 \\colon 0 \\le j< \\ell _2 \\rbrace ,\\quad \\ell _1, \\ell _2 \\in {\\mathbb {N}_0},$ where $\\mathbb {Z}_-$ is the set of negative integers.",
"For example, $\\operatorname{GH}(2,5)_+ = \\lbrace 2,3,4,5,6 \\rbrace ,\\qquad \\operatorname{O}(2,5)_+ = \\lbrace 1,2,4,5,8,11,14\\rbrace .$ We will call the former a Maya diagram of GH-type (Generalized Hermite), and the latter, a Maya diagram of O-type (Okamoto).",
"Before proceeding, we note the following degeneracies of this notation: $ \\operatorname{GH}(m,0) = \\operatorname{O}(0,0) = \\mathbb {Z}_-,\\quad \\operatorname{GH}(0,\\ell ) = \\mathbb {Z}_- + \\ell .$ The following results are proved in [27].",
"Proposition 6.3 Suppose that $M_i\\subset \\mathbb {Z},\\; i=1,\\ldots , 4$ is a chain of Maya diagrams such that $M_4=M_1+k$ for some $k\\in \\mathbb {Z}$ .",
"Then, up to translation, $M_1$ is either a Maya diagram of GH-type or of O-type.",
"In the first case, $M_1 = \\operatorname{GH}(m,\\ell ),\\quad M_4 = (M_1\\setminus m) \\cup \\lbrace 0, m+\\ell \\rbrace ,\\quad m,\\ell \\in {\\mathbb {N}_0}.$ In the second case, $M_1 = \\operatorname{O}(\\ell _1,\\ell _2),\\quad M_4 = M_1 \\cup \\lbrace 0, 3\\ell _1+1,3\\ell _2+2 \\rbrace ,\\quad \\ell _1,\\ell _2\\in {\\mathbb {N}_0}.$ Proposition 6.4 The rational solutions of $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ that arise from 3-cyclic chains of the harmonic oscillator fall into one of two classes.",
"The rational solutions of GH-type take the form: $y &= D_t \\left[\\log \\frac{H_{\\operatorname{GH}(m,\\ell )}}{H_{\\operatorname{GH}(m,\\ell +1)}}\\right]_{x=t},&& a =-(1+m+2\\ell ),\\; &&b = -2 m^2,\\\\y &= D_t \\left[\\log \\frac{H_{\\operatorname{GH}(m,\\ell )}}{H_{\\operatorname{GH}(m-1,\\ell )}}\\right]_{x=t},&& a=2m+\\ell -1,\\; &&b=-2\\ell ^2,\\; m>0,\\\\y &= D_t \\left[\\log \\frac{H_{\\operatorname{GH}(m,\\ell )}}{H_{\\operatorname{GH}(m+1,\\ell -1)}}\\right]_{x=t}-2t,&& a=\\ell -m-1,\\; && b = -2(m+\\ell )^2,\\; \\ell >0.$ The rational solutions of O-type take the form: $y &= -\\frac{2}{3}t+D_t \\left[\\log \\frac{H_{\\operatorname{O}(\\ell _1,\\ell _2)}}{H_{\\operatorname{O}(\\ell _1-1,\\ell _2-1)}}\\right]_{x=\\frac{t}{\\sqrt{3}}},&& a= \\ell _1+\\ell _2,\\; && b = -\\frac{2}{9}(1-3\\ell _1+3\\ell _2)^2 \\\\y &= -\\frac{2}{3}t+D_t\\left[ \\log \\frac{ H_{\\operatorname{O}(\\ell _1,\\ell _2)}}{ H_{\\operatorname{O}(\\ell _1+1,\\ell _2)}}\\right]_{x=\\frac{t}{\\sqrt{3}}},&& a= -1-2\\ell _1+\\ell _2,\\; && b = -\\frac{2}{9} (2+3\\ell _2)^2 \\\\y &= -\\frac{2}{3}t+D_t\\left[ \\log \\frac{ H_{\\operatorname{O}(\\ell _1,\\ell _2)}}{ H_{\\operatorname{O}(\\ell _1,\\ell _2+1)}}\\right]_{x=\\frac{t}{\\sqrt{3}}},&& a= -2-2\\ell _2+\\ell _1,\\;&& b = -\\frac{2}{9}(1+3\\ell _1)^2$ Since the rational solutions of $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ are linear combinations of log-derivatives of Hermite Wronskians, it makes sense to apply the shortest chain results of Section to obtain more economical descriptions of these rational solutions.",
"Proposition 6.5 If $m,\\ell >0$ , then the minimal order of a GH-type Maya diagram $M=\\operatorname{GH}(m,\\ell )$ is $\\min \\lbrace m,\\ell \\rbrace $ .",
"If $\\ell \\le m$ , then the minimal order determinant is just the usual Wronskian $ H_M = \\operatorname{Wr}[H_{m},\\ldots , H_{m+\\ell -1}].$ If $\\ell > m$ then the minimal-order determinant is the pseudo-Wronskian $ H_{M-m-\\ell } = \\begin{vmatrix}\\tilde{H}_{\\ell }&\\ldots & \\tilde{H}_{\\ell +m-1} \\\\\\vdots & \\ddots & \\vdots \\\\\\tilde{H}_{\\ell +m-1}& \\ldots & \\tilde{H}_{\\ell +2m-2}\\end{vmatrix}$ Example 6.1 For the rational solutions of $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ corresponding to the generalized Hermite class, take $m=2,\\ell =4$ and set $M_0&=\\operatorname{GH}(2,4)= \\mathbb {Z}_-\\cup \\lbrace 2,3,4,5\\rbrace \\\\M_1 &= \\operatorname{GH}(2,5) = \\mathbb {Z}_- \\cup \\lbrace 2,3,4,5,6\\rbrace \\\\M_2 &= \\operatorname{GH}(1,4) = \\mathbb {Z}_- \\cup \\lbrace 1,2,3,4\\rbrace \\\\M_3 &= \\operatorname{GH}(3,3) = \\mathbb {Z}_- \\cup \\lbrace 3,4,5 \\rbrace .$ The corresponding Wronskians are determinants of order $4,5,4,3$ , respectively.",
"The more economical description is in term of pseudo-Wronskian determinants of order $2,2,1,3$ , respectively.",
"By a direct calculation, $H_{M_0-6} &= \\begin{vmatrix}\\tilde{H}_5 & \\tilde{H}_6\\\\\\tilde{H}_4 & \\tilde{H}_5\\end{vmatrix} = -32(16x^8+64x^6+120x^4+45)\\\\H_{M_1-7} &= \\begin{vmatrix}\\tilde{H}_6 & \\tilde{H}_7\\\\\\tilde{H}_5 & \\tilde{H}_6\\end{vmatrix} = -64(32 x^{10}+240 x^8 + 720 x^6 + 600 x^4 + 450 x^2- 225)\\\\H_{M_2-5} &= \\tilde{H}_5 = 4 (4x^4+12x^2+3)\\\\H_{M_3-6} &= \\begin{vmatrix}\\tilde{H}_5 & \\tilde{H}_6 & \\tilde{H}_7\\\\\\tilde{H}_4 & \\tilde{H}_5 & \\tilde{H}_6\\\\\\tilde{H}_3 & \\tilde{H}_4 & \\tilde{H}_5\\end{vmatrix} = -512 x ( 16 x^8+72 x^4 - 135)$ Setting $x=t$ , the corresponding rational solutions (REF ) () () are $y_1 &= \\frac{32 \\left(4 t^7+12 t^5+15 t^3\\right)}{16 t^8+64t^6+120 t^4+45}-\\frac{20 \\left(16 t^9+96 t^7+216 t^5+120t^3+45 t\\right)}{32 t^{10}+240t^8+720 t^6+600 t^4+450 t^2-225} ,&&\\quad (a,b) = (-11,-8),\\\\y_2 &=- \\frac{32 \\left(4 t^7+12 t^5+15 t^3\\right)}{16 t^8+64t^6+120 t^4+45}-\\frac{8 \\left(2 t^3+3 t\\right)}{4 t^4+12t^2+3} ,&&\\quad (a,b) = (7,-32),\\\\y_3&=-\\frac{1}{t}-\\frac{32 \\left(4 t^7+9 t^3\\right)}{16 t^8+72t^4-135}+\\frac{32 \\left(4 t^7+12 t^5+15 t^3\\right)}{16 t^8+64t^6+120 t^4+45}-2 t,&&\\quad (a,b) = (1,-72).$ Proposition 6.6 The minimal order of an O-type Maya diagram $M=\\operatorname{O}(\\ell _1,\\ell _2),\\; \\ell _1,\\ell _2\\in \\mathbb {N}$ , is $\\max \\lbrace \\ell _1,\\ell _2\\rbrace $ .",
"If $\\ell _1\\le \\ell _2$ , then the minimal-order pseudo-Wronskian is $ H_{M-3\\ell _1} = \\pm \\begin{vmatrix}\\tilde{H}_{2} & \\tilde{H}_{3} & \\ldots & \\tilde{H}_{\\ell _2+1} \\\\\\tilde{H}_{5} & \\tilde{H}_{6} & \\ldots & \\tilde{H}_{\\ell _2+4} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\tilde{H}_{3\\ell _1-1} & \\tilde{H}_{3\\ell _1} & \\ldots & \\tilde{H}_{3\\ell _1+\\ell _2-2}\\\\H_2 & H_2^{\\prime } & \\ldots & H_2^{(\\ell _2-1)} \\\\H_5 & H_5^{\\prime } & \\ldots & H_5^{(\\ell _2-1)} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\H_{3\\ell _2-3\\ell _1-1} & H_{3\\ell _2-3\\ell _1-1}^{\\prime } & \\ldots &H_{3\\ell _2-3\\ell _1-1}^{(\\ell _2-1)}\\end{vmatrix}$ If $\\ell _1> \\ell _2$ , then the minimal-order pseudo-Wronskian is $ H_{M-3\\ell _2} = \\pm \\begin{vmatrix} \\tilde{H}_{2} & \\tilde{H}_{3} & \\ldots & \\tilde{H}_{\\ell _1+1} \\\\\\tilde{H}_{5} & \\tilde{H}_{6} & \\ldots & \\tilde{H}_{\\ell _1+4} \\\\ \\vdots & \\vdots &\\ddots & \\vdots \\\\ \\tilde{H}_{3\\ell _2-1} & \\tilde{H}_{3\\ell _2} & \\ldots &\\tilde{H}_{3\\ell _2+\\ell _1-2}\\\\H_1 & H_1^{\\prime } & \\ldots & H_1^{(\\ell _2-1)} \\\\H_4 & H_4^{\\prime } & \\ldots & H_4^{(\\ell _2-1)} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\H_{3\\ell _1-3\\ell _2-2} & H_{3\\ell _1-3\\ell _2-2}^{\\prime } & \\ldots &H_{3\\ell _1-3\\ell _2-2}^{(\\ell _2-1)}\\end{vmatrix}$ Suppose that $\\ell _1\\le \\ell _2$ , so that $ M_+ = \\lbrace 1,2, 4,5, \\ldots , 3\\ell _1-2, 3\\ell _1-1\\rbrace \\cup \\lbrace 3\\ell _1+2, 3\\ell _1+5,\\ldots , 3\\ell _2-1 \\rbrace .$ The inside corners are located at $0,3,6,\\ldots , 3\\ell _1$ and then $3\\ell _1+3,3 \\ell _1+6,\\ldots , 3\\ell _2$ .",
"The corresponding girths initially decrease: $\\ell _1+\\ell _2,+\\ell _2-1, \\ldots , \\ell _2+1, \\ell _2$ and increase by $+1$ thereafter.",
"Suppose that $\\ell _1> \\ell _2$ , so that $ M_+ = \\lbrace 1,2, 4,5, \\ldots , 3\\ell _2-2, 3\\ell _2-1\\rbrace \\cup \\lbrace 3\\ell _2+1, 3\\ell _2+5,\\ldots , 3\\ell _1-2 \\rbrace .$ The inside corners are $0,3,6,\\ldots , 3\\ell _2$ and then $3\\ell _2+2,3 \\ell _2+5,\\ldots , 3\\ell _1-1$ .",
"The corresponding girths initially decrease: $\\ell _1+\\ell _2,\\ell _1+\\ell _2-1, \\ldots , \\ell _1+1, \\ell _1$ and increase by $+1$ thereafter.",
"Figure REF contains a graphical illustration for the minimal order pseudo-Wronskians described by Propositions REF and REF for the Generalized Hermite and Okamoto polynomials, respectively.",
"We see thus that in the case of generalized Hermite polynomials, the minimal order occurs always for the partition itself or its conjugate partition, and therefore it is always a true Wronskian.",
"In the example of the figure for $GH(3,5)$ , the minimal order according to Proposition REF must be $\\min (3,5)=3$ : $ GH(3,5)=\\operatorname{Wr}(H_3,H_4,H_5,H_6,H_7)=\\frac{1}{18432}\\operatorname{Wr}(\\tilde{H}_5,\\tilde{H}_6,\\tilde{H}_7)$ On the contrary, for Okamoto polynomials, the minimal order is generically a pseudo-Wronskian.",
"In the example of the figure, for $O(3,5)$ that has order 8, the equivalent minimal order pseudo-Wronskian has order $\\max (3,5)=5$ and corresponds to the Durfee symbol $[6,4,2|4,2]_{3\\times 2}$ .",
"$ O(3,5)= \\operatorname{Wr}(H_1,H_2,H_4,H_5,H_7,H_8,H_{11},H_{14} ) \\propto {\\rm e}^{-3 x^2} \\operatorname{Wr}({\\rm e}^{x^2} \\tilde{H}_8, {\\rm e}^{x^2} \\tilde{H}_5, {\\rm e}^{x^2} \\tilde{H}_2, H_2,H_5 ) $ Figure: Partitions and minimal order pseudo-Wronskians for two generalized Hermite and Okamoto polynomials.",
"Minimal order representations for these polynomials are given by Durfee symbols [6,4,2|4,2] 3×2 [6,4,2|4,2]_{3\\times 2} for O(3,5)O(3,5) and [5,5,5|∅] 3×0 [5,5,5|\\emptyset ]_{3\\times 0} for GH(3,5)GH(3,5).Example 6.2 For the rational solutions of $\\mathrm {P}_{\\mathrm {IV}}\\,\\,$ corresponding to the Okamoto class, take $\\ell _1=1,\\ell _2=2$ and set $M_0 &= \\operatorname{O}(1,2) = \\mathbb {Z}_- \\cup \\lbrace 1,2,5\\rbrace \\\\M_1 &= \\operatorname{O}(0,1) = \\mathbb {Z}_- \\cup \\lbrace 2\\rbrace \\\\M_2 &= \\operatorname{O}(2,2) = \\mathbb {Z}_- \\cup \\lbrace 1,2,4,5\\rbrace \\\\M_3 &= \\operatorname{O}(1,3) = \\mathbb {Z}_- \\cup \\lbrace 1,2,5,8\\rbrace .$ The corresponding Wronskian determinants have order $3,1,4,4$ , respectively.",
"The minimal order description is in term of pseudo-Wronskian determinants of order $2,1,2,3$ , respectively.",
"By a direct calculation, $H_{M_0-3} &=\\begin{vmatrix} \\tilde{H}_2 & \\tilde{H}_2 \\\\ H_2 & H_2^{\\prime }\\end{vmatrix} = -8x (4x^4-5)\\\\H_{M_1} &= 2 = 2(2x^2-1)\\\\H_{M_2-6} &=\\begin{vmatrix} \\tilde{H}_5 & \\tilde{H}_6 \\\\ \\tilde{H}_2 & \\tilde{H}_3\\end{vmatrix} = -48(8x^6+20x^4+10x^2+5)\\\\H_{M_3-3} &=\\begin{vmatrix} \\tilde{H}_2 & \\tilde{H}_3 & \\tilde{H}_4 \\\\ H_2 & H_2^{\\prime } & H_2^{\\prime \\prime }\\\\ H_5 & H_5^{\\prime } &H_5^{\\prime \\prime }\\end{vmatrix} = 192(32x^{10}- 80x^8 - 80 x^6 + 200 x^4 - 150 x^2 - 25)$ Setting $x=t/\\sqrt{3}$ , the corresponding rational solutions (REF ) () () are $ y_1& = -\\frac{2 t}{3}+\\frac{16 t^3}{4t^4-45}+\\frac{1}{t}-\\frac{4 t}{2 t^2-3} ,& & \\quad (a,b) = \\left(3,-\\frac{32}{9}\\right),\\\\y_2&=-\\frac{2 t}{3}+\\frac{16 t^3}{4 t^4-45}+\\frac{1}{t}-\\frac{12\\left(4 t^5+20 t^3+15 t\\right)}{8 t^6+60 t^4+90 t^2+135},&&\\quad (a,b) = \\left(-1,-\\frac{128}{9}\\right),\\\\y_3&=-\\frac{2 t}{3}+\\frac{16 t^3}{4 t^4-45}+\\frac{1}{t}-\\frac{20\\left(16 t^9-96 t^7-216 t^5+1080 t^3-1215 t\\right)}{32t^{10}-240 t^8-720 t^6+5400 t^4-12150 t^2-6075},&&\\quad (a,b) = \\left(-5,-\\frac{32}{9}\\right).$"
],
[
"Acknowledgements",
"The research of DGU has been supported in part by Spanish MINECO-FEDER Grant MTM2015-65888-C4-3, the ICMAT-Severo Ochoa project SEV-2015-0554 and the BBVA Foundation Grant for researchers and cultural creators.",
"The research of RM was supported in part by NSERC grant RGPIN-228057-2009.",
"DGU and RM would like to thank Université de Lorraine for their hospitality during their visit in the summer of 2015 where many of the results in this paper where first obtained."
]
] | 1612.05514 |
[
[
"Well-posedness of Hamilton-Jacobi equations with Caputo's\n time-fractional derivative"
],
[
"Abstract A Hamilton-Jacobi equation with Caputo's time-fractional derivative of order less than one is considered.",
"The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions.",
"For this purpose comparison principle as well as Perron's method is established.",
"Stability with respect to the order of derivative as well as the standard one is studied.",
"Regularity of a solution is also discussed.",
"Our results in particular apply to a linear transport equation with time-fractional derivatives with variable coefficients."
],
[
"Introduction",
"Let $\\alpha \\in (0,1]$ and $0<T<\\infty $ be given constants.",
"We consider the initial-value problem for the Hamilton-Jacobi equation of the form $\\partial _{t}^{\\alpha }u+H(t,x,u,Du)=0\\quad \\text{in $(0,T]\\times \\mathbb {T}^{d}=:Q_T$}$ and $u|_{t=0}=u_0\\quad \\text{in $\\mathbb {T}^{d}$}.$ Here $\\mathbb {T}^{d}:=\\mathbb {R}^{d}/\\mathbb {Z}^{d}$ is the $d$ -dimensional torus, $u:\\overline{Q_T}\\rightarrow \\mathbb {R}$ is an unknown function and $H:\\overline{Q_T}\\times \\mathbb {R}\\times \\mathbb {R}^{d}\\rightarrow \\mathbb {R}$ is a given function called a Hamiltonian.",
"Moreover, $Du$ denotes the spatial gradient, i.e., $Du=(\\partial u/\\partial x_1,\\cdots ,\\partial u/\\partial x_d)$ and $\\partial _{t}^{\\alpha }u$ denotes Caputo's (time-)fractional derivative which is defined by $(\\partial _{t}^{\\alpha }f)(t):={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{1}{\\Gamma (1-\\alpha )}\\int _{0}^{t}\\frac{f^{\\prime }(s)}{(t-s)^{\\alpha }}ds\\quad &\\text{for $\\alpha \\in (0,1)$,}\\\\f^{\\prime }(t)\\quad &\\text{for $\\alpha =1$,}\\end{array}\\right.",
"}$ where $\\Gamma (\\cdot )$ is the usual gamma function.",
"Throughout this paper, a function $v$ on $\\mathbb {T}^{d}$ is regarded as a function defined on $\\mathbb {R}^{d}$ with $\\mathbb {Z}^{d}$ -periodically, i.e., $v(x+z)=v(x)$ for all $x\\in \\mathbb {R}^{d}$ and $z\\in \\mathbb {Z}^{d}$ .",
"Although some part of our arguments can be easily extended to other boundary conditions we now restrict ourselves only under periodic boundary conditions.",
"The goal of this paper is to find a proper notion of viscosity solutions so that (REF )-(REF ) is well-posed.",
"More specifically, we establish unique existence, stability and some regularity results of a viscosity solution for (REF )-(REF ).",
"Here we will consider only for $\\alpha \\in (0,1)$ since the case of $\\alpha =1$ has been well studied.",
"All results excepts for Section and Section will be established under the following assumptions: (A1) $H:\\overline{Q_T}\\times \\mathbb {R}\\times \\mathbb {R}^{d}\\rightarrow \\mathbb {R}$ is continuous, (A2) there is a modulus $\\omega :[0,\\infty )\\rightarrow [0,\\infty )$ such that $|H(t,x,r,p)-H(t,y,r,p)|\\le \\omega (|x-y|(1+|p|))$ for all $(t,x,y,r,p)\\in [0,T]\\times \\mathbb {R}^{d}\\times \\mathbb {R}^d\\times \\mathbb {R}\\times \\mathbb {R}^{d}$ , (A3) $r\\mapsto H(t,x,r,p)$ is nondecreasing for all $(t,x,p)\\in [0,T]\\times \\mathbb {T}^d\\times \\mathbb {R}^{d}$ , (A4) $u_0:\\mathbb {T}^{d}\\rightarrow \\mathbb {R}$ is continuous.",
"We emphasize that these assumptions are fairly standard for $\\alpha =1$ .",
"Of course there might be several generalizations but we do not touch them.",
"Note that we do not assume coercivity, i.e., $\\liminf _{r\\rightarrow \\infty }\\lbrace H(t,x,r,p) \\mid (t,x,r)\\in Q_T\\times \\mathbb {R},|p|\\ge r\\rbrace =+\\infty .$ Hence our results apply to a transport equation $\\partial _t^{\\alpha }u+b\\cdot Du=0$ for $b=b(t,x):\\overline{Q_T}\\rightarrow \\mathbb {R}^d$ .",
"Since a notion of viscosity solutions was introduced by Crandall and Lions [8], its theory has developed rapidly and by now there is a large number of literature.",
"The reader is referred to [1], [4] and [25] for basic theory and to [7] and [13] for more advanced theory.",
"The theory of viscosity solutions had been applied initially to local partial differential equations (pdes for short) and soon has been extended by Soner [40] to pdes with space-fractional derivatives which are defined non-locally.",
"See also [5], [2], [6], [17] and references therein.",
"In these papers the authors are commonly interested in Lévy operators, which can be represented (formally) as $g[f](x)=-\\int _{\\mathbb {R}^d}\\left(f(x+z)-f(x)-\\frac{Df(x)\\cdot z}{1+|z|^2}\\right)d\\mu (z),$ where $d\\mu $ is the Lévy measure.",
"An example of Lévy operators is the fractional Laplacian: $(-\\Delta )^{\\alpha }f(x)=C\\int _{\\mathbb {R}^d}\\frac{f(x)-f(y)}{|x-y|^{d+2\\alpha }}dy,$ where $C$ is a constant depending on $d$ and $\\alpha $ .",
"Above works are motivated from applied fields such as physics, engineering and finances.",
"Applicabilities of pdes with time-fractional derivatives has been discussed by many researchers in wide fields as well; [11], [12], [39] and [41] for instance.",
"We here refer to several mathematical works for pdes with Caputo's time-fractional derivative (CTFD for short) in order to motivate our research.",
"Although many definitions of a different kind of fractional derivatives have been suggested, we will not touch them in this paper and instead the reader is referred to [10], [14], [24], [23], [36], [38] and [45].",
"A typical example of pdes with CTFD is $\\partial _t^{\\alpha }u+L(u)=F,$ where $L$ consists of a symmetric uniformly elliptic operator and a transport term and $F=F(t,x)$ is a given function.",
"This can be considered as an equation describing diffusion phenomena in complex media like fractals and then is called anomalous diffusion or singular diffusion.",
"There seem to be several previous works for (REF ) (see [37] and references therein) and Luchko's works have a close relationship with ours.",
"He established a maximum principle for Caputo's fractional derivative in [31] and, based on it, proved a uniqueness of classical solutions for an initial-boundary value problem of (REF ) with the type of $L(u)=-\\operatorname{div}(p(x)Du)+q(x)u$ , where typically $p$ is smooth and uniformly positive with continuous $q$ .",
"In [32] he established an existence of classical solutions for same equations and gave some existence results of a continuous generalized solution as well.",
"His research has been continued in a work by Sakamoto and Yamamoto [37], which is a pioneer work in the theory of weak solutions for (REF ).",
"They defined a weak solution in the sense of distribution for a similar equation as one Luchko considered and established well-posedness in order to consider inverse problems.",
"Researches on this line have been growing rapidly; see, e.g., [30] for multi-term time-fractional derivatives and [33] for (REF ) with semilinear source terms.",
"Several pdes with strong nonlinearity have also been considered.",
"Anomalous diffusion equations are modeled by the continuous-time random walk (CTRW for short) introduced by Montroll and Weiss ([35]).",
"Kolokoltsov and Veretennikova ([26]) extended the notion of CTRW so that its processes can be controlled and then derived (heuristically) Hamilton-Jacobi-Bellman equations with CTFD, fractional Laplacian and some additional term.",
"We note that this does not include second order spatial-derivative.",
"In [27] they also defined mild solutions that belong to $C^{1}([0,T];C^{1}_{\\infty }(\\mathbb {R}^d))$ and proved well-posedness for an initial-value problem of $\\partial _t^{\\alpha }u=-a(-\\Delta )^{\\beta /2}u+H(t,x,Du).$ Here $C^{1}_{\\infty }(\\mathbb {R}^d)$ is a set of $C^1$ functions that decreasing rapidly at infinity and $\\beta \\in (1,2]$ , $a>0$ are given constants and $H$ is a given Lipschitz continuous function.",
"In [3] Allen extended the notion of viscosity solutions to the time-space nonlocal equation with CTFD of the form $\\partial _t^{\\alpha }u-\\sup _i\\inf _j\\left(\\int _{\\mathbb {R}^d}\\frac{u(t,x+y)-u(t,x)}{|y|^{d+2\\sigma }}a^{ij}(t,x,y)dy\\right)=f$ and discussed regularity problems of the solutions.",
"Here $a_{ij}$ is positive, bounded function that is symmetric with respect to the third variable and $f$ is a given function.",
"To the best of our knowledge, this seems to be the only results for pdes with CTFD by a viscosity approach.",
"He defined the solutions based on the idea of viscosity solutions by treating $K_0[f](t)&=\\frac{f(t)-f(0)}{t^\\alpha \\Gamma (1-\\alpha )}+\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _0^t\\frac{f(t)-f(s)}{(t-s)^{\\alpha +1}}ds\\\\&=\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _{-\\infty }^t\\frac{\\tilde{f}(t)-\\tilde{f}(s)}{(t-s)^{\\alpha +1}}ds;$ in [3] this quantity is denoted by $\\partial _t^\\alpha $ .",
"Here $\\tilde{f}:(-\\infty ,T]\\rightarrow \\mathbb {R}$ is an extension of $f:[0,T]\\rightarrow \\mathbb {R}$ by $\\tilde{f}(t)=f(0)$ for $t<0$ .",
"Note that $\\partial _t^\\alpha f=K_0[f]$ by integrating by parts if $f$ is smooth; see Proposition REF .",
"He dealt with CTFD by $K_0$ in the similar way as space-fractional derivatives mentioned above since the form of integration in $K_0$ is very close to fractional Laplacian.",
"However, the well-posedness remained to be unclear (at least for non-experts).",
"The operator $K_0$ is strictly different from the fractional Laplace operator since there are several different properties between their kernels such as symmetry.",
"In addition, the fractional Laplacian (or the Lévy operator) is suitable for boundary value problems, whereas Caputo's derivative is suitable for initial value problems.",
"From these facts, studying CTFD is not a simple adjustment of the space-fractional cases.",
"We explained so far second-order pdes or first-order pdes including fractional Laplacian with CTFD.",
"For first-order pdes with CTFD but without any higher-order terms than one such as (REF ), a formula of solution for (REF ) is given by Mainardi, Mura and Pagnini ([34]) for instance, but for equations with constant coefficients.",
"However, there seems to be no general frameworks.",
"In light of the above situations, we aim to construct the synthetic theory of viscosity solutions so that (fully nonlinear) pdes with CTFD mentioned above can be considered.",
"However, extensions to second order equations expects some technical issues, so we only treat first order equations in this paper as the first step.",
"For second order problems the reader is referred to one of forthcoming papers of the second author.",
"We motivate our definition of viscosity solutions (Definition REF ) by recalling the case of $\\alpha =1$ .",
"Let us suppose that $u$ is a classical subsolution of (REF ), that is, $(\\partial _t^\\alpha u)(t,x)+H(t,x,u(t,x),Du(t,x))\\le 0$ for all $(t,x)\\in Q_T$ and that $\\max _{[0,T]\\times \\mathbb {R}^d}(u-\\phi )=(u-\\phi )(\\hat{t},\\hat{x})$ for a test function $\\phi $ .",
"The classical maximum principle in space implies that $Du=D\\phi $ at $(\\hat{t},\\hat{x})$ .",
"With respect to time, the maximum principle for CTFD ([31]) implies that $\\partial _{t}^{\\alpha }u\\ge \\partial _{t}^{\\alpha }\\phi $ at $(\\hat{t},\\hat{x})$ .",
"Hence, that $u$ is a classical subsolution yields to $(\\partial _{t}^{\\alpha }\\phi )(\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x}))\\le 0.$ In the spirit of the case of $\\alpha =1$ , it is natural to define a weak subsolution of (REF ) by means of (REF ).",
"Let us call it a provisional subsolution in this paper.",
"Similarly, if $u$ is a classical supersolution and we replace the maximum by a minimum in (REF ), then the converse inequality of (REF ) is led.",
"Then let us call $u$ defined by means of the inequality a provisional supersolution of (REF ).",
"Let us call $u$ a provisional solution of (REF ) if it is a both provisional sub- and supersolution of (REF ).",
"The notion of provisional solutions looks easy to deal with but it is technically difficult to establish a comparison principle, so we do not know whether it is a proper notion of solution for (REF ) or not (see Section for some observations).",
"One reason is that the so-called doubling variable method (see, e.g., [13]) does not work and a main problem is, roughly speaking, that $(\\partial _t^{\\alpha }\\phi )(\\hat{t},\\hat{x})$ in (REF ) is not an appropriate substitute of $(\\partial _t^{\\alpha }u)(\\hat{t},\\hat{x})$ .",
"In a proof of comparison principle in the theory of viscosity solutions, we often aim to derive a contradiction by using the doubling variable method under a suitable supposition.",
"For provisional sub/supersolutions, we cannot derive a contradiction because of unnecessary values caused by $\\partial _t^{\\alpha }\\phi $ .",
"This fact makes us realize that it is necessary to find a function that has a closer value to $\\partial _{t}^{\\alpha }u$ at each point.",
"It becomes now clear that it is better to handle the operator $K_0$ considered by Allen than $\\partial _t^\\alpha $ .",
"Strictly speaking, Allen adopted (), but we adopt $K_0[u](t,x)=\\frac{u(t,x)-u(0,x)}{t^\\alpha \\Gamma (1-\\alpha )}+\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _0^t(u(t,x)-u(t-\\tau ,x))\\frac{d\\tau }{\\tau ^{\\alpha +1}}$ for technical reasons, which is derived from (REF ) by changing the variable of integration.",
"Here the integral is interpreted as an improper integral $\\int _{0}^{t}(u(t,x)-u(t-\\tau ,x))\\frac{d\\tau }{\\tau ^{\\alpha +1}}=\\lim _{r\\searrow 0}\\int _{r}^{t}(u(t,x)-u(t-\\tau ,x))\\frac{d\\tau }{\\tau ^{\\alpha +1}}.$ Since the convergence of (improper) integration (REF ) is not trivial, we apply test functions instead of unknown functions near the singular time, i.e., the lower end of interval of integration.",
"This idea is taken in our definition.",
"We note that our solution is the same as Allen's except for handling non-locality in the spatial direction since this idea is similar as for him (or space-fractional derivatives).",
"Let us give strategies of our proofs in this paper.",
"They are basically similar to space-fractional cases with integer-order time derivatives ([5], [2], [6] and [17]) although there are some differences depending on the role of time and space derivatives.",
"We will show that (REF ) exists as a finite number at points such that $u-\\phi $ attains a maximum/minimum (Lemma REF ), where $\\phi $ is a test function.",
"This is an analogy of [5] or [6] and the proof follows similar arguments.",
"This fact enables us to prove the comparison principle.",
"An idea of the proof of comparison principle is the same as usual ones, that is, doubling variable method under contradiction.",
"We note that the space-fractional case has an elliptic flavor while CTFD has an evolutional flavor, so the details of the proof are different from the space-fractional case.",
"Indeed, in the process of doubling variables we subtract the inequality for a viscosity supersolution from the inequality for a viscosity subsolution.",
"For space-fractional cases, we reach a contradiction from time derivative of test functions while all spatial quantities are canceled.",
"On the other hand, in our case the contradiction follows from the term $K_0$ , so we have to keep this term until the end.",
"More precisely, the contradiction is derived from terms having no integration in a difference of $K_0$ for viscosity sub- and supersolutions.",
"Note also that test functions like $\\eta /(T-t)$ or $\\eta t$ seem to be not helpful and even unnecessary in an auxiliary function, where $\\eta $ is a positive constant.",
"A proof of existence of viscosity solutions follows Perron's method, that is, a construction of maximal subsolutions.",
"Since (REF ) is nonlocal, we need some efforts to handle nonlocal terms compared with the case of local equations.",
"For this purpose we will employ the idea used in [17] for example.",
"We will establish stability results under limit operation from two perspectives.",
"One of them is for a family of solutions of (REF ) with a Hamiltonian depending on a parameter.",
"Our statement and proof are almost the same as for $\\alpha =1$ (see, e.g., [4]).",
"The other stability discussed here is the case when time-derivative's orders are regarded as parameters.",
"The latter can be proved under the same idea as the former by defining analogous functions of half-relaxed limits.",
"We will show that viscosity solutions are Lipschitz continuous in space and $\\alpha $ -Hölder continuous in time under some additional assumptions on $H$ and $u_0$ .",
"When the regularity problems for viscosity solutions are discussed, the coercivity condition is often assumed.",
"However, transport equations are not coercive.",
"In view of applications we will derive the above regularity results without the coercivity assumption.",
"Our proofs follow basically ones for $\\alpha =1$ ([1]) but a proof of the temporal regularity may be not standard.",
"We will construct a viscosity solution of (REF )-(REF ) that is $\\alpha $ -Hölder continuous in time by Perron's method for a family of viscosity subsolutions of (REF )-(REF ) that is $\\alpha $ -Hölder continuous in time.",
"In this argument we should be carefully for a dependence of the Hölder constant of viscosity subsolutions.",
"We will show that viscosity solutions are Hölder continuous in time at the initial time, where its Hölder constant depends only on $T$ , $\\alpha $ , $H$ and $u_0$ .",
"By restricting viscosity subsolutions to Hölder continuous functions with such a constant, we will obtain viscosity solution with the desired regularity.",
"Our results are new even for transport equations (REF ) with variable coefficients although a formula of solution for constant coefficients is known only in the one dimensional case (see Section ).",
"For this reason it is worth summarizing here.",
"Theorem 1.1 Let $b:\\overline{Q_T}\\rightarrow \\mathbb {R}^d$ be a continuous function.",
"Assume that there is constants $C_1>0$ and $C_2>0$ such that $|b(t,x)-b(t,y)|\\le C_1|x-y|$ and $|u_0(x)-u_0(y)|\\le C_2|x-y|$ for all $(t,x,y)\\in (0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ .",
"Then there exists at most one viscosity solution $u\\in C(\\overline{Q_T})$ of (REF )-(REF ).",
"Moreover there exists a constant $C_3>0$ such that $|u(t,x)-u(s,y)|\\le C_3(|t-s|^{\\alpha }+|x-y|)$ for all $(t,x,s,y)\\in ([0,T]\\times \\mathbb {R}^d)^2$ .",
"If one does not require (REF ), conditions (REF ) and (REF ) can be weakened so that Hamiltonian $H(t,x,p)=b(t,x)\\cdot p$ satisfies (A1) and (A2).",
"We finally compare with viscosity solutions with weak solutions in the sense of distribution (weak solution for short) and mention several open problems.",
"A weak solution for linear second order problems (REF ) given by Sakamoto and Yamamoto ([37]) was constructed by a Galerkin method.",
"Since approximate equations have no comparison principle, it is difficult to compare two notions under the current circumstances.",
"Of course, the case of $\\alpha =1$ has the same difficulty and, even such simple looking case, there seems to be few literatures ([16], [20], [21] and [22]).",
"In order to overcome such a difficulty, analyses for further regularities of weak solutions in the both senses will be needed.",
"As another direction of researches for weak solutions of pdes with CTFD, we should mention fractional derivative of the form $(D^\\alpha _tf)(t):=\\frac{1}{\\Gamma (1-\\alpha )}\\frac{d}{dt}\\int _0^t\\frac{f(\\tau )-f(0)}{(t-\\tau )^{\\alpha +1}}d\\tau .$ The original definition of Caputo's fractional derivative by himself was actually given as this form and hence this derivative is also called Caputo's fractional derivative.",
"We note that $D^\\alpha _t f=\\partial _t^\\alpha f$ almost everywhere on $[0,T]$ if $f$ is absolutely continuous on $[0,T]$ .",
"See [10] for a brief history of Caputo's fractional derivative and the above relationship between two definitions.",
"There are some works for weak solutions of pdes with (REF ).",
"Zacher ([42]) considered abstract evolutional equations of parabolic type including $D^\\alpha _tu-\\operatorname{div}(ADu)+b\\cdot Du+cu=0$ and, by introducing a notion of a weak solution, he established a unique existence.",
"Here, $A=A(t,x)$ is a symmetric and positive defined matrix-valued function with $L^\\infty $ elements and $b=b(t,x)$ and $c=c(t,x)$ are $L^\\infty $ functions.",
"See [43], [44] and [29] for related works.",
"An analysis of weak solutions for pdes with (REF ) involves the problem of the trace $u(0,\\cdot )$ of $u$ up to $t=0$ since $D_t^\\alpha u$ includes the value $u(0,\\cdot )$ .",
"This needs some regularity up to $t=0$ which forced as to restrict range of $\\alpha $ , say, for example $\\alpha >1/2$ or regularity of some of given functions $A$ , $b$ and $c$ compared with the case $\\alpha =1$ ([28]).",
"We note that such a trace problem was not considered in [42]; moreover, assumptions of [42] seem to be too weak to get necessary regularity.",
"In view of such restrictions, our viscosity solutions might look a better notion of weak solutions since we are able to obtain a continuous (viscosity) solution for every $\\alpha \\in (0,1)$ with no special assumptions on $H$ .",
"However, we cannot compare two notions since it is not guaranteed that our solution $u$ is absolutely continuous in time, so it is not clear whether or not $D_t^\\alpha u=\\partial _t^\\alpha u$ for our solution.",
"Even for this problem, further analyses from both aspects of viscosity solutions and weak solutions are needed.",
"This paper is organized as follows: In Section 2 we give a definition of viscosity solutions after and summarize some facts used in the other sections.",
"In Section 3 we prove a comparison principle and in Section 4 we establish an existence result.",
"In Section 5 we prove two types of stability results and in Section 6 we study regularity problem for (REF ).",
"Finally, in Section 7 we give a definition of provisional solutions as another possible notion of weak solutions and mention the technical difficulty for them."
],
[
"Definition and properties of solutions",
"In this section we assume that Hamiltonian $H$ is merely continuous on $\\overline{Q_T}\\times \\mathbb {R}\\times \\mathbb {R}^{d}$ ."
],
[
"Preliminaries",
"To give a definition of viscosity solutions we first introduce a function space of the type $\\mathcal {C}^{1}([a,b]\\times O):=\\lbrace \\phi \\in C^{1}((a,b]\\times O)\\cap C([a,b]\\times O) \\mid \\text{$\\partial _{t}\\phi (\\cdot ,x)\\in L^{1}(a,b)$ for every $x\\in O$}\\rbrace .$ Here $a,b\\in \\mathbb {R}$ are constants such that $a<b$ , $O$ is a domain in $\\mathbb {R}^d$ , $\\mathbb {T}^d$ and $\\mathbb {R}^{d}\\times \\mathbb {R}^d$ and $L^1(a,b)$ is the space of Lebesgue integrable functions on $(a,b)$ .",
"Note that $u\\in \\mathcal {C}^{1}([a,b]\\times O)$ may not be $C^1$ up to $t=a$ .",
"This space will be used as a space of test functions as well as of classical solutions of (REF )-(REF ).",
"Here we define classical solutions of (REF )-(REF ) as follows: Definition 2.1 (Classical solutions) A function $u\\in \\mathcal {C}^1(\\overline{Q_T})$ is called a classical solution of (REF )-(REF ) if $u(0,\\cdot )=u_0$ on $\\mathbb {T}^d$ and $(\\partial _t^{\\alpha }u)(t,x)+H(t,x,u(t,x),Du(t,x))=0$ for all $(t,x)\\in Q_T$ .",
"Note that $\\partial _t^{\\alpha }\\phi $ is bounded in $(0,T]\\times \\mathbb {R}^d$ if $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ ; see [10].",
"We are tempted to use $C^{1}([a,b]\\times O)$ as a space of classical solutions since the integrability condition for $\\partial _t\\phi (\\cdot ,x)$ is satisfied if $\\phi $ belongs to it: $C^{1}([a,b]\\times O)\\subset \\mathcal {C}^{1}([a,b]\\times O)$ .",
"However, the class $C^{1}([a,b]\\times O)$ is too narrow to define classical solutions since it is necessary to include functions that have a fractional power with respect to time at the initial time such as $t^{\\alpha }$ .",
"That is why we do not assume the differentiability at the initial time.",
"Example 2.2 As an example let us consider a simple ordinary differential equation of the form $\\partial _{t}^{\\alpha }f+f=0\\quad \\text{in $(0,\\infty )$}$ with prescribed data $f(0)=c\\in \\mathbb {R}$ .",
"According to [10] a solution of this equation is given as $f(t)=c E_{\\alpha }(-t^{\\alpha })$ , where $E_{\\alpha }$ is the Mittag-Leffler function defined by $E_{\\alpha }(z):=\\sum _{j=0}^{\\infty }\\frac{z^j}{\\Gamma (j\\alpha +1)}.$ In particular, $E_{1/2}(-\\sqrt{t})=e^t \\operatorname{\\text{erfc}}(\\sqrt{t})$ , where $\\operatorname{\\text{erfc}}$ is the complementary error function defined by $\\operatorname{\\text{erfc}}(z):=\\frac{2}{\\sqrt{\\pi }}\\int _{z}^{\\infty }e^{-t^2}dt.$ The function $f$ is not differentiable at $t=0$ though it is continuous up to $t=0$ ; we leave the verification to the reader.",
"Therefore classical solutions of equations with Caputo's (time-)fractional derivative are not always differentiable at the initial time even if an initial datum is smooth.",
"For a measurable function $f:[0,T]\\rightarrow \\mathbb {R}$ we define functions $J_r[f],K_r[f]:(0,T]\\rightarrow \\mathbb {R}$ by $J_r[f](t):=\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _0^r(f(t)-f(t-\\tau ))\\frac{d\\tau }{\\tau ^{\\alpha +1}}$ and $K_r[f](t):=\\frac{f(t)-f(0)}{t^{\\alpha }\\Gamma (1-\\alpha )}+\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _r^t(f(t)-f(t-\\tau ))\\frac{d\\tau }{\\tau ^{\\alpha +1}}$ with a parameter $r\\in (0,t)$ .",
"For a measurable function $f:[0,T]\\times \\mathbb {R}^\\ell \\rightarrow \\mathbb {R}$ with $\\ell \\ge 1$ we define $J_r[f],K_r[f]:(0,T]\\times \\mathbb {R}^\\ell \\rightarrow \\mathbb {R}$ by $J_r[f](t,x):=J_r[f(\\cdot ,x)](t)$ and $K_r[f](t,x):=K_r[f(\\cdot ,x)](t)$ for $(t,x)\\in (0,T]\\times \\mathbb {R}^\\ell $ .",
"Proposition 2.3 (Integration by parts) Let $f:[a,T]\\rightarrow \\mathbb {R}$ be a function such that $f\\in C^{1}((a,T])\\cap C([a,T])$ and $f^{\\prime }\\in L^{1}(a,T)$ , where $a<T$ .",
"Then $\\frac{1}{\\Gamma (1-\\alpha )}\\int _a^t\\frac{f^{\\prime }(\\tau )}{(t-\\tau )^{\\alpha }}d\\tau =\\frac{f(t)-f(a)}{(t-a)^{\\alpha }\\Gamma (1-\\alpha )}+\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _0^{t-a}(f(t)-f(t-\\tau ))\\frac{d\\tau }{\\tau ^{\\alpha +1}}.$ The left-hand side (multiplied by $\\Gamma (1-\\alpha )$ ) can be calculated as $\\int _a^t\\frac{f^{\\prime }(\\tau )}{(t-\\tau )^{\\alpha }}d\\tau &=\\int _a^t\\frac{\\frac{d}{d\\tau }(f(\\tau )-f(t))}{(t-\\tau )^{\\alpha }}d\\tau \\\\&=\\left[\\frac{f(\\tau )-f(t)}{(t-\\tau )^{\\alpha }}\\right]_a^t-\\alpha \\int _a^t\\frac{f(\\tau )-f(t)}{(t-\\tau )^{\\alpha +1}}d\\tau \\\\&=\\lim _{\\tau \\rightarrow t}\\frac{f(\\tau )-f(t)}{(t-\\tau )^{\\alpha }}+\\frac{f(t)-f(a)}{(t-a)^{\\alpha }}+\\alpha \\int _a^t\\frac{f(t)-f(\\tau )}{(t-\\tau )^{\\alpha +1}}d\\tau .$ Thanks to the smoothness of $f$ , the first term vanishes.",
"By the change of variable $s:=t-\\tau $ we obtain the desired result.",
"Let us share some words for an integral $I[f](t):=\\int _{a}^{b}f(t,\\tau )\\frac{d\\tau }{\\tau ^{\\alpha +1}}$ for constants $a,b\\in \\mathbb {R}$ with $0\\le a<b\\le T$ and a measurable function $f:[0,T]\\times [0,T]\\rightarrow \\mathbb {R}$ .",
"We say that the integral $I[f]$ makes sense if either $I[f^+]$ or $I[f^-]$ is finite (in the sense of Lebesgue integrals) and that $I[f]$ exists if both $I[f^{\\pm }]$ are finite.",
"Here $f^{\\pm }:=\\max \\lbrace \\pm f,0\\rbrace $ .",
"It is necessary to pay attention when $a=0$ .",
"Then we regard $I[f]$ as an improper integral by $I[f](t)=\\lim _{r\\searrow 0}I_r[f]$ , where $I_r[f](t)=\\int _r^bf(t,\\tau )\\frac{d\\tau }{\\tau ^{\\alpha +1}}.$ Thus $I[f]$ exists if $I_r[f^{\\pm }]$ are finite for each $r$ and $\\lim _{r\\searrow 0}I_r[f^{\\pm }]$ exist as a finite number.",
"Note that, if $\\tau \\mapsto |f(t,\\tau )|/\\tau ^{\\alpha +1}$ is integrable on $(0,b)$ , then $I[f]$ exits and it agrees with the Lebesgue integral; this is a direct consequence of the dominated convergence theorem.",
"We abuse above words not only for $J_r$ but also for $K_r$ including a non-integration term.",
"For a set $E\\subset \\mathbb {R}^{\\ell }$ with $\\ell \\ge 1$ , let $USC(E)$ and $LSC(E)$ be sets of real-valued upper and lower semicontinuous functions on $E$ , respectively.",
"Note that semicontinuous functions are measurable.",
"Proposition 2.4 (Properties of $J_r$ and $K_r$ ) Let $f\\in USC([0,T])$ (resp.",
"$LSC([0,T])$ ) and $g\\in C^1((0,T])$ .",
"Then (i) for each $t\\in (0,T]$ , $J_r[g](t)$ exists for all $r\\in (0,t)$ , (ii) for each $t\\in (0,T]$ , $K_r[f](t)$ makes sense and is bounded from below (resp.",
"above) for all $r\\in (0,t)$ , (iii) $K_0[f](\\hat{t})$ makes sense and is bounded from below (resp.",
"above) if $f-g$ attains a maximum (resp.",
"minimum) at $\\hat{t}\\in (0,T]$ over $(0,T]$ , i.e., $\\sup _{(0,T]}(f-g)=(f-g)(\\hat{t})\\quad (\\text{resp.",
"$\\inf _{(0,T]}(f-g)=(f-g)(\\hat{t})$}),$ Moreover for each $j\\ge 0$ let $t_j\\in (0,T]$ , $r_j\\in (0,t_j)$ and $\\alpha _j\\in (0,1)$ be sequences such that $\\lim _{j\\rightarrow \\infty }(t_j,r_j,\\alpha _j)=(\\hat{t},\\hat{r},\\alpha )\\in (0,T]\\times \\mathbb {R}^d\\times [0,\\hat{t})\\times (0,1)$ .",
"Let $J_r^{\\alpha _j}$ denote a function $J_r$ associated with $\\alpha =\\alpha _j$ .",
"Then (iv) $\\lim _{j\\rightarrow \\infty }J_{r_j}^{\\alpha _j}[g](t_j)=J_{\\hat{r}}^{\\alpha }[g](\\hat{t})$ .",
"(i) Fix $t\\in (0,T]$ and $r\\in (0,t)$ arbitrarily.",
"Since $g$ is Lipschitz continuous near $t$ due to the smoothness of $g$ , for some constant $C>0$ $\\int _0^r|g(t)-g(t-\\tau )|\\frac{d\\tau }{\\tau ^{\\alpha +1}}\\le \\int _0^rC\\tau \\frac{d\\tau }{\\tau ^{\\alpha +1}}=\\frac{Cr^{1-\\alpha }}{1-\\alpha }.$ Our assertion follows immediately from this.",
"(ii) Fix $t\\in (0,T]$ and $r\\in (0,t)$ arbitrarily.",
"Assume that $f\\in USC([0,T])$ .",
"Then $f$ attains a maximum and hence $f(t)-\\max _{[0,T]}f\\le f(t)-f(t-\\tau )$ for all $\\tau \\in (r,t)$ .",
"The left-hand side multiplied by $\\tau ^{-\\alpha -1}$ is integrable on $(r,t)$ since we integrate away from $\\tau =0$ .",
"Therefore the negative part $[f(t)-f(t-\\tau )]^{-}/\\tau ^{\\alpha +1}$ is integrable on $(r,t)$ .",
"This implies that $K_r[f](t)$ makes sense and is bounded from below.",
"The similar argument is applied for $f\\in LSC([0,T])$ .",
"The above yields our assertion.",
"(iii) Define $h:=g+(f-g)(\\hat{t})$ and $v(\\tau ):={\\left\\lbrace \\begin{array}{ll}h(\\hat{t})-h(\\hat{t}-\\tau )\\quad &\\text{for $\\tau \\in [0,\\hat{t}/2]$},\\\\f(\\hat{t})-f(\\hat{t}-\\tau )\\quad &\\text{for $\\tau \\in (\\hat{t}/2,\\hat{t}]$.}\\end{array}\\right.",
"}$ Since $f-h$ attains a maximum at $\\hat{t}$ over $(0,T]$ , we see $f\\le h$ on $(0,T]$ .",
"In addition, $(f-h)(\\hat{t})=0$ and thus $h(\\hat{t})-h(\\hat{t}-\\tau )\\le f(\\hat{t})-f(\\hat{t}-\\tau )$ on $(0,\\hat{t})$ .",
"By (i) and a similar argument as the proof of (ii) with $r=\\hat{t}/2$ it turns out that the negative part $v^-(\\tau )/\\tau ^{\\alpha +1}$ in integrable on $(0,\\hat{t})$ , so is $[f(\\hat{t})-f(\\hat{t}-\\tau )]^-/\\tau ^{\\alpha +1}$ since $v(\\tau )\\le f(\\hat{t})-f(\\hat{t}-\\tau )$ on $(0,\\hat{t})$ .",
"This yields our assertion for $f\\in USC([0,T])$ .",
"Another can be proved similarly.",
"(iv) Thanks to the smoothness of $g$ the dominated convergence theorem can be applied and ensures our assertion.",
"More precisely, since $\\inf _{j\\ge 0}(t_j-r_j)>0$ and $g\\in C^1((0,T])$ , there exists a constant $C_1>0$ such that $|g(t_j)-g(t_j-\\tau )|\\le C_1\\tau $ on $(0,r_j)$ .",
"In particular, we may assume that $C_1$ does not depend on $j$ since $\\lim _{j\\rightarrow \\infty }t_j=\\hat{t}>0$ .",
"Thus we have $\\sup _{j\\ge 0}(|g(t_j)-g(t_j-\\tau )|{1}_{(0,r_j)}(\\tau ))\\tau ^{-\\alpha -1}\\le C_1{1}_{(0,\\hat{r})}(\\tau )\\tau ^{-\\alpha }$ for all $\\tau \\in [0,T]$ .",
"Here ${1}_I$ is the indicator function on an interval $I$ , i.e., ${1}_I=1$ in $I$ and 0 elsewhere.",
"The right-hand side is integrable on $[0,T]$ .",
"It remains to check the convergence of $(g(t_j)-g(t_j-\\cdot )){1}_{[0,r_j]}(\\cdot )$ but this is obvious."
],
[
"Definition of solutions",
"We now give our definition of viscosity solutions for (REF ).",
"Definition 2.5 (Viscosity solutions) A function $u\\in USC(\\overline{Q_T})$ (resp.",
"$LSC(\\overline{Q_T})$ ) is called a viscosity subsolution of (REF ) if, for any constants $a,b\\in [0,T]$ with $a<b$ and an open ball $B$ in $\\mathbb {R}^d$ , $J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x}))\\le 0\\quad (\\text{resp.",
"$\\ge 0$})$ whenever $u-\\phi $ attains a maximum (resp.",
"minimum) at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ .",
"If $u\\in C(\\overline{Q_T})$ is both a viscosity sub- and supersolution of (REF ), then we call $u$ a viscosity solution of (REF ).",
"Remark 2.6 (i) For an arbitrary function $u:Q_T\\rightarrow \\mathbb {R}$ an upper semicontinuous envelope $u^*:\\overline{Q_T}\\rightarrow \\mathbb {R}\\cup \\lbrace \\pm \\infty \\rbrace $ and a lower semicontinuous envelope $u_*:\\overline{Q_T}\\rightarrow \\mathbb {R}\\cup \\lbrace \\pm \\infty \\rbrace $ are defined by $u^*(t,x):=\\lim _{\\delta \\searrow 0}\\sup \\lbrace u(s,y) \\mid (s,y)\\in Q_T\\cap \\overline{B_{\\delta }(t,x)}\\rbrace $ and $u_*:=-(-u)^*$ .",
"Here $B_{\\delta }(t,x)$ is an open ball of radius $\\delta $ centered at $(t,x)$ and $\\overline{B_{\\delta }(t,x)}$ is its closure.",
"As for $\\alpha =1$ a viscosity sub- and subsolution of (REF ) can be defined for arbitrary functions $u:Q_T\\rightarrow \\mathbb {R}$ by using $u^*$ (for a subsolution) and $u_*$ (for a supersolution) in Definition REF , where it is further assumed that $u^*<+\\infty $ and $u_*>-\\infty $ on $\\overline{Q_T}$ ; cf.",
"[13].",
"Note that functions $u^*$ and $u_*$ are upper semicontinuous and lower semicontinuous on $\\overline{Q_T}$ , respectively (see, e.g., [4]) so they are measurable.",
"(ii) Although we restrict ourselves for spatially periodic functions, our definition can be easily extended for $(0,T]\\times \\Omega $ , where $\\Omega $ is a domain in $\\mathbb {R}^d$ .",
"In fact, the comparison principle holds for a general bounded domain with necessary modifications.",
"If a viscosity subsolution (resp.",
"supersolution) $u$ of (REF ) satisfies $u(0,\\cdot )\\le u_0$ (resp.",
"$u(0,\\cdot )\\ge u_0$ ) on $\\mathbb {T}^{d}$ , $u$ is called a viscosity subsolution (resp.",
"viscosity supersolution) of (REF )-(REF ).",
"We often suppress the word “viscosity\" unless confusion occurs."
],
[
"Properties and equivalences of solutions",
"Proposition 2.7 (Replacement of test functions) A function $u\\in USC(\\overline{Q_T})$ (resp.",
"$LSC(\\overline{Q_T})$ ) is a subsolution (resp.",
"supersolution) of (REF ) if and only if, for any $a,b\\in [0,T]$ with $a<b$ and an open ball $B$ in $\\mathbb {R}^d$ , (REF ) holds whenever (i) $u-\\phi $ attains a zero maximum (resp.",
"minimum) at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ such that $(u-\\phi )(\\hat{t},\\hat{x})=0$ , or (ii) $u-\\phi $ attains a strict maximum (resp.",
"minimum) at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ , i.e., $&\\max _{[a,b]\\times \\overline{B}}(u-\\phi )=(u-\\phi )(\\hat{t},\\hat{x})>(u-\\phi )(t,x)\\\\(\\text{resp.",
"}&\\min _{[a,b]\\times \\overline{B}}(u-\\phi )=(u-\\phi )(\\hat{t},\\hat{x})<(u-\\phi )(t,x))$ for all $(t,x)\\in [a,b]\\times \\overline{B}$ .",
"(iii) $u-\\phi $ attains a maximum (resp.",
"minimum) at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^1([a,b]\\times \\overline{B})$ .",
"We only prove for a subsolution since a similar argument is applied for a supersolution.",
"It is enough to prove `only if' parts of both assertions since `if' parts are obvious.",
"(i) Set $\\psi :=\\phi +(u-\\phi )(\\hat{t},\\hat{x})$ .",
"Then $u-\\psi $ attains a maximum at $(\\hat{t},\\hat{x})$ over $[a,b]\\times \\overline{B}$ and $(u-\\psi )(\\hat{t},\\hat{x})=0$ .",
"Since $u$ is a subsolution of (REF ), $J_{\\hat{t}-a}[\\psi ](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\psi (\\hat{t},\\hat{x}))\\le 0.$ It is easy to verify from the definition of $J_r[\\phi ]$ that $J_{\\hat{t}-a}[\\psi ](\\hat{t},\\hat{x})=J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})$ .",
"Clearly, $D\\psi (\\hat{t},\\hat{x})=D\\phi (\\hat{t},\\hat{x})$ , so that (REF ) holds.",
"(ii) For $j\\ge 0$ we set $\\phi _j(t,x):=\\phi (t,x)+j^{-1}|t-\\hat{t}|^2+|x-\\hat{x}|^2$ on $(0,T]\\times \\mathbb {R}^d$ .",
"Then $\\phi _j\\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ and $u-\\phi _j$ attains a maximum at $(\\hat{t},\\hat{x})$ over $[a,b]\\times \\overline{B}$ .",
"Since $u$ is a subsolution of (REF ), $J_{\\hat{t}-a}[\\phi _j](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\phi _j(\\hat{t},\\hat{x}))\\le 0.$ By the definition of $\\phi _j$ we have $J_{\\hat{t}-a}[\\phi _j](\\hat{t},\\hat{x})=J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})-\\frac{\\alpha }{j\\Gamma (1-\\alpha )}\\int _0^{\\hat{t}}\\tau ^2\\frac{d\\tau }{\\tau ^{\\alpha +1}}$ The last integral in the right-hand side is clearly finite, so vanishes as $j\\rightarrow \\infty $ .",
"Since $D\\phi _j(\\hat{t},\\hat{x})=D\\phi (\\hat{t},\\hat{x})$ , we reach (REF ).",
"(iii) Choose $\\delta >0$ so that $2\\delta <\\hat{t}-a$ and $\\overline{B_{2\\delta }(\\hat{x})}\\subset B$ .",
"Let $\\xi _1,\\xi _2:[0,T]\\times \\mathbb {R}^d\\rightarrow [0,1]$ be $C^{\\infty }$ functions such that $\\xi _1+\\xi _2=1$ in $[0,T]\\times \\mathbb {R}^d$ , $\\xi _1=1$ on $[\\hat{t}-\\delta ,\\hat{t}]\\times B_{\\delta }(\\hat{x})$ and $\\xi _2=1$ on $([0,T]\\times \\mathbb {R}^d)\\setminus ([\\hat{t}-2\\delta ,\\hat{t}]\\times B_{2\\delta }(\\hat{x}))$ .",
"Set $\\psi :=\\xi _1\\phi +\\xi _2 M+(u-\\phi )(\\hat{t},\\hat{x})$ on $[0,T]\\times \\mathbb {R}^d$ , where $M:=\\max _{\\overline{Q_T}}u+1$ .",
"Then $\\psi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ and $u-\\psi $ attains a zero maximum $(\\hat{t},\\hat{x})$ over $[\\hat{t}-\\delta ,\\hat{t}]\\times \\overline{B}\\subset [a,b]\\times \\overline{B}$ .",
"Since $u$ is a subsolution of (REF ), $J_{\\delta }[\\psi ](\\hat{t},\\hat{x})+K_{\\delta }[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\psi (\\hat{t},\\hat{x}))\\le 0.$ It is easy that $J_{\\delta }[\\psi ](\\hat{t},\\hat{x})=J_{\\delta }[\\phi ](\\hat{t},\\hat{x})$ and $D\\psi (\\hat{t},\\hat{x})=D\\phi (\\hat{t},\\hat{x})$ .",
"Moreover, since $u(\\hat{t},\\hat{x})-u(\\hat{t}-\\tau )\\ge \\phi (\\hat{t},\\hat{x})-\\phi (\\hat{t}-\\tau ,\\hat{x})$ on $[0,\\hat{t}-a]$ , $K_{\\delta }[u](\\hat{t},\\hat{x})\\ge K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})+\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _{\\delta }^{\\hat{t}-a}(\\phi (\\hat{t},\\hat{x})-\\phi (\\hat{t}-\\tau ,\\hat{x}))\\frac{d\\tau }{\\tau ^{\\alpha +1}}.$ Thus we have $J_{\\delta }[\\psi ](\\hat{t},\\hat{x})+K_{\\delta }[u](\\hat{t},\\hat{x})\\ge J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})$ , which is nothing but (REF ).",
"Remark 2.8 By the similar way in the proof of (i) it turns out that, if $u$ is a subsolution (resp.",
"supersolution) of (REF ), then $u-C$ (resp.",
"$u+C$ ) is a subsolution (resp.",
"supersolution) of (REF ) for any positive constant $C>0$ .",
"This is valid even for sub/supersolutions of (REF )-(REF ).",
"Here a proof needs (A3).",
"In what follows we establish an equivalent definition of solution.",
"The similar fact is well known for pdes with space-fractional derivatives and integer-order time derivative, and it is utilized to obtain meaningful observations; see [2], [5], [6] and [17].",
"Even in our case the following fact is effective in obtaining results, especially the comparison theorem.",
"Lemma 2.9 (Equivalence) A function $u\\in USC(\\overline{Q_T})$ (resp.",
"$LSC(\\overline{Q_T})$ ) is a subsolution (resp.",
"supersolution) of (REF ) if and only if $K_0[u](\\hat{t},\\hat{x})$ exists and $K_0[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x}))\\le 0\\quad (\\text{resp.",
"$\\ge 0$})$ whenever $u-\\phi $ attains a maximum (resp.",
"minimum) at $(\\hat{t},\\hat{x})\\in (0,T]\\times \\mathbb {R}^d$ over $[0,T]\\times \\mathbb {R}^d$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ .",
"We only prove for subsolutions since the similar argument is applied for supersolutions.",
"We first prove the `if' part.",
"To do so, let $a,b\\in [0,T]$ with $a<b$ and an open ball $B$ in $\\mathbb {R}^d$ fix arbitrarily.",
"Assume that $u-\\phi $ attains a maximum at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^1((0,T]\\times \\mathbb {R}^d)$ .",
"Define $\\psi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ similarly as in the proof of Proposition REF (iii).",
"Then $u-\\psi $ attains a zero maximum $(\\hat{t},\\hat{x})$ over $[0,T]\\times \\mathbb {R}^d$ .",
"Thus $K_0[u](\\hat{t},\\hat{x})$ exists and $K_0[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\psi (\\hat{t},\\hat{x}))\\le 0.$ The relationship between $u$ and $\\phi $ implies that $u(\\hat{t},\\hat{x})-u(\\hat{t}-\\tau ,\\hat{x})\\ge \\phi (\\hat{t},\\hat{x})-\\phi (\\hat{t}-\\tau ,\\hat{x})$ for all $[0,\\hat{t}-a]$ , which further yields $J_{\\hat{t}-a}[u](\\hat{t},\\hat{x})\\ge J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})$ .",
"Since $D\\psi (\\hat{t},\\hat{x})=D\\phi (\\hat{t},\\hat{x})$ and $K_0[u](\\hat{t},\\hat{x})=J_{\\hat{t}-a}[u](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})$ , the assertion follows immediately.",
"To prove the `only if' part we assume that $u-\\phi $ attains a maximum at $(\\hat{t},\\hat{x})\\in (0,T]\\times \\mathbb {R}^d$ over $[0,T]\\times \\mathbb {R}^d$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ .",
"Set $\\psi :=\\phi +(u-\\phi )(\\hat{t},\\hat{x})$ on $(0,T]\\times \\mathbb {R}^d$ .",
"Let $r>0$ be a parameter such that $\\hat{t}-r>0$ .",
"Then $u-\\psi $ attains a zero maximum at $(\\hat{t},\\hat{x})$ over $[\\hat{t}-r,\\hat{t}]\\times \\overline{B(\\hat{x})}$ for all $r$ , where $B(\\hat{x})$ is an open ball centered at $\\hat{x}$ in $\\mathbb {R}^d$ .",
"Since $u$ is a subsolution of (REF ), $J_r[\\psi ](\\hat{t},\\hat{x})+K_r[u](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\psi (\\hat{t},\\hat{x}))\\le 0$ for all $r$ .",
"From Proposition REF and its proof we know that $J_r[\\psi ](\\hat{t},\\hat{x})$ and $K_r[u^-](\\hat{t},\\hat{x})$ exist for each $r$ and moreover $\\lim _{r\\rightarrow 0}J_r[\\psi ](\\hat{t},\\hat{x})=0$ .",
"Thus it is enough to show that $K_r[u^+]$ exists for each small $r$ and $\\lim _{r\\rightarrow 0}K_r[u^{\\pm }]=K_0[u^{\\pm }]$ exist as a finite number.",
"Indeed, if this is proved, it means that $K_0[u](\\hat{t},\\hat{x})$ exists and (REF ) follows by passing to the limit $r\\rightarrow 0$ in (REF ).",
"Define a function $v_r:[0,T]\\rightarrow \\mathbb {R}$ by $v_r(\\tau )={\\left\\lbrace \\begin{array}{ll}\\psi (\\hat{t},\\hat{x})-\\psi (\\hat{t}-\\tau ,\\hat{x})\\quad &\\text{for $\\tau \\in [0,\\hat{t}-r)\\times \\mathbb {T}^d$,}\\\\u(\\hat{t},\\hat{x})-u(\\hat{t}-\\tau ,\\hat{x})\\quad &\\text{for $\\tau \\in [\\hat{t}-r,\\hat{t}]\\times \\mathbb {T}^d$.}\\end{array}\\right.",
"}$ We rewrite (REF ) as $I[v_r]\\le \\frac{\\Gamma (1-\\alpha )}{\\alpha }\\left(-H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D\\psi (\\hat{t},\\hat{x}))-\\frac{u(\\hat{t},\\hat{x})-u(0,\\hat{x})}{\\hat{t}^{\\alpha }\\Gamma (1-\\alpha )}\\right)=:C,$ where $I[v_r](\\hat{t},\\hat{x})=\\int _0^{\\hat{t}}v_r(\\tau )\\frac{d\\tau }{\\tau ^{\\alpha +1}}.$ From the relationship between $u$ and $\\psi $ we see $\\psi (\\hat{t},\\hat{x})-\\psi (\\hat{t}-\\tau ,\\hat{x})\\le u(\\hat{t},\\hat{x})-u(\\hat{t}-\\tau ,\\hat{x})$ on $[0,\\hat{t}-r]$ .",
"Then it suffices to prove that $I[v_r^+]$ exists for each small $r$ and $\\lim _{r\\rightarrow 0}I[v_r^{\\pm }]=I[v_0^{\\pm }]$ exists as a finite number.",
"By the definition of $v_r$ and (REF ), $v_r^+$ is monotone increasing with respect to $r$ in the sense that $v_{r_1}^+\\le v_{r_2}^+$ on $[0,\\hat{t}]$ if $r_{1}\\ge r_2$ .",
"The monotone convergence theorem implies that$ \\lim _{r\\rightarrow 0}I[v_r^+]=I[v_0^+]$ .",
"It is verified similarly as for $v_r^+$ that $v_r^-$ is monotone decreasing with respect to $r$ in the sense that $v_{r_1}^-\\le v_{r_2}^-$ on $[0,\\hat{t}]$ if $r_1\\le r_2$ .",
"Thus we have from (REF ) $I[v_{r_1}^+](\\hat{t},\\hat{x})\\le I[v_{r_1}^-](\\hat{t},\\hat{x})+C\\le I[v_{r_2}^-](\\hat{t},\\hat{x})+C.$ This implies that $I[v_r^+]$ exists for each small $r$ and $I[v_0^+]$ exists (as a finite number) by passing to the limit $r_1\\rightarrow 0$ .",
"The monotone convergence theorem for $I[v_r^-]$ implies that $\\lim _{r\\rightarrow 0}I[v_r^-]=I[v_0^-]$ .",
"Therefore (REF ) ensures that $I[v_0^-]$ exists (as a finite number).",
"The proof is now complete.",
"Proposition 2.10 (Consistency) Assume that $u\\in \\mathcal {C}^{1}(\\overline{Q_T})$ .",
"Then $u$ is a classical solution of (REF )-(REF ) if and only if $u$ is a viscosity solution of (REF )-(REF ).",
"Assume that $u$ is a viscosity subsolution.",
"We may take $\\phi \\equiv u$ so that $u-\\phi $ attains a maximum at every point in $Q_T$ .",
"Since $u$ is a viscosity subsolution of (REF ), Lemma REF implies that $K_0[u](t,x)+H(t,x,u(t,x),Du(t,x))\\le 0$ for all $(t,x)\\in Q_T$ .",
"Similarly, we have the reverse inequality of (REF ) from an inequality by viscosity supersolution.",
"This shows that $u$ is a classical solution since $K_0[u]=\\partial _t^{\\alpha }u$ by Proposition REF (ii).",
"On the contrary we assume that $u$ is a classical solution and that $u-\\phi $ attains a maximum at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^{1}([0,T]\\times \\mathbb {R}^d)$ , where $a,b\\in (0,T]$ are constants and $B$ is an open ball in $\\mathbb {R}^d$ .",
"Since $(\\partial _t^\\alpha u)(\\hat{t},\\hat{x})=K_0[u](\\hat{t},\\hat{x})\\ge J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u](\\hat{t},\\hat{x})$ , to combine the maximum principle in space implies that $u$ is a viscosity subsolution.",
"It is similar for viscosity supersolutions.",
"Since an initial condition is easily verified, we obtain the conclusion."
],
[
"Comparison Principle",
"Theorem 3.1 (Comparison principle) Assume that (A1)-(A3).",
"Let $u\\in USC(\\overline{Q_T})$ and $v\\in LSC(\\overline{Q_T})$ be a subsolution and a supersolution of (REF ), respectively.",
"If $u(0,\\cdot )\\le v(0,\\cdot )$ on $\\mathbb {T}^{d}$ , then $u\\le v$ on $\\overline{Q_T}$ .",
"We shall prepare one lemma for a proof of Theorem REF ; see [9], [18] and [19] for similar results for $\\alpha =1$ .",
"To do so we invoke a limit inferior/superior inequality of product of constant sequences that one of sequences is allowed to be negative.",
"The statement looks fundamental and the proof is standard but we give for the reader's convenience.",
"Proposition 3.2 Let $\\lbrace f_{\\varepsilon }\\rbrace _{\\varepsilon >0}$ and $\\lbrace g_{\\varepsilon }\\rbrace _{\\varepsilon >0}$ be constant sequences.",
"Assume that $g_{\\varepsilon }$ is nonnegative, $\\liminf _{\\varepsilon \\rightarrow 0}f_{\\varepsilon }\\ge f_0$ (resp.",
"$\\limsup _{\\varepsilon \\rightarrow 0}f_{\\varepsilon }\\le f_0$ ) and $\\lim _{\\varepsilon \\rightarrow 0}g_{\\varepsilon }=g_0$ for some constants $f_0$ and $g_0$ .",
"Then $\\liminf _{\\varepsilon \\rightarrow 0}f_{\\varepsilon }g_{\\varepsilon }\\ge f_0g_0,\\quad (\\text{resp.",
"$\\limsup _{\\varepsilon \\rightarrow 0}f_{\\varepsilon }g_{\\varepsilon }\\le f_0g_0$}.",
")$ It is enough to prove the case when $\\liminf _{\\varepsilon \\rightarrow 0}f_{\\varepsilon }\\ge f_0$ since another case is proved by changing a sign of $f_{\\varepsilon }$ .",
"Then for any $\\delta >0$ there exists $\\varepsilon _{\\delta }>0$ such that $f_{\\varepsilon }\\ge f_0-\\delta $ for all $\\varepsilon <\\varepsilon _{\\delta }$ .",
"It is fundamental that $\\liminf _{\\varepsilon \\rightarrow 0}h_{\\varepsilon }g_{\\varepsilon }\\ge \\liminf _{\\varepsilon \\rightarrow 0}h_{\\varepsilon }\\liminf _{\\varepsilon \\rightarrow 0}g_{\\varepsilon }$ for nonnegative constants $h_{\\varepsilon },g_{\\varepsilon }$ .",
"Applying this fact as $h_{\\varepsilon }:=f_{\\varepsilon }-f_0-\\delta (\\ge 0)$ we see $\\liminf _{\\varepsilon \\rightarrow 0}f_{\\varepsilon }g_{\\varepsilon }&\\ge \\liminf _{\\varepsilon \\rightarrow 0}(f_{\\varepsilon }-f_0-\\delta )g_{\\varepsilon }+\\liminf _{\\varepsilon \\rightarrow 0}(f_0+\\delta )g_{\\varepsilon }\\\\&\\ge -\\delta g_0+(f_0+\\delta )g_0=f_0g_0.$ Lemma 3.3 Assume (A1).",
"Let $u\\in USC(\\overline{Q_T})$ and $v\\in LSC(\\overline{Q_T})$ be a subsolution and a supersolution of (REF ), respectively.",
"Assume that $(t,x,y)\\mapsto u(t,x)-v(t,y)-\\phi (t,x,y)$ attains a maximum at $(\\hat{t},\\hat{x},\\hat{y})\\in (0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ over $[0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d)$ .",
"Then $K_0[u](\\hat{t},\\hat{x})-K_0[v](\\hat{t},\\hat{y})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D_x\\phi (\\hat{t},\\hat{x},\\hat{y}))-H(\\hat{t},\\hat{y},v(\\hat{t},\\hat{y}),-D_y\\phi (\\hat{t},\\hat{x},\\hat{y}))\\le 0.$ We shall show that there exists a constant $C_r>0$ such that $C_r\\rightarrow 0$ as $r\\rightarrow 0$ and $\\begin{aligned}&-C_r+J_r[\\phi ](\\hat{t},\\hat{x},\\hat{y})+K_r[u](\\hat{t},\\hat{x})+K_r[v](\\hat{t},\\hat{y})\\\\&+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D_x\\phi (\\hat{t},\\hat{x},\\hat{y}))-H(\\hat{t},\\hat{y},v(\\hat{t},\\hat{y}),-D_y\\phi (\\hat{t},\\hat{x},\\hat{y}))\\le 0\\end{aligned}$ for all $r\\in (0,\\hat{t})$ .",
"If this is clarified, passing to the limit $r\\rightarrow 0$ in (REF ) yields the desired result by repeating the `only if' in the proof of Lemma (REF ).",
"Henceforth, let $r\\in (0,\\hat{t})$ fix arbitrarily.",
"For $\\varepsilon >0$ we consider a function $\\Phi :[0,T]\\times [0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\Phi (t,s,x,y)=u(t,x)-v(s,y)-\\phi (t,x,y)-\\frac{|t-s|^2}{2\\varepsilon }-|t-\\hat{t}|^2-|x-\\hat{x}|^2-|y-\\hat{y}|^2.$ Since $\\Phi \\rightarrow -\\infty $ as $|x|,|y|\\rightarrow +\\infty $ and $\\Phi $ is bounded from above, it attains a maximum at a point $(t_{\\varepsilon },s_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\in [0,T]\\times [0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ .",
"By following the standard argument of the theory of viscosity solutions we obtain ${\\left\\lbrace \\begin{array}{ll}(t_{\\varepsilon },s_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\rightarrow (\\hat{t},\\hat{t},\\hat{x},\\hat{y}),\\\\u(t_{\\varepsilon },x_{\\varepsilon })\\rightarrow u(\\hat{t},\\hat{x})\\text{ and $v(s_{\\varepsilon },y_{\\varepsilon })\\rightarrow v(\\hat{t},\\hat{y})$}\\end{array}\\right.",
"}$ as $\\varepsilon \\rightarrow 0$ by taking a subsequence if necessary; see [4] and [7] for detail.",
"Note that $t_{\\varepsilon }>0$ for sufficiently small $\\varepsilon $ since $\\hat{t}>0$ .",
"For such a small parameter $\\varepsilon $ , $(t,x)\\mapsto \\Phi (t,s_{\\varepsilon },x,y_{\\varepsilon })$ attains a maximum at $(t_{\\varepsilon },x_{\\varepsilon })\\in (0,T]\\times \\mathbb {R}^d$ over $[0,T]\\times \\mathbb {R}^d$ and $(s,y)\\mapsto -\\Phi (t_{\\varepsilon },s,x_{\\varepsilon },y)$ attains a minimum at $(s_{\\varepsilon },y_{\\varepsilon })\\in (0,T]\\times \\mathbb {R}^d$ over $[0,T]\\times \\mathbb {R}^d$ .",
"Since $u$ and $v$ are respectively a subsolution and a supersolution of (REF ), Lemma REF implies that $K_0[u](t_{\\varepsilon },x_{\\varepsilon }),K_0[v](s_{\\varepsilon },y_{\\varepsilon })$ exist for each $\\varepsilon $ and $K_0[u](t_{\\varepsilon },x_{\\varepsilon })+H(t_{\\varepsilon },x_{\\varepsilon },u(t_{\\varepsilon },x_{\\varepsilon }),D_x\\phi (t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })+2(x_{\\varepsilon }-\\hat{x}))\\le 0,\\\\K_0[v](s_{\\varepsilon },y_{\\varepsilon })+H(s_{\\varepsilon },y_{\\varepsilon },v(s_{\\varepsilon },y_{\\varepsilon }),-D_y\\phi (s_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })-2(y_{\\varepsilon }-\\hat{y}))\\ge 0.$ Thus, by subtracting () from (REF ), we see $\\begin{aligned}&K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](s_{\\varepsilon },y_{\\varepsilon })\\\\&+H(t_{\\varepsilon },x_{\\varepsilon },u(t_{\\varepsilon },x_{\\varepsilon }),D_x\\phi (t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })+2(x_{\\varepsilon }-\\hat{x}))\\\\&-H(s_{\\varepsilon },y_{\\varepsilon },v(s_{\\varepsilon },y_{\\varepsilon }),-D_y\\phi (s_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })-2(y_{\\varepsilon }-\\hat{y}))\\le 0\\end{aligned}$ for each $\\varepsilon $ .",
"We shall pass to the limit $\\varepsilon \\rightarrow 0$ in (REF ).",
"For Hamiltonians it is easily seen thanks to (A1), (REF ) and the smoothness of $\\phi $ that $&H(t_{\\varepsilon },x_{\\varepsilon },u(t_{\\varepsilon },x_{\\varepsilon }),D_x\\phi (t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })+2(x_{\\varepsilon }-\\hat{x}))\\\\&-H(s_{\\varepsilon },y_{\\varepsilon },v(s_{\\varepsilon },y_{\\varepsilon }),-D_y\\phi (s_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })-2(y_{\\varepsilon }-\\hat{y}))\\\\&\\rightarrow H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),D_x\\phi (\\hat{t},\\hat{x},\\hat{y}))-H(\\hat{t},\\hat{y},v(\\hat{t},\\hat{y}),-D_y\\phi (\\hat{t},\\hat{x},\\hat{y}))$ as $\\varepsilon \\rightarrow 0$ .",
"Let us focus on $K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](s_{\\varepsilon },y_{\\varepsilon })$ .",
"Assume hereafter that $\\varepsilon $ is so small that $r<\\min \\lbrace t_{\\varepsilon },s_{\\varepsilon }\\rbrace $ for all $\\varepsilon $ , which is possible since $(t_{\\varepsilon },s_{\\varepsilon })\\rightarrow (\\hat{t},\\hat{t})$ as $\\varepsilon \\rightarrow 0$ (see (REF )).",
"Set $&I_{1,\\varepsilon }:=\\frac{u(t_{\\varepsilon },x_{\\varepsilon })-u(0,x_{\\varepsilon })}{t_{\\varepsilon }^{\\alpha }\\Gamma (1-\\alpha )}-\\frac{v(s_{\\varepsilon },y_{\\varepsilon })-v(0,y_{\\varepsilon })}{s_{\\varepsilon }^{\\alpha }\\Gamma (1-\\alpha )},\\\\&I_{2,\\varepsilon }:=\\int _0^r(u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\tau ,x_{\\varepsilon }))\\frac{d\\tau }{\\tau ^{\\alpha +1}}-\\int _0^r(v(s_{\\varepsilon },y_{\\varepsilon })-v(s_{\\varepsilon }-\\tau ,y_{\\varepsilon }))\\frac{d\\tau }{\\tau ^{1+\\alpha }},$ and $I_{3,\\varepsilon }:=\\int _r^{t_{\\varepsilon }}(u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\tau ,x_{\\varepsilon }))\\frac{d\\tau }{\\tau ^{\\alpha +1}}-\\int _r^{s_{\\varepsilon }}(v(s_{\\varepsilon },y_{\\varepsilon })-v(s_{\\varepsilon }-\\tau ,y_{\\varepsilon }))\\frac{d\\tau }{\\tau ^{1+\\alpha }}$ so that $K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](s_{\\varepsilon },y_{\\varepsilon })=I_{3,\\varepsilon }+\\alpha (I_{1,\\varepsilon }+I_{2,\\varepsilon })/\\Gamma (1-\\alpha )$ .",
"First, for $I_{1,\\varepsilon }$ , Proposition REF with $f_{\\varepsilon }:=u(t_{\\varepsilon },x_{\\varepsilon })-u(0,x_{\\varepsilon })-v(s_{\\varepsilon },y_{\\varepsilon })+v(0,y_{\\varepsilon })$ and $g_{\\varepsilon }:=(t_{\\varepsilon }^{\\alpha }\\Gamma (1-\\alpha ))^{-1}$ implies that $\\liminf _{\\varepsilon \\rightarrow 0}I_{1,\\varepsilon }\\ge \\frac{u(\\hat{t},\\hat{x})-u(0,\\hat{x})}{\\hat{t}^{\\alpha }\\Gamma (1-\\alpha )}-\\frac{v(\\hat{t},\\hat{y})-v(0,\\hat{y})}{\\hat{t}^{\\alpha }\\Gamma (1-\\alpha )}.$ Next, since $&u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\tau ,x_{\\varepsilon })-(v(s_{\\varepsilon },y_{\\varepsilon })-v(s_{\\varepsilon }-\\tau ,y_{\\varepsilon }))\\\\&\\ge \\phi (t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })-\\phi (t_{\\varepsilon }-\\tau ,x_{\\varepsilon },y_{\\varepsilon })+|t_{\\varepsilon }-\\hat{t}|^2-|t_{\\varepsilon }-\\tau -\\hat{t}|^2$ for all $\\tau \\in [0,r]$ by the inequality $\\Phi (t_{\\varepsilon },s_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\ge \\Phi (t_{\\varepsilon }-\\tau ,s_{\\varepsilon }-\\tau ,x_{\\varepsilon },y_{\\varepsilon })$ , we see $\\frac{\\alpha }{\\Gamma (1-\\alpha )}I_{2,\\varepsilon }\\ge J_r[\\phi ](t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })+J_r[|t_{\\varepsilon }-\\hat{t}-\\cdot |^2](t_{\\varepsilon }).$ Proposition REF (iv) ensures that $\\lim _{\\varepsilon \\rightarrow 0}J_r[\\phi ](t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })=J_r[\\phi ](\\hat{t},\\hat{x},\\hat{y})$ .",
"Besides, $J_r[|t_{\\varepsilon }-\\hat{t}-\\cdot |^2](t_{\\varepsilon })$ can be calculated precisely as $J_r[|t_{\\varepsilon }-\\hat{t}-\\cdot |^2](t_{\\varepsilon })&=\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _0^r(|t_{\\varepsilon }-\\hat{t}|^2-|t_{\\varepsilon }-\\hat{t}-\\tau |^2)\\frac{d\\tau }{\\tau ^{1+\\alpha }}\\\\&=\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\int _0^r(2(t_{\\varepsilon }-\\hat{t})\\tau -\\tau ^2)\\frac{d\\tau }{\\tau ^{1+\\alpha }}\\\\&=\\frac{\\alpha }{\\Gamma (1-\\alpha )}\\left(\\frac{2(t_{\\varepsilon }-\\hat{t})r^{1-\\alpha }}{1-\\alpha }-\\frac{r^{2-\\alpha }}{2-\\alpha }\\right).$ Hence $\\lim _{\\varepsilon \\rightarrow 0}J_r[|t_{\\varepsilon }-\\hat{t}-\\cdot |^2](t_{\\varepsilon })=-\\frac{\\alpha r^{2-\\alpha }}{(2-\\alpha )\\Gamma (1-\\alpha )}=:-C_r.$ Note that $C_r\\rightarrow 0$ as $r\\rightarrow 0$ .",
"Therefore we know for $I_{2,\\varepsilon }$ that $\\liminf _{\\varepsilon \\rightarrow 0}\\frac{\\alpha }{\\Gamma (1-\\alpha )}I_{2,\\varepsilon }\\ge -C_r+J_r[\\phi ](\\hat{t},\\hat{x},\\hat{y}).$ Finally, for $I_{3,\\varepsilon }$ , we first see an existence of constants $C_1,C_2>0$ independent of $\\varepsilon $ such that $\\begin{aligned}&(u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\tau ,x_{\\varepsilon })){1}_{(r,t_{\\varepsilon })}(\\tau )\\ge -C_1{1}_{(r,T)}(\\tau ),\\\\&(v(s_{\\varepsilon },y_{\\varepsilon })-v(s_{\\varepsilon }-\\tau ,y_{\\varepsilon })){1}_{(r,s_{\\varepsilon })}(\\tau )\\le C_2{1}_{(r,T)}(\\tau )\\end{aligned}$ on $[0,T]$ .",
"Indeed, since $\\lim _{\\varepsilon \\rightarrow 0}u(t_{\\varepsilon },x_{\\varepsilon })=u(\\hat{t},\\hat{x})$ , there is a constant $C>0$ independent of $\\varepsilon $ $(u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\tau ,x_{\\varepsilon })){1}_{(r,t_{\\varepsilon })}(\\tau )&\\ge (u(\\hat{t},\\hat{x})-C-\\max _{\\overline{Q_T}}u){1}_{(r,t_{\\varepsilon })}(\\tau )\\\\&\\ge -|u(\\hat{t},\\hat{x})-C-\\max _{\\overline{Q_T}}u|{1}_{(r,T)}.$ This shows the above one of (REF ) and another is proved similarly.",
"Note that both right-hand sides of (REF ) multiplied by $\\tau ^{-\\alpha -1}$ is integrable on $[0,T]$ .",
"Proposition REF implies that $&\\liminf _{\\varepsilon \\rightarrow 0}(u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\cdot ,x_{\\varepsilon })){1}_{(r,t_{\\varepsilon })}(\\cdot )\\ge (u(\\hat{t},\\hat{x})-u(\\hat{t}-\\cdot ,\\hat{x})){1}_{(r,\\hat{t})}(\\cdot ),\\\\&\\limsup _{\\varepsilon \\rightarrow 0}(v(s_{\\varepsilon },y_{\\varepsilon })-v(s_{\\varepsilon }-\\cdot ,y_{\\varepsilon })){1}_{(r,s_{\\varepsilon })}(\\cdot )\\le (v(\\hat{t},\\hat{x})-v(\\hat{t}-\\cdot ,\\hat{x})){1}_{(r,\\hat{t})}(\\cdot )$ for each $\\tau \\in (0,T)$ .",
"Thus Fatou's lemma yields $\\liminf _{\\varepsilon \\rightarrow 0}I_{3,\\varepsilon }\\ge \\int _r^{\\hat{t}}(u(\\hat{t},\\hat{x})-u(\\hat{t}-\\tau ,\\hat{x}))\\frac{d\\tau }{\\tau ^{1+\\alpha }}-\\int _r^{\\hat{t}}(v(\\hat{t},\\hat{y})-v(\\hat{t}-\\tau ,\\hat{y}))\\frac{d\\tau }{\\tau ^{1+\\alpha }}.$ Summing up (REF ), (REF ) and (REF ) we reach $\\liminf _{\\varepsilon \\rightarrow 0}(K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](s_{\\varepsilon },y_{\\varepsilon }))\\ge -C_r+J_r[\\phi ](\\hat{t},\\hat{x},\\hat{y})+K_r[u](\\hat{t},\\hat{x})-K_r[v](\\hat{t},\\hat{x}).$ Consequently, taking the limit inferior to both sides of (REF ) yields the desired inequality (REF ).",
"Suppose that the conclusion were false: $\\max _{\\overline{Q_T}}(u-v)=:\\theta >0$ .",
"For $\\varepsilon >0$ we consider a function $\\Phi :[0,T]\\times \\mathbb {R}^{d}\\times \\mathbb {R}^{d}\\rightarrow \\mathbb {R}$ defined by $\\Phi (t,x,y):=u(t,x)-v(t,y)-\\frac{|x-y|^{2}}{2\\varepsilon }.$ Let $(t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\in [0,T]\\times \\mathbb {R}^{d}\\times \\mathbb {R}^{d}$ be a maximum point of $\\Phi $ .",
"Then there is a point $(\\hat{t},\\hat{x})\\in (0,T]\\times \\mathbb {R}^d$ such that ${\\left\\lbrace \\begin{array}{ll}(t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\rightarrow (\\hat{t},\\hat{x},\\hat{x}),\\\\|x_{\\varepsilon }-y_{\\varepsilon }|^{2}/\\varepsilon \\rightarrow 0,\\\\u(t_{\\varepsilon },x_{\\varepsilon })\\rightarrow u(\\hat{t},\\hat{x})\\text{ and $v(s_{\\varepsilon },y_{\\varepsilon })\\rightarrow v(\\hat{t},\\hat{x})$.}\\end{array}\\right.",
"}$ as $\\varepsilon \\rightarrow 0$ by taking a subsequence if necessary; see, e.g., [4].",
"The above permits to use Lemma REF and we know that $K_0[u](t_{\\varepsilon },x_{\\varepsilon }),K_0[v](t_{\\varepsilon },y_{\\varepsilon })$ exists for each $\\varepsilon $ and $K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](t_{\\varepsilon },y_{\\varepsilon })+H(t_{\\varepsilon },x_{\\varepsilon },u(t_{\\varepsilon },x_{\\varepsilon }),p_{\\varepsilon })-H(t_{\\varepsilon },y_{\\varepsilon },v(t_{\\varepsilon },y_{\\varepsilon }),p_{\\varepsilon })\\le 0.$ Here $p_{\\varepsilon }=(x_{\\varepsilon }-y_{\\varepsilon })/\\varepsilon $ .",
"Since $u(t_{\\varepsilon },x_{\\varepsilon })-u(t_{\\varepsilon }-\\cdot ,x_{\\varepsilon })-v(t_{\\varepsilon },y_{\\varepsilon })+v(t_{\\varepsilon }-\\cdot ,y_{\\varepsilon })\\ge 0$ on $[0,t_{\\varepsilon }]$ by the inequality $\\Phi (t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\ge \\Phi (t_{\\varepsilon }-\\tau ,x_{\\varepsilon },y_{\\varepsilon })$ , the term of integration in $K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](t_{\\varepsilon },y_{\\varepsilon })$ is estimated from below by zero, that is, $K_0[u](t_{\\varepsilon },x_{\\varepsilon })-K_0[v](t_{\\varepsilon },y_{\\varepsilon })\\ge \\frac{u(t_{\\varepsilon },x_{\\varepsilon })-v(t_{\\varepsilon },y_{\\varepsilon })-u(0,x_{\\varepsilon })+v(0,y_{\\varepsilon })}{t_{\\varepsilon }^{\\alpha }\\Gamma (1-\\alpha )}$ Since $u(t_{\\varepsilon },x_{\\varepsilon })>v(t_{\\varepsilon },y_{\\varepsilon })$ by the inequality $\\Phi (t_{\\varepsilon },x_{\\varepsilon },y_{\\varepsilon })\\ge \\theta >0$ , Hamiltonians in (REF ) are estimated as $&H(t_{\\varepsilon },x_{\\varepsilon },u(t_{\\varepsilon },x_{\\varepsilon }),p_{\\varepsilon })-H(t_{\\varepsilon },y_{\\varepsilon },v(t_{\\varepsilon },y_{\\varepsilon }),p_{\\varepsilon })\\\\&\\ge H(t_{\\varepsilon },x_{\\varepsilon },v(t_{\\varepsilon },y_{\\varepsilon }),p_{\\varepsilon })-H(t_{\\varepsilon },y_{\\varepsilon },v(t_{\\varepsilon },y_{\\varepsilon }),p_{\\varepsilon })\\ge -\\omega (|x_{\\varepsilon }-y_{\\varepsilon }|(1+|p_{\\varepsilon }|))$ by (A2) and (A3).",
"From these, (REF ) is led to $\\frac{u(t_{\\varepsilon },x_{\\varepsilon })-v(t_{\\varepsilon },y_{\\varepsilon })-u(0,x_{\\varepsilon })+v(0,y_{\\varepsilon })}{t_{\\varepsilon }^{\\alpha }\\Gamma (1-\\alpha )}\\le \\omega (|x_{\\varepsilon }-y_{\\varepsilon }|(1+|p_{\\varepsilon }|)).$ Taking the limit inferior $\\varepsilon \\rightarrow 0$ implies that $\\frac{\\theta -u(0,\\hat{x})+v(0,\\hat{x})}{\\hat{t}^{\\alpha }\\Gamma (1-\\alpha )}\\le 0$ by Proposition REF .",
"Since $u(0,\\cdot )\\le v(0,\\cdot )$ on $\\mathbb {T}^{d}$ and $\\theta >0$ , this is a contradiction.",
"Corollary 3.4 (Uniqueness) Assume (A1)-(A4).",
"Let $u\\in C(\\overline{Q_T})$ and $v\\in C(\\overline{Q_T})$ be solutions of (REF ).",
"Then $\\max _{(t,x)\\in \\overline{Q_T}}|u(t,x)-v(t,x)|\\le \\max _{x\\in \\mathbb {T}^d}|u(0,x)-v(0,x)|.$ Moreover, if $u$ and $v$ are solutions of (REF )-(REF ), then $u\\equiv v$ on $\\overline{Q_T}$ .",
"It suffices to prove (REF ).",
"Set $C:=\\max _{x\\in \\mathbb {T}^d}|u(0,x)-v(0,x)|$ .",
"Then $v-C$ and $v+C$ are a subsolution and a supersolution of (REF ), respectively; see Remark REF .",
"Moreover $v(0,\\cdot )-C\\le u(0,\\cdot )\\le v(0,\\cdot )+C\\quad \\text{on $\\mathbb {T}^d$}$ by the definition of $C$ .",
"Thus, from Theorem REF , we have $|u-v|\\le C$ on $\\overline{Q_T}$ .",
"The proof is complete by taking the maximum over $\\overline{Q_T}$ to both sides.",
"For the reader's convenience we give a statement of the comparison principle for a general bounded domain $\\Omega $ without a proof.",
"Theorem 3.5 Let $\\Omega $ be a bounded domain in $\\mathbb {R}^d$ .",
"Let $u\\in USC([0,T]\\times \\overline{\\Omega };\\mathbb {R})$ and $v\\in LSC([0,T]\\times \\overline{\\Omega };\\mathbb {R})$ be a subsolution and a supersolution of (REF ) on $(0,T]\\times \\overline{\\Omega }$ , respectively.",
"If $u\\le v$ on $(\\lbrace 0\\rbrace \\times \\overline{\\Omega })\\cup ([0,T]\\times \\partial \\Omega )$ , then $u\\le v$ on $[0,T]\\times \\overline{\\Omega }$ ."
],
[
"Existence result",
"Let denote by $S^{-}$ and $S^{+}$ a set of upper semicontinuous subsolutions and lower semicontinuous supersolutions of (REF ), respectively.",
"Note that $S^\\pm \\ne \\emptyset $ as will be observed in Corollary REF later.",
"Lemma 4.1 (Closedness under supremum/infimum operator) Assume (A1).",
"Let $X$ be a nonempty subset of $S^{-}$ (resp.",
"$S^{+}$ ).",
"Define $u(t,x):=\\sup \\lbrace v(t,x) \\mid v\\in X\\rbrace \\quad \\text{(resp.",
"$\\inf \\lbrace v(t,x) \\mid v\\in X\\rbrace $)}$ for $(t,x)\\in \\overline{Q_T}$ .",
"Assume that $u^*<+\\infty $ (resp.",
"$u_*>-\\infty $ ) on $\\overline{Q_T}$ .",
"Then $u^*$ (resp.",
"$u_*$ ) is a subsolution (resp.",
"supersolution) of (REF ).",
"We only prove for a subsolution since the argument for a supersolution is similar.",
"Fix $[a,b]\\times B\\subset (0,T]\\times \\mathbb {R}^d$ arbitrarily, where $a<b$ and $B$ is an open in $\\mathbb {R}^d$ .",
"Assume that $u^*-\\phi $ attains a maximum at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ .",
"Then we must show that $J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x})+K_{\\hat{t}-a}[u^*](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u^*(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x}))\\le 0.$ By Proposition REF we may assume that $(\\hat{t},\\hat{x})$ is a strict maximum point of $u^*-\\phi $ such that $(u^*-\\phi )(\\hat{t},\\hat{x})=0$ .",
"By arguing similarly as for $\\alpha =1$ we find sequences $\\lbrace (t_j,x_j)\\rbrace _{j\\ge 0}$ and $\\lbrace v_j\\rbrace _{j\\ge 0}\\subset X$ such that, for each $j\\ge 0$ , $v_j-\\phi $ attains a maximum at $(t_j,x_j)\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ and $(t_j,x_j,v_j(t_j,x_j))\\rightarrow (\\hat{t},\\hat{x},u^*(\\hat{t},\\hat{x}))$ as $j\\rightarrow \\infty $ .",
"Indeed it is enough to translate slightly the proof of [13] to the current situation.",
"This is not difficult, so the detail is safely omitted.",
"Since $v_j$ is a subsolution of (REF ), $J_{t_j-a}[\\phi ](t_j,x_j)+K_{t_j-a}[v_j](t_j,x_j)+H(t_j,x_j,v_j(t_j,x_j),D\\phi (t_j,x_j))\\le 0$ for each $j\\ge 0$ .",
"We shall pass to the limit $j\\rightarrow \\infty $ in (REF ).",
"The continuity of Hamiltonian (A1) ensures that $\\lim _{j\\rightarrow \\infty }H(t_j,x_j,v_j(t_j,x_j),D\\phi (t_j,x_j))=H(\\hat{t},\\hat{x},u^*(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x})).$ Proposition REF implies that $\\lim _{j\\rightarrow \\infty }J_{t_j-a}[\\phi ](t_j,x_j)=J_{\\hat{t}-a}[\\phi ](\\hat{t},\\hat{x}).$ Henceforth, let us focus on $K_{t_j-a}[v_j](t_j,x_j)$ .",
"Since $v_j\\le u\\le u^*$ on $\\overline{Q_T}$ by the definition of $u$ and $u^*$ , Proposition REF implies that $\\begin{aligned}\\liminf _{j\\rightarrow \\infty }\\frac{v_j(t_j,x_j)-v_j(0,x_j)}{t_j^{\\alpha }\\Gamma (1-\\alpha )}&\\ge \\liminf _{j\\rightarrow \\infty }\\frac{v_j(t_j,x_j)-u^*(0,x_j)}{t_j^{\\alpha }\\Gamma (1-\\alpha )}\\\\&\\ge \\frac{u^*(\\hat{t},\\hat{x})-u^*(0,\\hat{x})}{\\hat{t}^{\\alpha }\\Gamma (1-\\alpha )}.\\end{aligned}$ To handle the term of integration we first see the existence of a constant $C_2>0$ independent of $j$ such that $(v_j(t_j,x_j)-v_j(t_j-\\cdot ,x_j)){1}_{[t_j-a,t_j]}(\\cdot )\\ge -C_2{1}_{[r,T]}(\\cdot )$ on $[0,T]$ for sufficiently large $j$ , where $r:=\\min _{j\\ge 0}(t_j-a)>0$ .",
"Indeed, since $\\sup _{j\\ge 0}v_j\\le u\\le u^*$ and $u^*<+\\infty $ on $\\overline{Q_T}$ , there is a constant $C_3>0$ such that $\\sup _{j\\ge 0}v_j\\le C_3$ on $\\overline{Q_T}$ .",
"Since $v_j(t_j,x_j)\\rightarrow u^*(\\hat{t},\\hat{x})$ as $j\\rightarrow \\infty $ , for a constant $C_4>0$ (independent of $j$ ), $v_j(t_j,x_j)\\ge u^*(\\hat{t},\\hat{x})-C_4$ for large $j$ .",
"Thus, if we set $C_2:=|u^*(\\hat{t},\\hat{x})-C_4-C_3|$ , then $(v_j(t_j,x_j)-v_j(t_j-\\cdot ,x_j)){1}_{[t_j-a,t_j]}(\\cdot )&\\ge (u^*(\\hat{t},\\hat{x})-C_4-C_3){1}_{[t_j-a,t_j]}(\\cdot )\\\\&\\ge -C_2{1}_{[r,T]}(\\cdot )$ on $[0,T]$ for sufficiently large $j$ , which is the desired fact.",
"Note that $-C_2{1}_{[r,T]}(\\tau )/\\tau ^{\\alpha +1}$ is integrable on $[0,T]$ .",
"Proposition REF also implies that $&\\liminf _{j\\rightarrow \\infty }(v_j(t_j,x_j)-v_j(t_j-\\cdot ,x_j)){1}_{[t_j-a,t_j]}(\\cdot )\\\\&\\ge \\liminf _{j\\rightarrow \\infty }(v_j(t_j,x_j)-u^*(t_j-\\cdot ,x_j)){1}_{[t_j-a,t_j]}(\\cdot )\\\\&\\ge (u^*(\\hat{t},\\hat{x})-u^*(\\hat{t}-\\cdot ,\\hat{x})){1}_{[\\hat{t}-a,\\hat{t}]}(\\cdot ).$ Therefore Fatou's lemma can be applied and consequently $\\liminf _{j\\rightarrow \\infty }K_{t_j-a}[v_j](t_j,x_j)\\ge K_{\\hat{t}-a}[u^*](\\hat{t},\\hat{x})$ by combining with (REF ).",
"Taking the limit inferior $j\\rightarrow \\infty $ to both sides in (REF ) yields (REF ).",
"Theorem 4.2 (Existence) Assume (A1).",
"Let $u^-\\in USC(\\overline{Q_T})$ and $u^+\\in LSC(\\overline{Q_T})$ be a supersolution and a subsolution of (REF ) such that $(u^-)_*>-\\infty $ and $(u^+)^*<+\\infty $ on $\\overline{Q_T}$ .",
"Suppose that $u^-\\le u^+$ in $\\overline{Q_T}$ .",
"Then there exists a solution $u$ of (REF ) that satisfies $u^-\\le u\\le u^+$ in $\\overline{Q_T}$ .",
"Define $u(t,x):=\\sup \\lbrace v(t,x) \\mid v\\in X\\rbrace $ for $(t,x)\\in \\overline{Q_T}$ , where $X:=\\lbrace v\\in S^- \\mid \\text{$v\\le u^+$ on $\\overline{Q_T}$}\\rbrace .$ Note that $X\\ne \\emptyset $ since $u^-\\in X$ .",
"Also, since $u^-\\le u\\le u^+$ on $\\overline{Q_T}$ by the definition of $u$ , $-\\infty <(u^-)_*\\le u_*\\le u^*\\le (u^+)^*<+\\infty $ on $\\overline{Q_T}$ .",
"Our goal in this proof is to show that $u$ defined by (REF ) is actually a solution of (REF ).",
"Since we know that $u^*$ is a subsolution of (REF ) from Lemma REF , so it suffices to show that $u_*$ is a supersolution of (REF ).",
"Suppose by contradiction that $u_*$ were not a supersolution of (REF ).",
"Then there would be a function $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ , a point $(\\hat{t},\\hat{x})\\in (0,T]\\times \\mathbb {R}^d$ and a constant $\\theta >0$ such that $u_*-\\phi $ attains a minimum at $(\\hat{t},\\hat{x})\\in (0,T]\\times \\mathbb {R}^d$ over $(0,T]\\times \\mathbb {R}^d$ and $K_0[u_*](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u_*(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x}))<-2\\theta .$ Notice that $K_0[u_*](\\hat{t},\\hat{x})$ may be $-\\infty $ , while $K_0[u_*](\\hat{t},\\hat{x})$ makes sense and is bounded from above by Proposition REF .",
"Let $\\rho >0$ be a small parameter so that $\\rho <\\hat{t}$ and $\\overline{B_{2\\rho }(\\hat{x})}\\subset (\\hat{x}-\\frac{1}{2},\\hat{x}+\\frac{1}{2}]^d$ .",
"Define functions $w:(0,T]\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ and $U:[0,T]\\times (\\hat{x}-\\frac{1}{2},\\hat{x}+\\frac{1}{2}]^d\\rightarrow \\mathbb {R}$ by $w(s,y):=\\phi (s,y)+\\frac{\\rho ^{2}}{2}-|s-\\hat{t}|^2-|y-\\hat{x}|^{2}$ and $U={\\left\\lbrace \\begin{array}{ll}\\max \\lbrace u^*,w\\rbrace \\quad &\\text{in $((\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x}))\\cap ([0,T]\\times (\\hat{x}-\\frac{1}{2},\\hat{x}+\\frac{1}{2}]^d)$,}\\\\u^*\\quad &\\text{in $([0,T]\\times (\\hat{x}-\\frac{1}{2},\\hat{x}+\\frac{1}{2}]^d)\\setminus ((\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x}))$,}\\end{array}\\right.",
"}$ respectively.",
"We regard $B_{2\\rho }(\\hat{x})$ and, for each $s\\in [0,T]$ , $U(s,\\cdot )$ as a open ball in $\\mathbb {T}^d$ and a function on $\\mathbb {T}^d$ by extending it periodically, respectively.",
"We shall show that $U\\in X$ and that there exists a point $(s,y)\\in \\overline{Q_T}$ such that $U(s,y)>u(s,y)$ .",
"Once these were proved, we would obtain a contradiction due to the maximality of $u$ .",
"Set $\\Omega :=\\left\\lbrace (s,y)\\in (\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x}) \\mid |s-\\hat{t}|^2+|y-\\hat{x}|^2\\le \\frac{\\rho ^2}{2}\\right\\rbrace \\subset \\overline{Q_T}.$ Then $\\overline{\\Omega }\\subset (\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x})$ and $u^*(s,y)\\ge u_*(s,y)\\ge \\phi (s,y)=w(s,y)-\\frac{\\rho ^{2}}{2}+|s-\\hat{t}|^{2}+|y-\\hat{x}|^{2}> w(s,y)$ for all $(s,y)\\in ((\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x}))\\setminus \\Omega $ .",
"Thus $U$ is upper semicontinuous on $\\overline{Q_T}$ by its definition.",
"Assume that $U-\\psi $ attains a maximum at $(\\hat{s},\\hat{y})\\in (0,T]\\times \\mathbb {R}^d$ over $(0,T]\\times \\mathbb {R}^d$ for $\\psi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ .",
"We may assume that $(U-\\psi )(\\hat{s},\\hat{y})=0$ .",
"Case 1: Suppose that $U(\\hat{s},\\hat{y})=u^*(\\hat{s},\\hat{y})$ .",
"Then, since $U\\ge u^*$ on $\\overline{Q_T}$ , it turns out that $u^*-\\psi $ attains a maximum at $(\\hat{s},\\hat{y})$ over $(0,T]\\times \\mathbb {R}^d$ and that $U(\\hat{s},\\hat{y})-U(\\hat{s}-\\tau ,\\hat{y})\\le u^*(\\hat{s},\\hat{y})-u^*(\\hat{s}-\\tau ,\\hat{y})$ for all $\\tau \\in [0,\\hat{s}]$ .",
"Recall that $u^*$ is a subsolution of (REF ), so that $K_0[u^*](\\hat{s},\\hat{y})+H(\\hat{s},\\hat{y},u(\\hat{s},\\hat{y}),D\\psi (\\hat{s},\\hat{y}))\\le 0.$ Proposition REF (iii) with (REF ) ensures that $K_0[U](\\hat{s},\\hat{y})$ exists and simultaneously $K_0[U](\\hat{s},\\hat{y})\\le K_0[u^*](\\hat{s},\\hat{y})$ .",
"This implies that $U$ is a subsolution of (REF ).",
"Case 2: Suppose that $U(\\hat{s},\\hat{y})=w(\\hat{s},\\hat{y})>u(\\hat{s},\\hat{y})$ .",
"Then, from (REF ), we see $(\\hat{s},\\hat{y})\\in \\Omega $ , which yields $\\lim _{\\rho \\rightarrow 0}(\\hat{s},\\hat{y})=(\\hat{t},\\hat{x})$ .",
"By employing the idea in [17] for example, we shall show that $\\limsup _{\\rho \\rightarrow 0}K_0[U](\\hat{s},\\hat{y})\\le K_0[u_*](\\hat{t},\\hat{x}).$ Since $U\\ge u^*\\ge u_*$ on $\\overline{Q_T}$ the non-integration term is estimated as $\\frac{U(\\hat{s},\\hat{y})-U(0,\\hat{y})}{\\hat{s}^{\\alpha }\\Gamma (1-\\alpha )}\\le \\frac{w(\\hat{s},\\hat{y})-u_*(0,\\hat{y})}{\\hat{s}^{\\alpha }\\Gamma (1-\\alpha )}.$ Recalling that $\\lim _{\\rho \\rightarrow 0}w(\\hat{s},\\hat{y})=\\phi (\\hat{t},\\hat{x})=u_*(\\hat{t},\\hat{x})$ we see $\\limsup _{\\rho \\rightarrow 0}\\frac{U(\\hat{s},\\hat{y})-U(0,\\hat{y})}{\\hat{s}^{\\alpha }\\Gamma (1-\\alpha )}\\le \\frac{u_*(\\hat{t},\\hat{x})-u_*(0,\\hat{x})}{\\hat{t}^{\\alpha }\\Gamma (1-\\alpha )}.$ To handle the term of integration let us divide the term of integration in $K_0[U](\\hat{s},\\hat{y})$ multiplied by $\\Gamma (1-\\alpha )/\\alpha $ into two integrations as follows: $I_{1,\\rho }[U]:=\\int _0^{\\rho ^2}(U(\\hat{s},\\hat{y})-U(\\hat{s}-\\tau ,\\hat{y}))\\frac{d\\tau }{\\tau ^{\\alpha +1}}$ and $I_{2,\\rho }[U]:=\\int _{\\rho ^2}^{\\hat{s}}(U(\\hat{s},\\hat{y})-U(\\hat{s}-\\tau ,\\hat{y}))\\frac{d\\tau }{\\tau ^{\\alpha +1}}.$ By definitions of $U$ and $w$ we have $U(\\hat{s},\\hat{y})-U(\\hat{s}-\\tau ,\\hat{y})&\\le w(\\hat{s},\\hat{y})-w(\\hat{s}-\\tau ,\\hat{y})\\\\&=\\phi (\\hat{s},\\hat{y})-\\phi (\\hat{s}-\\tau ,\\hat{y})+\\tau ^2-2(\\hat{s}-\\hat{y})\\tau $ for all $\\tau \\in [0,\\rho ^2]$ .",
"Hence $I_{1,\\rho }[U]\\le I_{1,\\rho }[\\phi ]+C_{\\rho }$ with a constant $C_{\\rho }$ such that $\\lim _{\\rho \\rightarrow 0}C_{\\rho }=0$ .",
"By Proposition REF (iv), we see that $\\lim _{\\rho \\rightarrow 0}I_{1,\\rho }[\\phi ]=0$ , so that $\\limsup _{\\rho \\rightarrow 0}I_{1,\\rho }[U]\\le 0.$ Since $U\\ge u_*$ on $\\overline{Q_T}$ , $\\begin{aligned}U(\\hat{s},\\hat{y})-U(\\hat{s}-\\tau ,\\hat{y})&\\le w(\\hat{s},\\hat{y})-u_*(\\hat{s}-\\tau ,\\hat{y})\\\\&\\le \\phi (\\hat{s},\\hat{y})-u_*(\\hat{s}-\\tau ,\\hat{y})+\\frac{\\rho ^2}{2}\\end{aligned}$ on $[\\rho ^2,\\hat{s}]$ .",
"Moreover, the relationship between $u_*$ and $\\phi $ yields $\\phi (\\hat{s},\\hat{y})-u_*(\\hat{s}-\\tau ,\\hat{y})+\\frac{\\rho ^2}{2}\\le \\phi (\\hat{s},\\hat{y})-\\phi (\\hat{s}-\\tau ,\\hat{y})+\\frac{\\rho ^2}{2}.$ Since $\\phi (\\cdot ,x)$ is continuous on $[0,T]$ , we are able to find a large constant $C_1>0$ such that $\\phi (\\hat{s},\\hat{y})-\\phi (\\hat{s}-\\tau ,\\hat{y})\\le C_1\\tau $ for all $\\tau \\in [\\rho ^2,\\hat{s}]$ .",
"In addition we may assume that $C_1$ does not depend on $\\rho $ .",
"Notice that there exists a constant $C_2>0$ such that $C_1\\tau ^2+\\rho ^2/2\\le C_2\\tau $ for all $\\tau \\in [\\rho ^2,\\hat{s}]$ .",
"Consequently (REF ) is lead to $U(\\hat{s},\\hat{y})-U(\\hat{s}-\\tau ,\\hat{y})\\le C_2\\tau $ on $[\\rho ^2,\\hat{s}]$ .",
"The right-hand side with $\\tau ^{-\\alpha -1}$ is integrable on $[0,T]$ , so that Fatou's lemma yields $\\limsup _{\\rho \\rightarrow 0}I_{2,\\rho }[U](\\hat{s},\\hat{y})\\le I_{2,0}[u_*](\\hat{t},\\hat{x}).$ The above ensures (REF ) and thus we see $K_0[U](\\hat{s},\\hat{y})-K_0[u_*](\\hat{t},\\hat{x})\\le \\theta $ for sufficiently small $\\rho $ .",
"Notice that this means $K_0[U]$ and $K_0[u_*]$ actually exist.",
"Since the maximizer $(\\hat{s},\\hat{y})$ of $U-\\psi $ is of $w-\\psi $ (on $\\Omega $ ) as well, the classical maximum principle for $w-\\psi $ implies that $D\\phi (\\hat{s},\\hat{y})-2(\\hat{y}-\\hat{x})=D\\psi (\\hat{s},\\hat{y})$ .",
"Hence we see $\\lim _{\\rho \\rightarrow 0}D\\psi (\\hat{s},\\hat{y})=D\\phi (\\hat{t},\\hat{x})$ .",
"Moreover, $\\lim _{\\rho \\rightarrow 0}U(\\hat{s},\\hat{y})=\\lim _{\\rho \\rightarrow 0}w(\\hat{s},\\hat{y})=\\phi (\\hat{t},\\hat{x})=u_*(\\hat{t},\\hat{x})$ .",
"Therefore $H(\\hat{s},\\hat{y},U(\\hat{s},\\hat{y}),D\\psi (\\hat{s},\\hat{y}))-H(\\hat{t},\\hat{x},u_*(\\hat{t},\\hat{s}),D\\phi (\\hat{t},\\hat{x}))\\le \\theta $ if $\\rho $ is sufficiently small.",
"Summing up the above we obtain for sufficiently small $\\rho $ that $&K_0[U](\\hat{s},\\hat{y})+H(\\hat{s},\\hat{y},U(\\hat{s},\\hat{y}),D\\psi (\\hat{s},\\hat{y}))\\\\&\\le -2\\theta +K_0[U](\\hat{s},\\hat{y})-K_0[u_*](\\hat{t},\\hat{x})\\\\&\\quad +H(\\hat{s},\\hat{y},U(\\hat{s},\\hat{y}),D\\psi (\\hat{s},\\hat{y}))-H(\\hat{t},\\hat{x},u_*(\\hat{t},\\hat{s}),D\\phi (\\hat{t},\\hat{x}))\\le 0,$ which shows that $U$ is a subsolution of (REF ).",
"Theorem REF implies that $U\\le u^+$ .",
"Let $\\lbrace (t_j,x_j)\\rbrace _{j\\ge 0}$ be a sequence such that $(t_j,x_j,u(t_j,x_j))\\rightarrow (\\hat{t},\\hat{x},u_*(\\hat{t},\\hat{x}))$ as $j\\rightarrow \\infty $ .",
"Then $\\liminf _{j\\rightarrow \\infty }(U(t_j,x_j)-u(t_j,x_j))\\ge \\lim _{j\\rightarrow \\infty }(w(t_j,x_j)-u(t_j,x_j))=\\frac{\\rho ^{2}}{2}>0.$ In other words there exists a point $(s,y)$ such that $U(s,y)>u(s,y)$ .",
"Therefore the proof is complete.",
"Corollary 4.3 (Unique existence for (REF )-(REF )) Assume (A1)-(A4).",
"Then there exists at most one solution $u$ of (REF )-(REF ).",
"The uniqueness of a solution is guaranteed by Theorem REF .",
"Henceforth, it is enough to construct $u^-$ and $u^+$ in Theorem REF so that $u$ defined by (REF ) satisfies $u(0,\\cdot )=u_0$ on $\\mathbb {T}^d$ .",
"Set $\\omega (\\ell ):=\\sup \\lbrace |u_{0}(\\zeta )-u_{0}(\\eta )| \\mid \\zeta ,\\eta \\in \\mathbb {T}^d,|\\zeta -\\eta |\\le \\ell \\rbrace $ for $\\ell \\ge 0$ and $f_y(x):=\\sum _{i=1}^d(1-\\cos (2\\pi (x_i-y_i)))$ for $x,y\\in \\mathbb {T}^d$ , where $x_i$ and $y_i$ are $i$ -th components of each variable.",
"Then for each $\\varepsilon >0$ there exists a constant $C_{\\varepsilon }>0$ such that $\\omega (|x-y|)\\le \\varepsilon +C_{\\varepsilon }f_y(x)$ for all $x,y\\in \\mathbb {T}^d$ .",
"For $\\varepsilon \\in (0,1)$ and $y\\in \\mathbb {T}^{d}$ we define a function $u_{\\varepsilon ,y}^-:\\overline{Q_T}\\rightarrow \\mathbb {R}$ by $u_{\\varepsilon ,y}^{-}(t,x):=u_{0}(y)-\\varepsilon -C_{\\varepsilon }f_y(x)-\\frac{C t^{\\alpha }}{\\Gamma (1+\\alpha )},$ where $C>0$ is a large constant.",
"Then $u_{\\varepsilon ,y}^-\\in \\mathcal {C}^1(\\overline{Q_T})$ .",
"Moreover $u_{\\varepsilon ,y}^-\\le u_0(y)$ by the non-negativity of $f_y$ and $|Du_{\\varepsilon ,y}^-|$ is bounded on $Q_T$ .",
"It is well-known that $\\frac{1}{\\Gamma (1-\\alpha )}\\int _a^t\\frac{[(s-a)^{\\beta }]^{\\prime }}{(t-s)^{\\alpha }}ds=\\frac{\\Gamma (\\beta +1)}{\\Gamma (\\beta -\\alpha +1)}(t-a)^{\\beta -\\alpha }$ for given constants $a\\in \\mathbb {R}$ and $\\beta \\in (0,1)$ ; see [36] for the proof.",
"From this formula with $(a,\\beta )=(0,\\alpha )$ and the above, we see $-C+H(t,x,u_{\\varepsilon ,y}^-(t,x),Du_{\\varepsilon ,y}^-(t,x))\\le 0$ for all $(t,x)\\in Q_T$ .",
"Thus Proposition REF implies that $u_{\\varepsilon ,y}^-$ is a (viscosity) subsolution of (REF ).",
"We also see $u_{\\varepsilon ,y}^-(t,x)\\le u_0(x)+\\omega (|x-y|)-\\varepsilon -C_{\\varepsilon }f_y(x)-\\frac{Ct^{\\alpha }}{\\Gamma (1-\\alpha )}\\le u_0(x)$ for all $(t,x)\\in \\overline{Q_T}$ .",
"Therefore, Lemma REF ensures that $u^-(t,x):=(\\sup \\lbrace u_{\\varepsilon ,y}^-(t,x) \\mid \\varepsilon \\in (0,1),y\\in \\mathbb {T}^d\\rbrace )^*$ is a subsolution of (REF ) and satisfies $u^-(t,x)\\le u_0(x)$ for all $(t,x)\\in \\overline{Q_T}$ .",
"The definition of $u^-$ yields $u^-(0,\\cdot )\\ge u_0$ on $\\mathbb {T}^d$ , which guarantees that $(u^-)_*>-\\infty $ on $\\overline{Q_T}$ and $u^-(0,\\cdot )=u_0$ on $\\mathbb {T}^d$ .",
"Similarly, a supersolution with desired properties is constructed.",
"Moreover, it turns out that $u^\\pm $ satisfy $u_0(x)=\\lim _{(t,y)\\rightarrow (x,0)}u^\\pm (t,y)$ but we leave the verification to the reader; cf.",
"[13].",
"With $u^\\pm $ above, we obtain a solution $u$ by Theorem REF , and it satisfies $u(0,\\cdot )=u_0$ on $\\mathbb {T}^d$ .",
"The proof is now complete."
],
[
"Some stability results",
"Two main theorems in this section are in what follows: Theorem 5.1 (Stability I) Let $H_\\varepsilon $ and $H$ satisfy (A1)-(A3), where $\\varepsilon >0$ .",
"Let $u_\\varepsilon \\in USC(\\overline{Q_T})$ (resp.",
"$LSC(\\overline{Q_T})$ ) be a subsolution (resp.",
"supersolution) of $\\partial _t^\\alpha u_\\varepsilon +H_\\varepsilon (t,x,u_\\varepsilon ,Du_\\varepsilon )=0\\quad \\text{in $Q_T$.",
"}$ Assume that $H_\\varepsilon $ converges to $H$ as $\\varepsilon \\rightarrow 0$ locally uniformly in $(0,T]\\times \\mathbb {T}^d\\times \\mathbb {R}\\times \\mathbb {R}^d$ .",
"Assume that $\\lbrace u_{\\varepsilon }\\rbrace _{\\varepsilon >0}$ is locally uniformly bounded.",
"Then $u:=\\limsup {}^{*}u_{\\varepsilon }$ (resp.",
"$\\liminf {}_{*}u_{\\varepsilon }$ ) is a subsolution (resp.",
"supersolution) of $\\partial _t^\\alpha u+H(t,x,u,Du)=0\\quad \\text{in $Q_T$.",
"}$ Here $\\limsup {}^{*}u_{\\varepsilon }$ appears above is the upper relaxed limit defined by $(\\limsup _{\\varepsilon \\rightarrow 0}{}^{*}u_{\\varepsilon })(t,x):=\\lim _{\\delta \\searrow 0}\\sup \\lbrace u_{\\varepsilon }(s,y) \\mid (s,y)\\in Q_T\\cap \\overline{B_{\\delta }(t,x)},0<\\varepsilon <\\delta \\rbrace $ for $(t,x)\\in \\overline{Q_T}$ and $\\liminf {}_{*}u_{\\varepsilon }:=-\\limsup {}^{*}(-u_{\\varepsilon })$ is the lower relaxed limit.",
"Theorem 5.2 (Stability II) Assume that (A1)-(A4).",
"Let $u_{\\alpha }\\in C(\\overline{Q_T})$ be a solution of (REF )-(REF ) whose time-derivative's order is $\\alpha \\in (0,1)$ .",
"Then $u_{\\alpha }$ converges to $u_{\\beta }$ locally uniformly in $Q_T$ as $\\alpha \\rightarrow \\beta $ , where $u_{\\beta }$ is a solution of (REF )-(REF ) whose time-derivative's order is $\\beta \\in (0,1]$ .",
"A same idea as for the proof of [4] is used for Theorem REF and Theorem REF .",
"A deal for the term of time-derivative is the only difference between Theorem REF and [4] but it is similar between Theorem REF and Theorem REF .",
"For this reason we only prove Theorem REF .",
"As the analogy of the upper/lower relaxed limits, for $\\beta \\in (0,1]$ , we define functions $u^\\sharp $ and $u_\\sharp $ by $u^\\sharp (t,x):=\\lim _{\\delta \\searrow 0}\\sup \\lbrace u_{\\alpha }(s,y) \\mid (s,y)\\in \\overline{B_{\\delta }(t,x)}\\cap Q_T,\\alpha \\in (\\beta -\\delta ,\\beta +\\delta )\\cap (0,1)\\rbrace $ and $u_\\sharp :=-(-u)^\\sharp $ on $\\overline{Q_T}$ .",
"In Remark REF later we will mention that $\\lbrace u_{\\alpha }\\rbrace _{\\alpha \\in (0,1]}$ of (REF )-(REF ) is uniformly bounded on $\\overline{Q_T}$ .",
"Hence $u^\\sharp $ and $u_\\sharp $ are bounded on $\\overline{Q_T}$ .",
"Note also that $u^\\sharp $ is an upper semicontinuous function, so $u_\\sharp $ is a lower semicontinuous function.",
"We shall show that $u^\\sharp $ and $u_\\sharp $ are a subsolution and a supersolution of (REF )-(REF ) whose time-derivative's order is $\\beta $ .",
"It suffices to show that $u^\\sharp $ is a subsolution of (REF ) since the similar argument is applied for $u_\\sharp $ and it is clear that $u^\\sharp (0,\\cdot )\\le u_0$ and $u_\\sharp (0,\\cdot )\\ge u_0$ on $\\mathbb {T}^{d}$ .",
"Fix $[a,b]\\times B\\subset (0,T]\\times \\mathbb {R}^d$ arbitrarily, where $a<b$ and $B$ is an open set in $\\mathbb {R}^d$ .",
"Assume that $u^\\sharp -\\phi $ attains a maximum at $(\\hat{t},\\hat{x})\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ for $\\phi \\in \\mathcal {C}^{1}([0,T]\\times \\mathbb {R}^d)$ .",
"Let $\\lbrace \\alpha _j\\rbrace _{j\\ge 0}$ and $\\lbrace (t_j,x_j)\\rbrace _{j\\ge 0}$ be sequences such that $u_{\\alpha _j}-\\phi $ attains a maximum at $(t_j,x_j)\\in (a,b]\\times B$ over $[a,b]\\times \\overline{B}$ and $(\\alpha _j,t_j,x_j,u_{\\alpha _j}(t_j,x_j))\\rightarrow (\\beta ,\\hat{t},\\hat{x},u^\\sharp (\\hat{t},\\hat{x}))$ as $j\\rightarrow \\infty $ .",
"A proof of existence of such sequences is essentially same as for [4] and not difficult, so we omit it.",
"Case 1: $\\beta \\ne 1$ .",
"Since $u_{\\alpha _j}$ is a subsolution of (REF ), $J_{t_j-a}^{\\alpha _j}[\\psi _j](t_j,x_j)+K^{\\alpha _j}_{t_j-a}[u_{\\alpha _j}](t_j,x_j)+H(t_j,x_j,u_{\\alpha _j}(t_j,x_j),D\\phi (t_j,x_j))\\le 0.$ Here $J_r^{\\alpha _j}$ and $K_r^{\\alpha _j}$ are associated with $\\alpha =\\alpha _j$ .",
"By similar arguments in previous sections it can be turns out that $\\liminf _{j\\rightarrow \\infty }(J_{t_j-a}^{\\alpha _j}[\\psi _j](t_j,x_j)+K^{\\alpha _j}_{t_j-a}[u_{\\alpha _j}](t_j,x_j))\\ge J_{\\hat{t}-a}^{\\beta }[\\phi ](\\hat{t},\\hat{x})+K_{\\hat{t}-a}^{\\beta }[u^\\sharp ](\\hat{t},\\hat{x}).$ Since $\\lim _{j\\rightarrow \\infty }H(t_j,x_j,u_{\\alpha _j}(t_j,x_j),D\\phi (t_j,x_j))=H(\\hat{t},\\hat{x},u^{\\sharp }(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x})),$ we find that $u^\\sharp $ is a subsolution of (REF ).",
"Case 2: $\\beta =1$ .",
"There are similar sequences $\\lbrace \\alpha _j\\rbrace _j$ and $\\lbrace (t_j,x_j)\\rbrace _j$ even for $\\varphi \\in C^1((0,T]\\times \\mathbb {R}^d)$ instead of $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ .",
"Since $(t_j,x_j)\\rightarrow (\\hat{t},\\hat{x})\\in (a,b]\\times B$ as $j\\rightarrow \\infty $ , we may assume that $\\lbrace (t_j,x_j)\\rbrace _j\\subset (\\hat{t}-3\\delta ,\\hat{t}]\\times B_{2\\delta }(\\hat{x})$ by considering large $j$ , where $\\delta >0$ is a constant such that $[\\hat{t}-3\\delta ,\\hat{t}]\\times \\overline{B_{2\\delta }(\\hat{x})}\\subset (a,b]\\times B$ .",
"Let $\\xi _1,\\xi _2:[0,T]\\times \\mathbb {R}^d\\rightarrow \\mathbb {R}$ be $C^\\infty $ functions such that $\\xi _1+\\xi _2=1$ on $[0,T]\\times \\mathbb {R}^d$ , $\\xi _1=1$ on $[\\hat{t}-\\delta ,\\hat{t}]\\times B_\\delta (\\hat{x})$ and $\\xi _2=1$ on $([0,T]\\times \\mathbb {R}^d)\\setminus ([\\hat{t}-2\\delta ,\\hat{t}]\\times B_{2\\delta }(\\hat{x}))$ .",
"Since $\\lbrace u_\\alpha \\rbrace _{\\alpha \\in (0,1]}$ is uniformly bounded on $\\overline{Q_T}$ , there is a constant $C>0$ such that $\\max _{\\overline{Q_T}}|u_{\\alpha _j}|\\le C$ for all $j$ .",
"Set $\\psi =\\xi _1\\phi +\\xi _2 M$ , where $M:=C+1$ .",
"Then $\\psi \\in C^1([0,T]\\times \\mathbb {R}^d)\\subset \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ and $u_{\\alpha _j}-\\psi $ attains a maximum at $(t_j,x_j)\\in (0,T]\\times \\mathbb {R}^d$ .",
"Thus we have $K_0[u_{\\alpha _j}](t_j,x_j)+H(t_j,x_j,u_{\\alpha _j}(t_j,x_j),D\\psi (t_j,x_j))\\le 0.$ Note that $K_0[u_{\\alpha _j}](t_j,x_j)\\ge \\partial _t^{\\alpha _j}\\psi (t_j,x_j)$ .",
"According to [10] we notice that $\\lim _{j\\rightarrow \\infty }\\partial _{t}^{\\alpha _j}\\psi (t_j,x_j)=\\partial _{t}\\psi (\\hat{t},\\hat{x}).$ Thus estimating $K_0[u_{\\alpha _j}](t_j,x_j)$ by $\\partial _t^{\\alpha _j}\\psi (t_j,x_j)$ in (REF ) and then passing to the limit $j\\rightarrow \\infty $ implies that $\\partial _{t}\\psi (\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u^\\sharp (\\hat{t},\\hat{x}),D\\psi (\\hat{t},\\hat{x}))\\le 0.$ Since $\\partial _t\\psi (\\hat{t},\\hat{x})=\\partial _t\\phi (\\hat{t},\\hat{x})$ and $D\\psi (\\hat{t},\\hat{x})=D\\phi (\\hat{t},\\hat{x})$ , $u^\\sharp $ is a subsolution of (REF ) with $\\alpha =1$ .",
"The comparison principle implies that $u^\\sharp \\le u_\\sharp $ on $\\overline{Q_T}$ but $u_\\sharp \\le u^\\sharp $ by their definition.",
"We hence see that $u:=u^\\sharp =u_\\sharp $ is a solution of (REF ) and $u(0,\\cdot )=u_0$ on $\\mathbb {T}^{d}$ .",
"Corollary REF ensures that $u=u_{\\beta }$ , a conclusion."
],
[
"Regularity results",
"Let consider one-dimensional transport equations of the form $\\partial _t^\\alpha u+\\partial _xu=0\\quad \\text{in $(0,\\infty )\\times \\mathbb {R}$}$ with prescribed initial value $u|_{t=0}=u_0\\in C(\\mathbb {R})$ .",
"In [34] for instance, a solution of this equation was given as $u(t,x)=\\frac{1}{t^{\\alpha }}\\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }\\left(-\\frac{z}{t^{\\alpha }}\\right)u_0(x-z)dz$ through the Laplace and the inverse Laplace transformation.",
"Here $W_{-\\alpha ,1-\\alpha }$ is Wright function defined by $W_{-\\alpha ,1-\\alpha }(z):=\\sum _{j=0}^{\\infty }\\frac{z^j}{j!\\Gamma (-\\alpha j+1-\\alpha )}.$ For properties and formulae for Wright function, see [36] and references therein.",
"It can be verified that this solution is indeed a unique viscosity solution.",
"We leave the detail of calculations to the reader.",
"Let us assume that (A4') $u_0$ is a Lipschitz continuous function with the Lipschitz constant $\\operatorname{\\text{Lip}}[u_0]$ .",
"We shall prove a continuity of solutions with respect to each variable.",
"In space, since $W_{-\\alpha ,1-\\alpha }\\ge 0$ on $(0,\\infty )$ and $\\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }(-z)dz=1,$ we have $|u(t,x)-u(t,y)|&\\le \\frac{1}{t^{\\alpha }}\\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }\\left(-\\frac{z}{t^{\\alpha }}\\right)|u_0(x-z)-u_0(y-z)|dz\\\\&\\le \\frac{1}{t^{\\alpha }}\\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }\\left(-\\frac{z}{t^{\\alpha }}\\right)dz\\operatorname{\\text{Lip}}[u_0]|x-y|=\\operatorname{\\text{Lip}}[u_0]|x-y|$ for $(t,x,y)\\in [0,T]\\times \\mathbb {R}\\times \\mathbb {R}$ .",
"In time, since $\\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }(-z)zdz=\\frac{1}{\\Gamma (\\alpha +1)}$ and $u$ given by (REF ) is rewritten as $u(t,x)=\\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }(-z)u_0(x-t^{\\alpha }z)dz,$ we have $|u(t,x)-u(s,x)|&\\le \\int _0^{\\infty }W_{-\\alpha ,1-\\alpha }(-z)|u_0(x-t^{\\alpha }z)-u_0(x-s^{\\alpha }z)|dz\\\\&\\le \\int _{0}^{\\infty }W_{-\\alpha ,1-\\alpha }(-z)zdz\\operatorname{\\text{Lip}}[u_0]|t^{\\alpha }-s^{\\alpha }|\\le \\frac{\\operatorname{\\text{Lip}}[u_0]}{\\Gamma (\\alpha +1)}|t-s|^{\\alpha }$ for $(t,s,x)\\in [0,T]\\times [0,T]\\times \\mathbb {R}$ .",
"These regularity results hold for solutions of (REF )-(REF ) with some general Hamiltonians $H$ under some Lipschitz continuity in $x$ for $H$ as for $\\alpha =1$ .",
"Lemma 6.1 (Lipschitz preserving) Assume (A1), (A3), (A4') and that there exist constants $L_1\\ge 0$ and $L_2>0$ such that $|H(t,x,r,p)-H(t,y,r,p)|\\le L_1|x-y|+L_2|x-y||p|$ for all $(t,x,y,r,p)\\in (0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d\\times \\mathbb {R}\\times \\mathbb {R}^d$ .",
"Let $u\\in C(\\overline{Q_T})$ be a solution of (REF )-(REF ).",
"Then $|u(t,x)-u(t,y)|\\le L(t)|x-y|$ for all $(t,x,y)\\in [0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ with $L(t)=\\left(\\operatorname{\\text{Lip}}[u_0]+\\frac{L_1}{L_2}\\right)E_{\\alpha }(L_2 t^{\\alpha })-\\frac{L_1}{L_2}.$ For $\\delta $ we set $\\Phi _{\\delta }(t,x,y):=u(t,x)-u(t,y)-L_{\\delta }(t)|x-y|,$ where $L_{\\delta }(t)=\\left(\\operatorname{\\text{Lip}}[u_0]+\\frac{L_1+\\delta }{L_2}\\right)E_{\\alpha }(L_2 t^{\\alpha })-\\frac{L_1+\\delta }{L_2}.$ Note that $L_{\\delta }\\in C^{1}((0,T])\\cap C([0,T])$ and $L_{\\delta }^{\\prime }\\in L^{1}(0,T)$ .",
"Suppose by contradiction that there would exit $\\delta >0$ such that $\\max _{[0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d}\\Phi _{\\delta }>0$ .",
"Let $(\\hat{t},\\hat{x},\\hat{y})\\in [0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ be a maximum point of $\\Phi $ .",
"Note that $\\hat{t}>0$ and $\\hat{x}\\ne \\hat{y}$ ; otherwise $0<\\Phi (\\hat{t},\\hat{x},\\hat{x})=0$ or $0<\\Phi (0,\\hat{x},\\hat{y})\\le 0$ since $u(0,\\cdot )=u_0$ is Lipschitz continuous and $E_{\\alpha }(0)=1$ .",
"Moreover $u(\\hat{t},\\hat{x})\\ge u(\\hat{t},\\hat{y})$ from $\\Phi _{\\delta }(\\hat{t},\\hat{x},\\hat{y})>0$ .",
"Since $u$ is a solution of (REF ), we have from Lemma REF $K_0[u](\\hat{t},\\hat{x})-K_0[u](\\hat{t},\\hat{y})+H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),L_{\\delta }(\\hat{t})\\hat{p})-H(\\hat{t},\\hat{y},u(\\hat{t},\\hat{y}),L_{\\delta }(\\hat{t})\\hat{p})\\le 0,$ where $\\hat{p}:=(\\hat{x}-\\hat{y})/|\\hat{x}-\\hat{y}|$ .",
"It follows from the definition of $\\Phi $ that $K_0[u](\\hat{t},\\hat{x})-K_0[u](\\hat{t},\\hat{y})\\ge K_0[L_{\\delta }](\\hat{t})|\\hat{x}-\\hat{y}|=(\\partial _t^\\alpha L_{\\delta })(\\hat{t})|\\hat{x}-\\hat{y}|.$ Hamiltonians are estimates as $&H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{x}),L_{\\delta }(\\hat{t})\\hat{p})-H(\\hat{t},\\hat{y},u(\\hat{t},\\hat{y}),L_{\\delta }(\\hat{t})\\hat{p})\\\\&\\ge H(\\hat{t},\\hat{x},u(\\hat{t},\\hat{y}),L_{\\delta }(\\hat{t})\\hat{p})-H(\\hat{t},\\hat{y},u(\\hat{t},\\hat{y}),L_{\\delta }(\\hat{t})\\hat{p})\\\\&\\ge -L_1|\\hat{x}-\\hat{y}|-L_2 L_{\\delta }(\\hat{t})|\\hat{x}-\\hat{y}|$ by (A3) and (A2') with $|\\hat{p}|=1$ .",
"Thus (REF ) is led to $[(\\partial _t^{\\alpha }L_{\\delta })(\\hat{t})-L_2L_{\\delta }(\\hat{t})-L_1]|\\hat{x}-\\hat{y}|\\le 0.$ Since $L_{\\delta }$ satisfies $\\partial _t^\\alpha L_{\\delta }-L_2L_{\\delta }-L_1=\\delta $ according to [10], we obtain $\\delta |\\hat{x}-\\hat{y}|\\le 0,$ a contradiction.",
"Consequently for any $\\delta >0$ we see that $|u(t,x)-u(t,y)|\\le L_{\\delta }|x-y|$ for all $(t,x,y)\\in [0,T]\\times \\mathbb {R}^d\\times \\mathbb {R}^d$ .",
"Letting $\\delta \\rightarrow 0$ yields the conclusion.",
"Lemma 6.2 Assume (A1), (A2), (A3) and (A4').",
"Let $u\\in C(\\overline{Q_T})$ be a solution of (REF )-(REF ).",
"Then there exists a constant $M>0$ depending only on $H$ , $u_0$ , $\\alpha $ and $T$ such that $|u(t,x)-u_0(x)|\\le M t^{\\alpha }$ for all $(t,x)\\in \\overline{Q_T}$ .",
"We find that $u^-(t,x):=u_0(x)-Mt^{\\alpha }$ and $u^+(t,x):=u_0(x)+Mt^{\\alpha }$ are a subsolution and a supersolution of (REF )-(REF ), respectively, where the constant $M$ is chosen so large that $M\\ge \\sup \\lbrace \\Gamma ^{-1}(\\alpha +1)|H(t,x,\\max _{\\mathbb {T}^d}|u_0|,p)| \\mid (t,x)\\in Q_T,|p|\\le \\operatorname{\\text{Lip}}[u_0],\\alpha \\in (0,1)\\rbrace .$ In fact, if $u^- -\\phi $ attains a maximum at $(\\hat{t},\\hat{x})\\in (0,T]\\times \\mathbb {R}^d$ for $\\phi \\in \\mathcal {C}^1([0,T]\\times \\mathbb {R}^d)$ , then $|D\\phi (\\hat{t},\\hat{x})|\\le \\operatorname{\\text{Lip}}[u_0]$ and $K_0[u^-](\\hat{t},\\hat{x})=-M \\partial _t^\\alpha t^\\alpha |_{t=\\hat{t}}=\\Gamma (1+\\alpha )M$ by using the formula $\\partial _t^{\\alpha }t^{\\alpha }=\\Gamma (1+\\alpha )$ derived from (REF ).",
"Therefore $K_0[u^-](\\hat{t},\\hat{x})+H(\\hat{t},\\hat{x},u^-(\\hat{t},\\hat{x}),D\\phi (\\hat{t},\\hat{x}))\\le -\\Gamma (1+\\alpha )M+H(\\hat{t},\\hat{x},\\max _{\\mathbb {T}^d}|u_0|,D\\phi (\\hat{t},\\hat{x}))\\le 0.$ by (A3) and the choice of $M$ since $u^-(t,x)=u_0(x)-Mt^\\alpha \\le \\max _{\\mathbb {T}^d}|u_0|$ .",
"Similarly, it is verified that $u^+$ is a supersolution of (REF ).",
"Theorem REF (comparison principle) yields to $u_0(x)-Mt^{\\alpha }=u^-(t,x)\\le u(t,x)\\le u^+(t,x)=u_0(x)+Mt^{\\alpha }$ on $\\overline{Q_T}$ , which is noting but the desired estimate.",
"Remark 6.3 Let $u_{\\alpha }$ be a solution of (REF )-(REF ) whose time-derivative's order is $\\alpha \\in (0,1]$ .",
"For $u_0\\in C(\\mathbb {T}^{d})$ not necessarily Lipschitz continuous, we observe that $\\sup \\lbrace |u_\\alpha (t,x)| \\mid (t,x)\\in \\overline{Q_T},\\alpha \\in (0,1]\\rbrace \\le \\max _{\\mathbb {T}^{d}}|u_0|+C\\max \\lbrace 1,T\\rbrace .$ Here $C>0$ is a large constant so that $C\\ge \\sup \\lbrace \\Gamma ^{-1}(\\alpha +1)|H(t,x,\\max _{\\mathbb {T}^d}|u_0|,0)| \\mid (t,x)\\in \\overline{Q_T},\\alpha \\in (0,1]\\rbrace .$ Indeed, $\\max _{\\mathbb {T}^{d}}|u_0|-Ct^{\\alpha }$ and $-\\max _{\\mathbb {T}^{d}}|u_0|+Ct^{\\alpha }$ are a (classical) subsolution and a (classical) supersolution of (REF )-(REF ), respectively.",
"Thus the comparison principle implies (REF ) once one realizes that $t^{\\alpha }\\le T^{\\alpha }\\le \\max \\lbrace 1,T\\rbrace $ for all $t$ , $T$ and $\\alpha \\in (0,1]$ .",
"Lemma 6.4 (Temporal Hölder continuity) Assume (A1), (A2), (A3) and (A4').",
"Let $u\\in C(\\overline{Q_T})$ be a solution of (REF )-(REF ).",
"Then for the same constant $M>0$ as in Lemma REF $|u(t,x)-u(s,x)|\\le M|t-s|^{\\alpha }$ for all $(t,s,x)\\in [0,T]\\times [0,T]\\times \\mathbb {T}^{d}$ .",
"Let $X$ be a set of subsolutions $v$ of (REF )-(REF ) such that $v$ satisfy (REF ) and $u_0-Mt^\\alpha \\le v\\le u_0+Mt^\\alpha $ on $\\overline{Q_T}$ .",
"Notice that $X\\ne \\emptyset $ since $u_0(x)-Mt^\\alpha \\in X$ due to Lemma REF .",
"Define $u=\\sup \\lbrace v\\mid v\\in X\\rbrace $ .",
"We show by Perron's method that $u$ is a solution of (REF )-(REF ) satisfying (REF ).",
"In view of Corollary REF it is enough to prove that $u$ is a solution of (REF ) satisfying (REF ).",
"In this proof we use same notations associated to the above $u$ as in Theorem REF .",
"It is not hard to see that $|u(t,x)-u(s,x)|\\le \\sup \\lbrace |v(t,x)-v(s,x)|\\mid v\\in X\\rbrace $ for all $t,s\\in [0,T]$ and $x\\in \\mathbb {T}^d$ .",
"Since $M$ does not depend unknown functions and $v$ satisfies (REF ), we see that $u$ satisfies (REF ).",
"Let us prove that $u$ is a solution of (REF ).",
"To do so we must show that the function $U$ satisfies (REF ).",
"All processes except for this step work to the current situation.",
"We first show that the function $w$ satisfies (REF ) near $(\\hat{t},\\hat{x})$ .",
"Expanding $w$ by Taylor formula, we have $w(s,y)-w(\\hat{t},\\hat{x})&=\\phi (s,y)-\\phi (\\hat{t},\\hat{x})-|s-\\hat{t}|^2-|y-\\hat{x}|^2\\\\&=a(s-\\hat{t})+p\\cdot (y-\\hat{x})+o(|s-\\hat{t}|+|y-\\hat{x}|)-|s-\\hat{t}|^2-|y-\\hat{x}|^2$ for $Q_T\\ni (s,y)\\rightarrow (\\hat{t},\\hat{x})$ , where $a=\\partial _t\\phi (\\hat{t},\\hat{x})$ and $p=D\\phi (\\hat{t},\\hat{x})$ .",
"For every $\\eta >0$ there exists $\\delta >0$ such that $|w(s,y)-w(\\hat{t},\\hat{x})|\\le ((|a|+\\eta )|s-\\hat{t}|^{1-\\alpha }+|s-\\hat{t}|^{2-\\alpha })|s-\\hat{t}|^\\alpha +(|p|+\\eta )|y-\\hat{x}|+|y-\\hat{x}|^2$ for all $(s,y)\\in \\overline{B_\\delta (\\hat{t},\\hat{x})}$ .",
"Fix such $\\eta >0$ .",
"For $\\delta $ so that $(|a|+\\eta )\\delta ^{1-\\alpha }+\\delta ^{2-\\alpha }\\le M$ , $w$ satisfies (REF ) on $\\overline{B_\\delta (\\hat{t},\\hat{x})}$ .",
"Let $\\rho $ be taken so small that $\\rho \\le \\sqrt{2}\\delta $ .",
"Then $\\Omega \\subset B_\\delta (\\hat{t},\\hat{x})$ .",
"Since $u$ and $w$ satisfy (REF ) in $((\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x}))\\cap B_{\\delta }(\\hat{t},\\hat{x})$ , it turns out by a similar inequality for the function $\\max \\lbrace u^*,w\\rbrace $ as (REF ) that $\\max \\lbrace u^*,w\\rbrace $ satisfies (REF ) in the same region.",
"Since $u^*>w$ in $((\\hat{t}-\\rho ,\\hat{t}+\\rho )\\times B_{2\\rho }(\\hat{x}))\\setminus \\Omega $ , in consequence, $U$ satisfies (REF ) in $\\overline{Q_T}$ , a conclusion."
],
[
"Another possible definition",
"In this section, for simplicity, we only treat Hamiltonians independent of $t$ and $r$ , i.e., $H=H(x,p)$ and assume (A1)-(A2).",
"Following to the usual style for viscosity solutions we are also able to define weak solutions of (REF ) as follows: Definition 7.1 (Provisional solutions) For a function $u\\in C(\\overline{Q_T})$ we call $u$ a provisional subsolution (resp.",
"provisional supersolution) of (REF ) if $(\\partial _{t}^{\\alpha }\\phi )(\\hat{t},\\hat{x})+H(\\hat{x},D\\phi (\\hat{t},\\hat{x}))\\le 0\\quad \\text{(resp.",
"$\\ge 0$)}$ whenever $u-\\phi $ attains a maximum (resp.",
"minimum) at $(\\hat{t},\\hat{x})\\in Q_T$ over $\\overline{Q_T}$ for $\\phi \\in \\mathcal {C}^{1}([0,T]\\times \\mathbb {R}^d)$ .",
"If $u\\in C(\\overline{Q_T})$ is a both provisional sub- and supersolution of (REF ), then we call $u$ a provisional solution of (REF ).",
"It is no difficulties to prove that provisional solutions of (REF ) are consistent with classical solutions of (REF ) if they belong to $\\mathcal {C}^{1}(\\overline{Q_T})$ ; cf.",
"Proposition REF and [31].",
"Although Definition REF looks good, there are some technical difficulties to handle provisional solutions.",
"We conclude this paper by sharing a main part of such difficulties.",
"They occurs in a proof of comparison principle.",
"Let $u$ and $v$ be respectively a provisional subsolution and a provisional supersolution of (REF ) such that $u(0,\\cdot )\\le v(0,\\cdot )$ on $\\mathbb {T}^{d}$ .",
"Suppose that $\\max _{\\overline{Q_T}}(u-v)>0$ and aim to derive a contradiction.",
"There is a small constant $\\eta >0$ such that $\\max _{(t,x)\\in \\overline{Q_T}}((u-v)(t,x)-\\eta t^{\\alpha })=:\\theta >0.$ For $\\varepsilon >0$ and $\\delta >0$ we consider the function $\\Phi (t,s,x,y):=u(t,x)-v(s,y)-\\frac{|x-y|^{2}}{2\\varepsilon }-\\frac{|t-s|^{2}}{2\\delta }-\\eta t^{\\alpha }.$ on $[0,T]^{2}\\times \\mathbb {T}^{2d}$ .",
"Let $(\\bar{t},\\bar{s},\\bar{x},\\bar{y})$ be a maximum point of $\\Phi $ .",
"From inequalities for provisional sub- and supersolutions, we have $\\left(\\partial _t^{\\alpha }\\frac{|\\cdot -\\bar{s}|^{2}}{2\\delta }\\right)(\\bar{t},\\bar{s})+\\left(\\partial _{s}^{\\alpha }\\frac{|\\bar{t}-\\cdot |}{2\\delta }\\right)(\\bar{t},\\bar{s})+\\eta \\Gamma (1+\\alpha )+H(\\bar{x},\\bar{p})-H(\\bar{y},\\bar{p})\\le 0.$ Here $\\bar{p}:=(\\bar{x}-\\bar{y})/\\varepsilon $ .",
"A similar argument is found in Section of this paper.",
"The third term comes from the last term of $\\Phi $ , i.e, $\\eta t^{\\alpha }$ .",
"Let us focus on the first and second terms.",
"A direct calculation implies that $\\partial _{t}^{\\alpha }|t-s|^{2}=\\frac{2(t-(2-\\alpha )s)t^{1-\\alpha }}{\\Gamma (3-\\alpha )}$ by the formula (REF ).",
"By changing the role of $t$ and $s$ , consequently, we get $\\begin{aligned}\\left(\\partial _t^{\\alpha }\\frac{|\\cdot -\\bar{s}|^{2}}{2\\delta }\\right)(\\bar{t})+\\left(\\partial _{s}^{\\alpha }\\frac{|\\bar{t}-\\cdot |^2}{2\\delta }\\right)(\\bar{s})&=\\frac{(\\bar{t}-(2-\\alpha )\\bar{s})\\bar{t}^{1-\\alpha }+(\\bar{s}-(2-\\alpha )\\bar{t})\\bar{s}^{1-\\alpha }}{\\delta \\Gamma (1-\\alpha )}\\\\&=\\frac{\\bar{t}^{2-\\alpha }+\\bar{s}^{2-\\alpha }-(2-\\alpha )(\\bar{s}\\bar{t}^{1-\\alpha }+\\bar{t}\\bar{s}^{1-\\alpha })}{\\delta \\Gamma (3-\\alpha )}.\\end{aligned}$ When $\\alpha =1$ , (REF ) vanishes.",
"Thus estimating Hamiltonians suitably (see the proof of Theorem REF in this paper) and then passing to the limit $\\varepsilon ,\\delta \\rightarrow 0$ in (REF ) yields the contradiction thanks to the third term.",
"On the other hand, the situation for $\\alpha \\in (0,1)$ is completely different.",
"Indeed, (REF ) possibly does not vanish and it is hard to control $(\\bar{t},\\bar{s})$ so that (REF ) is sufficiently small comparing to $\\eta \\Gamma (1+\\alpha )$ as $\\delta \\rightarrow 0$ .",
"There is a possibility that (REF ) diverges as $\\delta \\rightarrow 0$ as well.",
"To solve above difficulties let us consider the following problem: Problem 7.2 Find a function $\\psi \\in C^1((0,T]^2;\\mathbb {R})\\cap C([0,T]^2)$ satisfying $\\partial _t\\psi (\\cdot ,s)\\in L^1(0,T)$ for every $s\\in [0,T]$ and $\\partial _s\\psi (t,\\cdot )\\in L^1(0,T)$ for every $t\\in [0,T]$ such that ${\\left\\lbrace \\begin{array}{ll}\\partial _t^{\\alpha }\\psi (\\cdot ,s)+\\partial _s^{\\alpha }\\psi (t,\\cdot )\\ge 0\\quad &\\text{on $(0,T)^{2}$,}\\\\\\psi =0\\quad &\\text{on $\\lbrace t=s\\rbrace $ and}\\\\\\psi >0\\quad &\\text{on $[0,T]^{2}\\setminus \\lbrace t=s\\rbrace $.}\\end{array}\\right.",
"}$ If we could find such a function, then the contradiction would be obtained by handling $\\Psi (t,s,x,y):=u(t,x)-v(s,y)-\\frac{|x-y|^{2}}{2\\varepsilon }-\\frac{\\psi (t,s)}{2\\delta }-\\eta t^{\\alpha }$ instead of $\\Phi $ .",
"However, such a modification unfortunately does not overcome the difficulty yet.",
"Proposition 7.3 There is no function $\\psi $ solving Problem REF .",
"Suppose by contradiction that there is a function $\\psi $ solves Problem REF .",
"Then $\\psi $ should satisfy $(\\partial _t^\\alpha \\psi )(t,t)+(\\partial _s^{\\alpha }\\psi )(t,t)\\ge 0,$ that is, $\\int _0^t\\frac{(\\partial _t\\psi )(\\tau ,t)}{(t-\\tau )^{\\alpha }}d\\tau +\\int _0^t\\frac{(\\partial _s\\psi )(t,\\tau )}{(t-\\tau )^{\\alpha }}d\\tau \\ge 0.$ Since $\\psi (t,t)=0$ and $\\psi (\\cdot ,t)\\in C^{1}(0,T)$ , integration by parts implies that $\\int _0^t\\frac{(\\partial _t\\psi )(\\tau ,t)}{(t-\\tau )^{\\alpha }}d\\tau =-\\frac{\\psi (0,t)}{t^{\\alpha }}-\\alpha \\int _0^t\\frac{\\psi (t,\\tau )}{(t-\\tau )^{\\alpha +1}}d\\tau .$ Thus (REF ) is rewritten as $\\frac{\\psi (0,t)+\\psi (t,0)}{t^{\\alpha }}+\\alpha \\int _0^t\\frac{\\psi (t,\\tau )+\\psi (\\tau ,t)}{(t-\\tau )^{\\alpha +1}}d\\tau \\le 0.$ However the left-hand side is positive since $\\psi >0$ on $[0,T]^2\\setminus \\lbrace t=s\\rbrace $ , a contradiction."
],
[
"Acknowledgements",
"The authors are grateful to Professor Masahiro Yamamoto and members of his group including Ms. Anna Suzuki for valuable information on this topic.",
"The authors would like to thank Professor Adam Kubica for careful reading of the manuscript and beneficial discussions.",
"The authors are grateful to the anonymous referee for improving presentation of the paper.",
"The first author is partly supported by the Japan Society for the Promotion of Science (JSPS) through grant No.",
"26220702 (Kiban S) and No.",
"16H03948 (Kiban B).",
"The second author is supported by Grant-in-aid for Scientific Research of JSPS Fellows No.",
"16J03422 and the Program for Leading Graduate Schools, MEXT, Japan."
]
] | 1612.05408 |
[
[
"Terminal Singularities, Milnor Numbers, and Matter in F-theory"
],
[
"Abstract We initiate a systematic investigation of F-theory on elliptic fibrations with singularities which cannot be resolved without breaking the Calabi-Yau condition, corresponding to $\\mathbb Q$-factorial terminal singularities.",
"It is the purpose of this paper to elucidate the physical origin of such non-crepant singularities in codimension two and to systematically analyse F-theory compactifications containing such singularities.",
"The singularities reflect the presence of localised matter states from wrapped M2-branes which are not charged under any massless gauge potential.",
"We identify a class of $\\mathbb Q$-factorial terminal singularities on elliptically fibered Calabi-Yau threefolds for which we can compute the number of uncharged localised hypermultiplets in terms of their associated Milnor numbers.",
"These count the local complex deformations of the singularities.",
"The resulting six-dimensional spectra are shown to be anomaly-free.",
"We exemplify this in a variety of cases, including models with non-perturbative gauge groups with both charged and uncharged localised matter.",
"The underlying mathematics will be discussed further in a forthcoming publication."
],
[
"Introduction",
"This article investigates a certain class of singularities in elliptically fibered Calabi-Yau threefolds which cannot be resolved without breaking the Calabi-Yau condition.",
"Singularities of this type appear frequently in compactifications of F-theory and require new techniques for the computation of the massless spectrum of the associated effective field theory.",
"Our systematic investigation of such singularities reveals new contributions to the localised matter spectrum, and we provide the mathematical methods to compute them.",
"Indeed, geometric singularities play a distinguished role in compactifications of string and M-theory.",
"String theory is the ideal framework to study compactification on singular spaces: It contains just the right type of extended BPS objects to oftentimes render the lower-dimensional physics completely well-defined despite the appearance of a singularity from the perspective of classical geometry.",
"Indeed, when a singularity arises as a cycle shrinks to zero volume, wrapped BPS branes become massless and their inclusion spectacularly resolves the seeming singularity in the Wilsonian effective action of the string compactification [1].",
"Relatedly, geometric singularities typically signal an enhancement of the symmetries governing the effective physics, as they usually sit at the intersection of a Coulomb and Higgs branch of the effective field theory.",
"This general lore is at the heart of F-theory [2], [3], [4] and its dual formulation via M-theory compactified on an elliptically fibered Calabi-Yau space.",
"The latter provides a beautiful dictionary between the geometric structure of singularities in the elliptic fiber and the effective physics governing the dynamics of compactifications with 7-branes.",
"The traditional way to deal with such singularities is to perform a resolution and to infer the physics associated with the original, singular model by taking a suitable limit.",
"This procedure works particularly well if the singularities allow for a crepant resolution of the singular Calabi-Yau, which by definition does not change the canonical bundle of the space.",
"In particular, since the crepant resolution of a singular Calabi-Yau space is still Calabi-Yau, supersymmetry is preserved along the way.",
"A crepant resolution of the fibral singularities corresponds to moving along a flat direction in the Kähler moduli space by giving a non-zero volume to the vanishing curves in the fiber whose shrinking has created the singularity.",
"In this way zero modes from M2-branes wrapped around the vanishing cycles become massive.",
"In the singular limit, the wrapped M2-branes at singularities in codimension one give rise to non-abelian gauge bosons [5], and the resolution hence corresponds to moving away from the origin of the Coulomb branch.",
"Here and in the sequel the Coulomb branch we are referring to is the one of the M-theory compactification, e.g.",
"to $\\mathbb {R}^{1,4}$ for a Calabi-Yau threefold, not of the dual F-theory vacuum in one dimension higher (see also Section REF for a review).",
"As a result, one can make an association between the resolved fibers in codimension one and the affine Dynkin diagrams of A-D-E type.In fact in codimension one the Calabi-Yau and the associated Jacobian are isomorphic and one considers the fiber components which do not intersect the section of the Jacobian.",
"As two such codimension-one strata intersect, or one self-intersects, in codimension two on the base, new vanishing cycles might arise in the fiber.",
"These are in 1-1 correspondence with the weight system of representations of the Lie algebra of the model.",
"This is because M2-branes wrapping the codimension-two fiber curves form charged matter states.",
"The possible crepant resolutions are in a beautiful match with the different phases of the Coulomb branch [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"We will briefly review this connection further in Section REF as it plays a key role for our analysis.Singularities in codimension three and codimension four encode cubic [16], [17], [18] and, respectively, quartic [19], [20] Yukawa interactions in F-theory compactifications to four and two spacetime dimensions.",
"There is, however, an important difference between the structure of singularities in codimension one and higher of a singular elliptic Calabi-Yau as above: The first type always admits a crepant resolution; in the second case, by contrast, the singularity type might defy a resolution which does not break the Calabi-Yau condition.",
"In this article we elucidate the physical origin of non-crepant resolvable singularities in codimension two and find a way to systematically work with F-theory compactifications containing such singularitiesIn related work to appear, the second author, with Halverson and Shaneson, will consider a local version of a similar situation using a different method.. We focus on six-dimensional F-theory models on elliptically fibered Calabi-Yau threefolds and their dual M-theory compactifications, whose most important properties relevant to our analysis are summarized in Section .",
"As stressed already, from a physics perspective, a crepant resolution represents a flat direction in the classical Coulomb branch of the dual M-theory along which localised charged matter acquires a mass.",
"Turning tables around, we will argue in Section REF that a singularity without a crepant resolution must host massless matter from wrapped M2-branes which cannot be rendered massive along any flat direction in Coulomb branch.",
"This means that the localised matter must be uncharged, at least under any massless gauge potential of the compactification.",
"Indeed, examples of such behaviour have already appeared in [21], in [22], which studies the singular Jacobian of a genus one-fibration, and more recently in [23].",
"The relation between singularities and the presence of matter charged only under discrete gauge groups has been stressed further in [24], [25], [26].",
"As we will see in this work, the phenomenon of uncharged localised matter is, however, much more general.",
"In Section we identify a class of ${\\mathbb {Q}}$ -factorial terminal singularities, which arise naturally from Weierstrass models.",
"Their mathematical properties [27] allow us to deduce the precise number of localised matter states despite the absence of a Calabi-Yau resolution.",
"The idea is to interpret the localised uncharged states as part of the space of complex structure deformations of the singular variety.",
"The latter can be computed by employing its definition via the complex structure moduli space of a nearby deformation.",
"In this way, we find a formula for the number of localised hypermultiplets in terms of the dimension of the space of the local (versal) deformations of each codimension-two singularity.",
"This dimension is the Tyurina-Milnor number of the terminal codimension-two singularity (Section REF ).",
"The complex deformations have a natural splitting (summarised in Figure REF ) in terms of the third Betti number $b_3(X)$ of the singular threefold $X$ and the Tyurina numbers of the singularities (Section REF ).",
"The Kähler deformations are computed by $b_2(X)$ as in the smooth case (Section REF , and more generally in [27]), which gives the tensor multiplets.",
"Another aspect of our analysis is to establish a relation between the topological Euler characteristic and the number of hypermultiplets in Section REF .",
"Note that on singular spaces Poincaré duality and the Hodge decomposition might not hold and half the topological Euler characteristic is not the difference between the Kähler and complex structure deformations.",
"We stress that the modified relation which we find is the same for all the types of singularities we consider; however, different arguments are needed in different cases, as we discuss in Section REF and further in [27].",
"To apply these results concretely, we review and extend the systematic computation of [28] of the topological Euler characteristic on Calabi-Yau threefolds in appendix .",
"We illustrate our general findings in a number of examples of singular elliptic fibrations over a two-dimensional base $B_2$ .",
"In Section we analyze two models with non-trivial gauge group, but with $\\mathbb {Q}$ -factorial terminal singularities in codimension two, given by the ${\\rm I}_1$ model which has already been studied in [21], [29] as well as a non-perturbative model with enhancements of the form type II $\\rightarrow $ III over isolated points.",
"The validity of the spectra we find is checked to satisfy the stringent six-dimensional anomaly cancellation conditions.",
"While, given the physical explanation sketched above, the lack of a crepant resolution for models without a gauge group might not come as a surprise, we exemplify a similar phenomenon for a family of models with non-trivial gauge algebra.",
"Specifically, Section is dedicated to a family of Weierstrass fibrations with codimension-one fibers of type III, corresponding to gauge group $SU(2)$ .",
"As we will see, the singularities in codimension-two can become precisely of $\\mathbb {Q}$ -factorial terminal Kleinian type $A_n$ and hence defy a crepant resolution.",
"This implies that at such loci, both charged and uncharged hypermultplets localize.",
"We provide a partial resolution of these geometries and verify consistency of our claims by establishing an anomaly-free spectrum in the six-dimensional F-theory compactification.",
"These two classes of models are generalized further in appendix .",
"In appendix we also provide a detailed resolution of a Weierstrass model with singularities of type IV in codimension-one which, although crepant resolvable, we find interesting in itself.",
"We conclude in Section with directions for future research."
],
[
"F-theory on Resolvable Elliptic Threefolds",
"F-theory compactifications to six dimensions are defined in terms of an elliptically fiberedMore generally, one can consider F-theory on torus-fibrations which lack a zero-section [22].",
"Calabi-Yau threefold $Y_3$ with base $B_2$ , $\\begin{aligned}\\pi :\\quad \\mathbb {E}_\\tau \\ \\rightarrow & \\ \\ Y_3 \\cr & \\ \\ \\downarrow \\cr & \\ \\ B_2\\end{aligned}$ given in Weierstrass form as a hypersurface $P_W = 0$ with $P_W := - y^2 + x^3+f \\, x^2\\, z^4+g \\, z^6.$ Here $f,g$ are sections of ${\\mathcal {O}} ( -4 K_B)$ and ${\\mathcal {O}}( -6 K_B)$ , respectively, with $K_B$ denoting the canonical bundle of $B_2$ .",
"The fiber over points on $B_2$ where the discriminant $\\Delta = 4 \\, f^3 + 27 \\, g^2$ vanishes is singular, indicating the presence of a 7-brane.",
"The perhaps simplest non-trivial class of models contains a specific 7-brane stack along a divisor $\\Sigma _1$ parametrised locally by the vanishing of the coordinate $z_1=0$ on $B_2$ , together with a single 7-brane over a divisor $\\Sigma _0$ , required in order for the 7-brane tadpole to be cancelled.For simplicity, throughout this paper we assume that the Mordell-Weil rank of the elliptic fibration is trivial.",
"This assumption does not affect our results and may easily be dropped.",
"Such a situation, depicted schematically in figure REF , is modeled by choosing $f = z^{\\mu _f}_1 f_0, \\quad g= z_1^{\\mu _g} g_0 \\quad {\\rm such\\, \\, that} \\quad \\Delta = z_1^m \\, \\sigma _0,$ where $\\sigma _0 = 0$ defines the residual discriminant $\\Sigma _0$ and $f_0$ and $g_0$ are generic.",
"The Kodaira type of the fiber over generic points on $\\Sigma _1$ is determined by the vanishing order $(\\mu _f, \\mu _g,m)$ of $f$ , $g$ , and $\\Delta $ , with two special cases: If $(\\mu _f, \\mu _g,m) = (0,0,1)$ , the fiber is of type ${\\rm I}_1$ , corresponding to a singular nodal $\\mathbb {P}^1$ , and if $(\\mu _f, \\mu _g,m) = (1,1,2)$ , the fiber is a cuspidal $\\mathbb {P}^1$ , denoted as fiber type II.",
"In both cases $Y_3$ is smooth as a fibration.",
"In all other cases, the singularity of the fiber is also a singularity of $Y_3$ , indicating the presence of a non-trivial gauge algebra with associated gauge group $G$ on the 7-brane along $\\Sigma _1$ .We will not be very careful distinguishing between the gauge algebra and the gauge group [30].",
"For simplicity we furthermore assume that there are no extra abelian gauge group factors, an assumption which can easily be relaxed.",
"In this case, a resolution of $Y_3$ exhibits multi-component fibers classified by their Kodaira type.",
"Some of the fiber types encountered in this article are shown in figure REF .",
"For the reader's convenience we also include the vanishing orders appearing in Kodaira's classification and the associated gauge algebra in codimension one in table REF .",
"Figure: Our notation for an elliptically fibered Calabi-Yau manifold with Δ=Σ 1 ∪Σ 0 \\Delta = \\Sigma _1 \\cup \\Sigma _0.The singularity type of the fiber enhances further in codimension two, i.e.",
"over points on $B_2$ .",
"In the above setup, there are two types of such points: The first corresponds to the intersection $\\Sigma _1 \\cap \\Sigma _0$ consisting of $B_i$ points of type $P_i$ , where the index $i$ labels the fiber type over $P_i$ .",
"The second type of points are given by the points $Q$ where the residual discriminant acquires a cuspidal singularity (as a divisor in $B_2$ ).",
"Incidentally, over the latter points the fiber type enhances to type II, while for $P_i$ the specific fiber type depends on the vanishing orders of $f$ , $g$ , and $\\Delta $ .",
"Being the intersection between two divisors, points of type $P_i$ carry localised matter in form of massless hypermultiplets.",
"Our interest in this paper is in particular in this localised matter.",
"With a few exceptions, discussed in more detail in Section REF , the literature has focused on singular $Y_3$ which allow for a crepant resolution $\\hat{Y}_3$ , i.e.",
"a resolution such that $\\hat{Y}_3$ is itself Calabi-Yau.",
"As will be explained in more detail in Section REF , the existence of a crepant resolution implies that the total gauge group $G$ must be non-trivial, and furthermore that the matter hypermultiplets localised at the intersection points $P_i$ are charged, transforming in some representation $R$ of $G$ .",
"The 6d effective action of F-theory on $Y_3$ is given by an ${\\cal N}=(1,0)$ supergravity theory, which is well-known to be subject to a number of non-trivial constraints from the cancellation of gauge and gravitational anomalies.",
"Of particular interest for us is the famous condition $n_H - n_V + 29 \\, n_T = 273$ for the cancellation of gravitational anomalies.",
"Here $n_V = {\\rm dim}(G)$ is the number of vector multiplets, $n_T = h^{1,1}(B_2) -1$ counts the number of tensor multiplets and $n_H$ counts the total number of hypermultiplets.",
"In models with crepant resolution, the origin of the hypermultiplets is one of the following four: Each point $P_i$ gives rise to a localised, charged hypermultiplet in some representation $R$ of $G$ , and from the bulk of $\\Sigma _1$ one finds in addition $g$ hypermultiplets in some representation of $G$ , with $g$ being the genus of $\\Sigma _1$ .",
"Apart from this charged matter, there is one universal hypermultiplet containing the overall volume modulus as well as $h^{2,1}(\\hat{Y}_3)$ uncharged hypermultiplets associated with the complex structure moduli, i.e.",
"$n_H = n_H^0 + n_H^c, \\qquad \\quad n_H^0 = 1 + h^{2,1}(\\hat{Y}_3),$ where $n_H^0$ and $n_H^c$ denote the number of neutral and charged hypermultiplets, respectively.",
"Figure: Some Kodaira fiber types appearing in this paper.The computation of $h^{2,1}(\\hat{Y}_3)$ , and thus of $n_H^0$ , is facilitated by the fact that it appears in the topological Euler characteristic $\\chi _{\\rm top}(\\hat{Y}_3)$ of the smooth resolution $\\hat{Y}_3$ , $\\chi _{\\rm top}(\\hat{Y}_3) := \\sum _{i=1}^6 (-1)^i b_i(\\hat{Y}_3) = 2 \\left(h^{1,1}(\\hat{Y}_3) - h^{2,1}(\\hat{Y}_3) \\right) \\,.$ Taking furthermore into account that by the Shioda-Tate-Wazir theorem $h^{1,1}(\\hat{Y}_3) = 1 + h^{1,1}(B_2) + {\\rm rk}(G)$ one finds $n_H^0 = 2 + h^{1,1}(B_2) + {\\rm rk}(G) - \\frac{1}{2} \\chi _{\\rm top}(\\hat{Y}_3).$ Together with the anomaly cancellation condition (REF ) this in particular allows us to compute $n_H^c$ .",
"Since we will make heavy use of the explicit form of $\\chi _{\\rm top}(\\hat{Y}_3)$ , let us briefly summarize its computation, following a general algorithm described in detail in [28], [31]: Away from the discriminant locus, the elliptic fibration is locally a product $ \\mathbb {E}_\\tau \\times (B_2 - \\Delta )$ .",
"The product formula $\\chi _{\\rm top} (A \\times B) = \\chi _{\\rm top} (A) \\, \\chi _{\\rm top} (B)$ , where $A$ and $B$ are topological spaces, implies that $\\chi _{\\rm top} (\\mathbb {E}_\\tau \\times (B_2 - \\Delta )) = 0$ because $\\chi _{\\rm top}(\\mathbb {E}_\\tau )=0$ for the generic, i.e.",
"smooth, elliptic fiber.",
"Therefore $\\chi _{\\rm top}(\\hat{Y}_3)$ receives contributions only from the degenerate fibers in codimension one and two, which must be added up carefully, avoiding double counting and correcting for potential singularities of the discriminant as a divisor on $B_2$ .",
"The result of this computation is the expression [28], [31] $\\begin{split}\\chi _{\\text{top}}(\\hat{Y}_3) &= \\Big ( \\sum _i B_i \\cdot \\chi _{\\text{top}}(X_{P_i}) \\Big ) + m\\, \\Big (2-2g-\\sum _i B_i\\Big ) \\\\&\\quad - 132 K_B^2 + m \\,K_B \\cdot \\Sigma _1 + 2 m \\,\\Sigma _0 \\cdot \\Sigma _1 + m^2 \\,\\Sigma _1^2 + 3 \\, C+ \\sum _i \\epsilon _i B_i .\\end{split}$ Here $X_{P_i} = \\pi ^{-1}(P_i)$ denotes the degenerate fiber of the resolution space $\\hat{Y}_3$ over $P_i$ , and $B_i$ counts the number of points $P_i$ of a given type.",
"$C$ is the number of cuspidal points $Q$ of the residual discriminant $\\Sigma _0$ and the coefficients $\\epsilon _i$ correct for singularities of the discriminant at $P_i$ .",
"The computation of $X_{P_i}$ , $C$ and $\\epsilon _i$ is detailed in appendix , where we also describe the generalization of (REF ) to situations with several discriminant components.",
"Table: Kodaira-Tate classification of singular fibers, monodromy covers and gauge algebras as given in .",
"If the monodromy cover does not factorise, the gauge algebra is monodromy reduced as indicated."
],
[
"F-theory in Presence of Non-Crepant Singularities",
"In this section we discuss the physics and mathematics of F-theory compactifications with non-crepant resolvable singularities in codimension two.",
"We begin in Section REF with a brief review of the standard relation between crepant resolutions and unobstructed directions in the classical Coulomb branch of the dual M-theory.",
"This classic material is included for completeness and the expert reader can safely jump ahead.",
"We then explain, in Section REF , the general meaning of codimension-two non-crepant resolutions from the physics perspective.",
"Section REF introduces the mathematical background to quantitatively analyze such singularities.",
"In particular we introduce the notion of Kleinian $\\mathbb {Q}$ -factorial terminal hypersurface singularities and their Milnor-Tyurina number.",
"In section REF we establish the presence of uncharged localised hypermultiplets at such singularities, counted by the Milnor-Tyurina number, and we explain the meaning of the topological Euler characteristic in the presence of singularities in section REF ."
],
[
"Crepant Resolutions and the M-theory Coulomb Branch",
"Consider F-theory on $\\mathbb {R}^{1,5} \\times Y_3$ with $Y_3$ a Calabi-Yau threefold elliptically fibered over base $B_2$ .",
"This setup is dual to M-theory on $\\mathbb {R}^{1,4} \\times Y_3$ .",
"More precisely if one compactifies the 6d effective action of F-theory on a circle $S_1$ of radius $R$ , the resulting theory is identified with the effective action of M-theory on $Y_3$ .",
"The radius $R$ is the inverse of the volume of the generic elliptic fibre of $Y_3$ , $R = 1/{\\rm vol}({\\mathbb {E}}_{\\tau })$ , all measured in natural units.",
"[2], [32].",
"The 6d F-theory effective action is recovered as ${\\rm vol}({\\mathbb {E}}_{\\tau }) \\rightarrow 0$ .",
"More details of this correspondence and how to recover the F-theory effective action from M-theory can be found in [6], [33], [34].",
"Suppose now, as in Section , that the elliptic fibration $Y_3$ degenerates over the vanishing locus of the discriminant $\\Delta = \\Sigma _1 \\cup \\Sigma _0$ , and that the generic fibre over the divisor $\\Sigma _1 \\subset B_2$ exhibits a singularity associated with gauge group $G$ .Generalisations to setups with several gauge branes are obvious.",
"Consider the Cartan subalgebra of its Lie algebra, $\\oplus _{i = 1}^{{\\rm rk}(G)} \\mathfrak {u}(1)_i$ , with associated 6d gauge potentials ${\\mathbb {A}}^i$ .",
"Under circle compactification to $\\mathbb {R}^{1,4}$ , the component ${\\mathbb {A}}_{5}^i =: \\xi ^i$ of the gauge potential along $S^1$ maps to a scalar field.",
"Together with the vector components of ${\\mathbb {A}}^{i}$ along the five extended directions it forms the bosonic part of a 5d vector multiplet $(A^i, \\xi ^i)$ .",
"Unlike the 6d F-theory effective action, the 5d M-theory effective action possesses a Coulomb branch, parametrized by the vacuum expectation values (VEVs) of the scalar fields $\\xi ^i$ .",
"Consider now a resolution $\\hat{Y}_3$ of $Y_3$ .",
"This replaces the fibral singularities with a chain of rational curves $\\mathbb {P}^1_i$ , $i=1, \\ldots , {\\rm rk}(G)$ .",
"Their fibration over $\\Sigma _1$ is denoted by the resolution divisors $E_i$ .",
"Expanding the M-theory 3-form $C_3$ and the Kähler form $J$ as $C_3 = \\sum _i A^i \\wedge [E_i] + \\ldots , \\qquad \\quad J = \\sum _i \\xi ^i \\, [E_i] + \\ldots $ gives rise to the Cartan $U(1)_i$ vector potentials $A^i$ in the 5d M-theory effective action.",
"The scalars $\\xi ^i$ are identified with the Kähler moduli associated with the size of the resolution curves.",
"By means of this identification, a resolution of the singularity in the fibre therefore corresponds to moving in the 5d Coulomb branch by allowing for a non-zero VEV $\\langle \\xi ^i \\rangle \\ne 0$ , whereas the singular limit corresponds to the origin of Coulomb branch $\\langle \\xi ^i \\rangle =0$ [6].",
"More precisely, if the resolution is crepant, the resolved elliptic fibration $\\hat{Y}_3$ is still Calabi-Yau and supersymmetry is unbroken.",
"This describes a flat direction in the Coulomb branch.",
"Along such a direction in Coulomb branch, all 5d states states which are charged under the Cartan factor $u(1)_i$ acquire a mass.",
"This matches with the described field theoretic perspective on F/M-theory duality as a circle reduction as follows: A massless state in 6d maps to a Kaluza-Klein (KK) zero mode in 5d together with a full tower of KK states.",
"The mass of the KK zero mode of Cartan charges $q_i$ is given by $m_0 = \\sum _i q_i \\, \\xi ^i.$ This quantity has a simple geometric meaning: The KK zero modes in M-theory arise from M2-branes wrapped on suitable curves in the fiber.",
"In particular this includes states localised in codimension two, where new curves in the fiber arise from the splitting of some of the resolution $\\mathbb {P}^1_i$ .",
"Their charges $q_i$ are computed via the intersection numbers of the wrapped curve with the resolution divisors $E_i$ .",
"In view of (REF ) we can therefore identify (REF ) with the volume of this curve in the resolved space $\\hat{Y}_3$ .",
"This is true up to a sign which is explained in [6], [7], but which plays no role for the general argument.",
"Since a crepant resolution corresponds to a non-trivial volume of all fibral curves, this implies that on $\\hat{Y}_3$ also the codimension-two matter states of the 5d theory become massive.",
"The possible inequivalent resolutions of the singularity are in 1-1 correspondence with the different Weyl chambers along the 5d Coulomb branch [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"Note furthermore that in the singular F-theory limit, one recovers massless matter states in the 6d effective action."
],
[
"The Physics of Non-Crepant Resolvable Singularities in Codimension Two",
"We can now elucidate the meaning of codimension-two singularities in F-theory which lack a crepant resolution.",
"As a result of what we said in the previous subsection, such singularities arise whenever there is localized massless matter in the 6d effective action of F-theory which cannot acquire a mass in a supersymmetric way along a Coulomb branch in the dual M-theory.",
"This in particular implies that the matter state is not charged under any massless Cartan or non-Cartan abelian gauge group factor in the dual M-theory.",
"From the previous discussion it is clear that in such a case all flat directions along the Coulomb branch leave the matter state in question massless in M-theory.",
"This points to the existence of vanishing cycles in the fibre of the elliptic fibration which cannot be resolved into a non-zero volume holomorphic curve without destroying the Calabi-Yau condition, i.e.",
"without breaking supersymmetry.",
"An interesting class of examples includes situations where the matter state in question carries charge only under a so-called massive $U(1)$ or, more generally, only under a discrete $\\mathbb {Z}_k$ symmetry in the M-theory effective action.The distinction between massless abelian gauge groups in M-theory and F-theory is important, as described in detail in the context of discrete symmetries in F-theory versus M-theory in [25], [26].",
"Indeed, suppose the matter field is charged under a 5d gauge multiplet $(A^{\\rm m},\\xi ^{\\rm m})$ with mass $m$ .",
"Clearly, the mass term of the scalar field $\\xi ^{\\rm m}$ obstructs the Coulomb branch and enforces $\\langle \\xi ^{\\rm m}\\rangle = 0$ .",
"Such massive vector multiplets are described in M-theory by expanding the 3-form $C_3$ along a pair of non-harmonic 2- and 3-forms $({\\rm w}_2, \\alpha _3)$ as [35], [36], [37] $C_3 = A^{\\rm m} \\wedge {\\rm w}_2 + c \\, \\alpha _3$ with the property $d {\\rm w}_2 = k \\alpha _3, \\qquad k \\in \\mathbb {Z}.$ This relation identifies $\\alpha _3$ as a torsional form, i.e.",
"as an element of ${\\rm Tor} (H^3( Y_3, \\mathbb {Z}))$ .",
"Dimensional reduction of the kinetic term for $C_3$ yields a Stückelberg coupling of the form $S \\simeq \\int _{\\mathbb {R}^{1,4}} (d c + k A^{\\rm m}) \\wedge \\ast (d c + k A^{\\rm m}),$ which signals the presence of a $\\mathbb {Z}_k$ symmetry in the 5d effective action: The shift symmetry enjoyed by the axionic field $c$ is gauged, $A^{\\rm m} \\rightarrow A^{\\rm m} + d \\chi , \\qquad c \\rightarrow c - k \\chi $ and $c$ becomes the longitudinal component of the massive vector field $A^{\\rm m}$ .",
"Note that the special case $k=1$ corresponds to a complete breaking of the $U(1)$ gauge symmetry associated with $A^{\\rm m}$ , whereas for $k >1$ a remnant $\\mathbb {Z}_k$ symmetry governs the 5d (and 6d) effective action.",
"Now, if some massless matter state localised in codimension two carries charge only under the remnant $\\mathbb {Z}_k$ symmetry, and is uncharged under any other massless $U(1)$ gauge symmetry, then by the above arguments the Calabi-Yau $Y_3$ will exhibit a non-crepant resolvable singularity in codimension two.",
"Examples of such singularities have already appeared in the literature: For the special case $k=1$ , the ${\\rm I}_1$ -model studied in [21] (and further in [29]) contains a non-crepant conifold singularity which, from a Type IIB perspective, is expected to host localised matter.",
"Indeed, ref.",
"[21] stressed the relation between the presence of a massive $U(1)$ symmetry and the occurrence of a conifold singularity in codimension-two which only admits a small non-Kähler resolution.",
"The cases $k>1$ correspond to the singular Weierstrass models [22] forming the Jacobian of torus-fibrations with a $k$ -section: In the Weierstrass model the location of matter charged only under $\\mathbb {Z}_k$ symmetry leads to a non-crepant singularity in codimension two.Note that, by contrast, the associated $k$ -section fibrations are smooth because these describe a different M-theory background in which the matter field in question does carry charge under a massless $U(1)$ symmetry [24], [25], [26] (see also [38]).",
"The models agree only in the F-theory limit, but the geometry as such is sensitive to the M-theory effective action as opposed to its F-theory uplift.",
"Geometries associated with discrete symmetries in F-theory have been studied intensively recently, including [22], [24], [38], [25], [26], [39], [40].",
"In the sequel we will find further examples non-crepant resolvable singularities in co-dimension two and establish the presence of massless matter responsible for the singularity."
],
[
"Mathematical Background",
"In this work we analyze mostly non-smooth Calabi-Yau threefolds.",
"While there are many examples of smooth Calabi-Yau varieties, more generally, elliptically fibered Calabi-Yau varieties have singularities.",
"We have already motivated the appearance of such singularities in our context of codimension-two singular fibers.",
"Independent of this, in compactifications on higher-dimensional singular spaces, new striking features appear in the associated physics, which illustrates the importance of studying such singularities even further: For instance [41] presents an example for which the crucial features of the physics are captured by a singular Calabi-Yau (with terminal singularities), but not by the smooth birationally equivalent minimal one.",
"In fact, in dimension higher than three, a minimal model can be smooth while other models in the same minimal class are singular.",
"Other recent appearances of non-resolvable singularities in F-theory with rather different physics interpretation include [42], [43], [23].",
"In this section we provide some mathematical background on the types of codimension-two singularities which we will study in this paper."
],
[
" ${\\mathbb {Q}}$ -Factoriality, Canonical and Terminal Singularities",
"Let $X$ be a complex algebraic (normal) variety.",
"A Weil divisor is a formal linear combination of codimension-one subvarieties.",
"A Cartier divisor on the other hand has the property that it can be locally written as the vanishing locus of a single function on $ X$ .",
"If $X$ is smooth, then all Weil divisors are Cartier; this is true more generally if $X$ is factorial, that is if every local ring is a unique factorization domain [44].",
"On the other hand, if $X$ is the singular quadric in ${\\mathbb {P}}^3$ of equation ${ x_0 \\cdot x_1- x_2^2=0 }$ , the weighted projective surface ${\\mathbb {P}}[1,1,2]$ , the divisors $D_0$ and $D_1$ , defined by the equations $x_0=x_2=0$ and $x_1=x_2=0$ are Weil divisors, but not Cartier.",
"Note that $2D_0$ and $2D_1$ are Cartier.",
"We say that $ X$ has ${\\mathbb {Q}}$ -factorial singularities if for every Weil divisor $D$ there exists an integer $r$ such that $rD$ is Cartier, or equivalently that every Weil divisor is $\\mathbb {Q}$ -Cartier.",
"Let $X$ be a complex algebraic variety.",
"A resolution of $X$ is a birational morphism $\\rho : \\widetilde{X} \\rightarrow X$ from a smooth variety $\\widetilde{X}$ to $X$ .",
"This means that there are (dense) open sets $\\mathcal {V} \\subset X$ and $\\mathcal {U} \\subset \\widetilde{X}$ such that $ \\rho _{\\mathcal {U}} : \\mathcal {U} \\simeq \\mathcal {V}$ .",
"The remaining locus $ {\\rm Ex} (\\rho )=\\widetilde{X} \\setminus \\mathcal {U}$ is the exceptional locus of $\\rho $ .",
"The components of codimension one in the exceptional locus are the exceptional divisors $E_i$ .",
"$X$ is ${\\mathbb {Q}}$ -Gorenstein if there exists some integer $r$ such that $rK_X$ is a line bundle (that is $K_X$ is ${\\mathbb {Q}}$ -Cartier).",
"When $K_X$ is Cartier $X$ is said to be Gorenstein.",
"In particular a Calabi-Yau variety is Gorenstein.",
"For a ${\\mathbb {Q}}$ -Gorenstein variety $X$ and its resolution $\\tilde{X}$ we can compare the bundles $rK_{\\widetilde{X}}$ and $rK_X$ , $rK_{\\widetilde{X}} = \\rho ^*rK_X + \\sum _i a_i r E_i \\,.$ The $a_i$ are called the discrepancies.",
"Such a resolution always exists [45], [46] and it is easy to see that the discrepancies are independent of the choice of the resolution.",
"If $a_i \\ge 0 \\ \\forall i$ , $X$ is said to have at worst canonical singularities.",
"If $a_i > 0$ for all $i$ , $X$ is said to have at worst terminal singularities.",
"Reid showed that if $X$ has at worst canonical singularities, then $H^0(mrK_X)=H^0(mrK_{\\widetilde{X}}), \\ \\forall \\ m \\in {\\mathbb {N}}$ .",
"Note that a smooth variety has at worst terminal singularities.",
"If $X$ is a surface it can be shown that $X$ has at worst terminal singularities if and only if it is smooth.",
"The canonical surface singularities are the A-D-E singularities (rational double points).",
"These are also the Gorenstein surface singularities, see for example [47], [48].",
"In particular if $\\dim (X)=2$ and $X$ is a Weierstrass model with singularities, then its singularities are canonical.",
"If for a particular resolution $a_i=0 \\ \\forall i$ , the resolution is called crepant, and we refer to these singularities as crepant resolvable.",
"A crepant resolution of a singular Calabi-Yau variety hence remains Calabi-Yau.",
"For example an elliptically fibered threefold $X$ given by a Weierstrass model which is equisingular along a smooth curve has canonical singularities and a crepant resolution.",
"More generally a Weierstrass model has canonical singularities in codimension one.",
"This means that the non-crepant resolvable singularities of such a model must be due to enhancements in codimension two or higher.",
"A morphism $ \\varphi : \\widetilde{X} \\rightarrow X$ where $\\widetilde{X}$ is smooth and all the components of the exceptional loci have codimension greater than one is called a small resolution.",
"In this case $\\widetilde{X}$ and $X$ are isomorphic in codimension one.",
"For example, the nodal quintic threefold $X \\subset {\\mathbb {P}}^3$ of equation $ x_0g_0 + x_1 g_1=0$ with general polynomials $g_0$ and $g_1$ has 16 nodal isolated singularities and a small resolution $ \\phi : \\widetilde{X} \\rightarrow X$ , obtained by blowing up the plane $x_0=g_1=0$ ; the exceptional loci consist of 16 disjoint ${{\\mathbb {P}}}^1$ s. Because small resolutions of singularities preserve the Calabi-Yau condition, we also refer to the corresponding singularities as resolvable by a crepant resolution.",
"Alternatively, we could resolve $X$ by a big resolution with exceptional divisors.",
"However, the nodal singularities are terminal in the sense that the appearing exceptional divisors have positive discrepancy.",
"A small algebraic resolution of the nodal quintic threefold above is possible because the singularities are not ${\\mathbb {Q}}$ -factorial, as we see in the following."
],
[
" ${\\mathbb {Q}}$ -factorial and analytic {{formula:891e0fa0-4e14-4862-a2b0-b134a4f80222}} -factorial singularities",
"In the class of birationally equivalent elliptic fibrations of Calabi-Yau threefolds there is a model $ X$ with $K_{X} \\simeq \\mathcal {O}_{X}$ where $X$ has terminal singularities, but $X$ is not necessarily smooth [49], [50].",
"The model $X$ has however ${\\mathbb {Q}}$ -factorial singularities, namely every Weil divisor is also $\\mathbb {Q}$ -Cartier.",
"Kawamata [51] showed that if the singularities are at worst canonicalKawamata proves the result assuming that the singularities are rational; canonical and terminal singularities are rational.",
"then the quotient of the Weil divisors by the Cartier divisors has finite dimension, and its rank is denoted by $\\sigma (X)$ .",
"$X$ is ${\\mathbb {Q}}$ -factorial if and only this group is torsion, that is if $\\sigma (X)=0$ .",
"Kawamata shows that if $X$ is not ${\\mathbb {Q}}$ -factorial, there exists a small projective (Kähler) birational morphism $\\phi : X_1 \\rightarrow X$ , where $X_1$ is ${\\mathbb {Q}}$ -factorial.",
"For example the nodal quintic threefold $X: \\ x_0g_0 + x_1 g_1=0$ is not ${\\mathbb {Q}}$ -factorial, as the divisor $x_0=g_1=0$ must be defined by two equations; the small birational morphism $\\phi $ provides a small projective resolution.",
"When the isolated singularity is toric, then there is a nice criterion: the singularity is ${\\mathbb {Q}}$ -factorial if and only if the maximal cone corresponding to the toric singular point is simplicial.",
"If the cone is not simplicial, the small resolution is achieved by a simplicial subdivision of the cone.",
"In many instances, a Calabi-Yau threefold is ${\\mathbb {Q}}$ -factorial, but after an analytic change of coordinates the local equation is $f(z,x_1,x_2,x_3) = z^2 + x_1^2 + x_2^2 + x_3^2.$ Such behaviour characterizes the singularities in the Weierstrass model of [21], studied in Section REF , as well as the singular Jacobians of [22].",
"Both singularities correspond to fiber enhancements ${\\rm I}_1 \\times {\\rm I}_1 \\rightarrow {\\rm I}_2$ .",
"Another example of this behaviour which we will study is a particular enhancement of type ${\\rm III} \\times {\\rm I}_1 \\rightarrow {\\rm I}_0^*$ in Section REF .",
"In these examples there exist two local independent analytic, but not algebraic, Weil divisors which are not $\\mathbb {Q}$ -Cartier.",
"This motivates the following definition: $(\\mathcal {U}, p)$ is locally analytically $\\mathbb {Q}$ -factorial if every analytic Weil divisor in a neighborhood of $p$ is $\\mathbb {Q}$ -Cartier.",
"More generally, an important class of isolated hypersurface singularities are the $A_{a-1}$ Kleinian singularities: In a neighborhood of the singular point $P$ , $(\\mathcal {U}, p)$ is (analytically) the zero-locus of $f(z,x_1,x_2,x_3) = z^a + x_1^2 + x_2^2 + x_3^2 \\qquad {\\rm with} \\qquad a \\ge 2 \\in \\mathbb {N} \\,.$ These are terminal (and non-canonical) singularities [52].",
"A local, possibly non-projective (non-Kähler) small resolution is possible if and only if $a$ is even, [52], [53], [54].",
"In particular if $a$ is odd, then no crepant or small resolution is possible and these singularities are also ${\\mathbb {Q}}$ -factorial and analytically $\\mathbb {Q}$ -factorial [55].",
"The isolated singularities in this paper happen to be Kleinian: These are the Kodaira fiber ${\\rm II} \\times {\\rm I}_1 \\rightarrow {\\rm III}$ and ${\\rm II} \\times {\\rm I}_2 \\rightarrow $ IV enhancements in the models of Section REF as well as certain types of Kodaira fiber ${\\rm III} \\times {\\rm I}_1 \\rightarrow {\\rm I}_0^\\ast $ enhancements in Section .",
"If $a$ is even, or more generally for other hypersurface equations, a careful global analysis is needed to determine if a projective (Kähler) small resolution exists, see for example [56].",
"Finally let us note an important point: Given a three-dimensional Calabi-Yau Weierstrass model, it is always possible to resolve the singularities in codimension one in a crepant way.",
"However, there may remain $\\mathbb {Q}$ -factorial terminal singularities in codimension two.",
"These are always analytic hypersurface singularities, see for example [48].",
"Technically, this means that a Weierstrass model can always be resolved into a terminal $\\mathbb {Q}$ -factorial model.",
"The Kleinian singularities we are studying in this paper are a special type of these $\\mathbb {Q}$ -factorial terminal hypersurface singularities."
],
[
"Milnor and Tyurina Numbers, and Versal Deformations",
"An important concept for us is the characterization of hypersurface singularities via their Milnor and Tyurina numbers.",
"Let $\\mathcal {U}$ be a neighborhood of an isolated hypersurface singularity $P$ , that is ${\\mathcal {U}}= f^{-1}(0)$ , where $f: \\mathbb {C}^{n+1} \\rightarrow \\mathbb {C}$ , and consider a local smoothing $\\mathcal {U}_t= f^{-1}(t)$ .",
"Let $D_\\epsilon $ denote a ball of radius $\\epsilon $ centered at $0 \\in \\mathbb {C}^{n+1}$ .",
"Milnor showed that for $\\epsilon >0$ small enough, $B_\\epsilon = {\\mathcal {U}}^t \\cap D_\\epsilon $ is homologically a bouquet of $n$ -spheres, where $B_\\epsilon $ is called the Milnor fiber of $P$ .",
"The Milnor number characterizing this singularity is $m_P = b_n(B_\\epsilon ),$ the $n$ th Betti number of the ordinary simplicial homology.",
"Equivalently [57], $m_P = \\dim _{\\mathbb {C}} \\mathcal {A}_f = \\dim _{\\mathbb {C}} \\left(\\mathbb {C}\\lbrace x_1, \\ldots , x_{n+1} \\rbrace / \\left\\langle \\frac{\\partial f}{\\partial x_1}, \\ldots \\frac{\\partial f}{\\partial x_{n+1}} \\right\\rangle \\right).$ For example, an $A_{a-1}$ Kleinian singularity as in (REF ) has Milnor number $m_P = a-1$ , since $\\mathcal {A}_f = \\mathbb {C}\\lbrace z,x_1, \\ldots , x_{3} \\rbrace / \\langle z^{a-1},x_1,x_2,x_3 \\rangle = \\langle 1,z,z^2,\\ldots ,z^{a-2} \\rangle $ and the number of generators of $\\mathcal {A}_f$ is $a-1$ .",
"A related concept, from the algebraic point of view, is the Tyurina number $\\tau _P$ , which counts the dimension of the space of versal deformations of the hypersurface singularity at $P$ in $\\mathcal {U}$ .",
"The Tyurina number is computed algebraically as $\\tau _P = \\dim _{\\mathbb {C}} \\mathcal {B}_f = \\dim _{\\mathbb {C}} \\left(\\mathbb {C}\\lbrace x_1, \\ldots , x_{n+1} \\rbrace / \\left\\langle f, \\frac{\\partial f}{\\partial x_1}, \\ldots \\frac{\\partial f}{\\partial x_{n+1}} \\right\\rangle \\right).$ In general $\\tau _P \\le \\mu _P$ .",
"Saito showed that $\\tau _P = \\mu _P$ if and only if $P$ is a weighted hypersurface singularity, that is if there exist weights $(d_1, \\ldots d_{n+1})$ and $d$ such that $f(\\lambda ^{d_1} x_1, \\ldots , \\lambda ^{d_{n+1}} x_{n+1} ) = \\lambda ^d f(x_1, \\ldots x_{n+1})$ for all $\\lambda \\in \\mathbb {C}$ [57].",
"A generalization of this result for complete intersections is proven by Greuel [58].",
"The Tyurina and Milnor number can be computed by SINGULAR [59] and Maple [60].",
"In particular, the $A_{a-1}$ Kleinian singularities of (REF ) are weighted hypersurface singularities, and hence the Tyurina and Milnor numbers agree, $\\mu _P = \\tau _P = a-1 \\qquad {\\rm for} \\qquad (\\ref {A_a-1sing}) \\,.$"
],
[
" Hypermultiplets in Presence of $\\mathbb {Q}$ -Factorial Terminal Singularities ",
"The presence of singularities makes the computation of the spectrum of massless moduli fields of a string compactification more involved.",
"As reviewed in Section , in F-theory on a smooth Calabi-Yau threefold $X$ , the number of tensor and uncharged hypermultiplets is related to the dimensions of the space of Kähler deformations and of complex structure deformations.",
"We now describe the situation in the presence of isolated hypersurface singularities and describe methods to compute in particular the hypermultiplet spectrum in the presence of ${\\mathbb {Q}}$ -factorial Kleinian terminal singularities of type $A_{a-1}$ with either $a$ odd or $a=2$ , as these are the type of singularities which occur in the examples we consider here .The methods developed in [27] are, however, more general, and in particular hold for rational homology manifolds."
],
[
"Kähler deformations of Singular Threefolds",
"For F-theory compactified on a smooth Calabi-Yau threefold $X$ , with zero Mordell-Weil rank, the number of tensor multiplets $n_T$ is given by $n_T+2 + \\operatorname{rk} (\\mathfrak {g})=\\mathrm {KaDef}(X) \\,,$ with $\\mathrm {KaDef}(X)$ the dimension of the space of Kähler deformations.",
"This formula continues to hold in the presence of isolated singularities on a threefold $X$ .",
"When $X$ is smooth then $ \\mathrm {KaDef}(X)= h^{1,1}(X)=b_2(X), $ where $b_2(X)$ is the second Betti number of the ordinary (simplicial) homology.",
"It turns out that if $X$ is a Calabi-Yau with terminal singularities, and under more general assumptions spelled out in the companion paper [27], the equalities $\\mathrm {KaDef}(X)= b_2(X)$ and $n_T+2 + \\operatorname{rk} (\\mathfrak {g})=b_2(X)$ continue to hold."
],
[
" Hypermultiplets and Complex Deformations",
"More subtle is the number $n_H^{0}$ of uncharged hypermultiplets.",
"On a smooth Calabi-Yau threefold $X$ , this is related to the dimension of the Kuranishi space of complex structure deformations $\\mathrm {CxDef}(X)$ via $n_H^{0} = 1 + \\mathrm {CxDef}(X) \\,.$ When $X$ is smooth, it is furthermore a classic result that $\\mathrm {CxDef}(X)= h^{2,1}(X)= \\frac{1}{2} b_3(X) -1$ .",
"Now, in the presence of isolated singularities on a threefold $X$ , the relation (REF ) is also still valid as each complex structure deformation corresponds to a massless modulus of the metric.",
"However, in the presence of singularities one cannot use the formula $\\mathrm {CxDef}(X)= h^{2,1}(X)= \\frac{1}{2} b_3(X) -1$ to compute the number of complex structure deformations.",
"The reason for this is explained in more detail in [27].",
"We show now how to calculate $\\mathrm {CxDef}(X)$ when $X$ is a singular Calabi-Yau with ${\\mathbb {Q}}$ -factorial terminal hypersurfaces singularities.",
"Note that the resolutions of the general singularities (in codimension two) of Weierstrass models have indeed isolated hypersurface singularities.",
"The proofs are presented in the companion paper [27].",
"Results of Namikawa and Steenbrink [61] imply that if $X$ is a ${\\mathbb {Q}}$ -factorial Calabi-Yau threefold with isolated terminal hypersurface singularities, $X$ admits a smoothing to a smooth Calabi-Yau $X^t$ .",
"The dimension of the complex deformation of $X$ is then given by the dimension of the complex deformation space of $X^t$ .",
"Since $\\mathrm {CxDef}(X^t) = h^{2,1}(X^t) = \\frac{1}{2} b_3(X^t) - h^{3,0}(X)$ , we then have the relation $\\mathrm {CxDef}(X)= \\mathrm {CxDef}(X^t) =\\frac{1}{2} b_3(X^t)-1.$ Namikawa and Steenbrink show more generally that if $X^t$ is a smooth deformation of a (normal projective) threefold $X$ with isolated hypersurface singularities and $h^2(X, \\mathcal {O}_X)=0$ , then $b_3(X^t) = b_3(X) + \\sum _P m_P - \\sigma (X) \\,.$ Here $\\sigma (X) $ is the rank of the quotient of the Weil divisors by the Cartier divisors, and $m_P$ is the Milnor number of the singular point $P$ , both defined in Section REF .",
"Combining equations (REF ) and (REF ) we obtain: $\\mathrm {CxDef}(X) = \\frac{1}{2} \\big ( b_3(X) + \\sum _P m_P - \\sigma (X) \\big ) - 1.$ Under the same hypothesis Namikawa and Steenbrink show also that [61] that $ \\sigma (X) = b_4(X) - b_2(X)$ ; in fact, Poincaré duality does not necessarily hold.",
"For example, we can use the above formula to calculate $\\mathrm {CxDef}(X)$ for Calabi-Yau varieties with conifold singularities which are not ${\\mathbb {Q}}$ -factorial; recall that $X$ is ${\\mathbb {Q}}$ -factorial if and only if $ \\sigma (X) =0$ .",
"The important point for applications in this paper is that if $X$ is a Calabi-Yau variety with ${\\mathbb {Q}}$ -factorial terminal singularities, then $\\mathrm {CxDef}(X) = \\frac{1}{2} \\big (b_3(X) + \\sum _P m_P \\big ) - 1 \\,.$ Note that only the sum $\\frac{1}{2} b_3(X) + \\frac{1}{2}\\sum _P m_P$ is guaranteed to be integer, whereas each individual term may fail to be so because the Hodge decomposition and Hodge duality might not hold; these points are discussed further in [27].",
"It follows that the number of uncharged hypermultiplets of equation (REF ) is given by $\\begin{aligned}n_H^{0} &= 1 + \\mathrm {CxDef}(X)\\\\& = \\frac{1}{2} ( b_3(X) + \\sum _P m_P ).\\end{aligned}$"
],
[
"Uncharged Localised Hypermultiplets ",
"In Section REF we had argued that in the presence of ${\\mathbb {Q}}$ -factorial terminal (or more generally non-crepant resolvable) codimension-two singularities we expect localised massless uncharged matter.",
"This implies a split of the total number of uncharged hypermultiplets into localised versus non-localised uncharged hypermultiplets of multiplicity $n_{H, l}^0$ and $n_{H,n-l}^0$ : $n_H^0 = n_{H,n-l}^0 + n_{H, l}^0 \\,.$ The uncharged localised hypermultiplets are to be interpreted as part of the Kuranishi space, i.e.",
"the space of complex structure deformations of the singular space $X$ .We count the universal hypermultiplet as part of the $n_{H,n-l}^0$ non-localised hypermultiplets.",
"The split (REF ) implies a natural decomposition of the Kuranishi space of $X$ into two spaces $K_{n-l}$ and $K_{l}$ [27].",
"The space $K_{l}$ is the space of complex structure deformations of $X$ which deform the isolated singularities, by changing their singularity into a milder singularity type (or completely smoothening them out).",
"These are precisely the versal deformations, and the dimension of this space is counted by the Tyurina number.",
"In fact, in our hypothesis $m_P= \\tau _P$ , see (REF ).",
"The remaining deformations of $X$ are deformations which do not change the location or form of the isolated singularities.",
"This suggests identifying the localised uncharged hypermultiplets with the metric moduli counted by the versal deformations such that $n_{H, l}^0 = \\sum _P \\tau _P= \\sum _P m_P,$ while the non-localised uncharged hypermultiplets are due to the remaining deformations of $X$ .",
"We therefore find thatNote also here that $\\frac{1}{2} ( b_3(X) - \\sum _P m_P)$ is an integer.",
"$n_H^0 &=& \\frac{1}{2} \\big ( b_3(X) - \\sum _P m_P\\big ) + \\sum _P m_P \\\\&=& n_{H,n-l}^0 + n_{H, l}^0.$ Figure: Origin of uncharged hyper multiplets in six-dimensional F-theory compactifications.The identification of the versal deformation moduli with localised hypermultiplets is indeed very natural given the general relation between deformations and Higgsings: Physically, the deformation of a singularity corresponds to a process where a massless hypermultiplet acquires a vacuum expectation value such that the singularity arises at the origin of a Higgs branch.",
"For singularities which allow a crepant resolution the localised hypermultiplets are charged and the Higgsing necessarily breaks part of the gauge group.",
"A classic example is the deformation of a codimension-two resolvable conifold singularity in the fiber of an F-theory elliptic fibration, corresponding to a Higgsing of a $U(1)$ gauge group under which the localised states are charged [62], [63], [64].",
"The novelty in absence of a crepant resolution is that the localised states are uncharged, at least under any massless gauge group."
],
[
"The Euler Characteristic of Singular Threefolds",
"In our examples $X$ is a Calabi-Yau variety with ${\\mathbb {Q}}$ -factorial Kleinian terminal singularities of type $A_{a-1}$ with either $a$ odd or $a=2$ .",
"It is a consequence of a result in [27] that in both cases $ \\chi _{\\rm top}(X) =2 + 2 b_2(X)- b_3(X),$ as in the smooth case.",
"It is important to stress that the arguments needed are more general than for Kleinian singularities, but different for $a$ odd (a rational homology manifold) and $a=2$ , and the final statement is the same.",
"By combining equations (REF ), (REF ) and (REF ) we find $\\tfrac{1}{2} \\, \\chi _{\\text{top}}(X) = \\mathrm {KaDef}(X) - \\mathrm {CxDef}(X) + \\tfrac{1}{2} \\, \\sum _P m_P.$ In particular, $\\begin{aligned}n_H^{0} &= 1 + \\mathrm {CxDef}(X)\\\\& = \\frac{1}{2} ( b_3(X) + \\sum _P m_P ) \\\\& = 1 + \\mathrm {KaDef}(Y_3) - \\tfrac{1}{2} \\, \\chi _{\\text{top}}(Y_3) + \\tfrac{1}{2} \\, \\sum _P m_P,\\end{aligned}$ and $n_H^0 &=& \\left(1 + \\mathrm {KaDef}(X) - \\tfrac{1}{2} \\, \\chi _{\\text{top}}(X) - \\tfrac{1}{2} \\, \\sum _P m_P\\right) + \\sum _P m_P \\\\&=& n_{H,n-l}^0 + n_{H, l}^0.$ These expressions will be successfully be applied in a number of examples in the remainder of this paper and we will verify that the resulting spectrum is free of gravitational anomalies.",
"In particular, the methods of [28] to compute the Euler characteristic of the elliptic threefold, reviewed in appendix , are still valid in the presence of singularities.",
"The crucial point, however, is that the topological Euler characteristic of the fibers entering (REF ) must be evaluated for the singular or partially resolved models.",
"We now turn to explaining this procedure in more detail."
],
[
"Terminal Singularities in Models With Trivial Gauge Group",
"In this section we exemplify the appearance of uncharged, localised matter at non-crepant singularities in two models with trivial gauge group.",
"The first model is perturbative with a terminal singularity of conifold type at an ${\\rm I}_1 \\rightarrow {\\rm I}_2$ enhancement locus, while the second class of models is inherently non-perturbative in nature due to a terminal singularity at a II $\\rightarrow $ III locus."
],
[
"The $\\text{I}_1$ Conifold Model",
"As our first example of a non-resolvable model, we consider a non-generic elliptic fibration with only ${\\rm I}_1$ singularities in codimension one.",
"This model has been discussed before in [28], [21], [29] and is most efficiently described as a specialization of a Tate model $y^2 + a_1\\, x y z + a_3 \\, y z^3 = x^3 + a_2 \\, x^2 z^2 + a_4\\, x z^4 + a_6 \\, z^6,$ where the $a_n$ are sections of $\\mathcal {O} (-n K_B)$ .",
"The parameters of the Tate form and the Weierstrass form are related via $f = -\\tfrac{1}{48} (b_2^2 - 24 \\,b_4), \\quad g = -\\tfrac{1}{864} (-b_2^3 + 36\\, b_2 b_4 - 216\\, b_6),$ where the $b_n$ are sections of $\\mathcal {O} (-n K_B)$ and take the form $b_2 = a_1^2 + 4 a_2, \\quad b_4 = a_1 a_3 + 2 a_4,\\quad b_6 = a_3^2 + 4 a_6.$ Let us now choose the vanishing orders ${\\rm ord}(a_1,a_2,a_3,a_4,a_6)|_{z_1 = 0} =: (k_1,k_2,k_3,k_4,k_6) = (0,0,1,1,1)$ along a divisor $\\Sigma _1: z_1 = 0$ on the base $B_2$ by setting $a_i = \\tilde{a}_i z_1^{k_i}$ with $\\tilde{a}_i$ generic.",
"As a result, the discriminant $\\Delta $ splits into two components $\\Sigma _1$ and $\\Sigma _0$ , $\\Delta = \\tfrac{1}{16} \\,z_1\\, \\Big ( \\tilde{a}_{6}\\, (\\tilde{a}_1^2+4\\,\\tilde{a}_2)^3 + z_1\\cdot (\\ldots ) \\Big )$ and in terms of the associated Weierstrass model the vanishing orders along generic points of the ${\\rm I}_1$ -locus $\\Sigma _1$ are ${\\rm ord}(f,g,\\Delta )|_{z_1=0} = (0,0,1).$ There are two types of codimension-two enhancement points from the intersection $\\Sigma _1 \\cap \\Sigma _0$ , $P_1:& \\lbrace z_1=0\\rbrace \\cap \\lbrace \\tilde{a}_1^2+4\\,\\tilde{a}_2=0 \\rbrace \\quad &{\\text{I}}_1 \\rightarrow {\\rm II} \\\\P_2:& \\lbrace z_1=0\\rbrace \\cap \\lbrace \\tilde{a}_{6}=0\\rbrace \\quad &{\\text{I}}_1 \\rightarrow {\\text{I}_2}.$ At $P_1$ the singular fiber takes the form of a singular cuspidal curve with $\\chi _{\\text{top}}(X_{P_1}) = 2$ even though the threefold $Y_3$ remains smooth.",
"At $P_2$ the fiber develops an $\\text{I}_2$ singularity, corresponding to a singularity of $Y_3$ .",
"This singularity admits no small resolution (see [28], [21]).",
"In the language of Section (REF ), it is in fact $\\mathbb {Q}$ -factorial terminal (but not analytically $\\mathbb {Q}$ -factorial).",
"Indeed the singularity is only locally of conifold form $z^2+x_1^2+x_2^2+x_3^2=0$ with higher order terms in $z$ obstructing a small resolution [21].",
"From a physical perspective the absence of a small, crepant resolution is owed to the fact that the gauge group in the M-theory compactification is trivial and hence no Coulomb branch is available to render the expected massless states localised at $P_2$ massive.",
"The Milnor number of this $A_1$ type singularity is $m_{P_2}=1$ .",
"In order to compute the number of complex structure deformations $\\mathrm {CxDef}$ of the singular threefold $Y_3$ we follow the formalism developed in Section REF and in particular apply (REF ), where $\\tfrac{1}{2} \\, \\sum _P m_P = \\tfrac{1}{2} \\, \\sum _{P_2} m_{P_2} = \\frac{1}{2} \\, (-6 K_B - [z_1]) \\cdot _{B_2} [z_1]$ , which is just $\\frac{1}{2}$ times the number of points $P_1$ .",
"For concreteness let us now take $B_2=\\mathbb {P}^2$ with homogenous coordinates $[z_0 : z_1 : z_2]$ and identify $\\Sigma _1: z_1 =0$ .",
"This is a rational curve and hence $g(\\Sigma _1) = 0$ .In particular there are no bulk matter states propagating along $\\Sigma _1$ .",
"There are then 6 points of type $P_1$ and 17 points of type $P_2$ .",
"Interestingly, $\\chi _{\\text{top}}(Y_3)$ must now be odd.",
"This reflects the fact that on the singular space $Y_3$ ordinary homology does no longer enjoy Poincaré duality, in agreement with the fact that $Y_3$ is only $\\mathbb {Q}$ -factorial, but not analytically $\\mathbb {Q}$ -factorial.",
"This is indeed confirmed by explicit computation of $\\chi _{\\text{top}}(Y_3)$ via (REF ) as follows: Since the fiber $X_{P_2}$ over $P_2$ is not resolved, we must take $\\chi _{\\text{top}}(X_{P_2}) = 1$ , corresponding to the value of the singular original fiber.",
"The intersection multiplicities of $f$ and $g$ at both points are given by $\\mu _{P_1}(f,g) = 2,\\, \\mu _{P_2}(f,g) = 0$ [28], and the parameters $\\epsilon _i$ correcting for singularities of $\\Delta $ at $P_i$ are given by $\\epsilon _1 = -1 = \\epsilon _2$ (see Table 4 in [28]).",
"All in all, this leads to $\\chi _{\\text{top}}(Y_3) = -523$ .",
"Taking into account that $h^{1,1}(Y_3) =2$ (since the gauge group is trivial), our expression (REF ) for the number of complex structure deformations yields $\\mathrm {CxDef}(Y_3) = 2+\\tfrac{1}{2} \\cdot 523 + \\tfrac{1}{2} \\cdot 17 \\cdot 1 = 272.$ Since the gauge group is trivial, there are no charged hypermultiplets at all, and we find $n_H = n_H^0 = 1 + \\mathrm {CxDef}(Y_3) = 273$ in agreement with condition (REF ) for cancellation of gravitational anomalies.",
"According to (REF ), the hypermultiplets from $\\mathrm {CxDef}(Y_3)$ split into $17 \\times 1$ localised uncharged hypermultiplets, while the remaining ones are unlocalized states.",
"This fits perfectly with the type IIB orientifold limit of the Tate model, as described in detail in [21]: On the Calabi-Yau twofold which is the Type IIB double cover of $B_2$ , $\\Sigma _1$ uplifts to two divisors $D_1 \\cup D_1^{\\prime }$ exchanged by the orientifold action, while the 7-brane along $\\Sigma _0$ uplifts to the O7-plane together with another 7-brane on an invariant divisor $D_0$ of Whitney type.",
"The 17 points $P_2$ correspond to the intersection points between $D_1$ and $D_0$ (which are identified with $D_1^{\\prime } \\cap D_0$ by the orientifold involution), each of which gives rise to one massless hypermultiplet from strings streched between both branes.The points $P_1$ uplift to the intersection points between $D_1$ and $D_1^{\\prime }$ on top of the orientifold plane; here no additional massless matter states reside as the 7-7' string zero modes are projected out by the orientifold action.",
"Since the $U(1)$ gauge symmetry from $D_1$ and $D_1^{\\prime }$ is massive by a Stückelberg mechanism, this matter appears as uncharged in F/M-theory, but still localised."
],
[
"The Type II Model",
"In this section we consider two non-crepant resolvable Weierstrass models with trivial gauge group which do not allow for a perturbative Type IIB orientifold limit.",
"Over the divisor $\\Sigma _1$ we engineer a type II (cuspidal) Kodaira fiber, which shall enhance in codimension-two to type III (model 1) or to type IV (model 2).",
"This is achieved by the following vanishing ordersMore generally, we can consider $f= z_1^n \\, f_0$ , $g = z_1 \\, g_0$ .",
"The models with $n > 2$ are similar to $n=2$ and are discussed in appendix REF .",
"in the Weierstrass model: ${\\rm model \\, 1:\\, } &\\qquad f = z_1 f_0, \\,\\quad g= z_1 g_0 \\qquad \\rightarrow &\\Delta = z_1^2 \\cdot \\big ( 27g_0^2 + 4 z_1 f_0^3 \\big ) \\\\{\\rm model \\, 2: \\, } &\\qquad f = z_1^2 f_0, \\,\\quad g= z_1 g_0 \\qquad \\rightarrow &\\Delta = z_1^2 \\cdot \\big ( 27g_0^2 + 4 z_1^4 f_0^3 \\big )$ There is one type of intersection points $P_1$ of $\\Sigma _1$ and $\\Sigma _0$ at $z_1= g_0 = 0$ : In model 1, the vanishing orders of $(f,g,\\Delta )|_{P_1}$ are $(1,2,3)$ , corresponding to type III, while in model 2, $(f,g,\\Delta )|_{P_1} = (2,2,4)$ , indicating an enhancement to type IV.",
"Since the gauge group is trivial, we expect the isolated singularities at $P_1$ not to allow for a crepant resolution.",
"Indeed, in both models the singularity at $P_1$ can be brought into the form of a hypersurface singularity $z^3+x_1^2+x_2^2+x_3^2=0 .$ To see this, note that locally near the singularity at $x=y=z_1=g_0=0$ we can set $f_0=1$ and $z=1$ and rewrite the Weierstrass polynomial as $P_W = -y^2 + x^ 3 + \\frac{1}{4}[z_1 + ( z_1^k \\, x + g_0)]^2 - \\frac{1}{4}[z_1 - (z_1^k \\, x + g_0)]^2 $ with $k=0$ and $k=1$ for model 1 and 2, respectively.",
"As described after equ.",
"(REF ), such a singularity is analytically $\\mathbb {Q}$ -factorial terminal, with Milnor number $\\mu _{P_1} = 2$ .",
"We can therefore use (REF ) to determine the number of localised and unlocalised neutral hypermultiplets.",
"To this end we first evaluate $\\chi _{\\rm top}(Y_3)$ via (REF ) with the help of the data summarized in table REF .",
"In model 1, the residual determinant $\\Sigma _0$ is smooth at $P_1$ and hence the parameter $\\epsilon _1$ defined in more detail in appendix REF , especially equ.",
"(REF ), is $\\epsilon _1 = -1$ .",
"In model 2, $\\Sigma _0$ at $P_1$ is locally of the form $x^2 + y^4 = 0$ so that $\\epsilon _1=2$ .",
"The intersection multiplicity $\\mu (f,g)$ vanishes since $f_0$ and $g_0$ are generic at $P_1$ .",
"The topological Euler characteristic of the fiber over the enhancement points is $\\chi _{\\text{top}}(X_{P_1}) = 2$ , corresponding to the value of $\\chi _{\\text{top}}$ of the type II fiber, since both models are not resolvable.",
"At this stage we restrict ourselves, for concreteness, to $B_2 = \\mathbb {P}^2$ and take again $\\Sigma _1: z_1=0$ with $z_1$ one of the homogeneous coordinates $[z_0 : z_1 : z_2]$ .",
"This implies that $g(\\Sigma _1)=0$ and the number of points of type $P_1$ is $B_1 = 17$ .",
"For both modelsThe difference in $\\epsilon _1$ is compensated by a different number $C$ of cuspidal points $Q$ appearing in (REF ) since $\\mu _f$ differs in both cases.",
"this leads to $\\chi _{\\text{top}}(Y_3) = -506$ .",
"Thus, the number (REF ) of complex structure deformations is $\\mathrm {CxDef}(Y_3) = 2+\\tfrac{1}{2} \\cdot 506 + \\tfrac{1}{2} \\cdot 17 \\cdot 2 = 272.$ As always there is also the universal hypermultiplet so that $n_H = n_H^0 = 273$ , as required by the gravitational anomaly condition.",
"This time, since $m_{P_1}=2$ , the number of localised uncharged hypermultiplets per terminally singular point is 2 - a statement which is of course independent of the choice of base space.",
"The 273 hypermultiplets thus split into $n^0_{H,l} = \\sum _{P_1} m_{P_1} = 17 \\cdot 2 = 34$ uncharged localised and $n^0_{H,n-l} = 1 + 272-34 = 1 + 238$ non-localised ones.",
"All results of this section are summarized in Table REF .",
"Table: Non-resolvable models with trivial gauge group.",
"The parameter aa characterises the form of the terminal codimension-two hypersurface singularity at P 1 P_1 via z a +x 1 2 +x 2 2 +x 3 2 =0z^a+ x_1^2+x_2^2+x_3^2=0.",
"m P m_P denotes the corresponding Milnor number."
],
[
"Terminal Singularities in Presence of Non-Trivial Gauge Group",
"In this section we present a family of models with $\\mathbb {Q}$ -factorial terminal codimension-two singularities in presence of a non-trivial gauge group.",
"According to our general logic the $\\mathbb {Q}$ -factorial terminal singularities should host both charged and in addition uncharged localised matter.",
"This expectation is indeed confirmed by our explicit analysis."
],
[
"A Family of Type III Models With $\\mathbb {Q}$ -Factorial Terminal Singularities",
"In the setup of interest the discriminant exhibits a (non-perturbative) type III singularity along a divisor $\\Sigma _1: z_1 = 0$ , corresponding to gauge group $G=SU(2)$ in codimension one.",
"In Weierstrass form this is achieved by setting $f = z_1 \\, f_0, \\qquad g= z_1^{\\mu _g} \\, g_0 \\quad {\\rm for} \\quad \\mu _g \\ge 2 \\qquad \\rightarrow \\quad \\Delta = z_1^3 \\, (4 f_0^3 + 27 z_1^{2 \\mu _g - 3} \\, g_0^2).$ The intersection points $P_1 = \\Sigma _1 \\cap \\Sigma _0$ , with $\\Sigma _0: 4 f_0^3 + 27 z_1^{2 \\mu _g - 3} \\, g_0^2=0$ , lie at $z_1 = f_0 = 0$ .",
"This gives rise to the following vanishing orders at $P_1$ : $\\begin{aligned}&\\mu _g= 2: \\qquad &{\\rm ord}(f,g,\\Delta )|_{P_1} = (2,2,4), \\qquad {\\rm III} \\rightarrow {\\rm IV}, \\cr &\\mu _g\\ge 3: \\qquad &{\\rm ord}(f,g,\\Delta )|_{P_1} = (2,\\mu _g,6), \\qquad {\\rm III} \\rightarrow {\\rm I}^*_0,\\end{aligned}$ where we also indicate the naively expected fiber type at the enhancement points from Kodaira's table.",
"As will be shown in detail in Section REF , for $\\mu _g =2$ and $\\mu _g =3$ a crepant resolution of the Weierstrass model exists which in particular completely resolves the codimension-two fibers over $P_1$ .",
"For $\\mu _g =2$ the resolved fiber is indeed of type IV, while for $\\mu _g =3$ it has three $\\mathbb {P}^1$ s deleted compared to the naively expected standard Kodaira fiber ${\\rm I}^*_0$ .",
"After performing this resolution one finds two localised hypermultiplets per enhancement point $P_1$ in representation ${\\bf 2}$ of the gauge algebra $SU(2)$ from wrapped M2-branes wrapping suitable fibral curves.",
"The appearance of two such hypermultiplets per point (as opposed to just one) is quite interesting by itself and discussed at the end of section REF .",
"By contrast, for certain values $\\mu _g \\ge 4$ , the singularity at $P_1$ turns out to be $\\mathbb {Q}$ -factorial terminal.",
"Concretely we have studied $\\mu _g = 4$ , $\\mu _g = 5$ , $\\mu _g = 7$ .",
"In these cases the (partial) resolution $\\hat{Y}_3$ presented in Section REF yields a monodromy reduced I$_0^*$ fiber, depicted in figure REF , with a residual terminal hypersurface singularity of type $z^a + x_1^2 + x_2^2 + x_3^2 = 0$ with $&a = 2 \\qquad {\\rm for} \\, &\\mu _g = 4, \\\\&a = 3 \\qquad {\\rm for} \\, &\\mu _g = 5 ,\\\\&a=5 \\qquad {\\rm for} \\, &\\mu _g = 7 .$ Figure: Affine Dynkin diagram of the partially resolved fiber over P 1 :z 1 =f 0 =0P_1: z_1 = f_0 =0.",
"The red cross denotes the intersection with the zero-section z=0z=0 of the Weierstrass model and the dashed lines symbolize the deleted ℙ 1 \\mathbb {P}^1s in the standard Kodaira fiber which are not realized in the actual fiber.",
"The resolution was obtained as resolution of a Tate model realising the vanishing orders ().Note that the conifold singularity for $\\mu _g = 4$ is indeed $\\mathbb {Q}$ -factorial terminal due to higher order terms; hence a small resolution does not exist.",
"The reason for the appearance of residual terminal singularities is the localisation of uncharged hypermultiplets at these points, in addition to the two localised hypermultiplets in representation 2 of $SU(2)$ present also in the resolvable cases with $\\mu _g =2,3$ .",
"Since the residual terminal singularity is either a $\\mathbb {Q}$ -factorial terminal double point (for $\\mu _g =4$ ) or of odd Kleinian type (for $\\mu _g = 5 ,7$ ), our general results of Section REF can be applied.",
"In particular, the number of localised uncharged hypermultiplets per point $P_1$ is given by $m_{P_1} =1$ for $\\mu _g = 4$ , $m_{P_1} = 2$ for $\\mu _g = 5$ and by $m_{P_1} = 4$ for $\\mu _g = 7$ .",
"Furthermore, we can follow the programme of computing the total number of uncharged hypermultiplets via (REF ).",
"The crucial step is again to evaluate $\\chi _{\\rm top}(\\hat{Y}_3)$ , where $\\hat{Y}_3$ denotes the partial resolution.",
"As can be deduced from the explicit form of the fiber in Figure REF following the procedure in appendix REF , the partially resolved I$_0^*$ fiber over $P_1$ contributes $\\chi _{\\rm top}(X_{P_1}) = (2-1) + (2-1) + 1 = 3.$ Furthermore, the value of the parameter $\\epsilon _1$ correcting for the singularities of $\\Sigma _0$ at $P_1$ follows readily from (REF ) as $\\epsilon _1 = 4 \\mu _g -9.$ Indeed $\\Sigma _0$ at $P_1$ takes the local form $x^3 +y^{2\\mu _g -3}=0$ .",
"If we specialise the base to $B_2 = \\mathbb {P}^2$ and let $\\Sigma _1$ wrap the curve $z_1=0$ as before, we find the values summarizedThe corresponding values for the two smooth models with $(\\mu _f, \\mu _g) = (1,2)$ and $(1,3)$ are listed in table REF in appendix .",
"in table REF for $\\chi _{\\rm top}(\\hat{Y}_3)$ and correspondingly for the numbers of localised and unlocalized uncharged hypers.",
"The spectrum is indeed consistent with the cancellation of all gravitational and gauge anomalies.",
"Table: Non-resolvable models with non-trivial gauge group.",
"The parameter aa characterises the form of the codimension-two singularity, given by z a +x 1 2 +x 2 2 +x 3 2 =0z^a+ x_1^2+x_2^2+x_3^2=0, and m P m_P denotes the corresponding Milnor number."
],
[
"Partial Resolution",
"The specific form of the fiber at $P_1$ and the local expression (REF ) for its terminal singularities can be achieved by a patchwise partial resolution of the most generic Weierstrass model subject to the restriction (REF ).",
"As an alternative we realize (REF ) as a Tate model as this allows us to construct a global resolution by a blowup of the toric ambient space into which hypersurface is embedded.",
"In the remainder of this section we present this model and its (partial) resolution.",
"In order to reproduce (REF ) the sections $a_i$ in the Tate model (REF ) must be restricted as $a_i = \\tilde{a}_i \\, z_1^{k_i},$ for the values of $k_i$ as collected in table REF .",
"That this indeed leads to the desired vanishing orders for $f$ and $g$ can be checked via (REF ).",
"Table: Vanishing orders k i k_i of the Tate sections a i a_i along the locus Σ 1 \\Sigma _1.To resolve the singularity in the fiber over $\\Sigma _1$ , located at $x=y=z_1=0$ , we perform the blow-up $x \\rightarrow e_1x, \\quad y \\rightarrow e_1y, \\quad z_1 \\rightarrow e_0 \\, e_1.$ The proper transform $PT$ of the Tate equation is then a hypersurface in a toric fiber ambient space with coordinates $x,y,z,e_0,e_1$ and toric weights displayed in table REF .",
"However, the hypersurface as such is not the most general hypersurface compatible with these scaling relations.",
"This most generic hypersurface would rather give rise to Kodaira fibers of type $\\mathrm {I}_2$ in codimension one, and not type III.",
"In particular, the dual polytope does not reproduce the monomials in $PT$ .",
"In this sense, this type III model cannot be analysed via the technology of tops [65], [66].",
"Nonwithstanding this fact we can still compute the Stanley-Reisner ideal (SRI) of the toric ambient space and analyse the hypersurface $PT$ by hand.",
"There exist two triangulations of the ambient space, and for concreteness we choose the triangulation with SRI given by ${\\rm SRI} = \\langle ze_1,xyz,xye_0 \\rangle \\, .$ Most of the potentially singular loci of our models where $PT = dPT=0$ are excluded by the SRI.",
"However, there remain some singularities which are displayed in table REF .",
"Note that in the $(1,3)$ -model the codimension of the singular locus is too high and therefore non-existent on Calabi-Yau threefolds, which is the case of interest in this paper.",
"For $\\mu _g \\ge 4$ , by contrast, the blow-up defines only a partial Calabi-Yau resolution $\\hat{Y}_3$ .",
"In fact, the remaining singularity is immediately identified as a terminal hypersurface singularity of the type advocated in (REF ), (), (): Near the singularity, we can set $z_1=e_0=\\tilde{a}_i=1$ for $i \\ne 4$ and write the proper transform as $PT = e_1^{k_6-2} + x ( \\tilde{a}_4 + e_1^{k_2} \\, x + e_1 x^2 - e_1^{k_1} y) - y (e_1^{k_3-1} + y),$ with $k_i$ the vanishing orders displayed in table REF .",
"The claim then follows by completing the square in $y$ , keeping only the leading monomials in $e_1$ , and by a simple coordinate change.",
"To analyse the structure of the fibers, and in particular to compute $\\chi _{\\rm top}$ for the critical fibers over $P_1$ , note first that the fiber over $z_1=0$ consists of two rational curves, given by the vanishing of $e_0$ and $e_1$ , respectively.",
"Concretely, $\\mathbb {P}^1_A:\\: PT|_{e_0 \\rightarrow 0} &= y^2-e_1 x^3,\\\\\\mathbb {P}^1_B:\\: PT|_{e_1 \\rightarrow 0} &= {\\left\\lbrace \\begin{array}{ll}y^2- \\tilde{a}_4 e_0\\, x z^4 +\\tilde{a}_3 e_0 \\,yz^3 -\\tilde{a}_6 e_0^2\\, z^6 & \\text{for} \\, \\, (\\mu _f,\\mu _g) = (1,2)\\\\y^2-\\tilde{a}_4e_0 \\, xz^4 & \\text{for} \\, \\, (\\mu _f,\\mu _g) = (1,i), \\, i = 3,\\ldots ,7.\\end{array}\\right.",
"}$ These two equations do not factorize and therefore define two $\\mathbb {P}^1$ s in the fiber called $\\mathbb {P}^1_A$ and $\\mathbb {P}^1_B$ .",
"They intersect at $\\lbrace e_0 \\rbrace \\cap \\lbrace e_1 \\rbrace \\cap \\lbrace y \\rbrace $ with order two and thus realise a type III fiber (see figure REF ).",
"Note that $\\mathbb {P}^1_A$ is intersected by the zero-section of the Weierstrass model.",
"The charge of $\\mathbb {P}^1_B$ under the Cartan of $SU(2)$ is given by minus the intersection product of the curve $\\mathbb {P}^1_B$ and the divisor $E_1$ locally defined by $e_1=0$ , $- \\mathbb {P}^1_B \\circ E_1 &= - [e_1] \\cdot \\left[y^2+\\ldots \\right] \\cdot [e_1] = [e_1] \\cdot [ y^2 ] \\cdot [e_0] = 2,$ where we used that $[e_1] = -[e_0]$ (see table REF ).",
"This identifies $\\mathbb {P}^1_B$ as the simple root of $SU(2)$ as expected.",
"Table: Fiber type III (SU(2)SU(2))From the discriminant of the Tate models, $\\Delta &= \\tfrac{1}{16} \\,z_1^3 \\cdot \\left( 64\\, \\tilde{a}_4^3 + \\mathcal {O} (z_1) \\right),$ we read off that the fiber type enhances at $z_1 =\\tilde{a}_4 =0$ , which corresponds to the location of the points $P_1$ .",
"In the $(1,2)$ -model the fiber over this point takes the following form, $PT|_{e_0,\\tilde{a}_4\\rightarrow 0} &= \\underbrace{y^2-e_1x^3}_{\\mathbb {P}^1_a},\\\\PT|_{e_1,\\tilde{a}_4\\rightarrow 0} &= \\underbrace{y^2+\\tilde{a}_3 e_0 \\,yz^3 -\\tilde{a}_6 e_0^2\\, z^6}_{\\mathbb {P}^1_B \\text{ at } \\tilde{a}_4 \\rightarrow 0}\\\\&= \\underbrace{\\left(y+\\tfrac{1}{2} \\tilde{a}_3e_0\\,z^3 + \\sqrt{\\tilde{a}_3^2+4\\tilde{a}_6}\\,e_0\\,z^3 \\right)}_{\\mathbb {P}^1_{b_+}} \\cdot \\underbrace{\\left(y+\\tfrac{1}{2}\\, \\tilde{a}_3e_0\\,z^3 - \\sqrt{\\tilde{a}_3^2+4\\tilde{a}_6}\\,e_0\\,z^3 \\right)}_{\\mathbb {P}^1_{b_-}}.$ At the codimension-two enhancement locus the curve $\\mathbb {P}^1_B$ therefore splits into two curves, called $\\mathbb {P}^1_{b_+}$ and $\\mathbb {P}^1_{b_-}$ .",
"All three curves meet at $\\lbrace e_0 \\rbrace \\cap \\lbrace e_1 \\rbrace \\cap \\lbrace y \\rbrace $ with multiplicity one, corresponding to a Kodaira fiber of Type IV as shown in figure REF .",
"This is in agreement with the vanishing orders of $f$ , $g$ and $\\Delta $ .",
"For the $(1,i)$ -models with $i>2$ the situation is slightly different: $\\mathbb {P}^1_a:\\: PT|_{e_0,\\tilde{a}_4\\rightarrow 0} &= y^2-e_1x^3,\\\\2\\, \\mathbb {P}^1_b:\\: PT|_{e_1,\\tilde{a}_4\\rightarrow 0} &= y^2.$ The second equation now describes a non-reduced curve of multiplicity two.",
"We denote the reduced curve defined by $e_1=\\tilde{a}_4 = y=0$ as $\\mathbb {P}^1_b$ .",
"Taking this into account, the fiber can indeed be interpreted as an I$_0^*$ Kodaira fiber with three multiplicity-one curves deleted, again in agreement with the vanishing orders (REF ).",
"The contribution of such a fiber to the topological Euler characteristic has already been computed in (REF ).",
"Table: Singular locus of proper transforms describing the Calabi-Yau threefold Y ^ 3 \\hat{Y}_3.",
"The singularity parameter aa describes the local form of the singularity: z a +x 1 2 +x 2 2 +x 3 2 z^a + x_1^2+x_2^2+x_3^2.Next, we need the Cartan weights of $\\mathbb {P}^1_{b_\\pm }$ and $\\mathbb {P}^1_b$ .",
"The curve $\\mathbb {P}^1_a$ is still intersected by the zero-section and does not play a role in our analysis.",
"We compute $- \\mathbb {P}^1_b \\circ E_1 &= - [e_1] \\cdot [y] \\cdot [e_1] = \\cdot [e_0] \\cdot [y] \\cdot [e_1] =1,\\\\- \\mathbb {P}^1_{b_\\pm } \\circ E_1 &= - [e_1] \\cdot \\left[ y+\\tfrac{1}{2} \\tilde{a}_3e_0\\,z^3 \\pm \\sqrt{\\tilde{a}_3^2+4\\tilde{a}_6}\\,e_0\\,z^3 \\right] \\cdot [e_1] = [e_1]\\cdot [y] \\cdot [e_0] = 1.$ The situation is hence completely equivalent in both cases.",
"The $SU(2)$ root $\\alpha $ is given by $\\mathbb {P}^1_B$ which has factorized into $\\mathbb {P}^1_{b_+} + \\mathbb {P}^1_{b_-}$ in the $(1,2)$ -case.",
"The highest weight of the fundamental representation of $SU(2)$ is $w=1$ which is either represented by $\\mathbb {P}^1_{b_+}$ or $\\mathbb {P}^1_{b_-}$ (or by $\\mathbb {P}^1_b$ ).",
"In this fashion we can build: $w = \\mathbb {P}^1_{b_+}:\\quad &\\begin{pmatrix}w-\\alpha \\\\w\\end{pmatrix} = \\begin{pmatrix}\\mathbb {P}^1_{b_+}-(\\mathbb {P}^1_{b_+}+\\mathbb {P}^1_{b_-})\\\\\\mathbb {P}^1_{b_+} \\end{pmatrix} = \\begin{pmatrix}-\\mathbb {P}^1_{b_-}\\\\\\mathbb {P}^1_{b_+} \\end{pmatrix} = \\begin{pmatrix}-1\\\\1\\end{pmatrix},\\\\w = \\mathbb {P}^1_{b_-}:\\quad &\\begin{pmatrix}w-\\alpha \\\\w\\end{pmatrix} = \\begin{pmatrix}\\mathbb {P}^1_{b_-}-(\\mathbb {P}^1_{b_+}+\\mathbb {P}^1_{b_-})\\\\\\mathbb {P}^1_{b_-} \\end{pmatrix} = \\begin{pmatrix}-\\mathbb {P}^1_{b_+}\\\\\\mathbb {P}^1_{b_-} \\end{pmatrix} = \\begin{pmatrix}-1\\\\1\\end{pmatrix}.$ Clearly both pairs of fibral curves each describe the weight vector of a fundamental representation ${\\bf 2}$ of $SU(2)$ , but they are not independent as they only differ by a minus sign.",
"An M2-brane can wrap each fibral curve with two orientations, and the sign difference above can be translated into the orientation of the wrapped M2.",
"This way one would naively conclude that there is one massless hypermultiplet in the ${\\bf 2}$ of $SU(2)$ localised at the enhancement point.",
"However, as observed already in [31], near the enhancement point the Weierstrass equation (with $z \\equiv 1$ ) can be written as $\\begin{aligned}P_W & = - y^2 + x^3 + f_0 z_1 x + z_1^2 + {\\cal O}(z_1^3) \\\\&= -y^2 + x^3 - \\frac{1}{4} f^2_0 x^2 + (z_1 + \\frac{1}{2} f_0 x)^2 + {\\cal O}(z_1^3).\\end{aligned}$ The enhancement point at $f_0 = z_1 =0$ corresponds to an $A_2$ singularity $-y^2 + x^3 + z_1^2 = 0$ .",
"The Weierstrass equation (REF ) is a deformation of this $A_2$ to an $A_1$ singularity with deformation parameter $t = \\frac{1}{4} f^2_0$ .",
"Since this parameter appears quadratically, the number of massless hypermultiplets in representation 2 of $SU(2)$ per enhancement point is given by 2 [67], [31] (see table REF ).Another way to see this is by interpreting the Type III model as a specialization of an ${\\rm I}_2$ model in which two enhancement points ${\\rm I}_2 \\rightarrow {\\rm I}_3$ , each carrying one hypermultiplet in the ${\\bf 2}$ of $SU(2)$ , coalesce to one ${\\rm Type \\, III} \\rightarrow {\\rm Type \\, IV }$ point.",
"This number is also required by cancellation of all gauge and gravitational anomalies."
],
[
"Conclusions and Outlook",
"We have investigated non-crepant resolvable codimension-two singularities in F-theory compactifications.",
"The physical interpretation of such singularities is very simple: Massless matter uncharged under any gauge group localizes in the fiber and cannot be rendered massive along a supersymmetric direction in the Coulomb branch, as would be required for a Calabi-Yau resolution to exist.",
"For a certain class of isolated terminal $\\mathbb {Q}$ -factorial hypersurface singularities (of Kleinian type $A_{a-1}$ with $a=2$ or $a$ odd) on elliptic Calabi-Yau threefolds we have shown how to compute both the unlocalised and the localised uncharged hypermultiplets in terms of the Milnor number of the singularity and the topological Euler characteristic of the singular variety.",
"These expressions have been put to a successful test in a number of examples and indeed produced an anomaly-free spectrum in the associated six-dimensional F-theory compactification.",
"The methods hold for more general singularities, which do not appear as examples in this paper [27].",
"There are many interesting directions for future investigations.",
"An obvious question concerns codimension-two terminal singularities on elliptic Calabi-Yau fourfolds: The physical intuition about the appearance of massless matter is independent of the dimension of the compactification.",
"Thus we clearly expect similarly localised uncharged hypermultiplets, now from terminal singularities in the fiber over a curve $C$ in base.",
"Extrapolating the counting in [64], [68] of massless matter in F-theory on smooth fourfolds a natural conjecture for the number of localised vectorlike pairs of $N=1$ multiplets would be $n = g(C) \\, m_P$ with $m_P$ the Milnor number of the singularity in the fiber and $g(C)$ the genus of $C$ .",
"This formula would hold in the absence of $G_4$ -flux.",
"Given the nature of the uncharged states as part of the complex structure moduli of the singular variety we expect the counting to be modified at best by horizontal fluxes $G_4 \\in H^{2,2}_{\\rm hor}(Y_4)$ , as opposed to fluxes in the vertical part of the middle cohomology.",
"Clearly the investigation of the middle cohomology of singular fourfolds might become equally challenging and exciting, both from a conceptual and a computational viewpoint.",
"While the terminal singularities studied in this paper do not allow for a crepant resolution, the singular varieties can be resolved into a non-Calabi-Yau space.",
"In the spirit of Section REF , in the dual M-theory the resolution breaks supersymmetry by moving along an obstructed direction in the Coulomb branch.",
"Nonetheless certain physical quantities might be robust enough to be computed from this non-Calabi-Yau phase.",
"Ideas along these lines have been put forward in [35], [37], [21].",
"Some of the examples studied in this paper will involve non-flat non-Calabi-Yau resolutions, and a preliminary analysis suggests that the counting of localised uncharged states is indeed reproduced.",
"We look forward to reporting on these questions in future work.",
"Acknowledgements We thank Paolo Aluffi, Markus Banagl, Jim Halverson, Craig Lawrie, Ling Lin, Laurentiu Maxim, Christoph Mayrhofer, Dave Morrison, Eran Palti, Michele Rossi, Sakura Schäfer-Nameki, Julius Shaneson, Vasudevan Srinivas and Washington Taylor for discussions.",
"AG and TW thank the Aspen Center for Physics and the Fields Institute, Toronto, for hospitality during part of this project.",
"The work of TW is partially supported by DFG under TR33 'The Dark Universe'."
],
[
"Computation of $\\chi _{\\text{top}}(\\hat{Y}_3)$",
"In this appendix, we summarize, for the reader's convenience, the computation of the topological Euler characteristic of an elliptically fibered Calabi-Yau threefold as presented in [28], and explain its use in the presence of terminal singularities over codimension-two points in the base.",
"Our notation for the (in general singular) Weierstrass model of an elliptically fibered Calabi-Yau threefold $Y_3$ has already been introduced in Section .",
"For simplicity, we first assume, as in [28], that the gauge group does not contain any abelian factors and that it has only one semi-simple factor, see figure REF .",
"Later, in Section REF , we will generalise the computation for $\\chi _{\\text{top}}$ to theories with two identical simple factors.",
"An important point to note is that the topological Euler characteristic we compute is then the Euler characteristic of a (partial) Calabi-Yau resolution $\\hat{Y}_3$ of this Weierstrass model.",
"If all singularities are crepant resolvable, as was the case for the geometries studied in [28], $\\hat{Y}_3$ is a smooth Calabi-Yau space.",
"In the presence of terminal singularities it is understood that $\\hat{Y}_3$ is a smooth Calabi-Yau with at worst those terminal singularities in codimension-two which cannot be resolved completely in a crepant way.",
"The topological Euler characteristic $\\chi _{\\text{top}}(\\hat{Y}_3)$ has two important properties.",
"First, one can split the space into smaller ones and compute their Euler characteristic and then sum all contributions up.",
"This is possible due to the Mayer-Vietoris sequence.",
"Second, for a product space the topological Euler characteristic can be expressed as a product of the Euler characteristics of the factors.",
"In our case the elliptic fibration is locally a product space and therefore we can compute the Euler characteristic of the base and multiply it with the Euler characteristic of the fiber.",
"But since the fibration is generally non-trivial this is only possible locally.",
"We can split the total space into five components: The fibers $ \\bigcup _i \\pi ^{-1} (P_i)$ over the intersection points $P_i$ contribute $\\sum _i \\chi _{\\text{top}}(X_{P_i}) \\cdot B_i \\, ,$ where $B_i$ denotes the number of points $P_i$ .",
"The generic fiber over $\\Sigma _1$ : $\\pi ^{-1} (\\Sigma _1 \\setminus \\bigcup _i P_i)$ contributes as follows: The Euler characteristic of $\\Sigma _1$ without the enhancement points is given by $\\chi _{\\text{top}}(\\Sigma _1) = 2-2g(\\Sigma _1)$ minus the number of points $P_i$ , $\\sum _i B_i$ .",
"It must be multiplied by the Euler characteristic of the fiber, and contributes $\\chi _{\\text{top}}(X_{\\Sigma _1}) \\cdot \\Big ( 2-2\\,g(\\Sigma _1) - \\sum _i B_i \\Big ).$ Analogously, there is a contribution from the general fiber over $\\Sigma _0$ given by $\\chi _{\\text{top}}\\Big ( \\pi ^{-1} (\\Sigma _0 \\setminus Q \\setminus \\bigcup _i P_i) \\Big ).$ Finally, the fibers over the cuspidal points $Q$ contribute $\\chi _{\\text{top}}(X_Q) \\cdot C,$ where $C$ is the number of such points.",
"The general fiber over points where the discriminant does not vanish does not contribute to the Euler characteristic since the $\\chi _{\\text{top}}$ of a torus is zero.",
"Let us compute the different contributions in turn.",
"(REF ) is already in its final form.",
"The fiber $X_{P_i}$ is in general given by a chain of rational curves intersecting each other at points.",
"We stress again that if the Weierstrass model $Y_3$ has non-crepant resolvable singularities, then $X_{P_i}$ is the fiber in a partial Calabi-Yau resolution $\\hat{Y}_3$ of $Y_3$ in which all codimension-one singularities are resolved and only the terminal singularities in codimension-two remain.",
"The computation of the topological Euler characteristic of a chain of $\\mathbb {P}^1$ s is standard, and we review it in Section REF .",
"In particular, the results of this computation show that $\\chi _{\\text{top}}(X_{\\Sigma _1}) = m$ as $X_{\\Sigma _1}$ is of standard Kodaira fiber type.",
"Similarly, in (REF ) we must set $\\chi _{\\text{top}}(X_Q) = 2$ for the cuspidal fibers (type II in Kodaira's classification).",
"What remains is to compute the number of cusps of $\\Sigma _0$ and to bring the third contribution into a nicer form."
],
[
"The number of cusps $C$ .",
"Cusps appear as soon as both $f$ and $g$ vanish along $\\Sigma _0 \\setminus \\bigcup _i P_i$ .",
"Since $f$ and $g$ are of the form (REF ) only $f_0$ and $g_0$ can vanish along $\\Sigma _0$ away from $\\Sigma _1$ .",
"The number of intersection points is naively $(-4 K_B - \\mu _f \\Sigma _1)\\cdot (-6K_B - \\mu _g \\Sigma _1)$ .",
"However, we have to correct it by the intersection multiplicity $\\mu _{P_i} (f,g)$ of $f_0$ and $g_0$ at the points $P_i$ .",
"All in all, the number of cusps is given by $C = 24 K_B^2 + \\Big (4\\mu _g + 6 \\mu _f \\Big ) K_B \\cdot \\Sigma _1 + \\mu _f \\mu _g \\Sigma _1^2 - \\sum _i \\mu _{P_i}(f,g) B_i .$"
],
[
"The third contribution.",
"Finally, to compute $\\chi _{\\text{top}}\\left(\\pi ^{-1} (\\Sigma _0 \\setminus Q \\setminus \\bigcup _i P_i ) \\right)$ , note first that along $\\Sigma _0$ (away from $Q$ and $P_i$ ) the vanishing order structure is $(f,g,\\Delta )|_{\\Sigma _0} = (0,0,1)$ , i.e.",
"the fiber develops a type ${\\rm I}_1$ singularity with Euler characteristic $\\chi _{\\text{top}}(X_{\\Sigma _0}) = 1$ .",
"If $\\Sigma _0$ was a smooth curve, its topological Euler characteristic would be given by $-(K_B + \\Sigma _0) \\cdot \\Sigma _0$ .",
"However, $\\Sigma _0$ may not be smooth at the intersection points $P$ , and it is definitely not smooth at the (cuspidal) self-intersection points $Q$ .",
"Besides, one has to exclude all these points because they are already taken into account in (REF ) and (REF ).",
"Therefore, we have to include correction terms for each point $P_i$ and $Q$ , $\\chi _{\\text{top}}\\Big (\\pi ^{-1} ( \\Sigma _0 \\setminus Q \\setminus \\bigcup _i P_i ) \\Big ) = \\Big ( -(K_B + \\Sigma _0) \\cdot \\Sigma _0 + \\sum _i \\epsilon _i B_i + \\epsilon _c C \\Big ) \\cdot \\underbrace{\\chi _{\\text{top}}(X_{\\Sigma _0})}_{=1}.$ Obviously, $\\epsilon $ has to be defined such that it is $-1$ if the considered point is a smooth point on the curve.",
"To evaluate $(K_B + \\Sigma _0) \\cdot \\Sigma _0 $ note that the Calabi-Yau condition for an elliptic fibration $\\Delta \\in \\mathcal {O} (-12 K_B)$ implies $\\Sigma _0 = -12 K_B - m \\Sigma _1$ .",
"It immediately follows that $\\Sigma _0 \\cdot \\Sigma _1 = -12 K_B \\cdot \\Sigma _1 - m \\Sigma _1^2.$ Applying these two relations several times gives $-(K_B + \\Sigma _0) \\cdot \\Sigma _0 = -132\\, K_B^2 + m \\,K_B \\cdot \\Sigma _1 + 2 m \\,\\Sigma _0 \\cdot \\Sigma _1 + m^2 \\,\\Sigma _1^2 .$ Finally, let us properly define the correction factors $\\epsilon _i$ .",
"As noted already, for a smooth point $\\epsilon $ must take the value $-1$ .",
"Let us consider a curve $D$ with singular point $P\\in D$ .",
"$\\phi _1: B_1 \\rightarrow B$ shall be the blow-up of the point $P$ with exceptional divisor $E$ .",
"We define the quantity $\\alpha _1$ via $D_1 = \\phi _1^* (D) - \\alpha _1 (P) E$ where $D_1$ is the strict transform of the curve.To illustrate the definition of $\\alpha _1$ let us look at a simple example: Let the curve $D$ be given by the equation $x^3+y^3=0$ .",
"It is singular at $(0,0)$ .",
"The blow-up $x\\rightarrow xy, y\\rightarrow y$ leads to $y^3\\cdot (x^3+1)=0$ .",
"Then $y$ is the exceptional divisor of the blow-up and appears with multiplicity $\\alpha _1 = 3$ .",
"Then the Euler characteristic of the blown-up curve is $ \\chi _{\\text{top}}(D_1) = -( K_{B_1} + D_1 ) \\cdot D_1 = - (K_B + D)\\cdot D - \\alpha _1 (P) \\cdot (\\alpha _1 (P) - 1).$ We perform successive blow-ups until the point $P$ is smooth.",
"Since we do not want to include the singular point itself in our calculation of $\\chi _{\\text{top}}$ we have to subtract the number of preimages of $P$ under the total blow-up $\\phi $ .",
"We combine the total correction of $\\chi _{\\text{top}}$ due to the singular point $P$ into the definition of $\\epsilon $ , $ \\epsilon _P := \\sum _{i} \\alpha _i (P) \\cdot \\Big (\\alpha _i (P) - 1\\Big ) - \\# \\phi ^{-1} (P),$ where $i$ runs over the successive blow-ups one has to perform until the singularity is smoothed out completely.In our above example $\\# \\phi ^{-1} (P) = 3$ and therefore $\\epsilon = 3\\cdot 2 - 3 = 3$ .",
"Let us look at another example: $x^3+y^5=0$ .",
"The first blow-up is $x\\rightarrow xy$ : $y^3 \\cdot (x^3+y^2) = 0$ .",
"So $\\alpha _1 = 3$ .",
"Then perform $y\\rightarrow xy$ : $x^2 \\cdot (x+y^2) = 0$ .",
"Since $x+y^2$ has only one solution $\\# \\phi ^{-1} (P) = 1$ and $\\epsilon = 3 \\cdot 2 + 2 \\cdot 1 - 1 = 7$ .",
"In Section REF we will explicitly compute $\\epsilon _i$ for all singularity types which appear in this paper."
],
[
"Final Result.",
"Putting everything together we arrive at the final expression [28] of the topological Euler characteristic of an elliptically fibered Calabi-Yau threefold $\\hat{Y}_3$ over $B_2$ with singular locus $\\Sigma _0 \\cup \\Sigma _1$ as specified above $\\begin{split}\\chi _{\\text{top}}(\\hat{Y}_3) &= \\chi _{\\text{top}}\\Big (\\bigcup _i \\pi ^{-1} (P_i) \\Big ) + \\chi _{\\text{top}}\\Big (\\pi ^{-1} (\\Sigma _1 \\setminus \\bigcup _i \\pi ^{-1} (P_i) ) \\Big ) +\\chi _{\\text{top}}\\Big (\\pi ^{-1} (\\Sigma _0 \\setminus Q \\setminus \\bigcup _i \\pi ^{-1} (P_i)) \\Big ) \\\\& \\quad + \\chi _{\\text{top}}\\Big (\\pi ^{-1} (Q) \\Big )\\\\&= \\Big ( \\sum _i B_i \\cdot \\chi _{\\text{top}}(X_{P_i}) \\Big ) + m\\, \\Big (2-2g-\\sum _i B_i\\Big ) \\\\&\\quad - 132 K_B^2 + m \\,K_B \\cdot \\Sigma _1 + 2 m \\,\\Sigma _0 \\cdot \\Sigma _1 + m^2 \\,\\Sigma _1^2 + 3C+ \\sum _i \\epsilon _i B_i ,\\end{split}$ with $C = 24 K_B^2 + \\Big (4\\mu _g + 6 \\mu _f \\Big ) K_B \\cdot \\Sigma _1 + \\mu _f \\mu _g \\Sigma _1^2 - \\sum _i\\mu _{P_i}(f,g) B_i.$"
],
[
"Computation of $\\epsilon _P$",
"In this article we encounter three different classes of points for which we have to evaluate the parameters $\\epsilon _P$ defined in (REF ): smooth points, singularities of the form $x^2+y^n=0$ for $n\\ge 2$ and singularities of the form $x^3+y^n=0$ for $n\\ge 3$ .",
"We now derive the general expression $\\epsilon _{\\mathrm {smooth}} = -1, \\quad \\quad \\epsilon _{x^2+y^n,\\: n\\,\\ge \\, 2} = n-2, \\quad \\quad \\epsilon _{x^3+y^n,\\: n\\,\\ge \\, 3} = 2n-3.$ for $\\epsilon _P$ in each case.",
"Since all singularities are located at $(x,y) = (0,0) $ we have suppressed the index $P$ .",
"Additionally, we drop the index $i$ which counts the number of successive blow-ups because it should be clear from the context."
],
[
"Smooth points:",
"As we already pointed out there is no need to blow up smooth points ($\\alpha = 0$ and $\\# \\phi ^{-1} (P) = 1 $ ).",
"Thus, $\\epsilon = \\alpha (\\alpha - 1) - \\# \\phi ^{-1} (P) = -1$ ."
],
[
"${\\bf x^2+y^n=0}$ for {{formula:00fc36b4-9a68-47fe-a568-7f110bfd463a}} : ",
"We prove by induction: First consider the curve $x^2 + y^2 = 0$ .",
"After the blow-up $x\\rightarrow xy$ it takes the form $y^2 (x^2+1) = 0$ , i.e.",
"$\\alpha = 2$ and $\\#\\phi ^{-1} = 2$ .",
"Thus, $\\epsilon = 0$ in this case.",
"Next, consider $x^2 + y^3 = 0$ .",
"The blow-up $x \\rightarrow xy$ leads to $y^2 (x^2+y) = 0$ , which means $\\alpha = 2$ and $\\#\\phi ^{-1} = 1$ .",
"Thus, $\\epsilon = 1$ .",
"Now, we are able to do the induction step.",
"The curve $x^2+y^{n+2} = 0$ is blown up ($x\\rightarrow xy$ ) to $y^2 (x^2+y^n) = 0$ .",
"Thus, $\\alpha = 2$ and $\\epsilon _{x^2+y^{n+2}} = 2 + \\epsilon _{x^2+y^n}$ , which proves the assertion."
],
[
"${\\bf x^3+y^n=0}$ for {{formula:1b578238-b595-4392-b723-550a4f864044}} :",
"We prove again by induction but this time we need three initial steps.",
"First, $x^3+y^3=0$ is blown up to $y^3(x^3+1)=0$ ($x\\rightarrow xy$ ), $\\alpha = 3$ , $\\# \\phi ^{-1} = 3$ and $\\epsilon = 3 \\cdot 2 - 3 = 3$ .",
"Second, $x^3+y^4=0$ is blown up to $y^3(x^3+y)=0$ ($x\\rightarrow xy$ ), $\\alpha = 3$ , $\\# \\phi ^{-1} = 1$ and $\\epsilon = 3 \\cdot 2 - 1 = 5$ .",
"Third, $x^3+y^5=0$ is blown up to $y^3(x^3+y^2)=0$ ($x\\rightarrow xy$ ) and $\\alpha = 3$ .",
"We have already shown that $\\epsilon _{x^2 + y^3} = 1$ .Obviously, we are free to swap $x$ and $y$ .",
"All in all, $\\epsilon = 3 \\cdot 2 + \\epsilon _{x^2 + y^3} = 7$ .",
"Finally, we perform the induction step $n \\rightarrow n+3$ .",
"Consider the curve $x^3 + y ^{n+3} = 0$ .",
"After the blow-up $x \\rightarrow xy$ , the exceptional divisor $y^3$ factors out: $y^3(x^3+y^n) = 0$ .",
"Thus, $\\epsilon _{x^3+y^{n+3}} = 3\\cdot 2 + \\epsilon _{x^3+y^n} = 6+2n-3 = 2(n+3) -3$ where we inserted the induction hypothesis $\\epsilon _{x^3+y^n} = 2n-3$ ."
],
[
"Computation of $\\chi _{\\text{top}}(X_{P_i})$",
"The last non-trivial element in formula (REF ) is the topological Euler characteristic of the (partially resolved) fiber in $\\hat{Y}_3$ over the enhancement points $P_i$ .",
"The (in general partial) resolution $\\hat{Y}_3$ leads to several $\\mathbb {P}^1$ s intersecting each other in points.",
"The intersection points are normal crossing singularities of the multi-component fiber.",
"Since the Euler characteristic of a single point is 1, the standard prescription to compute the Euler characteristic of such a variety is as follows: A smooth $\\mathbb {P}^1$ has $\\chi _{\\text{top}}= 2$ .",
"Subtract one for every singular point on the $\\mathbb {P}^1$ and sdd the contributions from all $\\mathbb {P}^1$ s up.",
"Finally, add $+1$ for every singular point.",
"Let us make the prescription more concrete by considering some examples.",
"The numbers in parenthesis denote the contributions to $\\chi _{\\text{top}}$ .",
"The type III fiber.",
"It has two components each of which has one singular point ($1+1$ ).",
"In total there is one singular point (1).",
"Thus, $\\chi _{\\text{top}}(\\text{type III}) = 1+1+1 = 3$ .",
"The type IV fiber.",
"It has three components all of which have a singular point ($1+1+1$ ).",
"These three singular points are coincident (1).",
"So, $\\chi _{\\text{top}}(\\text{type IV}) = 3+1 = 4$ .",
"The type $\\mathrm {I}_0^*$ fiber.",
"It has four components with one singular point ($1+1+1+1$ ), one component with four singular points ($2-4$ ) and all in all four singular points (4).",
"Therefore, $\\chi _{\\text{top}}(\\text{type I}_0^*) = 4-2+4 = 6$ ."
],
[
"Generalisation to Models With Three Discriminant Loci",
"Expression (REF ) applies to models with $\\Delta = \\Sigma _0 \\cup \\Sigma _1$ .",
"In this appendix we generalize it to models with $\\Delta = \\Sigma _0 \\cup \\Sigma _1 \\cup \\Sigma _2$ and identical gauge groups along $\\Sigma _1$ and $\\Sigma _2$ , which will be studied further in appendix .",
"Our starting points were the four contributions to the topological Euler characteristic (REF ) to (REF ).",
"By assumption, the vanishing orders of $\\Delta $ along $\\Sigma _1$ and $\\Sigma _2 $ are equal and are denoted again by $m$ and similarly for the vanishing orders $\\mu _f,\\mu _g$ .",
"Furthermore $B_1 = B_2$ for the number of enhancement points from intersection of $\\Sigma _i$ with $\\Sigma _0$ .",
"In addition there is a new type of enhancement locus ($\\lbrace z_1 = 0\\rbrace \\cap \\lbrace z_2=0 \\rbrace $ ) called $R$ .",
"This gives an extra contribution to the topological Euler characteristic, $\\chi _{top}(\\pi ^{-1}(R)) = 1 \\cdot \\chi _{top} (X_R)$ .",
"The second contribution (REF ), which took the topology of the brane without all enhancement points into account, becomes $\\chi _{top} \\left(\\pi ^{-1}\\big ((\\Sigma _1 \\cup \\Sigma _2) \\setminus P_1 \\setminus P_2 \\setminus R\\big ) \\right) &= 2 \\cdot \\chi _{top} (\\pi ^{-1}( \\Sigma _1 \\setminus P_1 \\setminus R))\\\\& = 2 \\cdot m \\cdot (2 - 2g-(B_1+1))$ The formula of the third contribution (REF ) relies on the fact that $\\Sigma _0 \\in -12 K_B -m \\Sigma _1$ in the old situation.",
"Here, $\\Sigma _0 \\in -12 K_B-m_1\\Sigma _1 - m_2 \\Sigma _2 = -12 K_B - 2m\\Sigma _1$ .",
"Therefore, we have to replace $m \\rightarrow 2m$ .",
"Similarly, one has to replace $\\mu _f\\rightarrow 2\\mu _f$ and $\\mu _g\\rightarrow 2\\mu _g$ in the formula for the number of cusps.",
"With the same argument as before the intersection multiplicity $\\mu _i (f,g)$ can be set to zero.",
"Then, the contributions to $\\chi _{top}(Y_3)$ are: $\\chi _{top} (\\pi ^{-1}(P_1 \\cup P_2)) =2B_1\\cdot \\chi _{top} (X_{P_1}) $ .",
"$\\chi _{top} (\\pi ^{-1}(R)) = \\chi _{top} (X_R)$ .",
"$\\chi _{top} (\\pi ^{-1}((\\Sigma _1 \\cup \\Sigma _2) \\setminus P_1 \\setminus P_2 \\setminus R)) = 2 m \\cdot (2 - 2g-(B_1+1))$ .",
"$\\chi _{top} (\\pi ^{-1} (\\Sigma _0 \\setminus Q \\setminus P_1 \\setminus P_2)) = -11 \\cdot 12 K_B^2 + 2mK_B \\cdot \\Sigma _1 + 4m^2 \\Sigma _1^2+4m\\Sigma _1\\cdot \\Sigma _0+ 2\\epsilon _1 B_1+C$ .",
"$\\chi _{top} (\\pi ^{-1} (Q)) =2C$ with the number of cusps $C = 24 K_B^2 + \\Big (8\\mu _g + 12 \\mu _f \\Big ) K_B \\cdot \\Sigma _1 + 4\\mu _f \\mu _g \\Sigma _1^2.$ All in all, $\\chi _{\\text{top}}$ is given by the more compact expression: $\\chi _{\\text{top}}(Y_3) &=& -540 + \\chi _{\\text{top}}(X_R) + 2 \\,B_1 \\, \\Big (\\chi _{\\text{top}}(X_{P_1}) + \\epsilon _1\\Big )+ m \\cdot \\Big (140 - 4m - 2B_1 \\Big ) - \\nonumber \\\\&& -72 \\mu _g - 108 \\mu _f + 12 \\mu _f \\mu _g.$"
],
[
"Toric Resolution of a Type IV Model",
"In the main text we have encountered Weierstrass models with Kodaira fiber of type II and type III in codimension-one.",
"Such models are interesting by themselves as they are inherently non-perturbative.",
"The remaining model in this list of Kodaira outliers is Kodaira type IV, corresponding to a non-perturbative realisation of gauge algebra $SU(3)$ (split) or $Sp(1)$ (non-split).",
"In this appendix we describe the resolution of such a split type IV model, which, to the best of our knowledge, has not been presented in the literature before.",
"A general type IV Weierstrass model is defined by arranging for vanishing orders of the following type along a divisor $\\Sigma _1: z_1 = 0$ , $f = z_1^{2} \\, f_0, \\qquad g = z_1^2 \\, g_0 \\qquad \\rightarrow \\quad \\Delta = z_1^4 (27 g_0^2 + 4 f_0^3 z_1^2).$ The codimension-two enhancement points $P_1$ at $\\Sigma _1 \\cap \\Sigma _0$ lie at $P_1: z_1 = g_0 = 0 \\quad {\\rm with} \\quad {\\rm ord}(f,g,\\Delta )|_{P_1} = (2,3,6),$ corresponding to an enhancement from type IV to type $I_0^*$ in the fiber.",
"We realize a split version as a Tate model with vanishing orders $a_i = \\tilde{a}_i\\, z^{k_i}_i, \\qquad k_i = (1,1,1,2,3) \\quad {\\rm for} \\,\\, i=1,2,3,4,6 \\,.$ Similarly to the procedure described in Section REF for the type III models, we resolve the singularities in the fiber by performing two blow-ups of the ambient space, $x \\rightarrow e_1\\, e_2 \\, x, \\quad y \\rightarrow e_1\\, e_2^2\\, y, \\quad z_1 \\rightarrow e_0\\, e_1\\, e_2 \\, .$ The proper transform $PT$ is now the non-generic hypersurface $PT = - e_1 x^3 + e_2 y^2 + \\tilde{a}_1 e_0 e_1 e_2 x y z - \\tilde{a}_2 e_0 e_1 x^2 z^2 + \\tilde{a}_3 e_0 y z^3 -\\tilde{a} _4 e_0^2 e_1 x z^4 - \\tilde{a}_6 e_0^3 e_1 z^6$ in the toric ambient space with scaling relations listed in table REF .",
"Table: Toric weights for the fiber ambient space of the type IV Tate model.This toric ambient space admits four different triangulations.",
"For definiteness, consider the phase with Stanley-Reisner ideal ${\\rm SRI} = \\langle ye_1,ze_1,ze_2,xyz,xye_0,xe_0e_2\\rangle .$ This time $PT$ is free of residual singularities and hence defines a crepant resolution to a smooth Calabi-Yau threefold $\\hat{Y}_3$ .In fact, even for higher dimensional bases no singularities remain due to the SRI constraints.",
"The fibral rational curves in codimension one, over $\\Sigma _1$ , are given by $\\mathbb {P}^1_A:\\: PT|_{e_0 \\rightarrow 0} &= y^2 e_2-x^3 e_1,\\\\\\mathbb {P}^1_B:\\: PT|_{e_1 \\rightarrow 0} &= y\\cdot (z^3 \\tilde{a}_3 e_0+y e_2)\\\\\\mathbb {P}^1_C:\\: PT|_{e_2 \\rightarrow 0} &= \\tilde{a}_3 e_0\\, y z^3 - x^3 e_1-\\tilde{a}_6 e_0^3 e_1 \\, z^6 - \\tilde{a}_4 e_0^2 e_1\\, x z^4- \\tilde{a}_2 e_0 e_1 \\, x^2 z^2 \\, .$ While the second equation factorises, the locus $\\lbrace e_1 \\rbrace \\cap \\lbrace y \\rbrace $ is forbidden by the SRI.",
"Hence one finds altogether three curves intersecting in one point $\\lbrace e_0 \\rbrace \\cap \\lbrace e_1 \\rbrace \\cap \\lbrace e_2 \\rbrace $ , as required for a type IV fiber.",
"The curve $\\mathbb {P}^1_A$ is intersected by the zero section $z=0$ , while $\\mathbb {P}^1_B$ and $\\mathbb {P}^1_C$ are identified with the simple roots $\\alpha _1$ and $\\alpha _2$ of $SU(3)$ , respectively.",
"As always, the Lie algebra roots correspond to minus the intersection numbers with the resolution divisors representing the Cartan generators.",
"In this sense, the simple roots are given by $\\alpha _1 = (2,-1) = (-E_1 \\cdot \\mathbb {P}^1_B, -E_2 \\cdot \\mathbb {P}^1_B), \\qquad \\alpha _1 = (-1,2) = (-E_1 \\cdot \\mathbb {P}^1_C, -E_2 \\cdot \\mathbb {P}^1_C).$ The next step is to take a look at the fiber enhancement in codimension two.",
"From the discriminant $\\Delta = \\tfrac{1}{16} \\,z_1^4 \\cdot \\left( 27\\, a_3^4 + \\mathcal {O} (z_1) \\right)$ one reads off that the fiber enhances at $z_1 = 0 = \\tilde{a}_3$ , corresponding to the points $P_1$ .",
"The fiber over this locus takes the form $PT|_{e_0 \\rightarrow 0 ,\\, \\tilde{a}_3 \\rightarrow 0} &= y^2e_2-x^3e_1, \\\\PT|_{e_1 \\rightarrow 0 ,\\, \\tilde{a}_3 \\rightarrow 0} &= y^2 e_2,\\\\PT|_{e_2 \\rightarrow 0 ,\\, \\tilde{a}_3 \\rightarrow 0} &= -e_1 \\cdot (x^3 + \\tilde{a}_2 x^2 (e_0 z^2) + \\tilde{a}_4 x (e_0 z^2)^2 + \\tilde{a}_6 (e_0 z^2)^3 )\\\\&= - e_1 \\cdot (x-e_0 z^2 \\cdot f_1(\\tilde{a}_i) )\\cdot (x-e_0 z^2 \\cdot f_2(\\tilde{a}_i) )\\cdot (x-e_0 z^2 \\cdot f_3(\\tilde{a}_i) ).$ Only the curve $\\mathbb {P}^1_C$ splits into four components.",
"If we denote the curves $\\mathbb {P}^1_A$ and $\\mathbb {P}^1_B$ over $z_1=\\tilde{a}_3=0$ by $\\mathbb {P}^1_a$ and $\\mathbb {P}^1_b$ , then altogether $\\mathbb {P}^1_A \\rightarrow \\mathbb {P}^1_a, \\qquad \\mathbb {P}^1_B \\rightarrow \\mathbb {P}^1_b, \\qquad \\mathbb {P}^1_C \\rightarrow \\mathbb {P}^1_b \\cup \\mathbb {P}^1_c \\cup \\mathbb {P}^1_d \\cup \\mathbb {P}^1_e.$ We observe five distinct components, one with multiplicity two, intersecting as expected for a $\\text{I}_0^*$ fiber, see figure REF .",
"Figure: Affine Dynkin diagram of the resolved {z 1 }∩{a ˜ 3 }\\lbrace z_1\\rbrace \\cap \\lbrace \\tilde{a}_3\\rbrace locus.",
"The red cross denotes the intersection with the zero-section z=0z=0 of the Weierstrass model.",
"The blue and red colour indicates the splitting of ℙ A,B,C 1 \\mathbb {P}^1_{A,B,C}.Computing $-1$ times the intersection numbers with the two resolution divisors $E_1$ and $E_2$ , we find the $SU(3)$ weights of these curves to be $\\mathbb {P}^1_a : (-1,-1), \\quad \\mathbb {P}^1_{b} : (2,-1), \\quad \\mathbb {P}^1_{c,d,e} : (-1,1).$ The highest weight of the fundamental representation is $w_1 = (1,0)$ .",
"There are three possibilities to represent the highest weight in terms of holomorphic fibral curves, $(\\mathbb {P}^1_b + \\mathbb {P}^1_c): (1,0), \\quad (\\mathbb {P}^1_b + \\mathbb {P}^1_d): (1,0), \\quad (\\mathbb {P}^1_b + \\mathbb {P}^1_e): (1,0).$ To construct the other states, we have to act with the simple roots on the highest weight vector.",
"For example, for starting with $\\mathbb {P}^1_b + \\mathbb {P}^1_c$ this gives $w_1 :& \\quad \\mathbb {P}^1_b + \\mathbb {P}^1_c,\\\\w_1 -\\alpha _1 :& \\quad (\\mathbb {P}^1_b + \\mathbb {P}^1_c) - \\mathbb {P}^1_b = \\mathbb {P}^1_c,\\\\w_1 - \\alpha _1 - \\alpha _2 :& \\quad \\mathbb {P}^1_c -( \\mathbb {P}^1_b + \\mathbb {P}^1_c + \\mathbb {P}^1_d + \\mathbb {P}^1_e) = - (\\mathbb {P}^1_b + \\mathbb {P}^1_d + \\mathbb {P}^1_e).$ Each these three curves can be wrapped by an M2-brane as well as an anti-M2-brane.",
"This gives rise to one hypermultiplet in the ${\\bf 3}$ of SU(3).",
"Repeating this for the remaining two highest weights, we conclude that we find in total three copies of the fundamental representation in the resolved fiber at each enhancement point $P_1$ .",
"This spectrum is in full agreement with the cancellation of both gravitational and SU(3) gauge anomalies.",
"For the special case of a base $B_2 = \\mathbb {P}^2$ we list the charged and uncharged spectrum, computed via $\\chi _{\\rm top}(\\hat{Y}_3)$ , in table REF .",
"Table: Smooth Weierstrass models over B 2 =ℙ 2 B_2 = \\mathbb {P}^2.",
"The first column corresponds to a generic Weierstrass model, the second and third column correspond to the smooth type III models discussed in Section ."
],
[
"Models With $\\Delta = \\Sigma _0 \\cup \\Sigma _1 \\cup \\Sigma _2$",
"In this appendix, the analysis of F-theory models with one brane is generalized to models with two identical branes in generic position to each other, i.e.",
"models with two identical gauge group factors.",
"This provides a richer structure of singularity types and geometry of the fibration.",
"We will consider models with two identical branes wrapped on the divisors $\\Sigma _1: z_1=0$ and $\\Sigma _2: z_2=0$ .",
"These models are defined via the requirement that $f$ and $g$ in the Weierstrass model vanish to certain orders along the divisors.",
"Since the two branes shall be identical we need two numbers to specify a model.",
"We employ the following notation: The model with $f = (z_1z_2)^{\\mu _f} f_0$ and $g = (z_1z_2)^{\\mu _g} g_0$ is called $[\\mu _f\\mu _g]$ -model."
],
[
"The $[n1]$ -Models: Type II {{formula:7a7f64d5-1329-452f-8625-a8a4a05df850}} Type II",
"Let us start out with the $[n1]$ -models generalizing the geometries discussed in Section REF by taking $f = (z_1z_2)^n f_0, \\qquad g = (z_1z_2) g_0 \\qquad \\Longrightarrow \\quad \\Delta = \\underbrace{z_1^2}_{\\Sigma _1} \\underbrace{z_2^2}_{\\Sigma _2} \\underbrace{(27 g_0^2+4 (z_1 z_2)^{3n-2} \\,f_0^3)}_{\\Sigma _0}.$ If we specialise these models to base $B_2 = \\mathbb {P}^2$ , where $f\\in \\mathcal {O} (12)$ , and identify $z_1$ and $z_2$ with two of the homogenous coordinates $[z_0 : z_1 : z_2]$ , the allowed range for $n$ is $1 \\le n \\le 6$ .",
"The fiber over $\\Sigma _1$ and $\\Sigma _2$ is type II with trivial gauge group.",
"As a novelty compared to the single brane models, we encounter now an intersection of two type II components at $\\Sigma _1 \\cap \\Sigma _2$ .",
"This locus is called $R$ in the notation of appendix REF , and characterized as follows: $R: z_1 = z_2= 0, \\qquad {\\rm ord}(f,g,\\Delta ) =(2n,2,4) \\qquad \\Longrightarrow \\quad {\\rm II} \\, {\\rm x} \\, {\\rm II} \\rightarrow {\\rm IV} \\, .$ In addition, over the $B_i=16$ points of type $P_i$ where $z_i = g_0 = 0$ , corresponding to the intersection of $\\Sigma _i$ with the residual discriminant $\\Sigma _0$ , the fiber enhances as follows: $&n=1: \\qquad B_i: z_i = g_0 = 0, \\qquad {\\rm ord}(f,g,\\Delta ) =(1,2,3) \\qquad &{\\rm II} \\, \\rightarrow \\, {\\rm III}, \\\\&n\\ge 2: \\qquad B_i: z_i = g_0 = 0, \\qquad {\\rm ord}(f,g,\\Delta ) =(n,2,4) \\qquad &{\\rm II} \\, \\rightarrow \\, {\\rm IV}\\, .$ Both the type IV enhancements over $R$ and $B_i$ (for $n \\ge 2$ ) and and the type III enhancements over $B_i$ (for $n=1$ ) are isolated terminal Kleinan singularities of local form $z^3+x_1^2+x_2^2+x_3^2=0$ , with Milnor number $m_{P_i} = m_R = 2$ .",
"Hence each of these enhancement points carries 2 massless uncharged hypermultiplets (independently of the choice of base).",
"To compute the Euler characteristic (REF ) we set $\\chi _{\\text{top}}(X_{P_i}) = 2 = \\chi _{\\text{top}}(X_R)$ , corresponding to the singular non-resolved fiber.",
"Over base $B_2 = \\mathbb {P}^2$ and with these values, the general expression (REF ) simplifies to $\\chi _{\\text{top}}(Y_3) = -346 - 96 \\,n + 32 \\,\\epsilon .$ The remaining task is to determine $\\epsilon $ .",
"The curve $\\Sigma _0$ takes the form $g_0^2+z_{1,2}^{3n-2} = 0 $ near an intersection locus with $\\Sigma _{1}$ or $\\Sigma _2$ .",
"Hence, $\\epsilon = 3n-4$ (see (REF )).",
"This cancels the $n$ -dependence of $\\chi _{\\text{top}}(Y_3)$ and we end up with the value $\\chi _{\\text{top}}( Y_3)= -474$ for all $n$ .",
"From equation (REF ) we obtain $239+\\frac{1}{2} \\sum m_P = 272$ complex structure deformations and hence 273 uncharged massless hypermultiplets, as required by anomaly cancellation.",
"Table: Scaling Relations for the fiber ambient space of the resolved [1n][1n]-models.One possible choice for the Stanley-Reisner ideal is: $\\langle ze_1, zf_1, e_0f_1, xyz, xye_0, xyf_0 \\rangle .$"
],
[
"The $[1n]$ -Models ({{formula:35d85e4b-e5be-4196-93e4-2618711b4d61}} ): Type III {{formula:f8269152-d667-4f3a-a70d-7d6baa785791}} Type III",
"These models generalize the geometries analyzed in Section REF in the sense that ${\\rm ord}(f,g,\\Delta )|_{z_i} = (1,n,3) \\quad \\Longrightarrow \\quad \\Delta = z_1^3z_2^3 \\,\\Big (4f_0^3 + 27\\, (z_1z_2)^{2n-3}\\, g_0^2\\Big )$ with $n >1$ , corresponding to gauge group $G = SU(2) \\times SU(2)$ .",
"The intersection of both 7-branes at the point R, $R: z_1 = z_2= 0, \\qquad {\\rm ord}(f,g,\\Delta ) =(2,2n,6) \\qquad \\Longrightarrow \\quad {\\rm III} \\, {\\rm x} \\, {\\rm III} \\rightarrow {\\rm I}_0^\\ast \\, ,$ is new compared to the models in Section REF , whereas the $B_i$ loci $\\Sigma _i = f_0 = 0$ behave as in eq.",
"(REF ).",
"The explicit resolution of a Tate model realisation of these fibrations given below confirms the structure of the resolved fiber over the locus $R$ as a monodromy reduced I$^*_0$ fiber with $\\chi _{\\rm top}(X_R)=4$ .",
"This fiber carries one massless hypermultiplet in representation $({\\bf 2},{\\bf 2})$ of $G = SU(2) \\times SU(2)$ .",
"Furthermore, as in Section REF , for $n \\ge 4$ the fiber over the points $B_i$ exhibit a residual Kleinan singularity of the form (REF ) - (REF ), which is responsible for the localisation of a corresponding number of uncharged hypermuliplets at these points in addition to the two hypermultiplets in the $({\\bf 2},1)$ or $(1,{\\bf 2})$ , respectively.",
"It can readily be checked that this charged spectrum leads to an anomaly free spectrum.",
"Furthermore, over base $B_2 = \\mathbb {P}^2$ , we can compute the topological Euler characteristic via (REF ) (see table REF ).",
"In this case, the number of points $B_i$ is 10, and the details of the computation parallel the analysis in Section REF .",
"Table: Singular locus of proper transforms.",
"The singularity parameter aa describes the local form of the singularity: z a +x 1 2 +x 2 2 +x 3 2 z^a + x_1^2+x_2^2+x_3^2.We resolve the $[1n]$ -models by realising them in Tate form, choosing the Tate vanishing orders listed in table REF along $z_1=0$ and $z_2=0$ .",
"Let us explicitly consider the cases $n = 2,3,4,5$ since all conceptional features appear already here.",
"The singularities at $z_i = x = y =0$ are resolved by the two blow-ups $x \\rightarrow e_1f_1x, \\quad y \\rightarrow e_1f_1y, \\quad z_1 \\rightarrow e_0e_1,\\quad z_2 \\rightarrow f_0f_1,$ which lead to the proper transform $PT$ of the Tate form $PT =& -e_1 f_1 \\, x^3 + y^2 + \\tilde{a}_1\\, e_0^n e_1^n f_0^n f_1^n\\, x y z -\\tilde{a}_2\\, e_0^n e_1^n f_0^n f_1^n\\, x^2 z^2 + \\\\& + \\tilde{a}_3\\, e_0^n e_1^{n-1} f_0^n f_1^{n-1} \\, y z^3 - \\tilde{a}_4\\, e_0 f_0 \\, x z^4 - \\tilde{a}_6 \\, e_0^n e_1^{n-2} f_0^n f_1^{n-2} \\, z^6 \\quad \\text{for } n \\ge 2.$ Again, we view $PT$ as a hypersurface in a toric fiber ambient space with coordinates $x$ , $y$ , $z$ , $e_1$ , $f_1$ , $e_0$ , $f_0$ .",
"The associated toric weights are displayed in table REF .",
"In complete analogy to the case studied in Section REF , for $n \\ge 4$ there remain singularities in codimension 2 as listed in table REF .",
"These residual singularities result in uncharged localised hypers.",
"Indeed, away from the point $z_1 = z_2=0$ the structure of fibers is identical to the pattern for a single type III brane detailed in Section REF .",
"Hence it only remains to analyze the fiber type at the intersection of the two type III singularities at $R: z_1 = z_2 =0$ .",
"Naïve application of Kodaira's classification predicts an $\\text{I}_0^*$ fiber.",
"Direct inspection reveals the following fibral curves over $R$ (note that $e_0 \\rightarrow 0, f_1\\rightarrow 0$ is forbidden by the SRI), $\\mathbb {P}^1_a:\\: PT|_{e_0 \\rightarrow 0,\\, f_0\\rightarrow 0} &= -e_1 f_1 x^3+y^2, \\\\\\mathbb {P}^1_{b_{1,2}}:\\: PT|_{e_1 \\rightarrow 0,\\, f_0\\rightarrow 0} &= y^2, \\\\\\mathbb {P}^1_c:\\: PT|_{e_1 \\rightarrow 0,\\, f_1\\rightarrow 0} &= {\\left\\lbrace \\begin{array}{ll}y^2-a_4 e_0 f_0 x z^4-a_6 e_0^2 f_0^2 z^6\\quad & \\text{for } a=2,\\\\y^2-a_4 e_0 f_0 x z^4\\quad & \\text{for } a>2.\\end{array}\\right.",
"}$ The second line indicates the presence of two copies of the rational curve $ e_1 = f_0= y = 0$ .",
"These two copies will be denoted by $\\mathbb {P}^1_{b_{i}}$ , $i=1,2$ .",
"We can interpret this fiber as a monodromy reduced $\\text{I}_0^*$ fiber with two nodes deleted, where $\\mathbb {P}^1_{b_{i}}$ , $i=1,2$ corresponds to the middle $\\mathbb {P}^1$ of multiplicity 2, intersecting the two other curves once (see figure REF ).",
"The negative of the intersection numbers with the resolution divisors $E_1$ and $F_1$ give the $U(1) \\times U(1)$ Cartan charges $\\mathbb {P}^1_{b_i} =(1,-1), \\qquad \\mathbb {P}^1_c: (0,2) \\,,$ while $\\mathbb {P}^1_a$ is intersected by the zero section and hence plays no role in determining the weight lattice.",
"If we denote the $w_0^{E/F}$ the highest weight of the fundamental representation of the two $SU(2)$ factors, and by $\\alpha _01^{E/F}$ their simple root, we can make the following identification between holomorphic and anti-holomorphic curves in the fiber and the weights of a bifundamental representation $({\\bf 2}, {\\bf 2})$ , $\\begin{pmatrix}(w_0^E+\\alpha _1^E,w_0^F)\\\\(w_0^E,w_0^F)\\\\(w_0^E,w_0^F+\\alpha _1^F)\\\\(w_0^E+\\alpha _1^E,w_0^F+\\alpha _1^F)\\\\\\end{pmatrix}=\\begin{pmatrix}\\mathbb {P}^1_{b_1}\\\\-(\\mathbb {P}^1_{b_2} + \\mathbb {P}^1_c)\\\\-\\mathbb {P}^1_{b_2}\\\\\\mathbb {P}^1_{b_1} + \\mathbb {P}^1_c\\end{pmatrix}=\\begin{pmatrix}(1,-1)\\\\(-1,-1)\\\\(-1,1)\\\\(1,1)\\end{pmatrix}.$ Here we explicitly distinguish between the two curves $\\mathbb {P}^1_{b_1}$ and $\\mathbb {P}^1_{b_2}$ .",
"M2 branes wrapping each of these four curves with positive and negative orientation give rise to a full hypermultiplet in the $({\\bf 2}, {\\bf 2})$ of $SU(2) \\times SU(2)$ .",
"This perfectly matches the prediction from the anomaly constraints.",
"Figure: Affine Dynkin diagram of the resolved {z 1 }∩{z 2 }\\lbrace z_1\\rbrace \\cap \\lbrace z_2\\rbrace locus.",
"The red cross denotes the intersection with the zero-section z=0z=0 of the Weierstrass model."
],
[
"Summary of All Models With $\\Delta = \\Sigma _0 \\cup \\Sigma _1$",
"The results for all models in this paper with enhancements along a single divisor on $\\mathbb {P}^2$ are summarized in table REF .",
"Table: Models over B 2 =ℙ 2 B_2 = \\mathbb {P}^2 with enhancements over a single divisor."
]
] | 1612.05646 |
[
[
"Towards Wide Learning: Experiments in Healthcare"
],
[
"Abstract In this paper, a Wide Learning architecture is proposed that attempts to automate the feature engineering portion of the machine learning (ML) pipeline.",
"Feature engineering is widely considered as the most time consuming and expert knowledge demanding portion of any ML task.",
"The proposed feature recommendation approach is tested on 3 healthcare datasets: a) PhysioNet Challenge 2016 dataset of phonocardiogram (PCG) signals, b) MIMIC II blood pressure classification dataset of photoplethysmogram (PPG) signals and c) an emotion classification dataset of PPG signals.",
"While the proposed method beats the state of the art techniques for 2nd and 3rd dataset, it reaches 94.38% of the accuracy level of the winner of PhysioNet Challenge 2016.",
"In all cases, the effort to reach a satisfactory performance was drastically less (a few days) than manual feature engineering."
],
[
"Introduction",
"With the rapid growth in the availability and size of digital health data and wearable sensors, along with the rise of newer machine learning methods, health care analytics has become a hot area of research today.",
"The main bottlenecks for solving a healthcare data analytics problem are: a) Effort required to build good models in terms of time, money and expertise b) Interpreting model features so that a healthcare expert can do a causality analysis and take preventable measures or derive meaningful insights backed by domain knowledge.",
"A typical analytics solution requires a) Pre-processing b) Feature Extraction c) Feature Selection d) Modeling such as Classification or Regression.",
"Among these steps, Feature Extraction and Feature Selection together form Feature Engineering (FE) and is the most time consuming and human expertise demanding among the rest.",
"Feature engineering can be broadly carried out in four ways: (a) manually selecting features guided by domain knowledge (b) recommending features by automated analysis - proposed method (c) feature transforms like Principal Component Analysis (PCA) (d) representation learning using deep architectures such as deep Multi-Layered Perceptron (MLP) and Convolutional Neural Network (CNN).",
"Through experiments on 3 different types of healthcare datasets including a recent challenge dataset and comparison of the approaches, the utility of our proposed method (b) has been shown.",
"Interpretation of features is not supported by deep learning and feature transform methods.",
"But, manual feature engineering and our proposed method yield interpretable features which is very helpful in prognostic domains like healthcare."
],
[
"Solution Approach",
"While in Deep Architectures, the different activation functions can be hierarchically stacked to form new structures, in our approach, this does not hold true.",
"For example, Wavelet transforms applied on Fourier transforms does not make sense.",
"Hence, here the emphasis is on creating a Wide Architecture with meaningful hierarchies so that lowest layer contains basic feature extraction techniques, and as we move up we keep adding more meaningful layers on top of what was extracted.",
"This helps in deriving physical interpretation of features (from bottom to top).",
"The dataset is partioned into p-folds of training, evaluation and testing sets (range of p is 5 to 10).",
"The performance is reported on the hidden testing set.",
"The proposed method consists of 3 steps: 1.",
"Feature Listing: We have organized commonly reported features (in the literature of sensor data analytics) in a hierarchical manner as shown in Figure 1.",
"The basic features (level 0) can be mainly categorized as: (i) time domain features (TD) (ii) fourier transformation based features (FD) like short-time fourier transform (STFT) (iii) discrete wavelet transformation based features (DWT).",
"One major challenge of using DWT features is the selection of suitable mother wavelet, as more than 100 different types of mother wavelets were reported in different papers.",
"The automated mother wavelet selection is done by measuring energy to entropy ratio [1].",
"In level 1, spectral, statistical and peak-trough features are extracted.",
"Level 2 includes different ratios and derivatives of the level 1 features.",
"The system has capability of easy plugging of new feature extraction algorithms that will lead to a collaborative ecosystem.",
"Hence, it is possible to get huge number (say, $N$ ) of features (including the transform domain coefficients) from the sensor signals.",
"This results in $2^N-1$ possible combinations of features, whose exploration is practically infeasible, thereby demanding usage of feature selection.",
"2.",
"Feature Selection: In our method, we followed an iterative feature selection where $k$ -features are selected (k$\\ll $ N) at each iteration and system performance (e.g.",
"classification accuracy) is checked for this feature set.",
"If the selected feature set results in expected performance, we return the feature set as the recommended one.",
"Otherwise, another $k$ -features are chosen in the next iteration and the same steps are repeated.",
"For checking the classification accuracy, we choose SVM (support vector machine) based classification with different kernels.",
"SVM was selected as a classifier as it generalizes well and converges fast.",
"Several values of $k$ are tried to choose an optimal value.",
"For a given value of $k$ , features are selected using two techniques namely, mRMR [2] and MRMS [3], described below: Minimum Redundancy and Maximum Relevance(mRMR): In order to select effective features, mRMR optimizes an objective function, either Mutual Information Difference (MID) or Mutual Information Quotient (MIQ), by minimizing the redundancy and maximizing the relevance of the features.",
"MID (additive) and MIQ (multiplicative) are defined as follows.",
"$MID = \\texttt {max} (V - W) \\quad \\quad , \\quad \\quad MIQ = \\texttt {max} (V/W)$ where $V$ minimizes redundancy by computing F-statistics and $W$ maximizes relevance by computing correlation between a pair of features.",
"Maximal Relevance Maximum Significance$ $ (MRMS): This technique uses fuzzy-rough set selection criteria to select relevant and non-redundant (significant) features.",
"The objective function is: $J = J_{rel} + \\beta J_{sig}$ where $J_{rel}$ computes relevance of a recommended feature with respect to a class label and $J_{sig}$ computes the significance of a pair of recommended features by computing their correlation, and $\\beta $ is the weight parameter.",
"Let, $x$ and $y$ be the sets of features recommended by mRMR and MRMS, respectively.",
"Then the recommended set of features R is $z$ , where $z = x \\cup y$ , where $|z|= k$ .",
"Note that mRMR and MRMS cover different aspects of feature selection.",
"For instance, mRMR is classifier independent where as MRMS is effective to reduce real valued noisy features which are likely to occur in large feature sets.",
"3.",
"Feature Recommendation: The system finds 2 feature sets for a particular performance metric (such as accuracy, sensitivity, specificity, precision, f-score): a) Fe1 - that produces the highest metric in any fold of cross-validation b) Fe2 - that is most consistent and performs well across all folds.",
"The above step of feature selection is done hierarchically - if layer 0 does not produce expected results set by pre-set threshold $\\tau $ or maximum possible value of a selected metric, then layer 1 is invoked.",
"Similarly if layer 1 does not yield expected results, layer 2 is invoked.",
"This follows the principle that if simple features can do the task, there is no need for complex features.",
"`c' is a regularizer for `k' and is dependent on the hardware capabilities of the system.",
"The intuition is that on a high-end machine (having higher valued `c'), feature combinations ($2^c-1$ ) can be carried in acceptable time.",
"Using the recommended feature sets, any classifier like SVM or Random Forest can be trained to see the results obtained.",
"Also by looking up the recommend features from the Feature Listing database, interpretation of the features can be easily obtained by a domain expert."
],
[
"Experiments and Results",
"Experiments were carried on 3 datasets: D1, D2, D3 in order to provide a comparison among the feature engineering ways (proposed method, manual, dimension reduction and deep learning).",
"D1: The Physionet 2016 Challenge dataset [4] consists of 3153 heart sounds, including 2488 normal and 665 abnormal recordings.",
"The ground truth label (normal or abnormal heart sound) of each record is manually annotated by expert doctors.",
"Raw PCG (phonocardiogram) is further down sampled to 1 KHz from 2 KHz, in order to segregate four cardiac states (S1, systole, S2 and diastole) using the logistic regression based HSMM approach [5].",
"The winner [6] of the challenge used 124 features and used deep learning for classification.",
"The challenge used their own modified metric for ranking participants, however for consistency of results across datasets, we have used accuracy score as the performance metric.",
"We participated in the challenge using manual features and got only 1% increase in performance compared to the proposed automated method.",
"D2: The second dataset is derived from MIMIC-II patients dataset [7].",
"A subset of the dataset containing PPG (photoplethysmogram) data was created after noise cleaning and the ground truth blood pressure (BP) was obtained from the simultaneously recorded arterial BP waveform, resulting in equally balanced 36 high (>140 mmHg reading) and 36 low BP patient waveform data instances.",
"D3: The third dataset (used to classify the emotion into happy and sad) records the fingertip pulse oximeter PPG data of 33 healthy subjects (Female: 13 and Male: 20) with average age 27 years.",
"We used standard video stimuli as ground-truth and time synchronization errors were minimized.",
"Table 1 lists the obtained result for a dataset along with the corresponding configuration and effort for each of the feature engineering approaches.",
"Experiments has been carried out using Theano Theano version 0.8.2 used from http://deeplearning.net/software/theano/ based Multi-Layer Perceptron with Dropout and varying number of layers to see if features can be automatically learned on the datasets under experimentation.",
"Different epochs (5 to 15) has been tried to see how the learning rate affects performance.",
"Different activation functions like rectified linear unit (relu), tanh, softmax, sigmoid, etc.",
"has been tried out at different layer level to get an ideal architecture for classification task for the given problems.",
"Table 1 shows that MLP based techniques fail when compared to the state of the art and the proposed method.",
"The problem with MLP and newer deep learning techniques like CNN is that they need a lot of data to train and there is no way to interpret the features.",
"Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to derive principal components representative of the features under consideration.",
"Experiments have been carried out with aforementioned datasets and Gaussian kernel is used for SVM based classification.",
"The different dimension reduction techniques used are Singular Value Decomposition (svd), Eigen Value Decomposition (eig) and Alternating Least Squares (als).",
"A varying number of principal components (like 5, 10, 15) are also tried out.",
"Table 1 shows that PCA based methods are outperformed by our proposed method.",
"Another drawback of PCA and similar feature reduction techniques is that the derived features are not interpretable.",
"It is seen that for 2nd and 3rd dataset, the proposed approach outperforms the state of the art (SoA) methods, and for the 1st dataset, 94.38% of the accuracy level of the winner was reached by this method.",
"In terms of effort taken to build the solution, the proposed method clearly beats others."
],
[
"Conclusion and Future Work",
"Interpretable Feature Engineering has been found to be the most demanding task among all the subtasks of health data analytics.",
"Hence, a system was built to automate this part of the process.",
"The system has been tested on three healthcare datasets and was found to give good results when compared to state of the art.",
"Apart from manual feature engineering, comparison has been made with MLP and PCA which are feature engineering approaches of different directions.",
"Interpretation of features is one of the strong points of the proposed method.",
"Another strong point of the proposed method is huge reduction in effort to develop a typical analytics solution.",
"Integration of knowledge bases for ease of interpreting features and automated causality analysis is also planned.",
"The work will be exteneded to other domains such as machine prognostics."
],
[
"References",
"[1] Ngui, W. K. et al.",
"(2013) Wavelet Analysis: Mother Wavelet Selection Methods, Applied Mechanics and Materials; Vol.",
"393, pp.",
"953-958.",
"[2] Peng, H et al.",
"(2005) Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy.",
"IEEE Transactions on Pattern Analysis and Machine Intelligence, 27.8: 1226-1238.",
"[3] Maji, P et al (2012) Fuzzy-Rough MRMS Method for Relevant and Significant Attribute Selection, Advances on Computational Intelligence: 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems.",
"[4] Liu, C et al.",
"(2016) An open access database for the evaluation of heart sound algorithms.",
"Physiological Measurement; 37(9).",
"[5] Springer et al.",
"(2016) Logistic regression-hsmm-based heart sound segmentation.",
"IEEE Transactions on Biomedical Engineering;63(4):pages 822–832.",
"[6] Potes, C et al.",
"(2016), Ensemble of Feature-based and Deep learning-based Classifiers for Detection of Abnormal Heart Sounds, CiNC 2016.",
"[7] Goldberger, AL et al.",
"(2000) PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals.",
"Circulation 101(23):e215-e220."
]
] | 1612.05730 |
[
[
"Detections of Planets in Binaries Through the Channel of Chang-Refsdal\n Gravitational Lensing Events"
],
[
"Abstract Chang-Refsdal (C-R) lensing, which refers to the gravitational lensing of a point mass perturbed by a constant external shear, provides a good approximation in describing lensing behaviors of either a very wide or a very close binary lens.",
"C-R lensing events, which are identified by short-term anomalies near the peak of a high-magnification lensing light curves, are routinely detected from lensing surveys, but not much attention is paid to them.",
"In this paper, we point out that C-R lensing events provide an important channel to detect planets in binaries, both in close and wide binary systems.",
"Detecting planets through the C-R lensing event channel is possible because the planet-induced perturbation occurs in the same region of the C-R lensing-induced anomaly and thus the existence of the planet can be identified by the additional deviation in the central perturbation.",
"By presenting the analysis of the actually observed C-R lensing event OGLE-2015-BLG-1319, we demonstrate that dense and high-precision coverage of a C-R lensing-induced perturbation can provide a strong constraint on the existence of a planet in the wide range of the planet parameters.",
"The sample of an increased number of microlensing planets in binary systems will provide important observational constraints in giving shape to the details of the planet formation scenario which has been restricted to the case of single stars."
],
[
"Introduction",
"Majority of stars reside in binary systems [1], [33].",
"The mechanism of planet formation around binary systems would be different from that around single stars not only because the environment of the protoplanetary disk would be affected by the binary companion but also because the binary companion would affect the long-term stability of the planet orbit.",
"Therefore, one of the most generic environments to be considered in the study of planet formation should be that of a binary.",
"However, the major planet-formation scenarios that have been developed over the past decades, e.g., core-accretion theory [35], [17], [23], [31] and disk instability theory [26], [5], [4], were mostly focused on the case of single stars.",
"Although a considerable work has been done about the effect of binary companions on the planet formation [40] and the long-term orbital stability [39], this work was restricted to mostly theoretical studies and thus many details about the planet formation scenario remain uncertain.",
"These details can be refined by the constraints provided by the sample of actually detected planetary systems.",
"Unfortunately, there exists only a total of 19 known planets in 17 binary systems and most of these planets exist under a similar environment, i.e.",
"circumbinary planets orbiting very close binaries.",
"To give details about the planet formation in binary systems, therefore, it is important to detect more of such planets residing under various environments.",
"When a gravitational microlensing event is caused by a very wide binary object, the lensing behavior in the region around each lens component is approximated by Chang-Refsdal (C-R) lensing, which refers to the gravitational lensing of a point mass perturbed by a constant external shear $\\gamma $ [7], [8].",
"In the low shear regime ($\\gamma < 1$ ), C-R lensing induces a small astroid-shape caustic around the lens.",
"The lensing behavior of a very close binary, on the other hand, can be approximated by a point-mass plus quadrupole lensing.",
"In this case, an astroidal caustic similar to the C-R lensing caustic is produced around the center of mass of the binary.",
"For this reason, the event caused by a very wide or a very close binary lens is often referred to as the C-R lensing event.",
"Due to the existence of the caustic in the central region, C-R lensing events are identified by a short-term anomaly that appears near the peak of a very high-magnification event.",
"Although C-R lensing events are routinely detected from microlensing surveys, not much attention is paid to them because they are thought to be simply one type of numerous binary-lens events and thus of little scientific importance.",
"In this work, we point out that C-R lensing events provide an important channel to detect planets in binaries, both in close and wide binary systems.",
"In order to demonstrate that dense and high-precision coverage of a C-R lensing-induced perturbation can provide a strong constraint on the existence of a planet in the wide range of the planet parameters, we present analysis of the actually observed C-R lensing event OGLE-2015-BLG-1319.",
"The paper is organized as follows.",
"In section 2, we briefly describe the lensing properties in the cases where the lens is composed of a single, binary, and triple masses.",
"We also describe the C-R lensing behavior.",
"In section 3, we estimate the detection efficiency of planets in binaries by conducting analysis of the lensing event OGLE-2015-BLG-1319.",
"We summarize the results and conclude in section 4."
],
[
"Chang-Refsdal Lensing Channel",
"When a point-mass lensing event occurs the lensing behavior is described by the lens equation $\\zeta = z - {1\\over \\bar{z}},$ where $\\zeta =\\xi + i\\eta $ and $z=x+iy$ denote the complex notations of the source and image positions, respectively, and $\\bar{z}$ represents the complex conjugate of $z$ .",
"Here all lengths are normalized to the angular Einstein ring radius $\\theta _{\\rm E}$ and the lens is positioned at the origin.",
"Solving the lens equation yields two solutions of image positions: one outside and the other inside the Einstein ring.",
"The magnification $A_j$ of each image $j$ is given by $A_j={1\\over {\\rm det} J_j}; \\qquad {\\rm det} J_j=\\left|1-{\\partial \\zeta \\over \\partial \\bar{z}}{\\partial \\bar{\\zeta }\\over \\partial z }\\right|_{z=z_j},$ where $J_j$ is the Jacobian of the lens equation evaluated at the image positions $z_j$ and ${\\rm det}J_j$ is the determinant of the Jacobian.",
"Since the individual microlensing images cannot be resolved, the observed lensing magnification is the sum of the magnifications of the individual images, i.e.",
"$A=\\sum _i A_i$ .",
"For a point mass, the magnification is represented analytically by $A={\\vert \\zeta \\vert ^2+2 \\over \\vert \\zeta \\vert \\sqrt{\\vert \\zeta \\vert ^2+4}},$ where $\\vert \\zeta \\vert =(t-t_0)/t_{\\rm E} + i u_0 $ is the lens-source separation with the length normalized to $\\theta _{\\rm E}$ , $t_{\\rm E}$ is the Einstein time scale, $t_0$ is the time of the closest lens-source approach, and $u_0$ is the lens-source separation at $t_0$ .",
"For a rectilinear relative lens-source motion, the lensing light curve is characterized by a smooth and symmetric shape [29].",
"When an event is produced by a lens composed multiple components, the lens equation is expressed as $\\zeta = z - \\sum _{i=1}^{N}{ \\epsilon _i\\over \\bar{z}-\\bar{z}_{L,i}},$ where $z_{L,i}$ and $\\epsilon _i=m_i/m_{\\rm tot}$ , and $m_i$ represent the location, mass fraction, and mass of each lens component, respectively.",
"The notation $N$ denotes the number of the lens components, and thus $N=2$ for a binary lens.",
"Here all lengths ($\\zeta $ and $z$ 's) are in units of the angular Einstein radius corresponding to the total mass of the lens, $\\theta _{\\rm E}$ .",
"One of the most important properties of a binary lens that differentiate from those of a single lens is the formation of caustics, which represent the sets of source positions at which ${\\rm det} J=0$ and thus the magnification of a point source becomes infinite.",
"As a result, a binary-lensing light curve can exhibit strong deviations when a source approaches close to or passes over the caustic.",
"Caustics of a binary lens form a single or multiple sets of closed curves and each curve composed of concave curves that meet at cusps.",
"The topology of the binary-lens caustic is broadly classified into 3 categories [14], [11].",
"In the case of a binary where the binary separation is greater than $(\\@root 3 \\of {\\epsilon _1} +\\@root 3 \\of {\\epsilon _2})^{3/2}$ (wide binary), there exist two sets of 4-cusp caustics that are located close to the individual lens components.",
"When the binary separation is smaller than $(\\@root 3 \\of {\\epsilon _1} +\\@root 3 \\of {\\epsilon _2})^{-3/4}$ (close binary), the caustic is composed of 3 pieces, where the central caustic with 4 cusps is formed around the center of mass of the binary lens, and the other 2 triangular caustics are located away from the center of mass.",
"In the intermediate separation region, there exists a single big caustic with 6 cusps.",
"For the visual presentation of the binary caustic topology, see Figure 3 of [12].",
"In the extreme case where the binary separation is much greater or less than $\\theta _{\\rm E}$ , the 4-cusp central caustic has an astroid shape which is symmetric with respect to the binary axis and the line vertical to the binary axis.",
"For the exact description of the lensing behavior produced by a lens system of a planet orbiting a binary object, one needs the triple lens equation, i.e.",
"Eq.",
"(REF ) with $N=3$ .",
"With the addition of a third lens component, the complexity of the lensing behavior greatly increases and caustics can exhibit self-intersection and nesting [34], [15], [11].",
"Figure: Sizes of the C-R lensing caustic (upper panel) and the planet-induced central causticas a function of the separation between the lens components.",
"Curves with different grey toneshow the variation of the caustic size depending on the mass ratio between binary lens componentsq b q_b (upper panel) and the mass ratio between the planet and the host q p q_p (lower panel).Both the caustic size and the binary separation are normalized to the angular Einstein radiuscorresponding to the total lens mass.A planetary lens corresponds to the extreme case of a binary lens where the mass of one of the lens components is much smaller than the other, i.e.",
"$\\epsilon _1\\sim 1$ and $\\epsilon _2\\ll 1$ .",
"In this case, the lens equation is approximated as $\\zeta \\sim z - {1\\over \\bar{z}} - {q_p\\over \\bar{z}-\\bar{z}_{p}},$ where $q_p=m_2/m_1$ is the mass ratio between the planet and the host and $z_{p}$ represents the position of the planet with respect to the host.",
"We note that the lengths of $\\zeta $ and $z$ 's in Eq.",
"(5) are normalized to the angular Einstein radius corresponding to the mass of the primary lens, i.e.",
"$\\theta _{{\\rm E},1}=\\theta _{\\rm E}/(1+q_p)^{1/2}\\sim (1-q_p/2)\\theta _{\\rm E}$ .",
"For a planetary lens, however, the mass of the primary dominates, i.e.",
"$q_p\\ll 1$ , and thus $\\theta _{{\\rm E},1}\\sim \\theta _{\\rm E}$ .",
"The planet induces two types of caustics, where one is located close to the host (central caustic) and the other is away from the host.",
"The central caustic has an arrowhead shape and its size is related to the star-planet separation $s_p=|z_{p}|$ and the mass ratio by [9] $\\Delta x_{\\rm p}={4 q_p \\over (s_{p}-s_{p}^{-1})^2}.$ Due to the location of the central caustic close to the host, perturbations induced by the central caustic of a planet always appear near the peak of the lensing light curve produced by the host of the planet [22].",
"Figure: Variation of the C-R lensing caustic (left panel) and the light curve (right panel)by the presence of planets with various separations s p s_p and the mass ratios q p q_p.",
"The C-Rlensing caustic is produced by a binary companion with s b =10s_b=10 and q b =0.5q_b=0.5.",
"The separationss p s_p and s b s_b are normalized to the angular Einstein radius corresponding to the total lensmass, θ E \\theta _{\\rm E}.",
"The light curve in each subpanel results from the source trajectory(line with an arrow) marked in the corresponding subpanel showing the caustic.",
"The notations(ξ,η)(\\xi ,\\eta ) represent the coordinates on the source plane where ξ\\xi is aligned with thebinary-lens axis and lengths are normalized to θ E \\theta _{\\rm E}.",
"The planet is located at(x p ,y p )=(x 1 +s p cosφ,y 1 +s p sinφ)(x_p,y_p)=(x_1+s_p\\cos \\phi ,y_1+s_p\\sin \\phi ), where (x 1 ,y 1 )(x_1,y_1) is the position of the primarylens (heavier binary component) and φ=60 ∘ \\phi =60^\\circ is orientation angle of the planet withrespect to the binary axis.The lens equation of C-R lensing is expressed as $\\zeta = z - {1\\over \\bar{z}} + \\gamma \\bar{z},$ which describes the lensing behavior around the primary lens with an external shear $\\gamma $ .",
"Here lengths are given in units of the Einstein radius corresponding to the primary mass.",
"The shear induces a caustic around the lens.",
"The shape of the caustic is similar to the central caustic of a very wide binary lens and thus the C-R lensing provides a good approximation in describing the binary lensing behavior in the region around the caustic.",
"For a very close binary lens, the lensing behavior around the center of mass is described by quadrupole lensing [12], which also induces an astroidal caustic similar to the C-R lensing caustic.",
"To the first order approximation, the caustic size in units of the angular Einstein radius of the binary-lens mass, $\\theta _{\\rm E}$ , is related to the separation and the mass ratio between the binary-lens components by $\\Delta x_{\\rm C-R} = {4\\gamma \\over \\sqrt{1-\\gamma }};\\qquad \\gamma ={q_b \\over s_b^{2}(1+q_b) },$ for a wide binary and $\\Delta x_{\\rm quad}=4Q \\left( 1+{9\\over 2}Q \\right);\\qquad Q={s_b^2 q_b\\over (1+q_b)^2}$ for a close binary.",
"A C-R lensing event can provide a channel to detect planets in binary systems.",
"This is because both of the C-R lensing caustic and the central caustic induced by the planet occur in the same region around the primary lens and the size of the planet-induced central caustic can be comparable to the size of the C-R lensing caustic [27].",
"In Figure REF , we present the sizes of the C-R lensing caustic and the planet-induced central caustic as a function of the primary-companion separation.",
"We note that a pair of caustics with separations $s$ and $s^{-1}$ have the same size to a linear order and thus we present distributions for only wide binaries.",
"The plot shows that the central caustic induced by planets located in the “lensing zone”The lensing zone represents the range of the planet-host separations where the probability of detecting the planet is high [18], [22].",
"The range is approximately $1/2 \\lesssim s \\lesssim 2$ , although the range varies depending on how the lensing zone is defined.",
"is of considerable size compared to the size of the C-R lensing caustic.",
"This suggests that the C-R lensing anomaly can be additionally affected by the planetary perturbation, enabling one to identify the presence of a planet in the binary system.",
"In Figure REF , we present the variation of the C-R lensing caustic and the lensing light curve affected by the presence of planets with various separations and mass ratios between the planet and the primary of the binary.",
"See [28] for more details about the caustic variation.",
"We note that the C-R lensing channel enables detections of planets not only in close binaries but also in wide binaries.",
"To be dynamically stable, a planet should be either in a circumbinary (or P-type) orbit, where the planet orbits the barycenter of the two stars of a close binary, or in a circumprimary (S-type) orbit, where the planet orbits just one star of a wide binary system.",
"This condition of the planet existence in the binary system matches the lens system configuration of the proposed C-R lensing channel of planet detections.",
"Planets in binary systems can be detected by various methods such as transit [13], eclipsing binary timing [32], and radial-velocity methods [10].",
"Due to the intrinsic nature of the methods, however, it is difficult to detect planets in circumprimary orbits and thus all planets in binaries detected by these methods, 19 in total, reside in circumbinary orbits.",
"On the other hand, C-R lensing events can be produced by both close and wide binary systems and thus the proposed C-R lensing channel provides a unique channel to detect planets both in circumbinary and circumprimary orbits.We note that there exist two known planets in wide binary systems.",
"These planets, OGLE-2013-BLG-0341LBb [21] and OGLE-2008-BLG-092LAb [30], were detected by using the microlensing method.",
"However, they were detected not through the C-R lensing channel but through a repeating channel where the perturbations induced by the planet and the binary companion were separately detected."
],
[
"Planet Detection Efficiency",
"In this section, we demonstrate the high efficiency of the proposed C-R lensing channel in detecting planets of binary systems.",
"Estimating the efficiency requires to consider various details of observational conditions such as photometric precision and cadence.",
"In order to reflect realistic observational conditions, we estimate the detection efficiency for an example C-R lensing event that was actually observed by lensing experiments.",
"Figure: Light curve of the microlensing event OGLE-2015-BLG-1319.",
"The lower and upperpanels show the whole view of the light curve and the zoom around the peak.",
"The insetin the lower panel shows the source trajectory with respect to the caustic.",
"The labelin the legend denote the lensing experiments and the telescopes used for observations.The event used for our efficiency estimation is OGLE-2015-BLG-1319.",
"The event was analyzed in detail by [37] and turned out to be an exemplary C-R lensing event caused either by a close or a wide binary lens.",
"Figure REF shows the light curve of the event reproduced based on the same data sets as those used in the previous analysis.",
"The model light curve superposed on the data points is obtained from binary-lensing modeling based on one of the 8 degenerate solutions (“++ wide” solution) presented in [37].",
"The inset in the lower panel shows the source trajectory with respect to the caustic, which has a very characteristic shape of a C-R lensing caustic.",
"The anomaly caused by the C-R lensing caustic in the peak of the light curve was densely and precisely observed.",
"This was possible because the event was predicted to be a high-magnification event before it reached the peak based on the light curve obtained from the survey observation conducted by the Optical Gravitational Lens Experiment [42] and the Microlensing Observations in Astrophysics [3], [38], enabling intensive observations of the peak by follow-up observation groups including the Microlensing Follow-Up Network [20] and the RoboNet [41] groups.",
"We note that the event was also observed in space using two space telescopes Spitzer and Swift.",
"This enabled the determinations of the lens mass and the distance, but we do not use the space-based data because these data are irrelevant to our scientific purpose.",
"The values of the binary separation and the mass ratio presented in [37] are $(s_b,q_b)\\sim (0.08,0.07)$ for the close binary solution and and $(s_b,q_b)\\sim (14,0.09)$ for the wide solution.",
"For other lensing parameters, see Table 1 of [37].",
"To show the constraint on the existence of a planet, we construct an “exclusion diagram”, which shows the probability of excluding the existence of a planet as a function of the planet separation and the mass ratio [2], [16], [25], [19], [6].",
"We construct the exclusion diagram following the procedure of [36].",
"In this procedure, we first introduce a planet with the parameters $(s_p,q_p,\\psi )$ to the binary lens with the parameters $(s_b,q_b)$ .",
"Here $(s_b,q_b)$ represent the separation and the mass ratio between the binary components, while $(s_p,q_p)$ are the separation and the mass ratio between the primary of the binary and the planet.",
"The angle $\\psi $ denotes the orientation angle of the planet with respect to the binary axis connecting the binary lens components.",
"We use $(s_b,q_b)$ that are determined from the binary-lensing modeling.",
"For a given set of $(s_p,q_p,\\psi )$ , we then search for other parameters that yield the best fit to the observed light curve and compute $\\chi ^2$ of the fit.",
"We repeat this process for many different orientation angles.",
"Then, the probability of excluding a planet for a given $(s_p,q_p)$ is estimated as the fraction of the angles $\\psi $ that result in fits with $\\Delta \\chi ^2 > \\Delta \\chi ^2_{\\rm th}$ , where $\\Delta \\chi ^2$ is the difference between the triple-lens (i.e.",
"binary + planet) and the binary-lens models.",
"As a criteria for the planet detection, we adopt a threshold value of $\\Delta \\chi ^2_{\\rm th}=500$ , which is a generally agreed value for planets detected through the high-magnification channel [19].",
"The probability of excluding a planet corresponds to the probability of detecting the planet, i.e.",
"planet detection efficiency.",
"Figure: Efficiency of detecting planets in the binary lens system responsible for thelensing event OGLE-2015-BLG-1319 as a function of the normalized separation s p s_p and themass ratio q p q_p between the planet and the primary of the binary lens.",
"The values markedin the upper xx axis and the right of yy axis represent the physical primary-planet separationin au and the mass of the planet in Jupiter masses, respectively.",
"The color coding representsthe regions of different efficiencies that are marked in the legend.",
"The planetary separationis expressed in units of the angular Einstein radius corresponding to the binary lens mass.Figure REF shows the constructed exclusion diagram which shows the planet detection efficiency as a function of the normalized separation $s_p$ and the mass ratio $q_p$ of the planet.",
"Since the microlens parallax of the event OGLE-2015-BLG-1319 was determined using the combined data taken from the ground and space, the physical sizes of the primary-planet separation and the mass of the planet were determined.",
"By adopting the lens mass of $M_{\\rm tot}=0.62\\ M_\\odot $ , the angular Einstein radius of $\\theta _{\\rm E}=0.63$ mas, and the distance to the lens of $D_{\\rm L}=4.93$ kpc, that were determined by [37], i.e.",
"(+,+) wide model, we convert $s_p$ and $q_p$ into the physical sizes of the primary-planet separation in au, i.e.",
"$a_\\perp =s_pD_{\\rm L}\\theta _{\\rm E}$ , and the mass of the planet in Jupiter masses ($M_J$ ), i.e.",
"$M_p=qM$ , and they are presented in the upper $x$ axis and the right of the $y$ axis, respectively.",
"lccc Range of planet detection 0pt 1cPlanet 3cEfficiency 1ctype 1c$> 90\\%$ 1c$> 50\\%$ 1c$> 10\\%$ Jupiter 1.0 – 10 au 0.5 – 11 au 0.4 – 18 au Saturn 2.0 – 5 au 1.0 – 7 au 0.9 – 9 au Neptune – 2.0 – 3.5 au 1.8 – 4.0 au In Table , we present the ranges of planet detection for 3 different planets with masses corresponding to those of the Jupiter, Saturn ($\\sim 0.3\\ M_J$ ), and Neptune ($\\sim 0.055\\ M_J$ ) of the Solar system.",
"It is found that the detection efficiency is greater than $90\\%$ for the Jupiter- and Saturn-mass planets located in the ranges of 1.0 – 10 au and 2.0 – 5 au from the host, respectively.",
"The ranges with $>10\\%$ probability are 0.4 – 18 au, 0.9 – 9 au, and 1.8 – 4 au for the Jupiter-, Saturn-, and Neptune-mass planets, respecively.",
"These ranges encompass wide regions around snow lines where giants are believed to form.",
"Despite the high efficiency of C-R lensing events in detecting planets of binary systems, there exists no report of planet detection yet.",
"The reasons for this can be (1) the rarity of C-R lensing events, (2) the unoptimized observational strategy for planet detections, and (3) the rarity of planets in the region of sensitivity.",
"Considering the existence of the already known microlensing planets detected through the repeating-event channel and those detected by other methods combined with the wide region of planet sensitivity of the C-R lensing channel, the reason (3) is unlikely to be the main reason of nondetection.",
"To check the possibilities (1) and (2), we investigate the lensing events reported by the OGLE and MOA surveys in 2015 season.",
"From the systematic analyses of all high-magnification events with anomalies near the peaks based on the online data of the surveys, we find 12 C-R lensing events including OGLE-2015-BLG-1319, MOA-2015-BLG-040/OGLE-2015-BLG-0318, OGLE-2015-BLG-0697/MOA-2015-BLG-148, OGLE-2015-BLG-0797, OGLE-2015-BLG-0812, OGLE-2015-BLG-0189 MOA-2015-BLG-085/OGLE-2015-BLG-0472, OGLE-2015-BLG-0313/MOA-2015-BLG-047, OGLE-2015-BLG-0033/MOA-2015-BLG-017, MOA-2015-BLG-047, OGLE-2015-BLG-0919, and OGLE-2015-BLG-0863.",
"This indicates that the rarity of C-R lensing events is not the reason for the nondetection, either.",
"However, we find that the coverage of the peak regions for all of the C-R lensing events except OGLE-2015-BLG-1319 was not dense enough to constrain the existence of a planet.",
"Therefore, it is likely that the main reason for the nondetection of planet is due to the unoptimized observational strategy.",
"In other words, this suggests that planets can be detected through the proposed channel in abundance with an aggressive strategy to densely cover the peak regions of high-magnification events, e.g., vigilant monitoring of high-magnification events and timely alerts of anomalies followed by prompt and intensive coverage of the anomalies by follow-up observations"
],
[
"Summary and Conclusion",
"We pointed out that C-R lensing events could provide one with an important channel to detect planets in binary systems.",
"We also pointed out that while other methods could detect planets only in circumbinary orbits the proposed C-R lensing channel could provide a unique channel to detect planets both in circumbinary and circumprimary orbits.",
"We demonstrated the high sensitivity of the C-R lensing channel to planets in a wide range of planet parameter space by presenting the exclusion diagram for an actually observed C-R lensing event.",
"We mentioned that an increased number of microlensing planets in binary systems could be detected with an aggressive strategy to densely cover the peak regions of high-magnification events.",
"The sample of an increased number of microlensing planets in binaries will make it possible to provide important observational constraints that can give shape to the details of the formation scenario which has been restricted to the case of single stars.",
"Work by C. Han was supported by the Creative Research Initiative Program (2009-0081561) of National Research Foundation of Korea.",
"We acknowledge the high-speed internet service (KREONET) provided by Korea Institute of Science and Technology Information (KISTI)."
]
] | 1612.05718 |
[
[
"Universal Bell Correlations Do Not Exist"
],
[
"Abstract We prove that there is no finite-alphabet nonlocal box that generates exactly those correlations that can be generated using a maximally entangled pair of qubits.",
"More generally, we prove that if some finite-alphabet nonlocal box is strong enough to simulate arbitrary local projective measurements of a maximally entangled pair of qubits, then that nonlocal box cannot itself be simulated using any finite amount of entanglement.",
"We also give a quantitative version of this theorem for approximate simulations, along with a corresponding upper bound."
],
[
"Introduction",
"A correlation box is a conceptual tool for reasoning about nonlocality: Definition 1 A (discrete, bipartite) correlation box is a map $\\mathsf {Cor}: X \\times Y \\rightarrow \\lbrace \\mathcal {D} : \\mathcal {D} \\text{ is a probability distribution over }A \\times B\\rbrace ,$ where $X, Y, A, B$ are countable (finite or countably infinite) alphabets.",
"We will abuse notation and write $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ .",
"Think of a correlation box $\\mathsf {Cor}$ as a kind of “channel” through which two separated parties, Alice and Bob, can interact.",
"Alice chooses $x \\in X$ and Bob chooses $y \\in Y$ .",
"A sample $(a, b)$ is drawn from $\\mathsf {Cor}(x, y)$ , and Alice is given $a$ and Bob is given $b$ .",
"The canonical example is the Popescu-Rohrlich box $\\mathsf {PR}: \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace \\rightarrow \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace $ , which is defined [18], [21] by $\\mathsf {PR}(x, y) \\stackrel{\\text{def}}{=} {\\left\\lbrace \\begin{array}{ll} (0, xy) & \\text{with probability $1/2$} \\\\(1, 1 - xy) & \\text{with probability $1/2$.}",
"\\end{array}\\right.",
"}$ Observe that PR boxes cannot be used to communicate, since the marginal distributions of $a$ and $b$ are uniform regardless of $x$ and $y$ .",
"But $\\mathsf {PR}$ is a nonlocal box, i.e.",
"given access to a PR box, Alice and Bob can perform tasks that would be impossible if they were isolated (even if they had shared randomness).",
"The standard example is winning the CHSH game [11] with certainty.",
"Qualitatively speaking, quantum entanglement is like a PR box: it can be used to generate nonlocal correlations, but it cannot be used to communicate.",
"Unfortunately, entanglement is not quantitatively equivalent to a PR box; the Tsierelson bound [12] implies that there is no quantum strategy for the CHSH game that wins with probability more than about $85\\%$ .",
"In this work, we show that there is no finite-alphabet correlation box that has exactly the same power as quantum entanglement."
],
[
"Distributed sampling complexity classes",
"We can think of a correlation box as a distributed sampling problem [16]: the problem of simulating the box.",
"That is, Alice is given $x \\in X$ and Bob is given $y \\in Y$ .",
"Alice is supposed to output $a \\in A$ and Bob is supposed to output $b \\in B$ such that $(a, b) \\sim \\mathsf {Cor}(x, y)$ .",
"We define $\\mathbf {SR}$ to be the class of all correlation boxes that can be simulated if Alice and Bob have unlimited shared randomness (but are otherwise isolated).",
"We define $\\mathbf {Q}$ to be the class of all correlation boxes that can be simulated if Alice and Bob have unlimited shared randomness and an arbitrary but finite amount of entanglement.",
"Clearly, $\\mathbf {SR} \\subseteq \\mathbf {Q}$ .",
"Bell's theorem [3] can be interpreted as stating that $\\mathbf {SR} \\ne \\mathbf {Q}$ .",
"For an upper bound on $\\mathbf {Q}$ , say that a correlation box $\\mathsf {Cor}$ is non-signaling if the marginal distribution of $a$ depends only on $x$ and the marginal distribution of $b$ depends only on $y$ , where $(a, b) \\sim \\mathsf {Cor}(x, y)$ .",
"Let $\\mathbf {NS}$ be the class of all non-signaling correlation boxes.",
"In this notation, the no-communication theorem states that $\\mathbf {Q} \\subseteq \\mathbf {NS}$ .",
"The PR box shows that $\\mathbf {Q} \\ne \\mathbf {NS}$ .",
"So to summarize, we have the proper inclusions $\\mathbf {SR} \\subsetneqq \\mathbf {Q} \\subsetneqq \\mathbf {NS}$ ."
],
[
"Our results",
"We define $\\mathbf {BELL}$ to be the class of all correlation boxes that can be simulated if Alice and Bob have unlimited shared randomness, each holds one of a pair of maximally entangled qubits, and they are only allowed to make projective measurements.",
"Understanding $\\mathbf {BELL}$ is a good first step toward understanding $\\mathbf {Q}$ .",
"Many previous results about simulating Bell correlations can be understood as reductions between correlation boxes.",
"A $k$ -query reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ is a protocol for simulating $\\mathsf {Cor}_1$ in which Alice and Bob have unlimited shared randomness and $k$ copies of $\\mathsf {Cor}_2$ .",
"(Taking a cue from quantum mechanics, we think of each correlation box as “single use only”.)",
"We will simply say that $\\mathsf {Cor}_1$ reduces to $\\mathsf {Cor}_2$ if there is a $k$ -query reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ for some $k$ .",
"(See Section REF for details.)",
"We say that $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ is binary if $X = Y = A = B = \\lbrace 0, 1\\rbrace $ .",
"Our main result: Theorem 1 Suppose $\\mathsf {Cor} \\in \\mathbf {Q}$ has countable input alphabets and finite output alphabets.",
"Then there is some binary correlation box in $\\mathbf {BELL}$ that does not reduce to $\\mathsf {Cor}$ .",
"As usual, we say that $\\mathsf {Cor}$ is $\\mathbf {C}$ -hard if every correlation box in $\\mathbf {C}$ reduces to $\\mathsf {Cor}$ .",
"We say that $\\mathsf {Cor}$ is $\\mathbf {C}$ -complete if $\\mathsf {Cor}$ is $\\mathbf {C}$ -hard and $\\mathsf {Cor} \\in \\mathbf {C}$ .",
"Corollary 1 There does not exist a finite-alphabet $\\mathbf {BELL}$ -complete correlation box.",
"Corollary 2 There does not exist a finite-alphabet $\\mathbf {Q}$ -complete correlation box.",
"Figure: Our result implies that the shaded region does not contain any finite-alphabet correlation boxes.Our result can be thought of as “bad news” for the project of understanding $\\mathbf {BELL}$ .",
"We also give a quantitative version of our result for approximate simulations.",
"An $\\varepsilon $ -error reduction is defined like an ordinary reduction except that we allow $\\varepsilon $ total variation error.",
"Theorem 2 Suppose $\\mathsf {Cor}_2: X \\times Y \\rightarrow A \\times B$ is a finite-alphabet correlation box in $\\mathbf {Q}$ .",
"Then there exists a binary correlation box $\\mathsf {Cor}_1 \\in \\mathbf {BELL}$ such that for every $k$ , if there is a $k$ -query $\\varepsilon $ -error reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ , then $k^4 \\cdot (2|X|)^{4|A|^k} \\cdot (2|Y|)^{4|B|^k} \\ge \\Omega (1/\\varepsilon ).$ Conversely, for any $\\varepsilon > 0$ , we give a simple construction of $\\mathsf {Cor}: [T] \\times [T] \\rightarrow \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace $ with $T \\le O(1/\\varepsilon ^2)$ such that $\\mathsf {Cor} \\in \\mathbf {BELL}$ and every correlation box in $\\mathbf {BELL}$ reduces to $\\mathsf {Cor}$ via a 1-query $\\varepsilon $ -error reduction.",
"Notice that for $|A| = |B| = 2, k = 1$ , Theorem REF implies that $|X| \\cdot |Y|$ must be at least $1/\\varepsilon ^{\\Omega (1)}$ .",
"On the other hand, when $|A|, |B|, k$ are large, our lower bound might be very far from tight."
],
[
"Related work",
"A long line of work [20], [2], [22], [9], [14], [13], [8], [23] investigated the problem of simulating Bell correlations using classical communication, culminating in a theorem by Toner and Bacon [23] that states that $\\mathbf {BELL}$ can be simulated using shared randomness and a single classical bit of one-way communication.",
"This result should be thought of as giving an upper bound on the power of $\\mathbf {BELL}$ .",
"Obviously it is a loose upper bound, since $\\mathbf {BELL} \\subseteq \\mathbf {NS}$ .",
"Cerf et al.",
"[10] improved on the Toner-Bacon theorem by showing that instead of a bit of communication, it suffices to have a single PR box.",
"In our terminology, Cerf et al.",
"showed that $\\mathsf {PR}$ is $\\mathbf {BELL}$ -hard with respect to 1-query reductions.",
"Part of what makes this result so appealing is that $\\mathsf {PR}$ has finite alphabets, making it an extremely explicit upper bound on $\\mathbf {BELL}$ .",
"(Similarly with the Toner-Bacon theorem before it.)",
"It is natural to hope to push even further and replace $\\mathsf {PR}$ with some finite-alphabet correlation box in $\\mathbf {Q}$ .",
"Our results dash this hope, even for the special case of simulating binary correlation boxes in $\\mathbf {BELL}$ .",
"In another direction, several works [6], [4], [15], [1], [17], [7], [24] have investigated the power of correlation boxes in their own right, apart from quantum entanglement.",
"Two such works are particularly relevant to the present paper.",
"First, Barrett and Pironio showed [6] that every correlation box in $\\mathbf {NS}$ with binary output alphabets reduces to $\\mathsf {PR}$ .",
"Our result shows that there is no corresponding phenomenon for $\\mathbf {BELL}$ .",
"Second, Dupuis et al.",
"[15] showed that no finite-alphabet correlation box is $\\mathbf {NS}$ -complete.",
"Our result can be thought of as a “scaled down” version of this second result."
],
[
"Proof overview",
"The biased CHSH game is a variant of the well-studied CHSH game.",
"In the biased game, Alice and Bob's input bits are not uniformly distributed.",
"We will consider the case that their inputs are independent, Alice's is uniform, and Bob's has bias $p \\in [1/2, 1]$ .",
"Alice and Bob know $p$ , i.e.",
"their strategy may depend on $p$ .",
"(See Section REF for details.)",
"We use a result by Lawson, Linden, and Popescu [19] that states that the optimal quantum strategy for the biased CHSH game can be implemented in $\\mathbf {BELL}$ and wins with probability $\\frac{1}{2} + \\frac{1}{2} \\sqrt{p^2 + (1 - p)^2}$ .",
"Throughout this paper, we will let $\\omega : \\mathbb {R}\\rightarrow \\mathbb {R}$ denote this optimal success probability: $\\omega (p) \\stackrel{\\text{def}}{=} \\frac{1}{2} + \\frac{1}{2} \\sqrt{p^2 + (1 - p)^2}.$ To prove Theorem REF , fix $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ .",
"Assume $\\mathsf {Cor}$ is $\\mathbf {BELL}$ -hard; then for any $p$ , there is some strategy for playing the biased CHSH game using finitely many copies of $\\mathsf {Cor}$ that wins with probability $\\omega (p)$ .",
"We can fix the shared randomness of the strategy without decreasing the probability of winning.",
"Assume that $\\mathsf {Cor} \\in \\mathbf {Q}$ ; then fixing the shared randomness must not have increased the probability of winning.",
"So the probability of winning is still exactly $\\omega (p)$ .",
"But it is easy to show that for any deterministic strategy, the probability of winning is some affine function of $p$ .",
"If $X, Y$ are countable and $A, B$ are finite, there are only countably many deterministic strategies, and hence there are only countably many affine functions floating around.",
"There must be some point $p$ where $\\omega (p)$ disagrees with all of these affine functions, a contradiction.",
"To prove our quantitative lower bound (Theorem REF ), we extend the preceding argument by analyzing the distance between $\\omega $ and any affine function at a randomly chosen point $p$ ."
],
[
"Outline of this paper",
"In Section , we provide more detailed definitions of $\\mathbf {SR}, \\mathbf {BELL}, \\mathbf {Q}$ and of our reduction model.",
"In Section , we prove our main, negative results.",
"In Section , we derive a simple consequence of our main result: there is an infinite chain of harder and harder finite-alphabet correlation boxes in $\\mathbf {BELL}$ .",
"In Section , we present our simple positive result.",
"Finally, in Section , we list some open problems."
],
[
"Quantum and classical simulations",
"In this section, we give the technical definitions of $\\mathbf {SR}, \\mathbf {BELL}, \\mathbf {Q}$ .",
"The reader who feels that these classes are intuitively clear may feel free to skip this section.",
"Suppose $\\mathcal {D}$ is a probability distribution over a class $\\mathbf {C}$ of correlation boxes $X \\times Y \\rightarrow A \\times B$ .",
"Then $\\mathcal {D}$ induces a single correlation box $\\mathsf {Cor}_\\mathcal {D} : X \\times Y \\rightarrow A \\times B$ defined by $ \\Pr [\\mathsf {Cor}_{\\mathcal {D}}(x, y) = (a, b)] = \\operatornamewithlimits{E}_{\\mathsf {Cor} \\sim \\mathcal {D}}[\\Pr [\\mathsf {Cor}(x, y) = (a, b)]].$ (Intuitively, $\\mathcal {D}$ models shared randomness.",
"The distribution $\\mathsf {Cor}_{\\mathcal {D}}(x, y)$ is determined by sampling $\\mathsf {Cor}$ from $\\mathcal {D}$ and then using “fresh randomness” to sample $(a, b)$ from $\\mathsf {Cor}(x, y)$ .)",
"Suppose $\\mathbf {C}$ is a class of correlation boxes.",
"We say that $\\mathbf {C}$ is closed under convex combinations if for every $X, Y, A, B$ , for every distribution $\\mathcal {D}$ over correlation boxes $X \\times Y \\rightarrow A \\times B$ in $\\mathbf {C}$ , the box $\\mathsf {Cor}_{\\mathcal {D}}$ is also in $\\mathbf {C}$ .",
"Definition 2 We define $\\mathbf {SR}$ to be the closure under convex combinations of the class of correlation boxes of the form $\\mathsf {Cor}(x, y) = (f(x), g(y))$ , where $f, g$ are (deterministic) functions.",
"Definition 3 We define $\\mathbf {BELL}$ to be the closure under convex combinations of the class of correlation boxes $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ of the following form.",
"For each $x \\in X, y \\in Y$ , let $U_x, V_y$ be associated $2 \\times 2$ unitary matrices.",
"Define binary random variables $S_{x, y}, T_{x, y}$ by $\\Pr [(S_{x, y}, T_{x, y}) = (s, t)] = |\\mathinner {\\langle {st|U_x \\otimes V_y|\\phi }\\rangle }|^2,$ where $\\mathinner {|{\\phi }\\rangle } = \\frac{1}{\\sqrt{2}}(\\mathinner {|{00}\\rangle } + \\mathinner {|{11}\\rangle })$ .",
"Let $\\mathsf {Cor}(x, y) = (f(S_{x, y}), g(T_{x, y}))$ , where $f, g$ are (deterministic) functions.",
"Definition 4 We define $\\mathbf {Q}$ to be the closure under convex combinations of the class of correlation boxes $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ of the following form.",
"Let $\\rho $ be a bipartite mixed state on $n \\otimes m$ for some finite $n, m$ .",
"For each $x \\in X, y \\in Y$ , let $\\lbrace A_x^a\\rbrace _{a \\in A}, \\lbrace B_y^b\\rbrace _{b \\in B}$ be associated POVMs, where each $A_x^a$ acts on $n$ and each $B_y^b$ acts on $m$ .",
"The output distribution $\\mathsf {Cor}(x, y)$ is defined by measuring $\\rho $ using the POVMs associated with $x$ and $y$ : $\\Pr [\\mathsf {Cor}(x, y) = (a, b)] = \\operatorname{Tr}((A_x^a \\otimes B_y^b) \\rho ).$ Notice that Definition REF allows for a $\\mathbf {Q}$ protocol in which the shared quantum state $\\rho $ is picked at random from some distribution over bipartite quantum states with finite Hilbert space dimensions."
],
[
"Details of the reduction model",
"Let $\\mathsf {Cor}_2: X_2 \\times Y_2 \\rightarrow A_2 \\times B_2$ be a correlation box.",
"In a deterministic $k$ -query $\\mathsf {Cor}_2$ -protocol $\\Pi : X_1 \\times Y_1 \\rightarrow A_1 \\times B_1$, Alice receives as input $x \\in X_1$ and Bob receives $y \\in Y_1$ .",
"The players make exactly $k$ queries $(x_1, y_1), \\dots , (x_k, y_k)$ and get exactly $k$ responses $(a_1, b_1), \\dots , (a_k, b_k)$ , where each $x_i \\in X_2, y_i \\in Y_2, a_i \\in A_2, b_i \\in B_2$ .",
"The queries may be chosen adaptively, i.e.",
"$x_i$ can be any deterministic function of $x, a_1, \\dots , a_{i - 1}$ and $y_i$ can be any deterministic function of $y, b_1, \\dots , b_{i - 1}$ .",
"The distribution of the responses is given by $\\Pr [(a_1, b_1), \\dots , (a_k, b_k)] = \\prod _{i = 1}^k \\Pr [\\mathsf {Cor}_2(x_i, y_i) = (a_i, b_i)],$ where $(x_i, y_i)$ is the $i$ th query made by $\\Pi $ when Alice and Bob see $(a_1, b_1), \\dots , (a_{i - 1}, b_{i - 1})$ as the first $i - 1$ responses.",
"At the end, Alice gives an output $a \\in A_1$ and Bob gives an output $b \\in B_1$ .",
"Here, $a$ is a deterministic function of $x, a_1, \\dots , a_k$ and $b$ is a deterministic function of $y, b_1, \\dots , b_k$ .",
"The distribution of $(a, b)$ as a function of $(x, y)$ defines a correlation box $\\mathsf {Cor}_1: X_1 \\times Y_1 \\rightarrow A_1 \\times B_1$ ; we say that $\\Pi $ is a deterministic $k$ -query reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ .",
"A randomized $k$ -query $\\mathsf {Cor}_2$ -protocol $\\Pi : X_1 \\times Y_1 \\rightarrow A_1 \\times B_1$ is a probability distribution $\\mathcal {D}$ over deterministic $k$ -query $\\mathsf {Cor}_2$ -protocols $\\Pi ^{\\prime }: X_1 \\times Y_1 \\rightarrow A_1 \\times B_1$ .",
"(This models “shared randomness”.)",
"This distribution induces a probability distribution $\\mathcal {D}^{\\prime }$ over correlation boxes $X_1 \\times Y_1 \\rightarrow A_1 \\times B_1$ .",
"Let $\\mathsf {Cor}_1 = \\mathsf {Cor}_{\\mathcal {D}^{\\prime }}$ , as defined in Equation REF .",
"We say that $\\Pi $ is a (randomized) $k$ -query reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ .",
"Suppose $\\mathsf {Cor}_1, \\mathsf {Cor}_1^{\\prime }$ are two correlation boxes on the same alphabets.",
"We say that $\\mathsf {Cor}_1$ is $\\varepsilon $ -close to $\\mathsf {Cor}_1^{\\prime }$ if for every $x, y$ , the distributions $\\mathsf {Cor}_1(x, y)$ , $\\mathsf {Cor}_1^{\\prime }(x, y)$ are $\\varepsilon $ -close in total variation distance.",
"An $\\varepsilon $ -error reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ is a reduction from $\\mathsf {Cor}_1^{\\prime }$ to $\\mathsf {Cor}_2$ for some $\\mathsf {Cor}_1^{\\prime }$ that is $\\varepsilon $ -close to $\\mathsf {Cor}_1$ ."
],
[
"Closure",
"Lemma 1 Suppose $\\mathsf {Cor}_1$ reduces to $\\mathsf {Cor}_2 \\in \\mathbf {Q}$ .",
"Then $\\mathsf {Cor}_1 \\in \\mathbf {Q}$ .",
"[Proof sketch] Say the reduction makes $k$ queries.",
"The protocol witnessing $\\mathsf {Cor}_2 \\in \\mathbf {Q}$ defines a probability distribution over bipartite quantum states.",
"To simulate $\\mathsf {Cor}_1$ , Alice and Bob share $\\rho _1 \\otimes \\dots \\otimes \\rho _k$ , where the $\\rho _i$ s are drawn independently at random from that distribution.",
"They run the reduction, using $\\rho _i$ to simulate the $i$ th query.",
"We remark that $\\mathbf {SR}$ and $\\mathbf {NS}$ are also easily seen to be closed under reductions; $\\mathbf {BELL}$ is closed under 1-query reductions."
],
[
"The biased CHSH game",
"For real numbers $p, q \\in [0, 1]$ , the biased CHSH game $\\mathrm {CHSH}[p, q]$ is a nonlocal game defined as follows [19]: The referee picks $x, y \\in \\lbrace 0, 1\\rbrace $ independently at random, with $\\Pr [x = 1] = p$ , $\\Pr [y = 1] = q$ .",
"Alice gets $x$ and Bob gets $y$ .",
"Alice outputs $a \\in \\lbrace 0, 1\\rbrace $ and Bob outputs $b \\in \\lbrace 0, 1\\rbrace $ .",
"The win condition is that $a + b = xy \\pmod {2}$ .",
"The standard CHSH game [11] is the case $p = q = \\frac{1}{2}$ .",
"We can think of a correlation box $\\mathsf {Cor}: \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace \\rightarrow \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace $ as a strategy for the biased CHSH game.",
"The probability that $\\mathsf {Cor}$ wins $\\mathrm {CHSH}[p, q]$ is just the probability that $a + b = xy \\pmod {2}$ , where $(a, b) = \\mathsf {Cor}(x, y)$ and the probability is over both the internal randomness of $\\mathsf {Cor}$ and the inputs $x, y$ .",
"(The inputs $(x, y)$ are independent of the internal randomness of $\\mathsf {Cor}$ .)",
"Lawson et al.",
"showed that like in the ordinary CHSH game, quantum entanglement gives an advantage in the biased CHSH game, at least in certain parameter regimes: Lemma 2 ([19]) If $\\frac{1}{2} \\le q \\le \\frac{1}{2p} \\le 1$ , then there exists a binary-alphabet correlation box $\\mathsf {S}_{p, q} \\in \\mathbf {BELL}$ that wins $\\mathrm {CHSH}[p, q]$ with probability $\\frac{1}{2} + \\frac{1}{2} \\sqrt{2} \\sqrt{q^2 + (1 - q)^2} \\sqrt{p^2 + (1 - p)^2}$ .",
"Conversely, Lawson et al.",
"also showed that Lemma REF is optimal: Lemma 3 ([19]) Suppose $\\frac{1}{2} \\le q \\le \\frac{1}{2p} \\le 1$ and $\\mathsf {Cor} \\in \\mathbf {Q}$ .",
"Then $\\mathsf {Cor}$ wins $\\mathrm {CHSH}[p, q]$ with probability at most $\\frac{1}{2} + \\frac{1}{2} \\sqrt{2} \\sqrt{q^2 + (1 - q)^2} \\sqrt{p^2 + (1 - p)^2}$ ."
],
[
"Reductions imply affine approximations",
"Lemma 4 Suppose $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ is a correlation box in $\\mathbf {Q}$ , and fix $k \\in \\mathbb {N}$ .",
"For each $p \\in [1/2, 1]$ , let $\\mathsf {S}_{p, 1/2}$ be the box of Lemma $\\ref {lem:biased-chsh-positive}$ .",
"There is some set $L_{\\mathsf {Cor}, k}$ of affine functions $\\mathbb {R}\\rightarrow \\mathbb {R}$ such that: For every $p \\in [1/2, 1]$ and every $\\varepsilon > 0$ , if there exists a $k$ -query $\\varepsilon $ -error reduction from $\\mathsf {S}_{p, 1/2}$ to $\\mathsf {Cor}$ , then there exists $\\ell \\in L_{\\mathsf {Cor}, k}$ such that $|\\ell (p) - \\omega (p)| \\le \\varepsilon $ .",
"If $X, Y, A, B$ are all finite, then $|L_{\\mathsf {Cor}, k}| \\le (2|X|)^{2|A|^k} \\cdot (2|Y|)^{2|B|^k}.$ If $X, Y$ are countable and $A, B$ are finite, then $L_{\\mathsf {Cor}, k}$ is countable.",
"For a deterministic $\\mathsf {Cor}$ -protocol $\\Pi $ , let $\\ell _{\\Pi }(p)$ be the probability that $\\Pi $ wins $\\mathrm {CHSH}[p, 1/2]$ .",
"Then $\\ell _{\\Pi }$ is an affine function, since it is just $\\frac{1 - p}{2} P_{00} + \\frac{p}{2} P_{10} + \\frac{1 - p}{2} P_{01} + \\frac{p}{2} P_{11},$ where $P_{xy}$ is the probability that $a + b = xy \\pmod {2}$ where $(a, b) = \\Pi (x, y)$ .",
"Let $L_{\\mathsf {Cor}, k}$ be the set of all $\\ell _{\\Pi }$ .",
"To prove the first item, let $\\Pi $ be a $k$ -query $\\varepsilon $ -reduction from $S_{p, 1/2}$ to $\\mathsf {Cor}$ .",
"Recall that $\\Pi $ is a distribution over deterministic $\\mathsf {Cor}$ -protocols $\\Pi ^{\\prime }$ .",
"Let $g(\\Pi ^{\\prime })$ be the probability that $\\Pi ^{\\prime }$ wins $\\mathrm {CHSH}[p, 1/2]$ .",
"By the correctness of the reduction, we know that $\\left|\\operatornamewithlimits{E}_{\\Pi ^{\\prime } \\sim \\Pi }[g(\\Pi ^{\\prime })] - \\omega (p)\\right| \\le \\varepsilon .$ The best case is at least as good as the average case, so there exists a deterministic $\\mathsf {Cor}$ -protocol $\\Pi ^{\\prime }_*$ such that $g(\\Pi ^{\\prime }_*) \\ge \\omega (p) - \\varepsilon $ .",
"Since $\\mathsf {Cor} \\in \\mathbf {Q}$ , by Lemma REF , $\\Pi ^{\\prime }_*$ implements a correlation box in $\\mathbf {Q}$ .",
"Therefore, by Lemma REF , $g(\\Pi ^{\\prime }_*) \\le \\omega (p)$ .",
"Therefore, $|g(\\Pi ^{\\prime }_*) - \\omega (p)| \\le \\varepsilon $ .",
"By the construction of $L_{\\mathsf {Cor}, k}$ , there is some $\\ell \\in L_{\\mathsf {Cor}, k}$ such that $g(\\Pi ^{\\prime }_*) = \\ell (p)$ , and hence $|\\ell (p) - \\omega (p)| \\le \\varepsilon $ .",
"To prove the second item, we bound the cardinality of $L_{\\mathsf {Cor}, k}$ simply by bounding the number of deterministic $k$ -query $\\mathsf {Cor}$ -protocols.",
"Such a protocol can be specified by: Functions $q_i: \\lbrace 0, 1\\rbrace \\times A^{i - 1} \\rightarrow X$ for each $1 \\le i \\le k$ , telling the $i$ th query that Alice makes as a function of her input and the query responses she has seen so far.",
"Corresponding functions $r_i: \\lbrace 0, 1\\rbrace \\times B^{i - 1} \\rightarrow Y$ for Bob.",
"A function $s: \\lbrace 0, 1\\rbrace \\times A^k \\rightarrow \\lbrace 0, 1\\rbrace $ , telling the output Alice gives as a function of her input and all query responses.",
"A corresponding function $t: \\lbrace 0, 1\\rbrace \\times B^k \\rightarrow \\lbrace 0, 1\\rbrace $ for Bob.",
"If $X, Y$ are countable and $A, B$ are finite, then there are only countably many possibilities for each of these functions, so there are countably many such protocols.",
"Suppose now that $X, Y, A, B$ are all finite and $|A|, |B| \\ge 2$ .",
"The number of possible functions $q_i$ is $|X|^{2|A|^{i - 1}}$ , and similarly for $r_i$ .",
"The number of possible functions $s$ is $2^{2|A|^k}$ , and similarly for $t$ .",
"Therefore, the number of affine functions is bounded by $\\left(\\prod _{i = 1}^k |X|^{2|A|^{i - 1}}\\right) \\left(\\prod _{i = 1}^k |Y|^{2|B|^{i - 1}}\\right) \\cdot 2^{2|A|^k} \\cdot 2^{2|B|^k} &= |X|^{2 \\sum _i |A|^{i - 1}} \\cdot |Y|^{2 \\sum _i |B|^{i - 1}} \\cdot 2^{2|A|^k} \\cdot 2^{2|B|^k} \\\\&\\le |X|^{2|A|^k} \\cdot |Y|^{2|B|^k} \\cdot 2^{2|A|^k} \\cdot 2^{2|B|^k} \\\\&= (2|X|)^{2|A|^k} \\cdot (2|Y|)^{2|B|^k}.$ Finally, if $A$ is a singleton set, the step above where we bounded the geometric series $\\sum _i |A|^{i - 1}$ by $|A|^k$ was not valid, but in this case the functions $q_i$ do not need to be specified anyway, so the final bound still holds.",
"Similarly if $B$ is a singleton set."
],
[
"Lower bounds on the error of affine approximations",
"For our qualitative negative result (Theorem REF ), the following trivial fact is sufficient.",
"Lemma 5 Suppose $L$ is a countable set of affine functions $\\mathbb {R}\\rightarrow \\mathbb {R}$ .",
"Then there is some $p \\in [1/2, 1]$ such that for every $\\ell \\in L$ , $\\ell (p) \\ne \\omega (p)$ .",
"Suppose that some value of $p$ satisfies $\\ell (p) = \\omega (p)$ , where $\\ell \\in L$ .",
"Rearranging, we find that $ p^2 + (1 - p)^2 = r(p)^2,$ where $r(p)$ is another affine function.",
"The quadratic expression on the left hand side of Equation REF has a nonzero discriminant of $-4$ .",
"Therefore, Equation REF must not be an identity, and hence it has at most two solutions $p$ .",
"So each $\\ell \\in L$ intersects $\\omega $ at most twice, and hence $L$ intersects $\\omega $ in at most countably many places.",
"For our quantitative negative result (Theorem REF ), we need to lower bound the error of any approximation of $\\omega $ by affine functions.",
"Lemma 6 Pick $p \\in [1/2, 1]$ uniformly at random.",
"Then for any affine function $\\ell : \\mathbb {R}\\rightarrow \\mathbb {R}$ and any $\\varepsilon > 0$ , $\\Pr \\left[|\\ell (p) - \\omega (p)| \\le \\varepsilon \\right] \\le O(\\sqrt{\\varepsilon }).$ Let $I = [1/2, 1]$ .",
"We first compute $ \\omega ^{\\prime \\prime }(x) = \\frac{1}{2}[x^2 + (1 - x)^2]^{-3/2} = \\frac{1}{2}\\left[2\\left(x - \\frac{1}{2}\\right)^2 + \\frac{1}{2}\\right]^{-3/2} \\ge \\frac{1}{2} \\quad \\text{on $I$.",
"}$ Hence $\\omega $ is uniformly convex on $I$ .",
"Without loss of generality we can assume that the graph of $\\ell $ intersects the graph of $\\omega $ twice (with a point of tangency counted as a double intersection).",
"After all, if $\\ell < \\omega $ on $I$ , translate $\\ell $ up until the first moment of equality with $\\omega $ , thus decreasing the pointwise error between $\\ell $ and $\\omega $ at every $x \\in I$ .",
"If $\\ell $ is then tangent to $\\omega $ , we are done.",
"Otherwise, $\\ell $ intersects $\\omega $ at an endpoint, so rotate $\\ell $ up about this point until it is tangent to $\\omega $ (no other intersections occur because $\\omega $ is uniformly convex).",
"Pointwise errors do not increase under this rotation, so the entire transformation only increases the probability in the lemma statement.",
"Similar considerations hold if initially $\\ell > \\omega $ or $\\ell $ intersects $\\omega $ at one point.",
"Thus we may assume that $\\ell $ linearly interpolates $\\omega $ .",
"Suppose $\\ell $ interpolates $\\omega $ at the (potentially coincident) points $x_1, x_2 \\in I$ .",
"We now claim that for all $x \\in I$ , there exists $\\xi _x \\in I$ such that $ \\omega (x) - \\ell (x) = \\frac{\\omega ^{\\prime \\prime }(\\xi _x)}{2} (x - x_1)(x - x_2).$ This follows from a standard argument in interpolation theory; we include the details here for completeness.",
"If $x = x_1$ or $x = x_2$ , Equation REF is trivial, since both sides are zero.",
"Otherwise, let $\\phi _x(t) = \\omega (t) - \\ell (t) - (\\omega (x) - \\ell (x)) \\cdot \\frac{(t - x_1)(t - x_2)}{(x - x_1)(x - x_2)}$ .",
"Then $\\phi _x$ is zero at $x$ , $x_1$ , and $x_2$ .",
"By Rolle's theorem, this implies that $\\phi _x^{\\prime }$ has at least two zeroes in $I$ (actually Rolle's theorem only gives one zero if $x_1 = x_2$ , but in this case $x_1 = x_2$ is another zero of $\\phi _x^{\\prime }$ , so either way $\\phi _x^{\\prime }$ has two distinct zeroes in $I$ ).",
"By another application of Rolle's theorem, there is some $\\xi _x \\in I$ such that $\\phi _x^{\\prime \\prime }(\\xi _x) = 0$ .",
"Equation REF follows.",
"By Equation REF , $|\\omega (x) - \\ell (x)| \\ge \\frac{1}{4} |x - x_1| |x - x_2|.$ In particular, when $\\min \\lbrace |x - x_1|, |x - x_2|\\rbrace > 2\\sqrt{\\varepsilon }$ , $|\\omega (x) - \\ell (x)| > \\varepsilon $ .",
"The probability that $p$ is within $2\\sqrt{\\varepsilon }$ of either $x_1$ or $x_2$ is $O(\\sqrt{\\varepsilon })$ by the union bound."
],
[
"Proofs of main results",
"[Proof of Theorem $\\ref {thm:main}$ ] Fix $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ , where $X, Y$ are countable, $A, B$ are finite, and $\\mathsf {Cor} \\in \\mathbf {Q}$ .",
"We will show that there is some choice of $p$ so that there is no reduction from $\\mathsf {S}_{p, 1/2}$ to $\\mathsf {Cor}$ ; since $\\mathsf {S}_{p, 1/2}$ is a binary correlation box in $\\mathbf {BELL}$ , this will complete the proof.",
"For each $k \\in \\mathbb {N}$ , let $L_{\\mathsf {Cor}, k}$ be the set of affine functions given by Lemma REF .",
"The alphabet bounds for $\\mathsf {Cor}$ imply that $L_{\\mathsf {Cor}, k}$ is countable.",
"Let $L = \\bigcup _{k \\in \\mathbb {N}} L_{\\mathsf {Cor}, k}$ , so that $L$ is still countable.",
"By Lemma REF , choose $p \\in [1/2, 1]$ so that for every $\\ell \\in L$ , $\\ell (p) \\ne \\omega (p)$ .",
"Then $\\mathsf {S}_{p, 1/2}$ does not reduce to $\\mathsf {Cor}$ , because if there were a $k$ -query (0-error) reduction for some $k$ , Lemma REF would imply that there was some $\\ell \\in L$ with $\\ell (p) = \\omega (p)$ .",
"[Proof of Theorem $\\ref {thm:lower-bound}$ ] Fix $\\mathsf {Cor}_2: X \\times Y \\rightarrow A \\times B$ , where $X, Y, A, B$ are finite and $\\mathsf {Cor}_2 \\in \\mathbf {Q}$ .",
"Let $L_{\\mathsf {Cor}_2, k}$ be the set of affine functions given by Lemma REF .",
"Pick $p \\in [1/2, 1]$ uniformly at random.",
"By Lemma REF and the union bound, for any $\\varepsilon _k > 0$ , the probability that some $\\ell \\in L_{\\mathsf {Cor}_2, k}$ satisfies $|\\ell (p) - \\omega (p)| \\le \\varepsilon _k$ is at most $O(\\sqrt{\\varepsilon _k} \\cdot |L_{\\mathsf {Cor}_2, k}|)$ .",
"Therefore, by the union bound over $k$ , $ \\Pr [\\exists k, \\exists \\ell \\in L_{\\mathsf {Cor}_2, k}, |\\ell (p) - \\omega (p)| \\le \\varepsilon _k] \\le O\\left(\\sum _{k = 1}^{\\infty } \\sqrt{\\varepsilon _k} \\cdot |L_{\\mathsf {Cor}_2, k}|\\right).$ Choose $\\varepsilon _k$ so that $\\sqrt{\\varepsilon _k} \\cdot |L_{\\mathsf {Cor}_2, k}| = c/k^2$ , where $c$ is a sufficiently small constant so that the bound in Equation REF is strictly less than 1.",
"(Such a $c$ exists because $\\sum _k 1/k^2$ is a convergent series.)",
"This can be achieved while maintaining $\\varepsilon _k \\ge \\Omega \\left(\\frac{1}{k^4 |L_{\\mathsf {Cor}_2, k}|^2}\\right),$ which implies by Lemma REF that $k^4 \\cdot (2|X|)^{4|A|^k} \\cdot (2|Y|)^{4|B|^k} \\ge \\Omega (1/\\varepsilon _k).$ By our choice of $\\varepsilon _k$ , there exists some $p$ so that for every $k$ , for every $\\ell \\in L_{\\mathsf {Cor}_2, k}$ , $|\\ell (p) - \\omega (p)| > \\varepsilon _k$ .",
"Choose $\\mathsf {Cor}_1 = \\mathsf {S}_{p, 1/2}$ .",
"By Lemma REF , if there is a $k$ -query $\\varepsilon $ -error reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ , then $\\varepsilon > \\varepsilon _k$ ."
],
[
"There's always a harder box",
"The proofs of our main results are done.",
"In this section, we make a simple observation that follows easily from our negative results.",
"Define a preorder on correlation boxes by saying that $\\mathsf {Cor} \\le \\mathsf {Cor}^{\\prime }$ if there is a reduction from $\\mathsf {Cor}$ to $\\mathsf {Cor}^{\\prime }$ .",
"Write $\\mathsf {Cor} < \\mathsf {Cor}^{\\prime }$ if $\\mathsf {Cor} \\le \\mathsf {Cor}^{\\prime }$ and $\\mathsf {Cor}^{\\prime } \\lnot \\le \\mathsf {Cor}$ .",
"Theorem 3 For any finite-alphabet correlation box $\\mathsf {Cor} \\in \\mathbf {BELL}$ , there is another finite-alphabet correlation box $\\mathsf {Cor}^{\\prime } \\in \\mathbf {BELL}$ such that $\\mathsf {Cor} < \\mathsf {Cor}^{\\prime }$ .",
"By Theorem REF , there is a binary correlation box $\\mathsf {Cor}_0 \\in \\mathbf {BELL}$ such that $\\mathsf {Cor}_0 \\lnot \\le \\mathsf {Cor}$ .",
"Write $\\mathsf {Cor}: X \\times Y \\rightarrow A \\times B$ .",
"By relabeling if necessary, we can assume that $0, 1 \\notin X, Y$ .",
"Define $\\mathsf {Cor}^{\\prime }: (X \\cup \\lbrace 0, 1\\rbrace ) \\times (Y \\cup \\lbrace 0, 1\\rbrace ) \\rightarrow (A \\cup \\lbrace 0, 1\\rbrace ) \\times (B \\cup \\lbrace 0, 1\\rbrace )$ by the following $\\mathbf {BELL}$ algorithm: If $x \\in X$ , then Alice does what she would have done in the protocol witnessing $\\mathsf {Cor} \\in \\mathbf {BELL}$ .",
"Otherwise, if $x \\in \\lbrace 0, 1\\rbrace $ , she does what she would have done in the protocol witnessing $\\mathsf {Cor}_0 \\in \\mathbf {BELL}$ .",
"Bob acts similarly.",
"By construction: If $x \\in X, y \\in Y$ , then $\\mathsf {Cor}^{\\prime }(x, y) \\sim \\mathsf {Cor}(x, y)$ .",
"This immediately implies that $\\mathsf {Cor} \\le \\mathsf {Cor}^{\\prime }$ .",
"If $x, y \\in \\lbrace 0, 1\\rbrace $ , then $\\mathsf {Cor}^{\\prime }(x, y) \\sim \\mathsf {Cor}_0(x, y)$ .",
"This immediately implies that $\\mathsf {Cor}_0 \\le \\mathsf {Cor}^{\\prime }$ , and hence by transitivity $\\mathsf {Cor}^{\\prime } \\lnot \\le \\mathsf {Cor}$ .",
"(Notice that if $x \\in X, y \\in \\lbrace 0, 1\\rbrace $ , the distribution $\\mathsf {Cor}^{\\prime }(x, y)$ has no clear interpretation, but that doesn't matter for us.",
"Similarly with the case $x \\in \\lbrace 0, 1\\rbrace , y \\in Y$ .)"
],
[
"Positive results",
"We now show how to construct a finite-alphabet correlation box that is approximately complete for $\\mathbf {BELL}$ .",
"The construction is simple, and just consists of an appropriate discretization of the Bloch sphere [5].",
"Theorem 4 For every $\\varepsilon > 0$ , there exists $\\mathsf {Cor}_2: [T] \\times [T] \\rightarrow \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace $ with $T \\le O(1/\\varepsilon ^2)$ such that $\\mathsf {Cor}_2 \\in \\mathbf {BELL}$ , and for every $\\mathsf {Cor}_1 \\in \\mathbf {BELL}$ , there is a 1-query $\\varepsilon $ -error reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ .",
"Let $c_1, c_2, \\dots , c_T \\in \\mathbb {R}^3$ be points on the unit sphere such that every point on the unit sphere is within $\\varepsilon $ of some $c_i$ in $\\ell _2$ distance.",
"Such a collection of points exists with $T \\le O(1/\\varepsilon ^2)$ .",
"We define $\\mathsf {Cor}_2$ by the following algorithm, simultaneously showing that $\\mathsf {Cor}_2 \\in \\mathbf {BELL}$ : Alice and Bob share a pair of qubits in the state $\\mathinner {|{\\phi }\\rangle } \\stackrel{\\text{def}}{=} \\frac{\\mathinner {|{01}\\rangle } - \\mathinner {|{10}\\rangle }}{\\sqrt{2}}$ .",
"(This can be obtained by applying local operations to $\\frac{\\mathinner {|{00}\\rangle } + \\mathinner {|{11}\\rangle }}{\\sqrt{2}}$ .)",
"On inputs $i, j$ : Alice finds a unitary matrix $U$ such that $U^{-1}\\mathinner {|{1}\\rangle }$ is represented by the point $c_i$ on the Bloch sphere.",
"She applies $U$ to her qubit, measures in the computational basis, and outputs the observed bit.",
"Bob finds a unitary matrix $V$ such that $V^{-1}\\mathinner {|{1}\\rangle }$ is represented by the point $c_j$ on the Bloch sphere.",
"He applies $V$ to his qubit, measures in the computational basis, and outputs the observed bit.",
"We now give the reduction.",
"From the definition of $\\mathbf {BELL}$ , it suffices to show how to approximately simulate applying some unitary matrix $U \\otimes V$ to $\\mathinner {|{\\phi }\\rangle }$ and then measuring in the computational basis, where Alice chooses $U$ and Bob chooses $V$ .",
"To do this, Alice finds $c_i$ that is closest to the Bloch sphere representation of $U^{-1} \\mathinner {|{1}\\rangle }$ in $\\ell _2$ distance, and Bob finds $c_j$ that is closest to the Bloch sphere representation of $V^{-1} \\mathinner {|{1}\\rangle }$ in $\\ell _2$ distance.",
"They query $\\mathsf {Cor}_2(i, j)$ .",
"We now prove correctness of this reduction.",
"A curiosity of the state $\\mathinner {|{\\phi }\\rangle }$ is that for any unitary $V$ , there is a scalar $\\lambda \\in such that $ (V V) | = |$.",
"Proof:{\\begin{@align*}{1}{-1}\\mathinner {\\langle {ij|V \\otimes V|\\phi }\\rangle } = \\frac{\\mathinner {\\langle {i|V|0}\\rangle } \\mathinner {\\langle {j|V|1}\\rangle } - \\mathinner {\\langle {i|V|1}\\rangle } \\mathinner {\\langle {j|V|0}\\rangle }}{\\sqrt{2}},\\end{@align*}}and hence $ 00|V V| = 11|V V| = 0$ and $ 01|V V| = -10|V V|$.$ Therefore, we can write $(U \\otimes V)\\mathinner {|{\\phi }\\rangle } &= (UV^{-1} \\otimes I) (V \\otimes V) \\mathinner {|{\\phi }\\rangle } \\\\&=\\lambda (UV^{-1} \\otimes I) \\mathinner {|{\\phi }\\rangle } \\\\&= \\frac{\\lambda }{\\sqrt{2}}((UV^{-1} \\mathinner {|{0}\\rangle }) \\mathinner {|{1}\\rangle } - (UV^{-1} \\mathinner {|{1}\\rangle }) \\mathinner {|{0}\\rangle }).$ It follows that when $(U \\otimes V) \\mathinner {|{\\phi }\\rangle }$ is measured, giving two bits $a, b$ , $\\Pr [a = b] &= \\frac{1}{2} |\\mathinner {\\langle {1| UV^{-1} | 0}\\rangle }|^2 + \\frac{1}{2} |\\mathinner {\\langle {0|UV^{-1}|1}\\rangle }|^2 \\\\&= 1 - |\\mathinner {\\langle {1 | UV^{-1} | 1}\\rangle }|^2.$ Let $x$ be the Bloch sphere representation of $U^{-1} \\mathinner {|{1}\\rangle }$ , and let $y$ be the Bloch sphere representation of $V^{-1} \\mathinner {|{1}\\rangle }$ .",
"Then $|\\mathinner {\\langle {1 | UV^{-1} | 1}\\rangle }|^2 = \\frac{1}{2} + \\frac{1}{2} x \\cdot y$ , where $\\cdot $ is the dot product.",
"So $\\Pr [a = b] = \\frac{1}{2} - \\frac{1}{2} x \\cdot y$ .",
"Let $(\\widehat{a}, \\widehat{b}) = \\mathsf {Cor}(i, j)$ .",
"Then $\\left|\\Pr [a = b] - \\Pr [\\widehat{a} = \\widehat{b}]\\right| &= \\frac{1}{2} |x \\cdot y - c_i \\cdot c_j| \\\\&\\le \\frac{1}{2} |x \\cdot y - x \\cdot c_j| + \\frac{1}{2}|x \\cdot c_j - c_i \\cdot c_j| \\\\&\\le \\varepsilon /2 + \\varepsilon /2.$ Since $a, b, \\widehat{a}, \\widehat{b}$ all have uniform marginal distributions, it follows that $(a, b)$ and $(\\widehat{a}, \\widehat{b})$ are $\\varepsilon $ -close in total variation distance.",
"Proposition 1 There exists $\\mathsf {Cor}_2: \\mathbb {N}\\times \\mathbb {N}\\rightarrow \\lbrace 0, 1\\rbrace \\times \\lbrace 0, 1\\rbrace $ such that $\\mathsf {Cor}_2 \\in \\mathbf {BELL}$ , and for every $\\mathsf {Cor}_1 \\in \\mathbf {BELL}$ and every $\\varepsilon > 0$ , there is a 1-query $\\varepsilon $ -error reduction from $\\mathsf {Cor}_1$ to $\\mathsf {Cor}_2$ .",
"[Proof sketch] Use a countable dense subset of the Bloch sphere."
],
[
"Open problems",
" We proved that there is no $\\mathbf {BELL}$ -complete correlation box with countable input alphabets and finite output alphabets.",
"Does there exist a $\\mathbf {BELL}$ -complete correlation box with countable alphabets?",
"(Our proof breaks down because there are uncountably many deterministic reductions to a correlation box with countably infinite output alphabets.)",
"Does there exist a minimal $\\mathbf {BELL}$ -hard finite-alphabet correlation box $\\mathsf {Cor}$ ?",
"(By minimal, we mean that if $\\mathsf {Cor}^{\\prime }$ is another $\\mathbf {BELL}$ -hard finite-alphabet correlation box, then $\\mathsf {Cor}$ reduces to $\\mathsf {Cor}^{\\prime }$ .)",
"What is the right relationship between $|X|, |Y|, |A|, |B|, k, \\varepsilon $ in Theorem REF ?"
],
[
"Acknowledgments",
"We thank Scott Aaronson and Ronald de Wolf for helpful comments and encouragement.",
"This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No.",
"DGE-1610403.",
"Cole Graham gratefully acknowledges the support of the Fannie and John Hertz Foundation."
]
] | 1612.05680 |
[
[
"Multi-partite entanglement speeds up quantum key distribution in\n networks"
],
[
"Abstract The laws of quantum mechanics allow for the distribution of a secret random key between two parties.",
"Here we analyse the security of a protocol for establishing a common secret key between N parties (i.e.",
"a conference key), using resource states with genuine N-partite entanglement.",
"We compare this protocol to conference key distribution via bipartite entanglement, regarding the required resources, achievable secret key rates and threshold qubit error rates.",
"Furthermore we discuss quantum networks with bottlenecks for which our multipartite entanglement-based protocol can benefit from network coding, while the bipartite protocol cannot.",
"It is shown how this advantage leads to a higher secret key rate."
],
[
"Multipartite QKD: protocol and security analysis",
"The entanglement-based Ekert protocol [7] can be generalised to $N$ parties as follows, see also [31].",
"The parties $A$ and $B_1$ , $B_2$ , ..., $B_{N-1}$ share an $N$ -partite entangled state and perform local projective measurements.",
"The best performance in the ideal (noiseless) case is ensured if one requires that the measurement outcomes of all parties are perfectly correlated for one set of local bases – which we can choose without loss of generality to be the $Z$ -bases – and occur with a uniform distribution.",
"The only pure $N$ -qubit quantum state that fulfils these requirements is the Greenberger-Horne-Zeilinger (GHZ) state [34]; however, for $N\\ge 3$ , the existence of perfect correlations in one set of bases forbids perfect correlations (even only pairwise) in any other bases, see Appendix .",
"We remark that other protocols with less than perfectly correlated resource states are possible, but will introduce intrinsic errors [35].",
"The protocol for N-party quantum conference key distribution (NQKD), with $N\\ge 2$ , consists of the following basic steps: 1) State preparation: The parties $A$ and $B_i$ , ${i=1,2,...,N-1}$ , share the $N$ -qubit GHZ state $| GHZ \\rangle = \\frac{1}{\\sqrt{2}}\\left(| 0 \\rangle ^{\\otimes N} + | 1 \\rangle ^{\\otimes N}\\right).$ 2) Measurement: There are two types of measurements.",
"First type: Party $A$ and parties $B_i$ measure their respective qubits in the $Z$ -basis.",
"Second type: They measure randomly, with equal probability, in the $X$ - or $Y$ -basis.",
"Similar to the standard bipartite QKD protocol [36], the latter case is much less frequent.",
"The parties know the type of the measurement from a short pre-shared random key.",
"3) Parameter estimation: The parties announce the measurement bases and outcomes for the second type and an equal number of randomly chosen rounds of the first type.",
"The announced data allows to estimate the parameters $Q_X$ and $Q_Z$ , which determine the secret key rate, see below.",
"4) Classical post-processing: As in the bipartite protocol, error correction and privacy amplification is performed, for the details see below.",
"Note that the state preparation in step 1) can be achieved by locally preparing the GHZ-state at Alice's site and sending qubits to the Bobs (see Fig.",
"REF ), or any suitable sub-protocol that achieves the same task.",
"We analyse the distribution via quantum repeaters [26] and quantum network coding [27] below.",
"In the following section we briefly discuss prepare-and-measure variants of conference key distribution.",
"Because the security proof of the NQKD protocol is done in the entanglement-based picture, this description is not necessary for understanding the rest of the paper."
],
[
"N-party prepare-and-measure schemes",
"We now sketch two different prepare-and-measure schemes for conference key distribution.",
"1) Preparing and measuring single qubits: Single qubits are experimentally easier to prepare and to distribute than entangled states.",
"Thus, establishing a conference key for $N$ parties by using single qubits is an interesting possibility, which has been studied for the case $N=3$ in [37].",
"The protocol proceeds in complete analogy to the case of $N=2$ , e.g.",
"for BB84 [5]: Alice prepares randomly $N-1$ copies of a state $| \\phi _k \\rangle , k=1,...4,$ taken from the set $S_{BB84}=\\lbrace | 0 \\rangle ,| 1 \\rangle ,| + \\rangle ,| - \\rangle \\rbrace $ , with $| \\pm \\rangle =1/\\sqrt{2}(| 0 \\rangle \\pm | 1 \\rangle )$ .",
"She sends each party $B_i$ , with $i=1,...,N$ , one of the copies.",
"Each party $B_i$ measures in the $Z$ - or $X$ -basis.",
"In the sifting step, the $N$ parties keep only those cases where all parties used the same basis, and thus establish a joint key.",
"We point out, however, that the secret key rate in this scenario decreases with increasing $N$ , even for perfect channels and measurements, and goes to zero for $N\\rightarrow \\infty $ : an eavesdropper can eavesdrop on all $N-1$ sent states at the same time, i.e.",
"she has to distinguish the four global states $ |\\phi _k>^{\\otimes (N-1)}$ , pairs of which have either overlap 0 or $(1/\\sqrt{2})^{N-1}$ , i.e.",
"the distinguishability increases with increasing $N$ .",
"In the limit of infinite $N$ the four global states are orthogonal and therefore perfectly distinguishable.",
"Thus, this prepare-and-measure-scheme is (for $N\\ge 3$ ) not equivalent to entanglement-based NQKD as described in the present article.",
"2) Prepare-and-measure equivalent of NQKD: The entanglement-based protocol NQKD described above can be formulated as a prepare-and-measure protocol, analogous to the six-state protocol [8].",
"Instead of producing the GHZ state of Eq.",
"(REF ) and measuring her qubit afterwards, Alice can directly produce the ($N-1$ )-qubit projection of the GHZ state according to her fictitious, random outcome.",
"Thus, for the $X$ -, $Y$ - and $Z$ -basis, the six different $(N-1)$ -qubit states she distributes among the Bobs, are $\\begin{aligned}| \\psi _{x,\\pm } \\rangle =&\\frac{1}{\\sqrt{2}}(| 00...0 \\rangle \\pm | 11...1 \\rangle ),\\\\| \\psi _{y,\\pm } \\rangle =&\\frac{1}{\\sqrt{2}}(| 00...0 \\rangle \\mp i| 11...1 \\rangle ),\\\\| \\psi _{z,0} \\rangle =&| 00...0 \\rangle \\text{ and }| \\psi _{z,1} \\rangle =| 11...1 \\rangle .\\end{aligned}$ This protocol is equivalent to NQKD, because it reproduces the correlations between $A$ , $B_1$ , ... $B_{N-1}$ .",
"Note that the six-state protocol is included as the special case $N=2$ .",
"The described protocol uses $(N-1)$ -partite entanglement for four of the sent states, which are, however, sent much less frequent than the two product states.",
"This fact renders an experimental implementation of our protocol more realistic than the entanglement-based description might suggest.",
"In the remainder of this article we use the equivalent entanglement-based description of the NQKD protocol."
],
[
"Security analysis of the N-party quantum key distribution",
"The composable security definition of the bipartite scenario [16], [17] can be generalised in an analogous way to the $N$ -partite case.",
"Our security analysis proceeds along analogous lines as the bipartite case in [33].",
"See Appendix for explicit details of these generalisations.",
"By employing this security definition and using one-way communication only, we prove secrecy of the key under the most general eavesdropping attack allowed by the laws of quantum mechanics, so-called coherent attacks [38], [39], independent of the context in which the key is used.",
"In the asymptotic limit, i.e.",
"for infinitely many rounds, the secret fraction $r_{\\infty }$ , i.e.",
"the ratio of secret bits and the number of shared states (without parameter estimation rounds), is given by $r_{\\infty } = \\sup _{U \\leftarrow K}\\inf _{\\sigma _{A\\lbrace B_i\\rbrace } \\in \\Gamma }[S (U|E) - \\max _{i\\in \\lbrace 1,...N-1\\rbrace } H( U|K_i)],$ where $U \\leftarrow K$ denotes a bitwise preprocessing channel on Alice's raw key bit $K$ , $S(U|E)$ is the conditional von-Neumann entropy of the (classical) key variable, given the state of Eve's system $E$ , $H(U|K_i)$ is the conditional Shannon entropy of $U$ given $K_i$ , which is $B_i$ 's guess of $K$ , and $\\Gamma $ is the set of all density matrices $\\sigma _{A\\lbrace B_i\\rbrace }$ of Alice and the Bobs which are consistent with the parameter estimation.",
"The secret key rate is $R=r_\\infty R_{\\mathrm {rep}},$ where the repetition rate $R_{\\mathrm {rep}}=\\frac{1}{t_{\\mathrm {rep}}}$ is given by the time $t_{\\mathrm {rep}}$ that one round (steps 1 and 2) takes.",
"For now we set $t_{\\mathrm {rep}}=1$ .",
"The secret key rate in Eq.",
"(REF ) as a figure of merit does not directly account for the amount of needed local randomness, classical communication, qubits and gates.",
"Depending on the context one might want to incorporate one or more of the former quantities into a cost-performance ratio as a figure of merit [40].",
"Note that we have not assumed any symmetry about the quality of the channels connecting $A$ and $B_i$ .",
"Therefore, the worst-case information leakage in the error correction step is determined by the noisiest channel, see the maximisation in the last term of Eq.",
"(REF ).",
"This is the main difference with respect to the bipartite case."
],
[
"The secret key rate",
"We now derive an analytical formula for the multipartite secret key rate based on a variant of the method of depolarisation [33].",
"In practice, the described depolarisation operations will be applied to the classical data only, as described in detail below.",
"Readers who are not interested in the technical details can skip to Eq.",
"(REF ).",
"Let us denote the GHZ basis of $N$ qubits as follows: $| \\psi _j^\\pm \\rangle = \\frac{1}{\\sqrt{2}}(| 0 \\rangle | j \\rangle \\pm | 1 \\rangle | \\bar{j} \\rangle )\\ ,$ where $j$ takes the values $0,...,2^{N-1}-1$ in binary notation, and $\\bar{j}$ denotes the binary negation of $j$ ; i.e.",
"for example if $j = 01101$ then ${\\bar{j}} = 10010$ .",
"Remember that any state of $N$ qubits can be depolarised to a state which is diagonal in the GHZ basis by a sequence of local operations [41], [42].",
"In our protocol we introduce the following extended depolarisation procedure.",
"The set of depolarisation operators is $\\mathcal {D}=\\lbrace X^{\\otimes N}\\rbrace \\cup \\lbrace Z_A Z_{B_j}|1\\le j \\le N-1 \\rbrace \\cup \\lbrace R_k |1\\le k \\le N-1 \\rbrace , $ where $X$ and $Z$ are Pauli operators and $R_k=\\mathrm {diag}(1,i)_A \\otimes \\mathrm {diag}(1,-i)_{B_k}.$ The parties apply each of these operators with probability $1/2$ or 1 else.",
"The operators from the first two sets of Eq.",
"(REF ) make the density matrix GHZ diagonal as in [41], [42].",
"We denote the coefficient in front of $\\left| \\psi _j^\\sigma \\right>\\left< \\psi _j^\\sigma \\right|$ by $\\lambda _j^\\sigma $ with $\\sigma \\in \\lbrace +,-\\rbrace $ and $j\\in \\lbrace 0,1,...,2^{N-1}-1\\rbrace $ .",
"The effect of $R_k$ is $R_k | \\psi _j^\\sigma \\rangle =&\\left\\lbrace \\begin{array}{ll}| \\psi _j^\\sigma \\rangle & \\text{if } j^{(k)} = 0\\\\-i | \\psi _j^{-\\sigma } \\rangle & \\text{if } j^{(k)} = 1\\\\\\end{array}\\right.",
",\\multicolumn{2}{l}{\\text{so applying this operator with probability $\\frac{1}{2}$ transforms}}\\\\\\lambda _j^\\sigma \\xrightarrow{}&\\left\\lbrace \\begin{array}{ll}\\lambda _j^\\sigma & \\text{if } j^{(k)} = 0\\\\\\frac{1}{2}(\\lambda _j^{-\\sigma }+\\lambda _j^{\\sigma }) & \\text{if } j^{(k)} = 1\\\\\\end{array}\\right.",
",$ where $j^{(k)}$ denotes the $k$ th bit of the string $j$ .",
"As this operation is applied for all $k=1,2,...,N-1$ , it achieves that $\\lambda _j^+=\\lambda _j^- \\text{ for all $j>0$.",
"}$ The resulting depolarised state reads $\\rho _\\text{dep} = \\lambda _0^+ \\left| \\psi _0^+\\right>\\left< \\psi _0^+\\right| + \\lambda _0^- \\left| \\psi _0^-\\right>\\left< \\psi _0^-\\right|+\\sum _{j=1}^{2^{N-1}-1}\\lambda _{j}(\\left| \\psi _j^+\\right>\\left< \\psi _j^+\\right|+\\left| \\psi _j^-\\right>\\left< \\psi _j^-\\right|).$ In our multipartite scenario we define the qubit error rate (QBER) $Q_Z$ to be the probability that at least one Bob obtains a different outcome than Alice in a $Z$ -basis measurement.",
"Note that this value is not the same as the bipartite qubit error rate $Q_{AB_i}$ , which is the probability that the $Z$ -measurement outcome of $B_i$ disagrees with the one of Alice.",
"$Q_Z$ can be read directly from the structure of the depolarised state in Eq.",
"(REF ) and is given by $Q_Z = 1- \\lambda _0^+-\\lambda _0^- \\ .$ For simplicity we neglect the possibility of increasing the key rate by adding pre-processing noise, i.e.",
"we set $q=0$ in the notation of [33] such that $\\mathbf {U}=\\mathbf {K}$ .",
"Because $\\begin{aligned}S(K|E)=&S(E|K)-S(E)+H(K)\\\\\\text{and }H(K|K_i)=&H(K_i|K)-H(K_i)+H(K)\\end{aligned}$ the asymptotic secret fraction is $r_{\\infty } = S(E|K) - S(E) - \\max _{1\\le i\\le N-1} (H(K_i|K) - H(K_i)).$ Note that we did not need to include the infimum over $\\Gamma $ , see Eq.",
"(REF ), here because, as we will see below, the measurement statistics completely determine all relevant quantities in our protocol.",
"The entropies involving the classical random variable $K$ are directly obtained from the measurement statistics in the parameter estimation phase.",
"They are given by $H(K|K_i)=h(Q_{AB_i}),$ with the binary Shannon entropy $h(p)=-p \\log _2 p-(1-p) \\log _2(1-p)$ and the bipartite error rate $Q_{AB_i}$ , given by $Q_{AB_i}=\\sum _{\\begin{array}{c}j\\\\ j^{(i)}=1\\end{array}}\\sum _{\\sigma =\\pm } \\lambda _j^\\sigma \\overset{Eq.~(\\ref {eq:equallambdas})}{=} 2\\sum _{\\begin{array}{c}j\\\\ j^{(i)}=1\\end{array}}\\lambda _j,$ where $j^{(i)}$ denotes the $i$ -th bit of $j$ and, because both outcomes are equiprobable, $H(K_i)=1.$ Giving Eve the purification of Eq.",
"(REF ), the von-Neumann entropies involving Eve's system in Eq.",
"(REF ) are given by $S(E|K)\\overset{\\phantom{Eq.~(\\ref {eq:equallambdas})}}{=}&\\frac{1}{2} S(E|K=0)+ \\frac{1}{2} S(E|K=1)\\nonumber \\\\\\overset{\\phantom{Eq.~(\\ref {eq:equallambdas})}}{=}&-\\sum _{i=0}^{2^{N-1}-1} (\\lambda _i^+ + \\lambda _i^-)\\log _2 (\\lambda _i^+ + \\lambda _i^-)\\nonumber \\\\\\overset{Eq.~(\\ref {eq:equallambdas})}{=}&-(1-Q_Z)\\log _2(1-Q_Z)-2\\sum _{i=1}^{2^{N-1}-1} \\lambda _i\\log _2 (\\lambda _i) - Q_Z$ and $S(E)\\overset{\\phantom{Eq.~(\\ref {eq:equallambdas})}}{=}&S(\\frac{1}{2}(\\sigma _E^0 + \\sigma _E^1))=-\\sum _{j,\\sigma =\\pm } \\lambda _j^{\\sigma } \\log _2 \\lambda _j^{\\sigma }\\nonumber \\\\\\overset{Eq.~(\\ref {eq:equallambdas})}{=}&-\\lambda _0^+\\log _2\\lambda _0^+-\\lambda _0^-\\log _2\\lambda _0^- -2 \\sum _{j>0} \\lambda _j \\log _2 \\lambda _j,$ i.e.",
"$S(E|K)-S(E)=&-Q_Z-(1-Q_Z)\\log _2(1-Q_Z)+\\lambda _0^+\\log _2\\lambda _0^++(1-Q_Z-\\lambda _0^+)\\log _2(1-Q_Z-\\lambda _0^+).",
"$ Now $\\lambda _0^+$ and $\\lambda _0^-$ can be obtained with the additional $X^{\\otimes N}$ measurement in the parameter estimation, because $\\lambda _0^++\\lambda _0^-=1-Q_Z=\\operatorname{tr}\\left(\\rho _{\\mathrm {dep}} (\\left| 0\\right>\\left< 0\\right|^{\\otimes N}+\\left| 1\\right>\\left< 1\\right|^{\\otimes N})\\right)$ is known from the QBER and $\\operatorname{tr}\\left( \\rho _{\\mathrm {dep}} X^{\\otimes N}\\right) = \\sum _j (\\lambda _j^+-\\lambda _j^-)=\\lambda _0^+-\\lambda _0^-$ .",
"In analogy to $Q_Z$ we denote the probability that the $X$ -measurement gives an unexpected result, i.e.",
"one that is incompatible with the noiseless state, by $Q_X$ .",
"Because $\\langle \\psi _j^\\sigma |X^{\\otimes N}| \\psi _j^\\sigma \\rangle =\\sigma $ this leads to $Q_X=\\frac{1-\\langle X^{\\otimes N}\\rangle _{\\mathrm {dep}}}{2},$ which can, as we will see in Section REF , be obtained from the measured data in the parameter estimation step.",
"We remark that $Q_X$ is not the probability that at least one Bob gets a different $X$ -measurement outcome than Alice, as the outcomes are not correlated, see Appendix .",
"Finally, inserting Eq.",
"(REF ) into Eq.",
"(REF ), and using Eq.",
"(REF ), we arrive at the achievable secret key rate, $\\begin{aligned}R=&\\hphantom{{}+{}}\\left(1- \\frac{Q_Z}{2} - Q_X\\right) \\log _2\\left(1- \\frac{Q_Z}{2} - Q_X\\right)+ \\left(Q_X-\\frac{Q_Z}{2}\\right) \\log _2\\left(Q_X-\\frac{Q_Z}{2}\\right) \\\\& + (1-Q_Z) (1 - \\log _2(1-Q_Z))- h(\\max _{1\\le i \\le N-1}Q_{AB_i}).\\end{aligned}$ Note that the parameters in this equation are obtained from the measured data and will depend on the number of parties $N$ ."
],
[
"Implementation and noise",
"In this section we compare the multipartite-entanglement-based protocol (NQKD) as introduced above to a protocol based on bipartite entanglement (2QKD), which we define in the following."
],
[
"Conference key distribution with bipartite entangled quantum states (2QKD)",
"A suitable protocol to establish a secret joint key for $N>2$ parties via bipartite entanglement proceeds as follows, see Fig.",
"REF : Party $A$ shares a Bell state with each of the $N-1$ parties $B_i$ and establishes a (different) secret bipartite key ${\\bf S}_i$ with each party $B_i$ .",
"For concreteness, we assume in our comparison that the six-state protocol [8] is used.",
"In general, the $N-1$ channels may be different and thus have individual QBERs.",
"Party $A$ then defines a new random key ${\\bf k}_c$ to be the conference key.",
"She sends the encoded conference key ${\\bf k}_i = {\\bf S}_i \\oplus {\\bf k}_c$ to party $B_i$ who performs ${\\bf k}_i \\oplus {\\bf S}_i = {\\bf k}_c$ and thus regains the conference key.",
"A comparison of the performance of the bipartite versus the multipartite entanglement-based strategy for $N$ parties is subtle and has to consider various aspects, as different resources are needed: on one hand only bipartite entanglement is needed for 2QKD, while multipartite entanglement is needed for NQKD.",
"(Note, however, that the number of necessary two-qubit gates for generation of the entangled states is in both cases $N-1$ .)",
"On the other hand, the number of resource qubits per round is $2(N-1)$ for 2QKD, while only $N$ qubits are needed for NQKD.",
"Finally, the 2QKD protocol requires to transmit $(N-1)$ additional classical bits (the encoded conference key).",
"Thus, each of the two strategies has its own advantages.",
"A quantitative comparison regarding imperfections in preparation and transmission is discussed below."
],
[
"Implementation of the NQKD protocol",
"We now describe how the depolarisation operations used in the security proof can effectively be implemented classically by adjusting the protocol.",
"For key generation and the $Q_Z$ estimation, the parties perform $Z^{\\otimes N}$ -measurements.",
"These are only affected by the $X^{\\otimes N}$ depolarisation operator, which flips the outcomes of all parties.",
"It can therefore be implemented on the classical data.",
"The other depolarisation operators are diagonal in the $Z$ -basis and thus do not change the $Z$ -measurement outcome.",
"Let us call the parameter estimation rounds, in which the parties measure $X^{\\otimes N}$ (after depolarisation), estimation rounds of the second type.",
"How the depolarisation step affects the $X^{\\otimes N}$ -measurement is not so obvious and is described in the following.",
"Note that the depolarisation operators $X^{\\otimes N}$ and $Z_AZ_{B_k}$ , $k=1,2,...,N-1$ (see Eq.",
"(REF )), commute with the $X^{\\otimes N}$ -measurement and thus these depolarisation operators do not have an effect in second type rounds.",
"But $R_k X_A X_{B_k} R_k^\\dagger = (-Y_A) Y_{B_k}$ i.e.",
"applying the depolarisation operator $R_k$ is equivalent to Bob $k$ measuring in $Y$ -basis.",
"Also note that $R_k (-Y_A) X_{B_k} R_k^\\dagger = (-X_A) Y_{B_k},$ so let $\\kappa _j$ be the number of Bobs measuring in $Y$ -basis in the $j$ -th round, then Alice measures in the basis $M_A(\\kappa ) = \\left\\lbrace \\begin{array}{cl}X_A & \\text{if } \\kappa _j \\bmod 4 = 0\\\\-Y_A & \\text{if }\\kappa _j \\bmod 4 = 1\\\\-X_A & \\text{if }\\kappa _j \\bmod 4 = 2\\\\Y_A & \\text{if }\\kappa _j \\bmod 4 = 3\\end{array}\\right.",
",$ where a minus sign corresponds to a flip of the measurement outcome.",
"Note that this measurement rule for Alice implies that always an even number of parties measures in $Y$ -basis and that the outcome of the measurement is flipped whenever it is not a multiple of four.",
"Each party measures in $X$ or $Y$ basis with probability $1/2$ .",
"Note that the rule for $M_A$ described above means that only half of all possible combinations of these measurement bases are actually measured.",
"However, in practice the parties can measure $X$ and $Y$ independently with probability $1/2$ and throw away half of their data (where an odd number of parties has measured in $Y$ -basis) and Alice still flips her measurement outcome whenever the number of parties measuring in $Y$ -basis was not a multiple of four.",
"This is not a problem, because in the parameter estimation rounds each party announces its measurement setting and outcome.",
"We thus arrive at the implementation described initially.",
"Let $\\tilde{\\kappa }_j$ be the number of parties measuring in $Y$ -basis in run $j$ , i.e.",
"$\\tilde{\\kappa }_j = \\left\\lbrace \\begin{array}{cl}\\kappa _j + 1 & \\text{if Alice measured in $Y$-basis}\\\\\\kappa _j & \\text{else}\\end{array} \\right.,$ then $\\langle X^{\\otimes N} \\rangle _{\\mathrm {dep}} =&\\lim _{\\text{\\#exp}\\rightarrow \\infty } \\frac{1}{\\text{\\#exp}} \\sum _{j=1}^{\\text{\\#exp}} f(\\tilde{\\kappa }_j) \\prod _{i=1}^N a_{i,j} \\\\=&\\lim _{\\text{\\#exp}\\rightarrow \\infty } \\frac{n_+-n_-}{n_++n_-} , $ where #exp is the number of experiments in the second type rounds with even $\\tilde{\\kappa }_j$ , $a_{i,j}$ is the outcome of party $i$ in experiment $j$ , $f(\\tilde{\\kappa })=\\left\\lbrace \\begin{array}{cl}0 & \\text{if $\\tilde{\\kappa }_j$ odd}\\\\1 & \\text{if }\\tilde{\\kappa }_j \\bmod 4 = 0\\\\-1 & \\text{else}\\end{array}\\right.",
"$ and $n_{\\pm } = \\frac{1}{2} \\text{\\#exp}\\pm \\frac{1}{2}\\sum _{j=1}^{\\text{\\#exp}} f(\\tilde{\\kappa })\\prod _{i=1}^N a_{i,j}.$ We remark that, in contrast to full tomography, the number of rounds needed to get sufficient statistics for estimating $\\langle X^{\\otimes N}\\rangle _{\\mathrm {dep}}$ does not increase with the number of parties $N$ .",
"Let us summarise the steps of an implementation of the NQKD protocol: Distribution of the state GHZ state $| \\psi _0^+ \\rangle $ .",
"$L\\cdot h(p_p)$ bits of pre-shared key are used to mark the second type rounds, where $L$ is the total number of rounds and $p_p$ is the probability for an $X^{\\otimes N}$ -round.",
"This amount of key suffices, because an $L$ -bit binary string with a 1 for each second type round can asymptotically be compressed to $L\\cdot h(p_p)$ bits.",
"In each second type round each party measures randomly in the X- or $Y$ -basis.",
"In all other cases all parties measure in $Z$ -direction.",
"Parameter estimation: Alice announces a randomly chosen small subset of size $L\\cdot h(p_p)$ of $Z$ -measurement rounds, in which all parties announce their $Z$ -measurement results.",
"From this data the QBER $Q_Z$ and the individual QBER's $Q_{AB_i}$ are estimated.",
"The parties announce the measurement results of the second type rounds together with the chosen measurement basis.",
"Alice flips her outcome if the number of parties who measured in $Y$ -basis is not a multiple of four (see Eq.",
"(REF )).",
"From the data where an even number of parties measured in $Y$ -basis (including zero), the parameter $Q_X$ is calculated according to Eq.",
"(REF ).",
"Alice announces which $Z$ -measurement results all parties have to flip (the probability for each bit is $1/2$ ).",
"This effectively implements the depolarisation with operator $X^{\\otimes N}$ .",
"Classical post-processing: Alice sends error correction information (for $\\max _i Q_{AB_i}$ ) to all Bobs, which perform the error correction.",
"In privacy amplification the parties obtain the key by applying a two-universal hash function, which was chosen randomly by Alice, to the error corrected data.",
"The achievable key rate is then given by Eq.",
"(REF )."
],
[
"Example of depolarising noise",
"In this section we assume that $\\rho _{AB_1...B_{N-1}}$ is a mixture of the GHZ-state and white noise, i.e.",
"the parties share the state $\\rho = \\lambda _0^+ \\left| \\psi _0^+\\right>\\left< \\psi _0^+\\right| + \\frac{1-\\lambda _0^+}{2^N-1} ({1}-\\left| \\psi _0^+\\right>\\left< \\psi _0^+\\right|).",
"$ Here all coefficients other than $\\lambda _0^+$ are equal, i.e.",
"$\\lambda _j^\\pm = \\lambda _0^- =Q_Z/(2^N-2)$ for $j=1,..., 2^{(N-1)}-1$ and $\\lambda _0^+ = 1-Q_Z\\frac{2^N-1}{2^N-2}$ .",
"The rate of unexpected results for the $X^{\\otimes N}$ -measurement is thus given by $Q_X=\\frac{2^{N-2}}{2^{N-1}-1}Q_Z.$ For the highly symmetric state of Eq.",
"(REF ) the key rate is then a function of $Q_Z$ and $N$ only.",
"The terms in Eq.",
"(REF ) are $Q_{AB_i}=& \\frac{2^{N-1}}{2^N-2} Q_Z,\\\\S(E|U)=& -(1-Q_Z) \\log _2(1-Q_Z) - Q_Z \\log _2 \\frac{2Q_Z}{2^N-2}\\\\\\text{and }S(E)=&-(1-Q_Z\\frac{2^N-1}{2^N-2})\\log _2 (1-Q_Z\\frac{2^N-1}{2^N-2})\\nonumber \\\\& - (2^N-1)\\frac{Q_Z}{2^N-2} \\log _2 (\\frac{Q_Z}{2^N-2})$ and inserting them into Eq.",
"(REF ) leads to the asymptotic secret key rate as function of $Q=Q_Z$ and $N$ , namely $R(Q,N)=1 + h(Q)-h\\left(Q \\frac{2^N - 1}{2^N - 2}\\right) - h\\left(Q \\frac{2^{N - 1}}{2^N - 2}\\right)+ \\left(\\log _2(2^{N - 1} - 1) - \\frac{2^N - 1}{2^N - 2} \\log _2 (2^N - 1)\\right) Q.$ This function is shown in Fig.",
"REF .",
"Figure: Key rates for N=2,3,4,...,8N=2,3,4,...,8 parties (right to left) as a function of the two-qubit gate failure probability f G f_G.For $N=2$ the key rate coincides with the one of the six-state protocol [8], [33], namely $R(Q,2)= 1-h\\left(\\frac{3}{2}Q\\right)-\\frac{3 \\log _2 3}{2} Q.$ In the limit of large $N$ the key rate simplifies to $R(Q,\\infty )=1-h\\left(\\frac{Q}{2}\\right)-Q.",
"$ We also numerically determined the threshold values for the QBER, i.e.",
"the value of $Q$ until which a non-zero secret key rate is achievable, for different numbers of parties $N$ , see Table REF .",
"Table: Threshold values of the multipartite entanglement based protocol (NQKD) without preprocessing noise for different numbers of Parties NN.The well-known bipartite case, i.e.",
"N=2N=2, is also given for comparison.A non-zero secret key can be distilled if the QBER is below the listed value.Note that for fixed $Q$ the key rate increases with the number of parties $N$ .",
"However, one might expect that in practice the QBER is not constant but increases with increasing number of parties $N$ (because the experimental creation of the $N$ -partite GHZ state becomes more demanding).",
"This intuition is discussed quantitatively in the following section."
],
[
"Noisy gates and channels",
"Let us compare the performance of NQKD and 2QKD when using imperfect two-qubit gates in the production of the entangled resource states.",
"We employ, for both 2QKD and NQKD, the model of depolarising noise, i.e.",
"if a two-qubit gate fails, which happens with probability $f_G$ , then the two processed qubits are traced out and replaced by the completely mixed state.",
"When the GHZ resource state is produced in the network of Fig.",
"REF , Alice starts with the state $| + \\rangle _A| 0 \\rangle ^{\\otimes N-1}$ and applies a controlled-NOT gate from $A$ to each of the other qubits.",
"The secret key rate is shown in Fig.",
"REF as a function of the gate error rate $f_G$ .",
"It captures the expectation that the demands on the gates for producing an $N$ -party GHZ state increase with the number of parties $N$ .",
"We mention that the GHZ state could also be produced using a single multi-qubit gate, e.g.",
"$C_{X^{\\otimes (N-1)}}=\\left| 0\\right>\\left< 0\\right|\\otimes {1}+\\left| 1\\right>\\left< 1\\right|\\otimes X^{\\otimes (N-1)}$ , which is locally equivalent to the controlled-Phase gate, see e.g. [43].",
"The QBER caused by this gate is $Q=\\frac{f_G}{2}$ .",
"Because the threshold $Q$ increases with $N$ (cf.",
"Fig.",
"REF ), so does the threshold gate failure probability in this case.",
"In addition to imperfect gates, noise might be introduced by the transmission channel.",
"Consider, for example, the situation when the qubit of each Bob is individually affected by a depolarising channel.",
"Let the probability of depolarisation be $f_C$ , then the QBER is $Q(f_C)=\\frac{2^N-2 }{2^N}\\left(1-(1-f_C )^N\\right) $ and the key rate can be calculated according to Eq.",
"(REF )."
],
[
"Quantum key distribution in networks",
"We will now show that in quantum networks with constrained channel capacity and with quantum routers, employing multipartite entanglement leads to a higher secret key rate than bipartite entanglement, when the gate quality is higher than a threshold value.",
"Beyond the simple network of Fig.",
"REF , the GHZ resource state can be distributed in many different networks.",
"Consider a fixed but general network as given via a graph with vertices and directed edges.",
"Let all channels have the same transmission capacity (also called bandwidth), which is associated with the direction of the corresponding edge.",
"For the sake of a simple presentation, we assume that this transmission capacity is one qubit per second.",
"Thus, the time $t_{\\mathrm {rep}}$ consumed in one round (steps 1 and 2 of the protocol) is proportional to the number of network uses in that round.",
"A generic network has some bottlenecks.",
"In this case the difference between the NQKD and 2QKD protocol becomes evident: Alice may send a single qubit in the NQKD scheme, while she has to transmit $N-1$ qubits in the 2QKD case.",
"As an example consider the quantum network where all parties are connected to a single central router $C$ , see Fig.",
"REF .",
"Figure: This quantum network with a central router CC, which is able toproduce and entangle qubits, exemplifies a network with a bottleneck.",
"The GHZ-like resource state used in the multipartite entanglement QKD protocol, see Eq.",
"(), can be distributed in a single use of the depicted network (i.e.",
"each channel transmits a single qubit only) , while N-1N-1 uses of the network are necessary in the 2QKD protocol.Because $C$ is not trusted we assume it to be in the control of Eve.",
"In this network the channel from $A$ to $C$ constitutes a bottleneck.",
"Note, however, that this network can be much more economical than the one of Fig.",
"REF if the distance between $A$ and $C$ is large.",
"The 2QKD protocol needs $N-1$ network uses, i.e.",
"$t_{\\mathrm {rep}}^{\\mathrm {(2QKD)}}=(N-1) \\,\\mathrm {s}$ , to distribute the Bell pairs.",
"In contrast to this the NQKD protocol can employ the quantum network coding [44], [45], [46], [47], [48], [49], [50] scheme of reference [27] to distribute the GHZ state in a single network use, i.e.",
"$t_{\\mathrm {rep}}^{\\mathrm {(NQKD)}}=1\\,\\mathrm {s}$ .",
"See Appendix for the explicit calculation.",
"Thus the key rate of the NQKD protocol is $(N-1)$ times larger than the one of the 2QKD protocol in the ideal case ($r_\\infty =1$ ).",
"When again using noisy two-qubit gates (the QBER calculation is analogous to the case of the network shown in Fig REF discussed above), the QBER for the NQKD protocol increases with $N$ .",
"These two effects lead to gate error thresholds below which the NQKD protocol outperforms 2QKD, see Fig.",
"REF .",
"Figure: Transmission noise, see Eq.",
"().For a fixed number of parties $N$ there is a maximal gate error probability below which the NQKD protocol outperforms the bipartite approach in the quantum network of Fig.",
"REF .",
"For $N=3$ already gate failure rates below $7.2\\,\\%$ imply that NQKD outperforms 2QKD.",
"More values are listed in the Appendix .",
"The exact same behavior can be observed when considering noisy channels.",
"In the ideal case NQKD outperforms 2QKD, while NQKD is more prone to channel noise.",
"The resulting threshold noise levels are shown in Fig.",
"REF .",
"We mention that the famous butterfly network [44] leads to a similar advantage, see Appendix for details."
],
[
"Conclusion",
"In this paper we analysed a quantum conference key distribution (QKD) protocol for $N$ parties which is based on multipartite entangled resource states.",
"We generalised the information theoretic security analysis of [16] to this $N$ -partite scenario.",
"Using the depolarisation method we derived an analytical formula for the secret key rate as a function of the quantum bit error rate (QBER).",
"For a fixed QBER the secret key rate is found to increase with the number of parties.",
"Accordingly, the threshold QBER until which a non-zero secret key can be obtained increases with the number of parties.",
"Furthermore, we presented an example where multipartite entanglement-based QKD outperforms the approach based on bipartite QKD links.",
"We found this advantage in networks with bottlenecks and showed that it holds above a certain threshold gate quality which depends on the number of parties.",
"We expect more interesting insights from analysing further aspects of the multipartite entanglement-based QKD protocol.",
"Regarding implementations the secret key calculation of the protocol for finite numbers of rounds will be beneficial.",
"Various examples of network layouts and the link to network coding schemes will deserve more detailed investigations.",
"Acknowledgments We acknowledge helpful discussions with Jan Börker and Norbert Lütkenhaus.",
"This work was financially supported by BMBF (network Q.com-Q) and ARL."
],
[
"The resource state and its properties",
"In this section we derive the form of a pure quantum state that fulfils the requirements of perfect correlations for one set of local measurement bases, with uniformly distributed random measurement outcomes.",
"(These local bases are used for the key generation.)",
"We also prove properties of the resource state regarding correlations of measurement outcomes in any other set of local bases.",
"A general normalized $N$ -qubit state reads $| \\phi \\rangle = \\sum _{i_1,i_2,...i_N=0}^1 a_{i_1,i_2,...i_N}| i_1,i_2,...i_N \\rangle \\ ,$ with complex coefficients $a_{i_1,i_2,...i_N}$ that satisfy $\\sum _{i_1,i_2,...i_N=0}^1 |a_{i_1,i_2,...i_N}|^2=1$ .",
"To achieve perfect correlations, we can assume without loss of generality that all parties measure in the $Z$ -basis and get the same outcome, as the choice of another local basis corresponds to a local rotation, and an opposite outcome could be flipped locally.",
"The requirement of perfect correlations in the $Z$ -basis is only fulfilled by a quantum correlated state of the form $| \\phi _{corr} \\rangle = a_{0,...,0}| 0,...,0 \\rangle +a_{1,...,1}| 1,...,1 \\rangle \\ .$ It turns out that this requirement of perfect correlations in one set of local bases forbids perfect correlations, even only pairwise, in any other local bases, for all $N\\ge 3$ .",
"Theorem 1 For $N$ qubits, with $N\\ge 3$ , the state $| \\phi _{corr} \\rangle = a_{0,...,0}| 0,...,0 \\rangle +a_{1,...,1}| 1,...,1 \\rangle $ leads to perfect classical correlations between any number of parties, if and only if each of them measures in the $Z$ -basis.",
"Measuring in the $Z$ -basis, perfect correlations follow trivially.",
"For the reverse implication, let us denote the direction of measurement for party $i$ by the vector $\\vec{M_i}$ , with components $M_{i}^{x}, M_{i}^{y}$ and $M_{i}^{z}$ .",
"An observable ${\\cal M}_{ij}$ of two parties $i$ and $j$ is given by ${\\cal M}_{ij} = (\\vec{M_i}\\cdot \\vec{\\sigma }) \\otimes (\\vec{M_j}\\cdot \\vec{\\sigma })= \\sum _{\\alpha ,\\beta \\in \\lbrace x,y,z\\rbrace }M_{i}^{\\alpha }M_{j}^{\\beta }\\sigma _i^\\alpha \\otimes \\sigma _j^\\beta ,$ where $\\vec{\\sigma }$ denotes the vector of Pauli matrices and the identity operators for the parties $\\ne i,j$ are omitted.",
"Observe that $\\langle \\phi _{corr} |\\sigma _i^\\alpha \\otimes \\sigma _j^\\beta | \\phi _{corr} \\rangle = 0 \\ \\ \\text{unless} \\ \\ \\alpha =\\beta =z ,$ because all other combinations of Pauli operators change $| \\phi _{corr} \\rangle $ to an orthogonal state.",
"Denoting by $p_i^{\\alpha }(\\pm )$ the probability that party $i$ finds eigenvalue $\\pm 1$ when measuring $\\sigma ^\\alpha $ , we also have $\\langle \\phi _{corr} |\\sigma _i^\\alpha \\otimes \\sigma _j^\\beta | \\phi _{corr} \\rangle = 2[p_i^{\\alpha }(+)p_j^{\\beta }(+)+p_i^{\\alpha }(-)p_j^{\\beta }(-)] -1$ , and thus $p_i^{\\alpha }(+)p_j^{\\beta }(+)+p_i^{\\alpha }(-)p_j^{\\beta }(-) \\ne 1$ , unless $\\alpha = \\beta = z$ .",
"Therefore, perfect correlations between two parties are not possible in any other than the $Z$ -basis.",
"This also excludes perfect correlations between any other number of parties.",
"- Note that the above argument, in particular Eq.",
"(REF ), does not hold for $N=2$ , which is special.",
"Thus, any state of the form (REF ) contains the resource of perfect multipartite correlations in the local $Z$ -bases.",
"In order to ensure uniformity of the outcome, i.e.",
"randomness of the resulting secure bit string, we choose for the key generation protocol $|a_{0,...,0}| = 1/\\sqrt{2} =|a_{1,...,1}|$ , i.e.",
"the unique perfect resource is a GHZ state [34]."
],
[
"Security analysis of the NQKD protocol",
"In this appendix we generalise the composable security definition of the bipartite scenario [16], [17] to the $N$ -partite case.",
"As mentioned in the main text, the security analysis proceeds along analogous lines as the bipartite case in [33], [51].",
"We assume that the parties $A$ and $B_i$ , for $i=1,...,N-1$ share $n$ multipartite states.",
"The eavesdropper $E$ is supposed to hold a purification of the global state.",
"The total quantum state after $Z$ -measurement of $A$ and all $B_i$ is described by the density operator $\\begin{aligned}\\rho ^n_{{\\bf K K_1...K_{N-1}}E} =& \\sum _{\\bf {x,x_1,...,x_{N-1}}}P_{{\\bf K K_1...K_{N-1}}}(\\bf {x,x_1,...,x_{N-1}}) \\\\&\\left| \\bf {x}\\right>\\left< \\bf {x}\\right|\\otimes \\bigotimes _{i=1}^{N-1}\\left| \\bf {x_i}\\right>\\left< \\bf {x_i}\\right|\\otimes \\rho _E^{\\bf x,x_1,...x_{N-1}} \\ ,\\end{aligned}$ where ${\\bf x}$ and ${\\bf x_i}$ describe the classical strings of parties A and $B_i$ , respectively.",
"Note that the classical post-processing is identical to the bipartite case: In an error correction step the parties transform their only partially correlated raw data into a fully correlated shorter string.",
"Party A pre-processes her random string ${\\bf K}$ according to the channel ${\\bf U}\\leftarrow {\\bf K}$ and sends classical error correction information ${\\bf W}$ to parties $B_i$ , who compute their respective guesses ${\\bf U_i}$ for ${\\bf U}$ from ${\\bf K_i}$ and ${\\bf W}$ .",
"The error correction information ${\\bf W}$ is the same for all Bobs, thus there is no additional information leakage compared to the bipartite case.",
"In a second step, the privacy amplification, Party A randomly chooses $f$ from a two-universal family of hash functions, computes her key ${\\bf S_A}= f({\\bf U})$ and sends the description of $f$ to all parties $B_i$ who also perform the privacy amplification to arrive at their respective keys ${\\bf S_{B_i}}=f({\\bf U_i})$ .",
"The total quantum state will then be denoted as $\\rho _{{\\bf S_A S_{B_1}...S_{B_{N-1}}}E`}$ .",
"The key tuple (${\\bf S_A, S_{B_1},..., S_{B_{N-1}}}$ ) is called $\\epsilon $ -secure, if it is $\\epsilon $ -close to the ideal state, i.e.",
"if $\\delta (\\rho _{{\\bf S_A S_{B_1}...S_{B_{N-1}}}E`}, \\rho _{\\bf SS...S}\\otimes \\rho _{E`}) \\le \\epsilon \\ ,$ where $\\delta (\\rho ,\\sigma )= \\text{tr}|\\rho - \\sigma |/2$ denotes the trace distance.",
"Note that we have not assumed any symmetry about the quality of the channels connecting A and $B_i$ .",
"The information leaking to the eavesdropper in the error correction step is determined by the amount of error correction information which the Bob with the noisiest channel requires.",
"This is the main difference with respect to the bipartite case.",
"Therefore we arrive at the following key length $\\ell ^{(n)}$ , generated from $n$ multipartite entangled states, in analogy to [33], [51]: $\\ell ^{(n)} = \\sup _{{\\bf U} \\leftarrow {\\bf K}}[S_2^\\epsilon ({\\bf U} E)-S_0^\\epsilon (E) - \\max _{i\\in \\lbrace 1,...N-1\\rbrace } H_0^\\epsilon ({\\bf U}|{\\bf K}_i)]\\ ,$ where the smooth Rényi entropy $S_\\alpha ^\\epsilon $ is defined as $S_{\\alpha }^{\\epsilon }(\\rho )=\\frac{1}{1-\\alpha } \\log _2 (\\inf _{\\sigma \\in \\mathbf {B}^\\epsilon (\\rho )}\\operatorname{tr}(\\sigma ^\\alpha )),$ which for $\\alpha \\in \\lbrace 0,\\infty \\rbrace $ is to be understood as $S_\\alpha ^\\epsilon (\\rho )=\\lim _{\\beta \\rightarrow \\alpha } S_\\beta ^\\epsilon (\\rho )$ .",
"The infimum is to be taken over all states $\\sigma $ in a ball with radius $\\varepsilon $ (w.r.t.",
"the trace distance) around $\\rho $ , denoted as $\\mathbf {B}^\\epsilon (\\rho )$ .",
"For a (classical) probability distribution $P$ the smooth Rényi entropy is $H_\\alpha ^\\epsilon (P)=\\frac{1}{1-\\alpha } \\inf _{\\begin{array}{c}Q\\\\\\bar{\\delta }(Q,P)\\le \\epsilon \\end{array}} \\log _2(\\sum _z Q(z)^\\alpha ),$ where the infimum is taken over all probability distributions $\\epsilon $ -close to $P$ in the sense of the statistical distance $\\bar{\\delta }$ (the classical analogon of the trace distance).",
"The conditional smooth Rényi entropy is $H_0^\\epsilon (P_X|P_Y)=\\max _y H_0^\\epsilon (P_{X|Y=y}).$ Note that, differently to the bipartite case [33], the worst of the $N-1$ channels influences the key length via the maximal leakage to the eavesdropper in the error correction step, see the last term of Eq.",
"(REF ).",
"In the following the symbols $K$ , $K_i$ and $U$ denote the single bit random variables corresponding to the respective bold-face strings.",
"For the limit $n\\rightarrow \\infty $ the secret fraction $r$ is given by $r_{\\infty } &=&\\lim _{n \\rightarrow \\infty }\\frac{\\ell ^{(n)}}{n} \\nonumber \\\\&=&\\sup _{ U \\leftarrow K}\\inf _{\\sigma _{A\\lbrace B_i\\rbrace } \\in \\Gamma }[S (U|E) - \\max _{i\\in \\lbrace 1,...N-1\\rbrace } H( U|K_i)]\\ , \\nonumber \\\\&&$ where $S(U|E)$ is the conditional von Neumann entropy, $H(U|K)$ is the conditional Shannon entropy and $\\Gamma $ is the set of all density matrices of Alice and the Bobs which are consistent with the parameter estimation."
],
[
"Details for the network coding example",
"Here we explicitly describe the distribution of the GHZ state in the network of Fig.",
"REF .",
"This is a special case of the quantum network coding scheme which some of the authors described in [27].",
"Let $| + \\rangle =\\frac{1}{\\sqrt{2}}(| 0 \\rangle +| 1 \\rangle )$ and $| - \\rangle =\\frac{1}{\\sqrt{2}}(| 0 \\rangle -| 1 \\rangle )$ .",
"Alice produces two qubits $C$ and $A$ , each in the state $| + \\rangle $ .",
"She then applies a controlled-Phase gate $C_Z=\\left| 0\\right>\\left< 0\\right|\\otimes {1} + \\left| 1\\right>\\left< 1\\right|\\otimes Z$ to produce the Bell state $| \\includegraphics [width=6ex]{bellpair} \\rangle _{CA}=\\frac{1}{\\sqrt{2}}(| 0+ \\rangle +| 1- \\rangle ).$ Alice sends the qubit $C$ to the router station.",
"The router produces $(N-1)$ qubits $B_i$ , $i=1,2,...,N-1$ , in the state $| + \\rangle $ and entangles each of them with the qubit $C$ using $(N-1)$ $C_Z$ gates.",
"At this stage the total state is $| \\psi _C \\rangle =&\\frac{1}{\\sqrt{2}}(| 0+...+ \\rangle +| 1-...- \\rangle )\\\\=&\\frac{1}{\\sqrt{2}}(| + \\rangle _C| GHZ^{\\prime } \\rangle +| - \\rangle _CX_{B_1}| GHZ^{\\prime } \\rangle ),$ where $| GHZ^{\\prime } \\rangle =&\\frac{1}{\\sqrt{2}}(| ++...+ \\rangle +| --...- \\rangle )$ is the GHZ state in the $X$ -basis.",
"The router measures $C$ in $X$ basis.",
"If the outcome is $-1$ , i.e.",
"$| - \\rangle _C$ , then it applies $X_{B_1}$ .",
"The state is now $| \\pm \\rangle _C| GHZ^{\\prime } \\rangle $ .",
"The router now distributes the qubits $B_1$ , $B_2$ , ..., $B_{N-1}$ to the corresponding parties.",
"Up to a local basis choice (Hadamard gate), the resource state of the main text has been distributed and the multipartite entanglement based quantum key distribution (NQKD) protocol can be performed.",
"To see that it is impossible to create $N-1$ Bell pairs by sending a single qubit from Alice to the router, let us group the router and all Bobs into a single party $B$ .",
"When Alice sends one qubit across the channel, the entropy of entanglement $E_{A|B}\\le 1$ .",
"The $N-1$ Bell pairs, however, have entropy of entanglement $E_{A|B}=N-1$ , so they cannot be created from the received state by local operations on $B$ .",
"Instead, $N-1$ network uses are necessary and the key rate decreases accordingly."
],
[
"Gate error rates and the QBER",
"In this appendix we give details for the key rate calculations regarding the quantum networks of Figs.",
"REF and REF with imperfect gates.",
"We start with the simple network of Fig.",
"REF .",
"The GHZ resource state is prepared as follows.",
"Alice starts with the state $| + \\rangle _A| 0 \\rangle ^{\\otimes N-1}$ and applies a controlled-Not gate from $A$ to each of the other qubits, see Fig.",
"REF .",
"Figure: The GHZ state that is to be distributed across the network of Fig.",
"can be produced by Alice using controlled-Not gates as depicted in this quantum circuit diagram.When a controlled-Not gate acts on qubits $i$ (control) and $j$ (target) we denote it by $C_X^{(i,j)}=(\\left| 0\\right>\\left< 0\\right|_i \\otimes {1}_j+\\left| 1\\right>\\left< 1\\right|_i \\otimes X_j)\\otimes {1}_{\\mathrm {rest}},$ where $X=| 0 \\rangle \\langle 1 |+| 1 \\rangle \\langle 0 |$ is a Pauli matrix.",
"We use a depolarizing noise model for the gate errors.",
"The action of the imperfect gate on the density matrix is $C_{X,f_G}^{(i,j)}(\\rho )=& (1-f_G) C_X^{(i,j)} \\rho C_X^{(i,j)} + f_G \\operatorname{tr}_{i,j}(\\rho ) \\otimes {1}_{ij}\\\\=&(1-f_G) C_X^{(i,j)} \\rho C_X^{(i,j)} + \\sum _{a,b\\in \\sigma } a_i b_j \\rho a_i b_j,$ where $\\sigma =\\lbrace {1},X,Y,Z\\rbrace $ contains Pauli matrices.",
"It will be convenient to extend the notation of the GHZ basis to include the number of parties as a subscript, i.e.",
"$| \\psi _{j,N}^\\pm \\rangle = \\frac{1}{\\sqrt{2}}(| 0 \\rangle _{1}| j \\rangle _{2...N}\\pm | 1 \\rangle _{1}| \\bar{j} \\rangle _{2...N}).",
"$ The initial state is $\\rho _{\\mathrm {in}} = \\left| \\psi _{0,1}^+\\right>\\left< \\psi _{0,1}^+\\right| \\otimes (\\left| 0\\right>\\left< 0\\right|)^{\\otimes (N-1)}.$ The first gate turns it into $\\rho _{1,\\mathrm {out}} = ((1-f_G) \\left| \\psi _{0,2}^+\\right>\\left< \\psi _{0,2}^+\\right|+f_G \\frac{{1}}{4})\\otimes (\\left| 0\\right>\\left< 0\\right|)^{\\otimes N-2},$ the second into $\\rho _{2,\\mathrm {out}} = &((1-f_G)^2 \\left| \\psi _{0,3}^+\\right>\\left< \\psi _{0,3}^+\\right|\\nonumber \\\\+&(1-f_G) f_G \\frac{1}{2} (\\left| 0\\right>\\left< 0\\right|\\otimes \\frac{{1}}{2}\\otimes \\left| 0\\right>\\left< 0\\right|+\\left| 1\\right>\\left< 1\\right|\\otimes \\frac{{1}}{2}\\otimes \\left| 1\\right>\\left< 1\\right|)\\nonumber \\\\+&f_G \\frac{{1}}{8})\\otimes (\\left| 0\\right>\\left< 0\\right|)^{\\otimes N-3},$ and the third into $\\rho _{3,\\mathrm {out}}=&((1-f_G)^3\\left| \\psi _{0,4}^+\\right>\\left< \\psi _{0,4}^+\\right|\\nonumber \\\\&+f_G(1-f_G)^2 \\frac{1}{2}(\\frac{{1}}{2}\\otimes \\left| 00\\right>\\left< 00\\right| \\otimes \\frac{{1}}{2}+\\frac{{1}}{2}\\otimes \\left| 11\\right>\\left< 11\\right| \\otimes \\frac{{1}}{2})\\nonumber \\\\&+(1-f_G)^2 f_G \\frac{1}{2} (\\left| 0\\right>\\left< 0\\right|\\otimes \\frac{{1}}{2}\\otimes \\left| 00\\right>\\left< 00\\right|+\\left| 1\\right>\\left< 1\\right|\\otimes \\frac{{1}}{2}\\otimes \\left| 11\\right>\\left< 11\\right|)\\nonumber \\\\&+(1-f_G) f_G \\frac{1}{2}(\\left| 0\\right>\\left< 0\\right|\\otimes \\frac{{1}}{4} \\otimes \\left| 0\\right>\\left< 0\\right|+\\left| 1\\right>\\left< 1\\right|\\otimes \\frac{{1}}{4} \\otimes \\left| 1\\right>\\left< 1\\right|)\\nonumber \\\\&+(2-f_G)f_G^2 \\frac{{1}}{16}) \\otimes (\\left| 0\\right>\\left< 0\\right|)^{\\otimes N-4}.$ One may deduce the following observation.",
"Let us denote the pattern of actual gate successes/failures as the binary representation of an $(N-1)$ -bit number $\\mathbf {x}$ , where a 0 at position $i$ indicates the failure of gate $i$ and 1 means the corresponding gate was successful.",
"The number of connected blocks of ones in the bit string $\\mathbf {x}1$ plus the number of zeros, $b(\\mathbf {x}1)$ , is the number of subsets of parties that are correlated amongst each other.",
"This gives the prefactor $c_\\mathbf {x}=\\left\\lbrace \\begin{array}{cl}1 & \\text{if } \\mathbf {x}=11...1\\\\2^{-b(\\mathbf {x}1)} &\\text{else } \\end{array}\\right.$ in front of the corresponding term in $\\rho $ .",
"These prefactors determine the overlap between $| \\psi _{j,0}^{\\pm } \\rangle \\langle \\psi _{j,0}^{\\pm } |$ and $\\rho $ , i.e.",
"the coefficients $\\lambda _0^\\pm (f_G)$ of $\\rho $ in the GHZ basis.",
"They read $\\begin{aligned}\\lambda _0^+(f_G)=&\\sum _{\\mathbf {x}=0}^{2^{N-1}-1} c_\\mathbf {x} f_G^{N-1-|\\mathbf {x}|_H} (1-f_G)^{|\\mathbf {x}|_H}\\\\\\text{and\\hspace{22.76228pt}}\\lambda _0^-(f_G)=&\\lambda _0^+-(1-f_G)^{N-1},\\end{aligned}$ where $|\\mathbf {x}|_H$ is the Hamming weight of $\\mathbf {x}$ .",
"After some combinatorics $\\sum _{\\mathbf {x}} c(\\mathbf {x})$ for a given weight $w=|\\mathbf {x}|_H$ (unequal to $N-1$ ) can be expressed in a more compact form by summing over all possible “subset counts” $\\beta $ as $\\sum _{\\begin{array}{c}\\mathbf {x}\\\\|\\mathbf {x}|_H=w\\end{array}} c(\\mathbf {x})= \\sum _{\\beta =N-w}^{N}{w \\atopwithdelims ()N-\\beta } {N-w-1 \\atopwithdelims ()\\beta -N+w} 2^{-\\beta },$ which leads to the relevant coefficients in the GHZ basis, $\\begin{aligned}\\lambda _0^-(f_G)=&\\sum _{w=0}^{N-2} c^{\\prime }(w) f_G^{N-1-w} (1-f_G)^{w}\\\\\\text{and }\\lambda _0^+(f_G)=&(1-f_G)^{N-1}+\\lambda _0^{-}(f_G)\\end{aligned}$ with $c^{\\prime }(w)=\\sum _{\\beta =N-w}^{N}{w\\atopwithdelims ()N - \\beta } {N - w - 1\\atopwithdelims ()\\beta - N + w} 2^{-\\beta }.$ From Eq.",
"(REF ) one obtains the QBER using Eq.",
"(REF ).",
"We show it in Fig.",
"REF .",
"Figure: The QBER as a function of f G f_G for the circuits described in the text, with N=2,3,4,...,8N=2,3,4,...,8 (bottom to top).The secret key rate is calculated using Eq.",
"(REF ) with $Q_{AB_i} =& \\frac{1}{N-1}\\sum _{k=1}^{N-1} \\frac{1}{2}(1-(1- f_G)^k)\\\\=& \\frac{(1 - f_G)^N + f_G N-1}{2 f_G (N-1)},$ which is the average $Q_{AB_i}$ for one to $N-1$ gates, because we use a random order of the gates.",
"This effectively mixes all $\\lambda _j^\\pm $ with $j$ of same Hamming weight and accomplishes that all $Q_{AB_i}$ are equal.",
"Compared to a fixed gate order it improves the key rate and removes the maximum in Eq.",
"(REF ).",
"In the case of the network shown in Fig.",
"REF , $N-1$ gates are performed at $C$ and one additional gate is performed at $A$ .",
"The initial state at $C$ depends on whether the gate of $A$ was successful, i.e.",
"it is $\\rho _{\\mathrm {in,QNC}}=&(1-f_G) \\rho _{\\mathrm {in}} + f_G \\frac{{1}}{2}\\otimes \\left| 0\\right>\\left< 0\\right|^{\\otimes (N-1)}\\\\=& (1-f_G) \\rho _{\\mathrm {in}} + f_G \\frac{1}{2} (\\rho _{\\mathrm {in}}+Z_1 \\rho _{\\mathrm {in}} Z_1),$ i.e.",
"$\\lambda _{0,QNC}^+(f_G)=&(1-f_G) \\lambda _{0}^+(f_G) + \\frac{f_G}{2}(\\lambda _0^+(f_G)+\\lambda _0^-(f_G))\\\\\\lambda _{0,QNC}^-(f_G)=&(1-f_G) \\lambda _{0}^-(f_G) + \\frac{f_G}{2}(\\lambda _0^+(f_G)+\\lambda _0^-(f_G)),$ and the previous results can be used to obtain the key rate in this case.",
"Note that while the final density matrix depends on whether a router was used or not, the QBER (and $Q_{AB_i}$ ) does not, because the additional phase error does not contribute to it.",
"For a fixed number of parties $N$ there is a threshold gate error probability below which the NQKD protocol outperforms the bipartite approach in the quantum network of Fig.",
"REF .",
"These values are listed in Table REF .",
"Table: The multipartite entanglement based QKD protocol is more prone to gate errors, but requires Alice to send one qubit only.",
"These two competing effects lead to a threshold value of the gate error probability f G f_G below which it outperforms the bipartite approach.The key rate as a function of $N$ is shown for different values of the gate error rate $f_G$ in Fig.",
"REF .",
"Figure: The secret key rate for the multipartite entanglement based (NQKD) protocol (solid lines) in the quantum network shown in Fig.",
"as a function of the number of parties NN for different values of the gate error probability f G =0%,1%and5%f_G=0\\,\\%,\\, 1\\,\\% \\text{ and }\\,5\\,\\% (top to bottom).",
"The key rate decreases with increasing NN, because more imperfect gates are applied.",
"The key rate of the bipartite entanglement based (2QKD) protocol (dashed lines), plotted for the same values of f G f_G, shows the 1/(N-1)1/(N-1) scaling which is due to the bottleneck between AA and CC."
],
[
"Key distribution in the butterfly network",
"We sketch how the NQKD protocol can be employed in the butterfly network shown in Fig.",
"REF .",
"As usual, the rate constraints on the channels are one, i.e.",
"each channel can send a single qubit per time step.",
"The quantum network code corresponding to the linear code shown in FIG.",
"SREF is employed to produce two GHZ states shared by $A$ , $B_1$ and $B_2$ (FIG.",
"SREF ).",
"See Thm.",
"1 of [27].",
"These two GHZ states allow to perform two rounds of the NQKD protocol in a single time step.",
"In contrast the bipartite entanglement based (2QKD) protocol (also in its prepare and measure formulation) can only do a single round, because only two Bell pairs can be distributed (due to the outgoing capacity at A).",
"Thus the key rate of the NQKD protocol is twice as high as in the “standard approach”.",
"From the construction of this example it is clear how it generalizes: If the network allows A to multicast $n$ bits, then a single use of the corresponding quantum network will produce $n$ GHZ states.",
"Thus the NQKD protocol can be performed $n$ times per time step.",
"However, the 2QKD protocol can only perform $\\frac{n}{N-1}$ rounds in the same time."
]
] | 1612.05585 |
[
[
"Heat convection and radiation in flighted rotary kilns: A minimal model"
],
[
"Abstract We propose a minimal model aiming to describe heat transfer between particles (i.e.",
"grains) and gases in a model of flighted rotary kilns.",
"It considers a channel in which a convective gas interacts with a granular suspension and a granular bed.",
"Despite its simplicity it captures the main experimental findings in the case of dilute suspension of heavy grains typical of what can be observed in many industrial rotary kilns.",
"Energy balance between each phase takes into account the main heat transfer mechanisms between the transverse granular motion and the convective gas.",
"In the absence of radiation heat transfer, the model predicts exponential variations of the temperatures characterized by a length which depends on the granular and gas heat flow rates as well as on the exchange areas.",
"When radiation is taken into account, the model can be solved numerically.",
"For this case, the temperature variations can be fitted by stretched exponentials whose parameters are found to be independent of the studied phases.",
"Finally, an efficiency criterion is proposed to optimize the length of the system."
],
[
"INTRODUCTION",
"In recent years, there has been a great deal of interest in the understanding of the behaviour of granular materials as they are used in a large number of engineering processes and present in many geophysical systems.",
"Accordingly, many progresses have been made on the theoretical description of these systems and on the understanding of the physical mechanisms that govern their behaviour when they are exhibiting a gaseous, a liquid or a solid-like behavior.",
"For instance, dilute granular flows are well described by a kinetic theory [1], [2] and unidirectional dense flows properties are well captured by a Druker-Parger-like rheology [3] or by extensions of the aforementioned kinetic theory [4].",
"The behavior of such materials in the vicinity of jamming [5], [6], [7], [8], [9], [10], [11], close to destabilization [12], [13] or submitted to aging processes [14], [15], [16], [17] is also the subject of active current research.",
"Heat transfer in such systems [18], [19] and in particularly in gas-grains mixtures [20], [21], [22], [23], [24] is an important physical phenomenon which may govern a wide variety of natural systems (e.g.",
"volcanic eruption) and industries (rotary kilns devoted to asphalt or cement production [24], torrefaction of biomass [25], fertilizers production, waste treatment, manufacturing of nuclear fuel [26]...).",
"This phenomenon is quite complex for several reasons [27].",
"Among them, we can mention that heat transfer may involve three different physical phenomena: (i) conduction (ii) convection and (iii) radiation.",
"The case of conduction is also quite complicated by itself since it is influenced by the nature of the contacts between grains which is controlled by the nano-asperities present at the surface of the grains [17].",
"Recent and important progresses have been made in the numerical simulation of fluid interacting with moving particles using Discrete Element Methods (DEM) coupled to computational fluid dynamics (CFD) –see for example [22], [28], [29], [30] or [20], [31], [32] for systems similar to ours–.",
"However, their practical use to simulate industrial facilities is partially limited because they require important computational resources.",
"Therefore, it is still necessary to develop physical models [33], [34], [35], [36] capable to reproduce and predict the heat transfer between a gas and a granular medium.",
"Here, we (i) present a minimal physical model which aims to quantify the heat transfer in the case of wall-bounded gas flow through dilute suspension of heavy grains and (ii) use this model to analyze the thermal efficiency of rotary kilns.",
"In particular, we discuss the relative importance of radiation heat transfer with respect to convection and show that, the presented model captures the main experimental findings obtained in industrial facilities.",
"Granular heat transfer modeling has a great importance in the material processing field.",
"For example, rotary kilns are the most popular equipment dedicated to the drying, heating and coating of granular materials.",
"A better understanding of the physical mechanisms [37], [38], [39] occurring within a rotary kiln is therefore crucial to optimize the industrial plants [40] in the context of sustainability, efficient energy use and reduction of toxic gases like polycyclic aromatic hydrocarbons.",
"Here, we aim to model such a typical industrial equipment.",
"A rotary kiln is a cylinder slightly tilted (a few degrees) with respect to the horizontal and which rotates around its axis (Figure REF ).",
"Figure: Sketch of a typical rotary kiln.",
"The heat is provided by a direct fire natural burner.",
"Shape metal slats called blades or flights are attached to the interior surface of the kiln.",
"They lift particles located at the bottom of the kiln to the gas.",
"The kiln is inclined (a few degrees) with respect to the horizontal.The material to be processed (typically grains) is fed into the upper end of the cylinder (the inlet).",
"Due to gravity and the rotation of the kiln, the material gradually moves down towards the lower end (the outlet).",
"In the case of grains, they are heated by a hot gas passing along the axis of the cylinder from the upper end to the lower end (co-current kiln) or from the lower end to the upper end (counter-current kiln).",
"The hot gases are generally provided by an external furnace, or by a flame inside the kiln.",
"In such a system, the grains are initially located at the bottom of the cylinder.",
"However, L-shaped baffles (also called flights) located along the cylinder parallel to its axis on its inner diameter lift grains from the bed, bring them up to the top of the cylinder where they fall down due to gravity [41].",
"A granular curtain is thus created [41].",
"It ensures a proper mixing between grains and hot gases, and so enhances the heat exchange.",
"It is thus convenient to consider that the physical system is made of four phases: (i) the gas, (ii) the dense granular bed, (iii) a dilute granular suspension i.e.",
"a granular curtain and (iv) the walls bounding the system, and to study the heat exchange between those four phases.",
"Although the analysis presented in this work can be easily extended to the counter-current case, we will restrict ourselves to the co-current configuration.",
"This choice may be surprising because it is reasonable to think that the maximum amount of heat transfer that can be obtained is more important with a counter-current kiln than with its co-current counterpart.",
"Indeed, the former maintains a slowly declining temperature gradient whereas in the latter, the strong thermal difference at the inlet induces a higher initial gradient which falls off quickly, leading to potential heat loss.",
"However, co-current rotary drum are still predominantly used in the industry for practical reasons: they are more easy to build, the fuel cost is nowadays not expensive enough to justify the replace old co-current kilns by new counter-current ones.",
"In our study case, the rotary kiln is approximated by a drum where heat is exchanged between different phases that will be defined below.",
"The drum is a cylinder made in stainless steel with an inner diameter, $D_D$ a length, $L_D$ and a thickness, $d_D$ .",
"The convective hot gas flows from one boundary of the drum (the inlet) to the other (the outlet) along its axis.",
"To simplify the problem we model the drum by two infinite and insulated horizontal walls separated by a distance $D_D$ (see Figure REF ).",
"Figure: Sketch of the system.",
"Two horizontal walls are separated by a distance D D D_D.A hot gas flows in the system in parallel with the walls.",
"A granular bed (height D b D_b) rests at the lower wall.",
"L-shaped baffles (not shown) lift grains from the granular bed to the upper wall leading to an homogeneous suspension of grains which falls down due to gravity.",
"The whole is inclined at an angle θ\\theta with respect to the horizontal, leading to a global motion of the grains along the drum's axis.This simplification neglects the action of rotation which is justified by the very small rotation speed used in industrial facilities.",
"A gas is convected in parallel with those two walls and between the inlet of the system, defined by $z=0$ , and the outlet, defined by $z=L_D$ .",
"A packing of grains (diameter $d_s$ , volumic mass density $\\rho _s$ , thermal conductivity $k_s$ ) whose height is denoted by $D_b$ , is located on the lower wall.",
"The whole is inclined with respect to the horizontal by a small angle $\\theta $ (few degrees).",
"We set the vertical origin at the level at the lower wall ($x=0$ ).",
"The upper wall is therefore located at $x=D_D$ .",
"The action of the L-shaped baffles is implicitly taken into account by the presence of a uniform suspension (i.e.",
"the volume fraction of the suspension depends neither on $x$ nor on $z$ ) of the same type of grains which interact with the convected gas.",
"In the granular suspension, the net transfer of thermal energy is potentially enhanced by an effective conduction from collisions between grains.",
"However, in most practical cases the collision times between grains are too small to lead to a significant heat transfer between particles [42] and the volume fraction of the granular suspension is small enough to consider that the average number of collisions per grain is very weak.",
"Moreover, we also assume that the grain size is large enough to neglect the velocity fluctuations of grains.",
"In other words, the granular temperature and thus the self-diffusion of grains are negligible.",
"Our model is therefore fundamentally different from that of Chen and Louge [21] and clearly not valid for highly agitated granular systems where the granular temperature is far from being negligible.",
"It should be pointed out that such assumptions are justified by practical reasons.",
"Our aim is not to derive a general and complex model that captures the behavior of a large variety of configurations but to focus on an applied point of view by deriving a minimal model devoted to thermal rotary kilns which are currently and widely used in the industry.",
"Of course, we will also study the predictions and the limits of the aforementioned model.",
"Another assumption consists in neglecting the momentum transfer between the gas and the grains.",
"In other words the grains are too large and heavy to be influenced by the gas (Stokes number $\\gg 1$ ), and the granular suspension is dilute enough ($<10 \\%$ ) to neglect its influence on the gas.",
"The only interaction between grains and gas is therefore related to the heat transfer.",
"The motion of grains is induced by gravity through the inclination of the kiln and through the action of baffles, whose size is negligible compared to that of the drum.",
"The former and the latter assumptions prevent the use of our work to model fluidized beds [22], [43] of other systems where the grains are highly agitated and where fluid grains interactions govern significantly the dynamics.",
"We also assume that the Biot number of grains $B_i = h(d_s/2)/k_s$ , where $h$ is the heat transfer coefficient, is small with respect to $0.1$ .",
"Thus, their internal temperature can be considered as uniform.",
"Finally our last assumption is the following: for a given $z$ -position along the axis of the drum, the grains belonging to the granular bed and to the granular suspension have the same temperature, $T_s$ .",
"Therefore the heat transfer between grains of the suspension and those belonging to the bed is not considered.",
"Such an assumption, which has been widely used previously in the literature [23], [44] is an important simplification.",
"Yet, in the case of a rotary kiln, the presence of baffles ensures the solid phase motion in order to renew the exchanges surfaces [45] and thus a reasonable homogeneity of the grain temperature.",
"The relevancy of this simplification has been recently confirmed by numerical simulations coupling DEM with CFD [46] which reports, in a counter-current kiln, a reasonably weak variation of temperature of the solid phase (the maximal standard deviation is approximately 20 K for an average temperature of approximately 320 K).",
"Moreover such an assumption is fully consistent with our aforementioned aim, that is, to derive the minimal model which can be used at the industrial scale.",
"Our model is thus therefore a reasonable alternative to heavier approaches like numerical simulations coupling DEM and CFD.",
"Thus, the grain temperature $T_s$ is assumed to be only a function of $z$ .",
"Similarly, the gas and the wall temperatures (respectively $T_g$ and $T_w$ ) are also assumed to depend only on $z$ .",
"In such a system, several types of convective heat transfer should be taken into account: The convective heat transfer between the granular bed and the gas given by: $h_{bg}\\,dS_{bg}\\left(T_{g}-T_s\\right)$ , where $h_{bg}$ is the convective exchange coefficient between the granular bed and the gas, and $dS_{bg}$ , the differential surface area between the granular bed (or more precisely its free surface) and the gas.",
"The convective transfer between the gas and the granular suspension (i.e.",
"the granular curtain): $h_{cg} dS_{cg}\\left(T_g-T_s\\right)$ , where $h_{cg}$ and $dS_{cg}$ are respectively the convective exchange coefficient and the differential surface area between the granular suspension and the gas.",
"The convective transfer between the granular bed and the contacting wall: $h_{sw}\\, dS_{sw}\\left(T_w-T_s\\right)$ .",
"Similarly, $h_{sw}$ is the convective exchange coefficient and $dS_{sw}$ , the differential surface area between the granular bed and the inner contacting walls.",
"The convective transfer between the gas and the walls: $h_{gw}\\,dS_{gw}\\left(T_w -T_g \\right),$ where $h_{gw}$ is the gas-wall convective exchange coefficient and $dS_{gw}$ the corresponding differential surface area.",
"From the differential surfaces $dS_{ij}$ we can define length $l_{ij}$ by $l_{ij}=dS_{ij}/dz$ where $dz$ is the differential of $z$ .",
"The heat conduction between grains belonging to the granular bed is assumed to be negligible.",
"It should be pointed out that, by assuming a constant grain diameter whereas real systems are polysized, we underestimate the heat exchange between the convected gas and the granular suspension.",
"Yet, in view of industrial applications such an underestimation is better than an overestimation.",
"The exchange surfaces between each phase are determined following the work of Piton et al. [47].",
"Using the same kiln geometry than ours and results from the literature, they provide expressions for the heat transfer coefficients which are empirical or semi-empirical.",
"For the sake of clarity we also reported those expressions in tables REF , REF , REF and REF .",
"As we will show below, one of the key parameter is the fraction of grains in the granular suspension, $\\alpha _{cg}$ , i.e.",
"the volume of the grains belonging to the suspension divided by the total volume of grains.",
"The latter quantity is directly linked to the number of grains contained in the granular suspension per unit of length $dN/dz$ through: $\\alpha _{cg}=\\frac{2}{3}\\frac{dN}{dz}\\frac{d_s^3}{D_D^2\\, F_t},$ where $F_t$ is the ratio of the volume of grains to the volume of the drum, i.e.",
"the kiln volume fraction.",
"Then, the convective heat transfer coefficients are calculated from correlations available in the literature [47].",
"Thermal radiation is modeled using the following heat transfers: between the gas and the granular suspension: $dS_{cg} E_{cg} \\left(T_g^4-T_s^4 \\right)$ , between the gas and the free surface of the granular bed: $dS_{bg}E_{bg} \\left(T_g^4-T_s^4\\right)$ , between the gas and the wall of kiln: $dS_{gw}E_{gw} \\left(T_w^4-T_g^4\\right)$ , between the granular bed and the contacting wall of the kiln $dS_{sw}E_{sw} \\left(T_w^4-T_s^4\\right)$ .",
"In the latter expressions, the quantities $E_{ij}$ with $i,j=\\lbrace c,b,g,w\\rbrace $ are the radiative heat transfer coefficients defined as $E_{i,j}=\\sigma \\varepsilon _{ij}$ where $\\sigma $ is the Stefan-Boltzmann constant and $\\varepsilon _{ij}$ the corresponding emissivity.",
"The heat exchange surfaces $dS_{ij}$ are those used in the convective heat transfer.",
"So, at equilibrium, using the aforementioned types of heat transfer, the energy balances (i.e.",
"the first law of thermodynamics) for the solid and gas phases as well as the insulated wall condition are respectively given by: $\\delta \\dot{Q}_s &= &\\dot{m_s} \\,C_{p,s}\\,dT_s \\nonumber \\\\& = &\\left(h_{bg}\\,dS_{bg} \\nonumber + h_{cg}\\,dS_{cg}\\right) \\left(T_g -T_s\\right) \\nonumber \\\\& & + E_{gs} \\left(dS_{bg} + dS_{cg}\\right) \\left(T_g^4 -T_s^4\\right) \\nonumber \\\\ & & + h_{sw}\\,dS_{sw}\\left(T_{w}-T_s\\right) \\nonumber \\\\& & + E_{sw} \\left(dS_{bg} + dS_{cg}\\right)\\left(T_w^4 -T_s^4\\right),$ $\\delta \\dot{Q}_g &= &\\dot{m_g}\\,C_{p,g}\\,dT_g \\nonumber \\\\& = &\\left(h_{bg}\\,dS_{bg} + h_{cg}\\,dS_{cg}\\right)\\left(T_s -T_g\\right)\\nonumber \\\\& & +E_{gs} \\left(dS_{bg} + dS_{cg}\\right) \\left(T_s^4 -T_g^4\\right) \\nonumber \\\\& & +h_{gw}\\,dS_{gw}\\left(T_{w}-T_g\\right)\\nonumber \\\\& & +E_{gw} dS_{gw}\\left(T_w^4 -T_g^4\\right),$ and $h_{sw}\\,dS_{sw}\\left(T_s -T_w\\right) +E_{sw} \\left(dS_{bg} +dS_{cg}\\right)\\left(T_s^4 -T_w^4\\right) \\nonumber \\\\ \\,+h_{gw}\\,dS_{gw}\\left(T_g -T_w\\right) + E_{gw} dS_{gw}\\left(T_g^4 -T_w^4\\right) = 0.",
"$ In the latter equations, $\\delta \\dot{Q}_i$ (with $i=s,g$ respectively for the solid and the gas phase) is the heat transfer per unit time (the dot notation is used for the time derivative), $\\dot{m_i}$ the mass flow rate of the phase $i$ , and $C_{p,i}$ the corresponding heat capacity.",
"The set of Equations (), (REF ) and (REF ) composes a system with the following boundary values: $T_s\\left(0 \\right)=T_{s,0}$ and $T_g\\left(0 \\right)=T_{g,0}$ are respectively the inlet temperature of solids (i.e.",
"grains) and the inlet temperature of hot gases.",
"The analytical resolution of previous model is not possible due to the presence of $T^4$ -terms.",
"However, neglecting the heat transfer by radiation leads to a classical system of equations which can be solved analytically.",
"The first law of thermodynamics applied to the grains and to the gas gives the respective heat flow rates : $\\dot{m_s} \\,C_{p,s}\\,dT_s&=&h_{bg}\\,dS_{bg}\\left(T_g -T_s\\right)\\nonumber \\\\& & +h_{cg}\\,dS_{cg}\\left(T_g-T_s\\right) \\nonumber \\\\& &\\,+h_{sw}\\,dS_{sw}\\left(T_{w}-T_s\\right),$ and $\\dot{m_g}\\,C_{p,g}\\,dT_g&=&h_{bg}\\,dS_{bg}\\left(T_s -T_g\\right) \\nonumber \\\\& &+h_{cg}\\,dS_{cg}\\left(T_s-T_g\\right) \\nonumber \\\\& &+h_{gw}\\,dS_{gw}\\left(T_{w}-T_g\\right).$ Since the system is insulated, the wall is in thermal equilibrium with the gas and the solid phase in such way that the energy balance is given by : $h_{sw}\\,dS_{sw}\\left(T_{w}-T_s\\right)+h_{gw}\\,dS_{gw}\\left(T_{w}-T_g\\right)=0.$ The latter equation leads to the identification of the inner wall temperature: $T_{w}=\\frac{h_{sw}\\,dS_{sw}T_s+h_{gw}\\,dS_{gw}\\,T_g}{ h_{sw}\\,dS_{sw}+h_{gw}\\,dS_{gw}}.$ Combining Equations (REF ), (REF ) and (REF ) and using $dS_{i,j}=dz\\,l_{i,j}$ and $\\Psi _{ij}=h_{ij}\\,l_{ij}$ , with $i,j=\\lbrace c,g,w,b\\rbrace $ we obtain the following first order differential system $\\begin{array}{ccc}\\displaystyle {\\frac{d\\,T_g}{dz}}&=&\\displaystyle {-\\left[ \\frac{\\Psi _{bg}+\\Psi _{cg}+\\Psi _{gw}}{\\dot{m}_g C_{p,g}} -\\frac{\\Psi _{gw}^2}{(\\Psi _{sw} + \\Psi _{gw})\\dot{m}_g C_{p,g}}\\right]T_g}\\\\&\\ & \\displaystyle {+\\left[\\frac{\\Psi _{gw} + \\Psi _{cg}}{\\dot{m_g}C_{p,g} }+\\frac{\\Psi _{gw}\\,\\Psi _{sw}}{\\left(\\Psi _{sw} + \\Psi _{gw} \\right)\\dot{m_{g}}C_{p,g}}\\right]T_s,}\\\\\\displaystyle {\\frac{d\\,T_s}{dz}}&=&\\displaystyle {-I\\left[ \\frac{\\Psi _{gw} + \\Psi _{cg} +\\Psi _{gw}}{\\dot{m}_g C_{p,g}} - \\frac{\\Psi _{gw}^2}{(\\Psi _{sw}+\\Psi _{gw})\\dot{m}_g C_{p,g}}\\right]T_s}\\\\& \\ &\\displaystyle { + I\\left[\\frac{\\Psi _{bg}+\\Psi _{cg}}{\\dot{m_g}C_{p,g} }+\\frac{\\Psi _{gw}\\,\\Psi _{sw}}{\\left(\\Psi _{sw}+\\Psi _{gw} \\right)\\dot{m_{g}}C_{p,g}}\\right]T_g.",
"}\\end{array}$ where $I$ is a dimensionless number defined as ${\\dot{m_g}C_{p,g}}/{\\dot{m_s}C_{p,s}}$ .",
"After resolution, the latter system leads to: ${\\left\\lbrace \\begin{array}{ccc}T_g &=& \\displaystyle {\\frac{T_{g,0}-T_{s,0} }{I+1 }\\exp \\left(-z/\\Lambda \\right)+\\frac{T_{g,0}I+T_{s,0}}{I+1},}\\\\T_s&=& \\displaystyle {-I\\frac{T_{g,0}-T_{s,0} }{I+1 }\\exp \\left(-z/\\Lambda \\right)+\\frac{T_{g,0}I+T_{s,0}}{I+1}.}\\end{array}\\right.",
"}$ In those solutions, $\\Lambda $ is a characteristic length defined as: $\\Lambda =\\frac{E}{I+1}\\frac{1}{A+B+CD/(D+C)}$ with $A=h_{cg}l_{cg}$ , $B=h_{bg}l_{bg}$ , $C=h_{sw}l_{sw}$ , $D=h_{gw}l_{{gw}}$ and $E=\\dot{m_g}C_{p,g},$ Thus, in the absence of thermal radiation the temperature profiles along the drum are given by exponential functions.",
"In the remainder of the paper, we will consider the following nominal case.",
"Grains are spheres of diameter $d_s=0.005\\,\\rm {m}$ and the curtain density is characterized by $\\alpha _{cg}=3\\%$ .",
"The solid mass flow rate is $\\dot{m}_s = 33.98\\,\\rm {kg.s}^{-1}$ .",
"The gas is defined by its mass flow rate $\\dot{m}_g=3.74\\,\\rm { kg.s}^{-1}$ (unless otherwise specified), its density $\\rho _g = 0.84\\,\\rm {kg.m}^{-3}$ and its viscosity $\\mu _g = 3.59\\,10^{-5}\\,\\rm {Pa.s}^{-1}$ .",
"For the sake of simplicity, we do not take into account the variations of the two latter quantities with the temperature.",
"However, in our model, they only influence $h_{cg}$ and $h_{gw}$ through the Reynolds number.",
"Yet, in presence of radiation, the variations of $\\mu _g$ and $\\rho _g$ are really important only when radiation is negligible, i.e.",
"for a gas temperature below $600\\rm { K}$ .",
"Within this range temperature, the effect on the heat transfer coefficients is negligible.",
"The convective heat transfer coefficients corresponding to the nominal case are given by the following values (see [47]): $h_{bg}=102.83~\\rm {W.m}^{-2}\\rm {.K}^{-1}$ , $h_{cg}=112.80~\\rm {W.m}^{-2}\\rm {.K}^{-1}$ , $h_{gw}=35.23~\\rm {W.m}^{-2}\\rm {.K}^{-1}$ and $h_{sw}=242.96~\\rm {W.m}^{-2}\\rm {.K}^{-1}$ .",
"Note that, those values are obtained by semi-empirical relations (see tables REF , REF , REF and REF ) obtained from the study of industrial rotary kilns used to process various types of granular matter [47] and that the heat transfer coefficients depend on both $\\alpha _{cg}$ and $\\dot{m}_g$ .",
"Although we focus here on the case of asphalt plants, our approach is valid for any type of granular material as long as the assumption of the model are valid.",
"It should be the case, for example, for kilns used to dry fertilizers.",
"For the nominal case, the exchange lengths, which depend on the grain distribution in the kiln cross-section (see table REF and [47]), are given by $l_{bg}=2.320~\\rm {m}$ , $l_{cg}=9.71~\\rm {m}$ , $l_{gw}=3.55~\\rm {m}$ and $l_{sw}=1.79~\\rm {m}$ .",
"Those values depend on $\\alpha _{cg}$ and will also be modified accordingly when the effect of that parameter will be studied.",
"Table: The coefficient h bg h_{bg} depends on A g A_g (the cross section of ”hot gas” phase) and m ˙ g \\dot{m}_g (the mass flowrate of hot gases in the kiln).",
"The constants H bg H_{bg}, σ bg \\sigma _{bg} and η bg \\eta _{bg} are respectively equal to0.4W.m -2 .K -1 0.4~\\mbox{W.}\\mbox{m}^{-2}\\mbox{.K}^{-1}, 3600kg -1 .m 2 .s3600~\\mbox{kg}^{-1}\\mbox{.m}^2\\mbox{.s} and 0.620.62.Table: The coefficient h cg h_{cg} depends onk s k_s, thethermal conductivity of solids, d s d_s the grain size, P r P_r the Prandtl number i.e.",
"the ratio ofkinematic viscosity to the thermal diffusivity and the particle Reynolds number R ep R_{ep} defined as v g d s /ν g v_g d_s /\\nu _g, where v g v_g and ν g \\nu _g are respectively the gas velocity and the gas kinematic viscosity.The coefficients A cg A_{cg}, B cg B_{cg}, aa, bb and cc depend on the range of the particle Reynolds number R ep R_{ep} .Table: The coefficient h gw h_{gw} depends on k g k_g, thethermal conductivity of gas, R e R_e the Reynolds number R e =D D v g /ν g R_e=D_D v_g/\\nu _g, on therotational Reynolds number R e,ω =D D 2 ω/2ν g R_{e,\\omega }=D_D^2 \\omega / 2\\nu _g (ω\\omega is the angular velocity of the kiln)and on D h D_h an effective diameter.",
"The latter quantity is determined from D D D_D and ε B \\varepsilon _B, the angle whose sinus is equal to the ratio 2δ/D D 2\\delta /D_D, where δ\\delta is the distance between the center of the kiln and the free surface of the granular bed (see the two last sketches of the table).",
"The constants A gw A_{gw}, η gw \\eta _{gw}, B gw B_{gw} and η ω,gw \\eta _{\\omega ,gw} are respectively equal to 0.020.02, 0.930.93, 8.5×10 -6 8.5\\times 10^{-6} and 1.451.45.Table: The coefficient h sw h_{sw} depends on k s k_s thethermal conductivity of solids D D D_D the kiln diameter, ε B \\varepsilon _B, ω\\omega and a s a_s, the thermal diffusivity of the solid phase, i.e.",
"k s /(ρ s C p,s ).k_s/(\\rho _s\\,C_{p,s}).",
"The constants K sw K_{sw} and η sw \\eta _{sw} are respectively equal to 11.611.6 and 0.30.3Table: The lengths l pw l_{pw}, l sp l_{sp}, l cg l_{cg} and l bg l_{bg} depend on the number of baffles, n F n_{F}, the kiln diameter D D D_D and on the angle of the bed in the kilnε B \\varepsilon _B (see ) which is determined using α B F t =(ε B -sinε B cosε B )/π\\alpha _{B} F_t = (\\varepsilon _B-\\sin \\varepsilon _B\\cos \\varepsilon _B)/\\pi where α B \\alpha _B is the fraction of grain in the bed and F t F_t the kiln volume fraction.Regarding the heat transfer capacities we will use $C_{p,s}=830\\,\\rm { J.kg}^{-1}\\rm {.K}^{-1}$ and $C_{p,g}=1100\\,\\rm { J.kg}^{-1}\\rm {.K}^{-1}$ which also correspond, as mentioned above, to the asphalt plant case.",
"Figure: Profiles of the gas (a), wall (b) and grain (c) temperatures versus the position zz along the drum axis depend on the density of grains within the granular suspension.",
"The curves reported have been obtained for α cg =1%\\alpha _{cg}= 1\\%, 2%2\\%, 3%3\\%, 4%4\\%, 5%5\\% and 6%6\\%.",
"The arrows indicate increasing α cg \\alpha _{cg}.Figure REF reports variations of the temperatures versus $z$ for several values of $\\alpha _{cg}$ .",
"The model predicts that the temperature reached at equilibrium (for $z$ tends towards $+\\infty $ ) is the average of the gas and grain inlet temperatures weighted by the corresponding heat flow rates.",
"Note that this equilibrium temperature does not depend on the exchange surfaces.",
"The latter quantities have only an influence on the dynamics and, obviously, the distance necessary to reach the equilibrium temperature are smaller for larger exchange surfaces.",
"This result can be obtained by a simple application of the first law of thermodynamics.",
"In order to simplify our analysis, it is reasonable to consider a kiln with a good efficiency, i.e.",
"a kiln for which the thermal transfers with the walls are much lower than those between the gas and the grains.",
"This assumption is fully valid for kilns having baffles which ensure a good heat transfer between the grains and the gas.",
"In such a case, which corresponds to recent experimental measurements [52], the term corresponding to the thermal exchange with the walls ($C\\,D/(C+D)$ ) in Equation (REF ) is between 2 and 30 times smaller than those corresponding to the transfer between the gas and the grains ($A$ and $B$ ) and the characteristic length can be reduced to the following: $\\Lambda _{app}\\approx \\frac{E}{(I+1)(A+B)}=\\frac{\\kappa }{h_{cg}l_{cg}+h_{bg}l_{bg}}, $ where $\\kappa $ is the effective mass flow rate $\\kappa =\\dot{m_g}C_{p,g} \\dot{m_s}C_{p,s} /(\\dot{m_g}C_{p,g} + \\dot{m_s}C_{p,s})$ .",
"Figure: The approximated characteristic length Λ app \\Lambda _{app} [Equation ()] is found to be very close to the exact value Λ\\Lambda [Equation ()] demonstrating the relevancy of the latter approximation for well-insulated kilns.",
"The points correspond to simulations obtained with α cg =1%\\alpha _{cg}=1\\%, 2%2\\%, 3%3\\%, 4%4\\%, 5%5\\% and 6%6\\% and m ˙ g =1.1\\dot{m}_g= 1.1, 1.661.66, 2.062.06, 2.562.56, 3.123.12, 3.743.74 and 3.903.90 kg.s -1 ^{-1}.",
"The lengths are made dimensionless by Λ max \\Lambda _{max}, the maximal value of Λ\\Lambda .",
"The dashed line corresponds to Λ app /Λ max =Λ/Λ max \\Lambda _{app}/\\Lambda _{max}=\\Lambda /\\Lambda _{max}.Figure REF reports the evolution of the characteristic length determined by neglecting the heat transfer with walls $\\Lambda _{app}$ [Equation (REF )] versus its full expression [Equation (REF )].",
"As expected both quantities are close to each other (less that 15% of difference) showing the relevancy of Equation(REF ) in the case of efficient kilns.",
"Note that the approximation is especially good for large values of $\\alpha _{cg}$ i.e.",
"for an important heat transfer between the granular suspension and the gas (with respect to heat transfer with the walls) and thus for small values of $\\Lambda $ .",
"Obviously, and in agreement with Figure REF , increasing $\\alpha _{cg}$ and thus the surface exchanges leads to a decrease of the characteristic length of the temperature variation along the drum.",
"However, one should keep in mind that the present model is only valid for dilute granular suspensions, i.e.",
"low values of $\\alpha _{cg}$ .",
"Otherwise, the momentum exchange between the gas and the grains cannot be neglected.",
"The characteristic length $\\Lambda $ thus depends on the gas and solid flow rates, $\\dot{m_g}$ and $\\dot{m_s}$ .",
"If one of the flow rates is increased while keeping the other flow rate and all the other parameters constant, $\\kappa $ and consequently $\\Lambda $ increase.",
"Thus a longer drum is required to reach the same equilibrium temperatures.",
"Such a result can be understood easily.",
"For example, if $\\dot{m_s}$ is increased while the other parameters are kept constant, the amount of heat transferred from the hot gas is shared by a larger number of grains, thus their temperature at the outlet of the drum is lower.",
"Figure: When radiation is not taken into account, the evolution of the temperatures along the kiln are given by exponential functions whose characteristic length varies with α cg \\alpha _{cg} and m ˙ g \\dot{m}_g.The effect of both $\\alpha _{cg}$ and $\\dot{m}_g$ is highlighted on Figure REF where we have reported the evolution of the characteristic length $\\Lambda $ versus those two parameters.",
"As expected, $\\Lambda $ decreases with (i) increasing $\\alpha _{cg}$ and (ii) decreasing $\\dot{m}_g$ .",
"Thus, the smaller characteristic lengths are obtained for important values of $\\alpha _{cg}$ and small values of $\\dot{m}_g$ .",
"Note however, that the effect of $\\dot{m_s}$ , contrary to that of $\\dot{m_g}$ , might be more complex since its modification might also induce a variation of $\\alpha _{cg}$ and thus of the exchange lengths.",
"To fully understand the effect of $\\dot{m_s}$ on $\\Lambda $ , an investigation of variety of parameters (velocity of the drum, number of grains present in the baffles...) which depend on the details of the kiln geometry is required.",
"Such a study is beyond the scope of this paper.",
"In previous analytical solution of the model, we have neglected thermal radiation.",
"However, it is possible to numerically solve the full model (including thermal radiation, see Equations (REF ) and ()) following the work of Piton et al. [47].",
"Figure: Profiles of the gas (T g T_g), wall (T w T_w) and grain (T s T_s) temperatures versus the position along the drum axis in the presence of radiation radiation (symbols).",
"The initial gas and solid temperatures are respectively equal to 1873K1873\\,\\rm K and 298.15K298.15\\,\\rm K. The curves have been obtained using α cg =3%\\alpha _{cg}=3\\%.",
"The dashed lines correspond to the fits obtained using stretched exponentials [Equation ( )].Figure REF reports the numerical results obtained for $\\alpha _{cg}=3\\%$ .",
"Note that, as expected, the heat supplied to the system at its outlet is the same with and without radiation as long as the length of the drum is large enough to reach the equilibrium temperature.",
"Indeed, since the system is insulated the temperatures at the equilibrium, i.e.",
"$T_g(\\infty )=T_{g,\\infty }$ , $T_s(\\infty )=T_{s,\\infty }$ and $T_w(\\infty )=T_{w,\\infty }$ , are also the same, independently of the fact that radiation is taken –or not– into account.",
"As mentioned above, the general equations which include thermal radiation terms cannot be solved analytically due to the $T^4$ -terms.",
"In such a case, to define a characteristic length of the phenomena and thus obtain an efficiency criterion it is necessary to fit the numerical results by an adequate function.",
"The $z$ -profiles of the temperatures obtained numerically cannot be fitted by an exponential function whose variation is too rapid.",
"Thus, following what is classically done in such a case, we try to fit our results by a stretched exponential given by Equation (REF ) and obtained satisfactory results.",
"First used by Kohlrausch in 1854 [53] to describe mechanical creep, this expression was far later popularized by Williams and Watts [54] who described dielectric relaxation in polymer as being stretched exponential functions.",
"It is now frequently applied to a large range of relaxations in disordered thermal systems as glasses or granular materials [55], [56] and is often called the KWW law.",
"Its expression is given by: $T_\\gamma (z) = T_{\\gamma ,\\infty } + \\left(T_{\\gamma ,0} - T_{\\gamma ,\\infty }\\right)\\exp \\left(-(z/\\lambda _\\gamma )^{\\beta _\\gamma }\\right),$ where index $\\gamma $ stands for the studied phase (i.e.",
"solid, gas or wall).",
"Note that the stretched exponential is also known as the complementary cumulative Weibull distribution.",
"To explain the use of such a fit function, one can consider a small region around of the drum $z=z_i$ for which the radiative terms $E_{ij}l_{ij}(T_g^4-T_s^4)$ can be approximated by $E_{ij}l_{ij}(T_g-T_s)(T_{g,i}+T_{s,i})(T_{g,i}^2+T_{s,i}^2)$ where $T_{g,i}$ and $T_{s,i}$ are respectively the solid and gas temperature at $z=z_i$ .",
"This assumption leads to a linear system of differential equations around $z_i$ .",
"The solution drives exponential functions whose characteristic length depends on $T_{g,i}$ and $T_{s,i}$ and increases with $z_i$ .",
"The full solution may thus be seen as a combination of simple exponential decays which is known as a stretched exponential function.",
"As mentioned above the temperatures reached for $z\\rightarrow +\\infty $ are the same with and without radiation : $T_{g,\\infty }=T_{s,\\infty }=(T_{g,0}I+T_{s,0})/(I+1).$ The expressions of the gas and solid temperatures can thus be respectively written as : $\\left\\lbrace \\begin{array}{ccc}T_g &=& \\frac{T_{g,0}-T_{s,0} }{I+1 }\\exp \\left(- \\left(z/\\lambda _g\\right)^{\\beta _g}\\right)+\\frac{T_{g,0}I+T_{s,0}}{I+1},\\\\\\ & & \\\\T_s&=&-I \\frac{T_{g,0}-T_{s,0} }{I+1 }\\exp \\left(- \\left(z/\\lambda _s\\right)^{\\beta _s}\\right)+\\frac{T_{g,0}I+T_{s,0}}{I+1}.\\end{array}\\right.$ It should be pointed out that, for the solid phase, the stretched exponential fit is not able to reproduce correctly the data close the equilibrium value of the temperature.",
"This is potentially problematic for very long drum and/or very low values of $\\alpha _{cg}$ (i.e.",
"$\\alpha _{cg} \\le 0.1\\%$ ).",
"However, close to the inlet of the kiln, the fit is fully valid.",
"The two free parameters used for the fit are $\\lambda _\\gamma $ , the characteristic length of the stretched exponential and $\\beta _\\gamma $ , its power.",
"Interestingly, our numerical results show that the dependence of the phase is very weak, i.e.",
"$\\beta _s\\approx \\beta _g \\approx \\beta _w = \\beta $ and $\\lambda _s\\approx \\lambda _g \\approx \\lambda _w = \\lambda $ are reasonable approximations as long as we restrict ourselves to the region of the kiln for which the stretched exponential fit is valid.",
"Figure: When radiation is taken into account, the temperatures are well fitted by stretched exponential functions whose parameters (the characteristic length λ\\lambda (a) and the power β\\beta (b)) are both found to be independent of the phase but vary with m ˙ g \\dot{m}_g and α cg \\alpha _{cg}.Therefore the profiles of the temperatures within the system are determined by the two parameters of the stretched exponential which depend on $\\alpha _{cg}$ and $\\dot{m}_g$ (see Figure REF ).",
"Two important remarks should be made.",
"Firstly, both $\\lambda $ and $\\beta $ increase with the gas flow rate $\\dot{m}_g$ .",
"This result indicates that increasing the mass flow rate of the gas requires to use longer kilns to reach equilibrium.",
"Secondly, an increase of $\\alpha _{cg}$ leads to a decrease of the exponent of the stretched exponential and of the characteristic length.",
"The latter result is expected since an increase of $\\alpha _{cg}$ increases the surface exchange between the granular suspension and the gas and thus improves the heat exchange between the gas and the grains.",
"Note that the relation between the two parameters $\\beta $ and $\\lambda $ is not one-to-one.",
"In other words one cannot deduce one of the parameters from the knowledge of the other.",
"However, an interesting result should be pointed out: the relation between the two parameters $\\lambda $ and $\\beta $ are one-to-one for each value of $\\alpha _{cg}$ , i.e., the two parameters $\\beta $ and $\\lambda $ are correlated if $\\alpha _{cg}$ is kept constant.",
"Interestingly, the correlation is linear (see Figure REF c) but, to date, we have no explanation for this linearity which warrants further studies.",
"Our analysis raises another question: is the characteristic length in presence of radiation, (i.e.",
"$\\lambda $ ) linked to that obtained in absence of radiation, (i.e.",
"$\\Lambda $ )?",
"Figure REF b reports the evolution of the former quantity versus the latter.",
"Clearly and surprisingly, they are linearly correlated for each value of $\\alpha _{cg}$ .",
"Naturally, $\\beta $ and $\\lambda $ being correlated for each value of $\\alpha _{cg}$ , $\\beta $ is also linked to $\\Lambda $ , the characteristic length in absence of radiation (Figure REF c).",
"The origin of those linear correlations is not established.",
"Figure: In presence of radiation, the characteristic length obtained by fitting the temperature evolution, λ\\lambda and its power β\\beta are linked linearly for each value of α cg \\alpha _{cg}, the fraction of grains in the granular suspension (a).",
"Similarly, for each value α cg \\alpha _{cg}, the same characteristic length is also linked linearly to Λ\\Lambda the characteristic length in absence of radiation (b).",
"Consequently the power β\\beta is also linearly linked to λ\\lambda , for each value of α cg \\alpha _{cg}.",
"The arrows indicate increasing m g ˙\\dot{m_g}.In our system, the gas temperatures range from a few hundreds of Kelvins to a few thousands, bringing us to the question of whether radiation should be neglected (i.e.",
"for low temperatures) or not (i.e.",
"for high temperatures).",
"Thus, we compare the temperatures obtained with the analytical solution of the model without radiation to those obtained by the numerical resolution of the full model.",
"Figure REF a reports the numerical (i.e.",
"with radiation) and analytical (i.e.",
"without radiation) temperature profiles for a fraction of grains belonging to the granular suspension $\\alpha _{cg}=3\\%$ .",
"As expected, the influence of the radiation is significant especially at the early stages of the heating, i.e.",
"when the gas temperature $T_g$ , and thus $T_g^4$ , are important.",
"Similarly, the influence of the radiation becomes smaller when the gas temperature $T_g$ decreases.",
"Radiation might therefore be neglected only at lower temperature values.",
"To illustrate this purpose, we show in Figure REF b the analytical and numerical relative variations of $T_g$ (i.e.",
"the quantity $(T_g - T_{g,\\infty })/T_{g,0}$ ) for $\\alpha _{cg}=3\\%$ and for several initial temperatures (see caption).",
"As an example, for $\\alpha _{cg}=3\\%$ and $T_{g,0}=798\\mbox{ K}$ the maximum relative error is found to be around $4\\%$ .",
"Figure: (a) Profiles of the gas (T g T_g), wall (T w T_w) and grain (T s T_s) temperatures versus the position along the drum axis with (numerical) and without (analytical) radiation.",
"The initial gas and solid temperatures are respectively equal to 1873K1873\\, K and 298.15K298.15\\, K. The curves have been obtained using α cg =3%\\alpha _{cg}=3\\%.",
"As expected the variations are more important with radiation and, at thermal equilibrium, all the temperatures are identical.",
"(b) Relative difference between the numerical and analytical temperatures along the drum axis at different initial gas conditions for α cg =3%\\alpha _{cg}=3\\%.",
"At the inlet, the grain temperature is fixed to 298.15K298.15\\, \\rm K. The curves have been obtained for398.15K398.15\\, \\rm K, 598.15K598.15\\, \\rm K, 698.15K698.15\\, \\rm K, 798.15K798.15\\, \\rm K, 998.15K998.15\\, \\rm K,1198.15K1198.15\\, \\rm K, 1398.15K1398.15\\, \\rm K, 1598.15K1598.15\\, \\rm K, and 1898.15K1898.15\\, \\rm K. The red arrow indicates increasing T g,0 T_{g,0}.The validity of our model is then tested by fitting experimental measurements of the gas temperature data obtained experimentally in an industrial flight rotary kiln devoted to the asphalt materials manufacturing [40] by simple exponential functions (Figure REF ).",
"Figure: The temperatures obtained experimentally in an industrial rotary kiln devoted to asphalt production (symbols) are well fitted by simple exponentials sharing the same characteristic length and equilibrium temperature.",
"The rate of the production is 140 Tons per hour.The characteristic length obtained is close to that obtained by Equation (REF ) (roughly $10\\%$ difference).",
"The grain and wall temperatures are also well reproduced by simple exponentials for which the characteristic lengths and the equilibrium temperatures are set to the values obtained with the fit of the gas temperature.",
"It should be pointed out that stretched exponentials also fit correctly experimental data, without being significantly better.",
"Unfortunately, the number of experimental measurements is too small to differentiate the two fits.",
"The above study predicts the spatial evolution of the grain temperature within a model kiln with or without radiation.",
"It is thus possible to estimate the optimal length of a rotating kiln allowing the grains to reach a given temperature, $T_\\nu $ .",
"In the present case, calculation of the requested temperature, $T_\\nu $ , is obtained for a given length $L_\\nu $ , in such way that ${T_\\nu }={T_{s}(z=L_\\nu )}$ .",
"Of course, for a given grain flow rate, the gas flow rate or the initial temperatures have to be chosen in order that the equilibrium temperature $T_{s,\\infty }=\\left(T_{g,0}I+T_{s,0}\\right)/\\left(I+1\\right)$ is larger or equal to $T_\\nu $ .",
"Otherwise, $T_\\nu $ is unreachable.",
"If $T_\\nu $ can be reached, the simplest solution consists in using a very long kiln to be sure that the outlet temperature of the grains is equal to or larger than $T_\\nu $ .",
"However, a too long drum is not efficient since energy is used to increase the grain temperature up to an unnecessarily high value.",
"A criterion is therefore necessary to quantify the efficiency of the kiln relative to its length.",
"From the above analytical resolution (i.e.",
"without radiation) our empirical expressions (i.e.",
"with radiation) of $T_g$ , the length $L_\\nu $ at which the grain temperature reaches $T_\\nu $ can be easily obtained.",
"If the length of the kiln $L_D$ is lower than $L_\\nu $ , the requested temperature cannot be reached.",
"The efficiency can be therefore defined as the following ratio: $\\varepsilon =\\frac{\\int _0^{L_D} T_s(z)\\,dz}{\\int _0^{L_\\nu } T_s(z)\\,dz}\\mbox{ if }L_D<L_\\nu .$ In such a case, the efficiency increases from zero when the length of the drum is increased and reaches its maximal value ($\\varepsilon =1$ ) when the length of the drum is equal to $L_\\nu $ .",
"The criterion can be seen as the ratio of the energy used to set the grain temperature to its value at the outlet of the drum over the energy necessary to achieve the requested temperature.",
"Figure: (color online)(a) Efficiency criteria.",
"The requested temperature for the grains is T ν T_\\nu and the length at which that temperature is reached is L ν L_\\nu .",
"This temperature must be lower than the equilibrium temperature reached for an infinite drum otherwise it is unreachable for any length.If the size of the drum L D L_D is lower to L ν L_\\nu (blue or light grey drum), the temperature T s,ν T_{s,\\nu } cannot be reached.",
"The efficiency ε\\varepsilon increases towards 1 when L D L_D tends towards L ν L_\\nu which corresponds to the ideal length of the drum (green or medium grey drum).On the contrary if L D >L ν L_D>L_\\nu (red or dark grey drum) the grain temperature at the outlet is greater than T s,ν T_{s,\\nu }, leading to a waste of energy.",
"(b) The efficiency of the kiln depends on the drum's length.",
"The present curve has been obtained using Equation () with β=0.76\\beta =0.76 and λ/L ν =0.47\\lambda /L_\\nu =0.47.",
"Efficiency rises to reach its maximum value at L D =L v L_{D}=L_{v}.",
"Above L v L_{v}, the energy waste increases and the efficiency decreases progressively.",
"The inset reports the same data in a lin-log scale.For a drum whose length is larger than $L_\\nu $ , the temperature of the grains at the outlet of the device is larger that the requested temperature.",
"Thus the final temperature of the grains is higher than the requested temperature and part of the used energy is wasted.",
"Therefore, the efficiency of the process decreases with drum's length.",
"It can be defined as: $\\varepsilon =1- \\frac{\\int _{L_\\nu }^{L_D} T_s(z)\\,dz}{\\int _0^{L_D} T_s(z)\\,dz}\\mbox{ if }L_D>L_\\nu .$ In the latter equation, the ratio of the two integrals can be seen as the ratio of the wasted energy over the total energy.",
"The definition of our efficiency criterion is sketched in Figure REF a and an example of the dependence on the efficiency with the length of the drum is given on Figure REF b.",
"Note that radiation, by increasing heat exchange between the gas and the granular suspension, improves the efficiency of the kiln for a given $z$ -position.",
"However, the shape of the curve is not significantly influenced by the presence (or the absence of) radiation as long the length of the kiln is made dimensionless by $L_\\nu $ .” Such kind of approach can also be applied to existent kilns, i.e.",
"kilns for which $L_D$ is fixed.",
"In this case, the gas temperature at the inlet can be chosen to obtain the requested solid temperature at the outlet (Equations (REF ) or (REF )).",
"Yet, if for practical reasons, it is not possible to set the inlet temperature at its optimal value, the efficiency can be determined using Equations (REF ) and (REF ) in which $L_\\nu $ is the length corresponding to the requested temperature.",
"In this paper, we report a theoretical approach aiming to model heat transfer between a convective gas and a model granular system.",
"The studied system is typical of a rotary kiln commonly used in industry to dry grains (asphalt production, fertilizers production, waste treatment...) i.e.",
"a dilute suspension of heavy grains interacting with a convected gas only through heat exchange.",
"We divided the system into different “phases” (gas, granular suspension, granular bed, walls) which interact through exchange surfaces.",
"The granular phases are treated as effective continuous media.",
"The model is solved analytically in the absence of radiation and it is shown that the temperature profiles along the drum are governed by one characteristic length whose expression depends on the heat transfer properties and on the exchange surfaces.",
"In the presence of radiation, the profiles of the temperatures along the axis of the drum are well fitted by stretched exponentials whose parameters depend on the surface exchanges but are the same for each phase of the system.",
"As expected, the presence of radiation does not modify the equilibrium temperature but reduces the length required to reach equilibrium.",
"Finally a criterion quantifying the efficiency of the process is proposed.",
"It is based on an estimation of the optimal length of the kiln as a function of the temperature of the grains at the outlet of the kiln.",
"Similar analysis should be applied to the counter current case in order to obtain rigorous energetic diagnostics of different rotary kiln systems.",
"It should be pointed out that several assumptions have been made to derive the present model.",
"Although they may appear to be somewhat crude, they are justified in the case of a convected gas moving through dilute suspension of heavy grains, which corresponds to the majority of application in the field of civil engineering.",
"However, the large variety of granular materials present in other industrial fields deserves consideration of a more detailed model which should take into account a diffusional time within the grain for larger Biot number and agitation of grains within the suspension (i.e.",
"granular temperature).",
"Note also that, although in the case of kiln devoted to asphalt production the moisture of the grains evaporates quickly and in vicinity of the outlet, it should probably be considered for other types of kilns (e.g.",
"kilns devoted to fertilizer production).",
"Our simple model can be used in industry to help the design of efficient kilns.",
"Indeed it gives an efficiency criterion which estimates the optimal kiln length, i.e.",
"the length of the kiln which allows to reach a given final temperature.",
"Thus it helps the design kilns which are really adapted to the intended application.",
"We thank Bogdan Cazacliu, Yannick Descantes, Nicolas Roquet, Riccardo Artoni and Erwan Hamard for fruitful discussions and Jean-Marc Paul for technical assistance.",
"We are indebted to Andrew Hobbs for a critical reading of the manuscript and for kindly providing us with unpublished numerical results."
]
] | 1612.05762 |
[
[
"Simultaneous microscopic description of nuclear level density and\n radiative strength function"
],
[
"Abstract Nuclear level density (NLD) and radiative strength function (RSF) are described simultaneously within a microscopic approach, which takes into account the thermal effects of the exact pairing as well as the giant resonances within the phonon-damping model.",
"The good agreement between the results of calculations and experimental data extracted by the Oslo group for $^{170, 171, 172}$Yb isotopes shows the importance of exact thermal pairing in the description of NLD at low and intermediate excitation energies and invalidates the assumption based on the Brink-Axel hypothesis in the description of the RSF."
],
[
"Simultaneous microscopic description of nuclear level density and radiative strength function N. Quang Hung$^{1}$ [email protected] N. Dinh Dang$^{2,3}$ [email protected] L.T.",
"Quynh Huong$^{4,5}$ 1) Institute of Research and Development, Duy Tan University, K7/25 Quang Trung, Danang City, Vietnam 2) Quantum Hadron Physics Laboratory, RIKEN Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, Wako City, 351-0198 Saitama, Japan 3) Institute for Nuclear Science and Technique, Hanoi, Vietnam 4) Department of Natural Science and Technology, University of Khanh Hoa, Nha Trang City, Khanh Hoa Province, Vietnam 5) Faculty of Physics and Engineering Physics, Ho Chi Minh University of Science, Ho Chi Minh City, Vietnam Nuclear level density (NLD) and radiative strength function (RSF) are described simultaneously within a microscopic approach, which takes into account the thermal effects of the exact pairing as well as the giant resonances within the phonon-damping model.",
"The good agreement between the results of calculations and experimental data extracted by the Oslo group for $^{170, 171, 172}$ Yb isotopes shows the importance of exact thermal pairing in the description of NLD at low and intermediate excitation energies and invalidates the assumption based on the Brink-Axel hypothesis in the description of the RSF.",
"21.60.-n, 21.60.Jz, 21.10.-k, 24.10.PaSuggested keywordsThe rapid decrease in level spacing between the excited states as the excitation energy increases to several MeV leads to an exponential increase in the level densities and transition probabilities between the excited levels in the medium and heavy nuclei.",
"In this condition it is impractical to deal with an individual state.",
"Instead, it is meaningful and convenient to consider the average properties of nuclear excitations.",
"Two main quantities, which are often employed to describe these properties, are the nuclear level density (NLD) and radiative $\\gamma $ -ray strength function (RSF).",
"The NLD is defined as the number of excited levels per unit of excitation energy $E^{*}$ , whereas the RSF is the average transition probability per $\\gamma $ -ray energy $E_\\gamma $ .",
"The NLD provides the information on several properties of an atomic nucleus, namely the pairing correlations and nuclear thermodynamic properties such as temperature, entropy, heat capacity, etc. [1].",
"The RSF reveals the characteristics of average nuclear electromagnetic properties [2].",
"These two quantities have important contributions in the study of low-energy nuclear reactions and nuclear astrophysics such as the calculation of the stellar reaction rates and the description of nucleosynthesis in stars [3], [4].",
"The study of NLD and RSF has therefore been one of the most important topics in nuclear structure physics.",
"It became particularly attractive after the recent developments of the experimental technique proposed by Oslo's group (the Oslo method), which is able to extract simultaneously both NLD and RSF from the primary $\\gamma $ -decay spectrum of the residual compound nucleus created in the transfer and/or inelastic scattering reactions [5], [6], [7].",
"These experimental data also serve as a good testing ground for all the present theoretical approaches to NLD and RSF.",
"Although the concepts of NLD and RSF are rather old [8], [2], a unified theory, which can describe simultaneously and microscopically both the NLD and RSF is still absent so far.",
"The NLD can be described quite well within the finite-temperature shell model quantum Monte-Carlo method [9], but this method is time consuming when it is applied to heavy nuclei.",
"Regarding the $\\gamma $ -strength functions, which involve giant resonances and the related RSF, they are beyond the scope of this method.",
"The Hartree-Fock BCS [10] and Hartree-Fock-Bogolyubov plus combinatorial method (HFBC) [11] have provided a global description of NLD and might be considered as the most microscopic theories for the NLD up to date.",
"However, they both violate the particle number.",
"Consequently, to fit the experimental data, the NLD predicted by these theories has to be renormalized by using two parameters, whose values are extracted from the experimental analysis of the cumulative number of levels and s-wave neutron spacing at the neutron binding energy [Eq.",
"(9) of [12]].",
"For those nuclei, whose experimental data are not available, the predictive power of these theories is questionable.",
"Concerning the RSF, there have been few phenomenological models such as the Kadmenskij-Markushev-Furman model (KMF) [13] and the generalized Lorentzian (GLO) [14] model, and only one microscopic approach, which is the quasiparticle random-phase approximation (QRPA) [15].",
"The KMF and GLO use several parameters such as the energy, cross section, width, centroid of $E1$ , $E2$ , and $M1$ resonances, whose values are found by fitting to the experimental RSF.",
"Within the QRPA, the $\\gamma $ -strength function is calculated based on the normalized Lorentzian distribution, from which the resonance width and energy of the giant dipole resonance (GDR) are extracted.",
"The $E1$ -strength functions for 3,317 nuclei were extensively calculated within the QRPA and the results have been uploaded on RIPL-3 database [16].",
"Because the QRPA calculations were performed only for the $E1$ strengths, the results obtained within this model have not been adjusted to the experimental RSF, which consists of $E1$ as well as $E2$ , and $M1$ strengths.",
"Moreover, the predictions within the KFM and GLO models have shown that, to fit the experimental data of the RSF at the low $E_\\gamma $ , the width of the GDR should depend on temperature [13], [14].",
"Since the GDR width obtained within the QRPA is temperature-independent, the predicted $\\gamma $ -strength functions cannot describe the experimental data unless a normalization is applied for data fitting.",
"It is therefore highly desirable to develop a unified microscopic theoretical approach, which can simultaneously describe both the NLD and RSF.",
"This approach should employ only the parameters taken over from previous calculations without introducing new parameters.",
"It has been shown that thermal pairing is crucial in the description of the NLD [17], [18] and $E1$ strength function at the excitation energies below the particle-threshold energy [19], [20].",
"Moreover, as mentioned above, the temperature dependence of the GDR width is also important for the description of the RSF.",
"In the present paper we propose, for the very first time, a theoretical approach, which takes into account both the effects of exact thermal pairing and temperature-dependent resonance width.",
"Within our approach, thermal pairing is treated based on the eigenvalues ${\\cal E}_S$ , obtained by diagonalizing the pairing Hamiltonian [21] $H = \\sum _k \\epsilon _k(a_{+k}^\\dagger a_{+k} + a_{-k}^\\dagger a_{-k}) - G\\sum _{kk^{\\prime }} a_{+k}^\\dagger a_{-k}^\\dagger a_{-k^{\\prime }}a_{+k^{\\prime }} $ at zero temperature and different numbers of unpaired particles (seniorities) $S$ .",
"Here, $a_{\\pm k}^\\dagger (a_{\\pm })$ are the creation (annihilation) operators of a nucleon with angular momentum $k$ (in the deformed basis), projection $m_{\\pm k}$ , and energy $\\epsilon _k$ , the total seniorities $S$ are equal to $0, 2,..., \\Omega $ (number of single-particle levels) for a system with an even number of particles and $1, 3,...,\\Omega $ for a system with an odd number of particles.",
"These exact eigenvalues are then used to construct the partition function of the canonical ensemble (CE) (See, e.g., Eq.",
"(7) of Ref. [22]).",
"Knowing the partition function, one can easily calculate all the thermodynamic quantities such as free energy ${\\cal F}$ , total energy ${\\cal E}$ , entropy ${\\cal S}$ , heat capacity ${\\cal C}$ , and thermal pairing gap $\\Delta $ [17], [19].",
"Because of the limitation by the size of the matrix to be diagonalized, the exact solutions of the pairing Hamiltonian are limited to the levels around the Fermi surface (truncated levels).",
"To find the total partition function of the whole system, the exact CE partition function of the truncated levels is combined with those obtained within the independent-particle model (IPM) [23] for the levels beyond the truncated space, where the independent motion of nucleons is assumed (that is without pairing).",
"The total partition function is then given as the sum of the exact CE partition function for the truncated levels and the IPM partition function for the levels beyond the truncated region.",
"The latter is obtained as the difference between the partition function of the entire single-particle spectrum (from the bottom of the single-particle potential to the closed shell $N=126$ ) and that of the truncated levels, for which exact pairing is taken into account [17], [18].",
"By using the inverse Laplace transformation of the partition function [1], one obtains the density of state $\\omega (E^*)$ at excitation energy $E^*$ as $\\omega (E^*) = {e^{\\cal S}}/(T\\sqrt{2\\pi {\\cal C}})$ [24].",
"The total NLD $\\rho (E^*)$ is obtained from the state density $\\omega (E^*)$ via the relation $\\rho (E^*) = {\\omega (E^*)}/(\\sigma \\sqrt{2\\pi })$ [25], where $\\sigma $ is the spin cut-off parameter.",
"In axially deformed nuclei, there are two spin cut-off parameters, namely the perpendicular $\\sigma _{\\bot }={\\cal I}_{\\bot }T/\\hbar ^2$ and parallel $\\sigma _{\\Vert }={\\cal I}_{\\Vert }T/ \\hbar ^2$ ones, associated with the moments of inertia perpendicular (${\\cal I}_{\\bot }$ ) and parallel (${\\cal I}_{\\Vert }$ ) to the nuclear symmetry axis.",
"Based on the limit of rigid body with the same density distribution as of the nucleus, $\\sigma _\\bot $ is empirically given in the form $\\sigma _\\bot ^2 \\approx 0.015A^{5/3}T$ [26], whereas $\\sigma _{\\Vert }$ is expressed in terms of $\\sigma _\\bot $ as $\\sigma _{\\Vert }=\\sigma _\\bot \\sqrt{(3-2\\beta _2)/(3+\\beta _2)}$ [27] with $\\beta _2$ and $A$ being the quadrupole deformation parameter and mass number, respectively.",
"The collective vibrational and rotational excitations, not included in the pairing Hamiltonian, also significantly increase the NLD.",
"These increases are expressed in terms of the vibrational $k_{\\text{vib}}$ and rotational $k_{\\text{rot}}$ enhancement factors, defined as the ratio between the “correct\" NLD including all degrees of freedom and the NLD where the collective vibration and rotation are respectively absent [27], [28], [29].",
"Their explicit forms are given based on the empirical formulas as $k_{\\text{vib}}={\\exp }[0.0555 A^{2/3}T^{4/3}]$ [29] and $k_{\\text{rot}}=(\\sigma _\\bot ^2-1)/[1+e^{(E^*-U_C)/D_C}]+1$ , where $E^*$ is the excitation energy obtained within the exact CE of the pairing Hamiltonian plus the IPM (EP+IPM), whereas $D_C$ and $U_C$ are given as $D_C=1400\\beta _2^2 A^{-2/3}, U_C=120\\beta _2^2 A^{1/3}$ [27].",
"An alternative treatment of $k_{\\text{vib}}$ based on the generalized boson partition function has been reported in Ref.",
"[12], where the coherent particle-hole ($ph$ ) configurations forming the collective phonons are separated from the incoherent ones to avoid double counting.",
"The distribution of $k_{{\\rm vib}}$ found in this way in the region of $E^* < $ 30 MeV is quantitatively equivalent to the empirical formula used in the present paper.",
"The final total NLD, including the effects of vibrational and rotational enhancements, is given as $\\rho (E^*) = k_{\\text{rot}}k_{\\text{vib}}{\\omega (E^*)}/{(\\sigma _{\\Vert }\\sqrt{2\\pi })}$ [27], [30].",
"The RSF $f_{X\\lambda }(E_\\gamma )$ for the electric ($X=E$ ) or magnetic ($X=M$ ) excitations with multipolarity $\\lambda $ is calculated via the $X\\lambda $ strength function $S_{X\\lambda }(E_\\gamma )$ .",
"In the phenomenological models, a Lorentzian is used for the strength function $S_{X\\lambda }(E_\\gamma )$ with an approximated resonance width for $E1$ excitations (the KMF) as a function of $T^2$ , whereas the widths for $M1$ and $E2$ excitations take their values at $T=$ 0 as $T$ varies (See Eqs.",
"(14) – (17) in Ref.",
"[6] or Eqs.",
"(9) – (11) in Ref.",
"[31] ).",
"These assumptions are generally incorrect because the giant resonance widths are known to be temperature-dependent, but the $T^2$ dependence of the $E1$ resonance width, which the KMF borrows from the collisional damping model, is a good approximation only up to $T\\sim $ 1 MeV (See Fig.",
"10 in Ref.",
"[33]).",
"Moreover, the effect of thermal pairing at low $T$ was completely neglected in these phenomenological models.",
"In the present work, we calculate the strength function $S_{X\\lambda }(E_\\gamma )$ within the Phonon Damping Model (PDM), where the temperature-dependent resonance width $\\Gamma _{X\\lambda }(T)$ is obtained microscopically, including the effect of non-vanishing thermal pairing [32].",
"Moreover, instead of using the approximate pairing as in Ref.",
"[32], we employ the exact CE pairing mentioned above.",
"The formalism of the PDM with exact pairing at zero and finite temperatures has been reported in Ref.",
"[20].",
"The PDM has also been discussed in a series of papers, whose most recent review is given in Sec.",
"3.5 of [33].",
"The resonance width in the PDM is the sum of the quantal width $\\Gamma _Q$ caused by coupling the collective giant excitations to the non-collective $ph$ configurations at zero and non zero $T$ , and the thermal width $\\Gamma _T$ caused by coupling of giant resonances to $pp$ and $hh$ configurations at $T\\ne $ 0 (See Eqs.",
"(1a) – (1c) in [32]).",
"The model has two parameters $F_1^{(\\lambda )}$ and $F_2^{(\\lambda )}$ for the couplings to $ph$ , and $pp$ ($hh$ ) configurations, respectively.",
"The value of $F_1^{(\\lambda )}$ is chosen to reproduce the resonance width $\\Gamma _{X\\lambda }(T=0)$ , whereas $F_2^{(\\lambda )}$ is selected at $T=$ 0 so that the resonance energy $E_{X\\lambda }$ does not changes significantly as $T$ varies.",
"In numerical calculations in the present work, the small fluctuation of the resonance peak is neglected by setting the resonance energies $E_{X\\lambda }$ for $E1, M1$ , and $E2$ excitations at their corresponding experimental value extracted at $T=$ 0.",
"The numerical calculations are carried out for $^{170, 171, 172}$ Yb isotopes, whose single-particle spectra are taken from the axially deformed Woods-Saxon potential [34].",
"The quadrupole deformation parameters $\\beta _2$ are 0.295 for $^{170, 171}$ Yb and 0.296 for $^{172}$ Yb, whereas other parameters of the Woods-Saxon potential are the same as those reported in Refs.",
"[17], [18].",
"The values of the pairing interaction parameter $G$ for neutrons and protons are chosen so that the exact neutron and proton pairing gaps obtained at $T=0$ reproduce the corresponding experimental values extracted from the odd-even mass formulas.",
"The diagonalization of the pairing Hamiltonian is carried out for 12 doubly degenerate single-particle levels with 6 levels above and 6 levels below the Fermi surface.",
"A set of total 73,789 eigenstates for each type of particles is obtained and employed to construct the exact CE partition function.",
"By using Eqs.",
"(11) and (12) of Ref.",
"[19], the exact CE chemical potential and pairing gap are calculated, from which one obtains the quantities that mimic the “exact\" quasiparticle energy $E_k$ , the coefficients $u_k$ and $v_k$ of the Bogolyubov transformation between particles and quasiparticles, as well as the quasiparticle occupation numbers $n_k$ based on their conventional definitions (See, e.g., Eqs.",
"(3), (4), and (13) of Ref.",
"[19]).",
"These quantities are used as inputs in the RSF calculations within the PDM for the levels with pairing around the Fermi surface, whereas for the remaining spectrum, where $u_k=$ 1 (0) and $v_k=$ 0 (1) for $k= p$ $(h)$ according to the IPM, one has $E_k = |\\epsilon _k - \\epsilon _F|$ , $n_p = f_p$ and $n_h = 1-f_h$ with $\\epsilon _F$ and $f_k$ being the Fermi energy and the single-particle occupation number described by the Fermi-Dirac distribution at finite $T$ , respectively.",
"Figure: (Color online) Neutron and proton pairing gaps Δ\\Delta [(a) – (c)] as functions of TT and total level densities ρ\\rho [(d) – (f)] as functions of E * E^{*} obtained within the EP+IPM in comparison with predictions of HFBC calculations for the positive and negative parities and the experimental data for 170,171,172 ^{170, 171, 172}Yb nuclei.The results of the exact neutron (solid lines) and proton (dotted lines) gaps as functions of $T$ are plotted in Figs.",
"REF (a) – REF (c).",
"It is seen in these figures that the exact gaps decrease with increasing $T$ and remain finite up to $T$ as large as 3 MeV, well above the critical temperature $T_c\\sim 0.57\\Delta (T=0)$ , where the pairing gap collapses within the approximate theories such as the BCS one.",
"A slight increase in the exact neutron gap at low $T < 0.5$ MeV is seen in $^{171}$ Yb because of the blocking effect from the odd neutron [35].",
"Owing to this non-vanishing pairing gaps, the NLDs obtained within the EP+IPM (solid lines) agree well with the experimental data for all nuclei considered in the present paper as seen in Figs.",
"REF (d) – REF (f).",
"These panels also show that the NLDs obtained within the EP+IPM almost coincide with results of the global microscopic calculations within the HFBC for both negative (dashed lines) and positive (dotted lines) parities, whose values are taken from the RIPL-3 database [16].",
"However, as has been mentioned above, to have a good description of the experimental data the NLDs obtained within the HFBC have to be renormalized based on two phenomenological parameters, spoiling their microscopic nature.",
"Moreover, since the HFBC was derived based on the partition function of the incoherent $ph$ states built on top of the HFB single-particle spectra, it is certainly not able to predict the NLD in the region of high excitation energy, where the contributions of the $pp$ , $hh$ , as well as of higher states like $2p2h, 3p3h$ , etc.",
"become significant.",
"Meanwhile, within the EP+IPM, the exact CE partition function is obtained from the direct diagonalization of the matrix elements of the Hamiltonian, which consist of all possible couplings between the $ph$ , $pp$ and $hh$ states.",
"Therefore, this exact CE partition, when combined with that of the IPM, is capable to describe the NLD up to high $E^*$ region.",
"The insets of Figs.",
"REF (d) – REF (f), where the NLDs obtained within the EP+IPM are compared with those obtained within the HFBC in the region 10 $\\le E^*\\le $ 30 MeV, clearly show that the former are significantly higher than the latter.",
"The merit of the EP+IPM is also in the fact that, beyond Woods-Saxon mean field, it uses only two parameters, namely the monopole pairing strength parameters $G$ for protons and neutrons, which are adjusted to fit the corresponding experimental gaps at $T=0$ .",
"Figure: (Color online) Radiative strength functions [(a) – (c)] obtained within the EP+PDM in comparison with experimental data for 170,171,172 ^{170, 171, 172}Yb nuclei, and the corresponding total strength functions [(d) – (f)] together with their components for E1E1, E2E2, and M1M1 excitations as functions of E γ E_{\\gamma } at different temperatures.Shown in Fig.",
"REF are the RSF [(a) – (c)] and the sum ${\\cal S}_{PDM}(E_{\\gamma })$ of the strength functions $S_{X\\lambda }(E_{\\gamma })$ calculated within the PDM for $E1$ , $M1$ , and $E2$ resonances at several values of $T\\le $ 0.7 MeV [(d) – (f)].",
"These strength functions have been multiplied by the corresponding cross sections $\\sigma (X\\lambda )$ at their maxima and normalized by $(2\\lambda +1)$ , namely ${\\cal S}_{PDM}(E_{\\gamma })=\\sigma (E1(I))S_{E1(I)}(E_{\\gamma })/3+\\sigma (E1(II))S_{E1(II)}(E_{\\gamma })/3+\\sigma (M1)S_{M1}(E_{\\gamma })/3+\\sigma (E2)S_{E2}(E_{\\gamma })/5$ , where $E1(I)$ and $E1(II)$ correspond to the two components of the GDR determined from the photoabsorption experiments [6].",
"The values of resonance energies $E_{X\\lambda }$ , their FWHM $\\Gamma _{X\\lambda }$ , and cross sections $\\sigma (X\\lambda )$ at $T=$ 0 for $^{170, 171, 172}$ Yb are taken from Table I of Ref.",
"[6].",
"The GDR with the largest values of $\\sigma (X\\lambda )$ ($X\\lambda = E1(I), E1(II)$ ) gives the largest contribution the total strength function [Figs.",
"REF (d) – REF (f)].",
"The widths of its two components remain nearly constant at $T \\le $ 0.4 MeV and increase with $T$ at $T>$ 0.4 MeV, resulting in a significant increase in the total RSF at low $E_\\gamma < 4$ MeV as seen in Figs.",
"REF (a) – REF (c).",
"The RSFs obtained within the PDM at $T=$ 0.7 MeV agree well with the experimental data for all nuclei under consideration.",
"This value of $T$ is higher than that obtained from the fitting by using the KFM model in Ref.",
"[6], which is always below 0.4 MeV.",
"This result is very important as it invalidates the assumption of the Brink-Axel hypothesis [36], which states that the GDR built on an excited state should be the same as that built on the ground state, and based on which the experimental RSFs were extracted.",
"Based on the fitting by using the KMF model, Ref.",
"[6] has also suggested that there should appear a two-component pygmy dipole resonance (PDR) in the region 2.1 $<E_{\\gamma }<$ 3.5 MeV in $^{171}$ Yb and $^{172}$ Yb.",
"Although not reproduced in any microscopic models so far, this two-component PDR was added on top of the GDR in fitting the experimental RSF in Ref.",
"[6].",
"Within the PDM, it has been found in Ref.",
"[20] that exact pairing enhances the $E1$ strength function in the region $E_\\gamma <$ 5 MeV.",
"Including this exact pairing, which does not vanish at $T>T_c$ , the RSFs, calculated within the PDM, agree well with the experimental data [thick solid lines in Figs.",
"REF (e) and REF (f)].",
"In this way, the enhancement of the experimental RSF at low $E_\\gamma $ , which was suggested to be caused by the PDR, is explained microscopically by the effect of exact thermal pairing within the PDM.",
"In conclusion, we propose for the very first time a microscopic approach, which is able to describe simultaneously the nuclear level density and radiative $\\gamma $ -ray strength function.",
"This approach used the exact solutions of the pairing problem to construct the partition function to calculate the NLD and thermal pairing gap at finite temperature.",
"The latter is included in the PDM to calculate the RSF.",
"The good agreement between the results obtained within this approach and the experimental data for NLD and RSF in $^{170, 171, 172}$ Yb has shown that exact thermal pairing is indeed very important for the description of both NLD and RSF in the low and intermediate region of excitation and $\\gamma $ -ray energies.",
"Moreover, to have a good description of the RSF the microscopic strength function with the temperature-dependent width for the giant resonances should be used instead of the Brink-Axel hypothesis.",
"The merits of this approach are its microscopic nature and the use of only the parameters taken over from previous calculations.",
"It does not consume much computing time either as the calculation takes less than five minutes even for a heavy nucleus, and therefore can be performed on a PC.",
"The numerical calculations were carried out using the FORTRAN IMSL Library by Visual Numerics on the RIKEN supercomputer HOKUSAI-GreatWave System."
]
] | 1612.05391 |
[
[
"Phonon Bottleneck Identification in Disordered Nanoporous Materials"
],
[
"Abstract Nanoporous materials are a promising platform for thermoelectrics in that they offer high thermal conductivity tunability while preserving good electrical properties, a crucial requirement for high- effciency thermal energy conversion.",
"Understanding the impact of the pore arrangement on thermal transport is pivotal to engineering realistic materials, where pore disorder is unavoidable.",
"Although there has been considerable progress in modeling thermal size effects in nanostructures, it has remained a challenge to screen such materials over a large phase space due to the slow simulation time required for accurate results.",
"We use density functional theory in connection with the Boltzmann transport equation, to perform calculations of thermal conductivity in disordered porous materials.",
"By leveraging graph theory and regressive analysis, we identify the set of pores representing the phonon bottleneck and obtain a descriptor for thermal transport, based on the sum of the pore-pore distances between such pores.",
"This approach provides a simple tool to estimate phonon suppression in realistic porous materials for thermoelectric applications and enhance our understanding of heat transport in disordered materials."
],
[
"Introduction",
"The efficient and inexpensive conversion of heat directly into electricity is a long-sought goal with enormous potential in the clean-energy technology landscape [1], [2].",
"The engineering of thermoelectric materials, however, is particularly challenging because of the interrelation of key physical properties constituting the thermoelectric figure of merit ZT, defined as $ZT = \\frac{T\\sigma S^2}{\\kappa }$ where $\\sigma $ is the electrical conductivity, $\\kappa $ is the lattice thermal conductivity, $S$ is the Seebeck coefficient, and $T$ the temperature.",
"Nanostructuring offers a powerful way to decouple the electrical and thermal transport.",
"In most semiconductors, the numerator of ZT, also referred to as “power factor,” is maximized at relatively high carrier concentrations, so the dominant electron mean free path (MFP) can be as small as a few nanometers [3].",
"Conversely, phonons may have much larger MFPs, even on the order of microns [4].",
"Properly engineered nanostructures are therefore able to scatter phonons more effectively than electrons.",
"Porous materials offer a highly tunable platform thanks to their great degree of structural tunability including pore size, shape, and arrangement, as well as the potential for controllable uniform thin films, high temperature resilience and robust contacts.",
"As an example, the thermal conductivities of nanoporous Si have been measured in many studies with the common finding of a strong suppression of thermal transport, leading to a significant improvement in experimentally measured ZT [5], [6], [7], [8], [9], [10], [11], [12].",
"On the computational level, several models based on the Boltzmann transport equation (BTE) also have shown low thermal conductivities and revealed significant features of phonon-boundary scattering and fundamental thermal transport in nanoporous materials [13], [14], [15].",
"Preliminary attempts aiming at tuning thermal conductivity in nanoporous Si have shown that, even within ordered configurations and with pores of the same size, the pattern of the pores can have a large influence on the resulting thermal transport [16].",
"Although aligned configurations offer a robust platform for controllable experiments, pore disorder is unavoidable, especially at smaller length scales [17].",
"Recent Monte Carlo calculations [18], [19] investigated thermal transport in disordered porous materials with circular pores and concluded that the density of pores along the heat flux direction has a significant influence on thermal conductivity.",
"In this paper, we expand on this concept by developing a method that identifies the actual set of pores representing the highest local resistance to phonon transport.",
"To this end, we use the recently developed first-principles BTE solver [20] to perform thermal transport calculations in random-pore configurations with pores of circular and square shapes.",
"Then, we establish a correlation between the phonon suppression and the pore arrangement within a given configuration, leading to the identification of the pores constituting the phonon bottleneck.",
"Upon introducing a simple descriptor representing the strength of this collection of pores, we find a correlation between such a parameter and the effective thermal conductivity $\\kappa _{eff}$ .",
"This work can be potentially used to estimate the degree of phonon suppression in realistic nanoporous samples while avoiding the computational burden of solving the BTE.",
"Our computational approach is based on our recent implementation of the BTE for phonons, which under the relaxation time approximations, reads as [15] $\\begin{split}\\Lambda \\mathbf {\\hat{s}}(\\Omega ) \\cdot \\nabla T(\\mathbf {r},\\Omega ,\\Lambda ) + T(\\mathbf {r},\\Omega ,\\Lambda )=\\\\ \\gamma \\int \\frac{K(\\Lambda ^{\\prime })}{\\Lambda ^{\\prime 2}}<T(\\mathbf {r},\\Omega ^{\\prime },\\Lambda ^{\\prime })>d\\Lambda ^{\\prime },\\end{split}$ where $K(\\Lambda )$ is the bulk MFP distribution, $T(\\mathbf {r},\\Omega ,\\Lambda )$ is the effective temperature associated to phonons with MFP $\\Lambda $ and direction, $\\mathbf {\\hat{s}}(\\Omega )$ , denoted by the solid angle $\\Omega $ .",
"The term $\\gamma = \\left[\\int K(\\Lambda )/\\Lambda ^2 d\\Lambda \\right]^{-1}$ is a bulk material property, and $<.>$ is an angular average.",
"The RHS of Eq.Eq:1 is the effective lattice temperature, a quantity describing the average phonon energy.",
"The term $K(\\Lambda )$ is obtained by using harmonic and anharmonic forces in connection with density functional theory [21], [4].",
"The spatial discretization of Eq.Eq:1 is achieved by the finite-volume (FV) method.",
"The simulation domain is discretized by means of an unstructured mesh, generated by GMSH [22].",
"The phonon BTE requires the solid angle discretization to account for different phonon directions.",
"We use the discrete ordinate method (DOM), a technique that solves the BTE for each phonon direction independently and then combines the solutions by an angular integration [23].",
"As Si is a nongray material, i.e., has a broad MFP distribution, we need to discretize the MFP space, as well.",
"We reach convergence with 30 MFPs (uniformly distributed in log space) and 576 phonon directions.",
"The algorithm is detailed in Ref. [24].",
"The overall solution of Eq.",
"1 requires solving the BTE thousands of times, leading to an increase in the computational time.",
"However, our solver has been conveniently parallelized and each configuration takes only a few minutes with a cluster of 32 nodes.",
"The walls of the pores are assumed diffusive, a condition that translates into $\\begin{split}T_b = - \\frac{\\int _{\\Omega ^+}\\int \\left(K(\\Lambda )/\\Lambda \\right) T(\\mathbf {r},\\Omega ,\\Lambda )\\mathbf {\\hat{s}}(\\Omega )\\cdot \\mathbf {\\hat{n}}\\,d\\Omega d\\Lambda }{\\int _{\\Omega ^-}\\int \\left(K(\\Lambda )/\\Lambda \\right) \\mathbf {\\hat{s}}(\\Omega )\\cdot \\mathbf {\\hat{n}} \\, d\\Omega d\\Lambda },\\end{split}$ where $\\Omega ^-$ and $\\Omega ^+$ are the solid angle for incoming and outgoing phonons with respect to the contact with normal $\\mathbf {\\hat{n}}$ .",
"Once Eq.Eq:1 is solved, thermal flux is computed via $\\mathbf {J}(\\mathbf {r}) = 3\\int K(\\Lambda )/\\Lambda <T(\\mathbf {r},\\Omega ,\\Lambda )\\mathbf {\\hat{s}}(\\Omega )> d\\Lambda $ .",
"The effective thermal conductivity is obtained by using Fourier's law, i.e., $\\kappa _{bte} = (L/\\Delta T) \\int _{hot} <\\mathbf {J}(\\mathbf {r},\\Omega ,\\Lambda )\\cdot \\mathbf {\\hat{n}}> dS $ , where $\\Delta T$ = 1 K is the applied temperature and K is the distance between the hot and cold contacts (or the size of the unit cell).",
"To focus on phonon size effects, we normalize the thermal conductivity by its diffusive value, i.e., $\\kappa _{eff} = \\kappa _{bte}\\left(\\kappa _{bulk}/\\kappa _{fourier}\\right)$ , where $\\kappa _{fourier}$ is the thermal conductivity computed by the diffusive heat equation and $\\kappa _{bulk}$ = 156 Wm$^{-1}$ K$^{-1}$ is the bulk thermal conductivity [25].",
"Our approach has been validated against experiments on porous silicon [6], [26] and, more recently, on silicon labyrinths [27].",
"We first compute thermal transport in aligned configurations, which we will refer to as “aligned circular” (AC) and “aligned square” (AS).",
"The unit cell comprises a single pore and is a square with size L = 10 nm.",
"Heat flux is enforced by applying a difference of temperature $\\Delta T$ = 1 K along the $x$ direction.",
"The porosity is fixed at $\\phi = 0.25$ , and periodic boundary conditions are applied throughout.",
"The computed values for $\\kappa _{eff}$ are in both cases around 10 W m$^{-1}$ K $^{-1}$ , considerably lower than $\\kappa _{bulk}$ .",
"The magnitude of heat flux, shown in Fig.",
"*Fig:10a and in Fig.",
"*Fig:10b for AC and AS, respectively, indicates that phonon travel mostly near the spaces between pores perpendicular to the applied temperature gradient.",
"Figure: Normalized magnitude of thermal flux for the (a) AC, (b) AS, (c) DC and (d) DS cases.",
"The temperature gradient is imposed along the xx direction.",
"Phonons prefer to travel in the spaces between the pores, as highlighted by the red areas.",
"In all the configurations the pores arrangement is periodic in both xx and yy directions.",
"The blue line represents the phonon bottleneck.For random-pore (or disordered) configurations, the size of the unit cell is chosen to be L = 40 nm, four times as large as that for the aligned cases, in order to generate significant disorder in the pores arrangement.",
"Sixteen nonoverlapping pores are randomly placed while keeping the porosity fixed to $\\phi $ = 0.25, thus allowing a direct comparison with the aligned counterparts.",
"We note that the material is still periodic in that pores crossing the border of the unit cell are repeated in the adjacent unit cells.",
"We compute $\\kappa _{eff}$ for two hundred arrangements, one hundred for each shape, which we refer to as “disordered circle” (DC) and “disordered square” (DS).",
"The magnitude of thermal flux for two configurations is shown in Fig.",
"*Fig:10c and Fig.",
"*Fig:10d, respectively.",
"We note that the formation of high-flux regions is irregular as it depends on the pore configuration.",
"According to Fig.",
"*Fig:30a and Fig.",
"*Fig:30b, respectively, the DC and DS cases are found to have average $\\kappa _{eff}$ values 15 $\\%$ and 30 $\\%$ lower than that of their aligned counterparts.",
"Intuitively, the combined effect of small bottleneck and vanishing view factor significantly lowers $\\kappa _{eff}$ .",
"In the next section, we will analyze in detail the correlation between the pores arrangement and $\\kappa _{eff}$ .",
"In previous work [16], we reported that $\\kappa _{eff}$ in nanoporous materials is dictated by the view factor and the pore-pore distances.",
"We note that the view factor is a geometrical feature that describes the ability of a ray to travel across the simulation domain without intercepting the pores [28].",
"In random-pore configurations, the view factor vanishes because of the disordered pores blocking all the direct paths.",
"It is natural, therefore, to speculate whether the average pore-pore distance in the disordered configurations is correlated with $\\kappa _{eff}$ .",
"However, after a regression analysis, we conclude that unlike for the ordered case, such a parameter has only a marginal role for the disordered systems.",
"In fact, rigorously speaking, only the interpore spaces perpendicular to heat flux matter.",
"In order to identify the phonon bottleneck we then analyze the pores configuration in terms of graphs.",
"To this end, we first compute the pores first-neighbor map, as elaborated in the following.",
"A given periodic configuration has a finite set of pores $P=\\lbrace P_0,P_1,...\\,P_{N-1}\\rbrace $ , where $N$ is the number of pores.",
"Given two pores $P_{\\alpha }$ and $P_{\\beta }$ , we define them to be neighbors if, when moving $P_{\\alpha }$ toward $P_{\\beta }$ , there is no collision with the surrounding pores.",
"The intersection among polygons is computed by the package PyClipper [29].",
"After repeating this procedure for all pore pairs, we obtain a first-neighbor map as shown in Fig. *Fig:20a.",
"We then build the set of edges $E=\\lbrace E_0,E_1,...\\,E_{M-1}\\rbrace $ , where $M$ is the number of edges.",
"Each edge connects two neighbor pores, say $P_\\alpha $ and $P_{\\beta }$ , and points toward increasing $y$ coordinates, i.e., $P_{\\alpha }$ is connected to $P_{\\beta }$ only if $\\left( \\mathbf {C}_\\beta - \\mathbf {C}_\\alpha \\right)\\cdot \\mathbf {y} > 0 $ , where $\\mathbf {C}_n$ is the circumcenter of the pore $P_n$ .",
"The resulting graph, $G(P,E)$ , is directed in that its edges are unidirectional.",
"We define a path in $G(P,E)$ as a sequence of vertices $p_{\\mu \\nu }=\\lbrace v_0= \\mu ,v_1,...\\,v_{K-1}=\\nu \\rbrace $ such that $\\lbrace v_i,v_{i+1}\\rbrace \\in E$ for $0\\le i<K-1$ , where $K$ is the length of the path.",
"An elementary circuit is a path where the only repeating vertexes are the first and the last ones, i.e, $\\mu =\\nu $ .",
"In a complete directed graph, the number of distinct elementary circuits, simply referred to as cycles, is $\\begin{split}S=\\sum _{i=1}^{N-1}\\binom{N}{N-i+1}(N-i)!,\\end{split}$ which grows faster than $2^N$ .",
"Although in our case $G(P,E)$ is not complete, the number of cycles can easily reach a few thousand.",
"Here we identify all possible cycles by using Johnson's algorithm, which has a time bound $O((N+M)(C+1))$ [30], where $C$ is the number of cycles.",
"As the pores are identified uniquely within the unit cell, every pore shares the same label with its periodic counterpart.",
"Consequently, the first and last nodes of a cycle, although having the same identifier, belong to two different unit cells.",
"For our purposes, we select only cycles whose extreme nodes share the same $y$ coordinate, as exemplified in Fig. *Fig:20b.",
"By doing so, we guarantee that the cycles are perpendicular to heat flow and, therefore, are suitable for the identification of a descriptor of thermal conductivity, as explained in the next section.",
"Figure: (a) Example of a first-neighbor map.",
"Each pore in the unit cell is uniquely labeled.",
"The bottleneck is highlighted by the orange line and, in (b), is represented by an elementary circuit, or cycle.To identify the bottleneck for each configuration we develop the following algorithm: For each cycle, $\\lbrace C\\rbrace =C_0, C_s ... C_{S-1}$ , we compute the interpore distances of its constituting pores, $\\lbrace R\\rbrace =R_0,R_k ... R_{K-1}$ .",
"Then, we compute the sum of such distances, i.e., $D_s=\\sum _k R_k$ .",
"From the previous point, we have the set $\\lbrace D\\rbrace =D_0, D_s ... D_{S-1}$ .",
"The bottleneck is then $g = \\mathrm {min}\\lbrace D\\rbrace $ .",
"Figure: Distribution of κ eff \\kappa _{eff} for the (a) DC and (b) DS cases.",
"The straight, horizontal lines represent the aligned counterparts.",
"Distribution of gg for the (c) DC and (d) DS cases.",
"The vertical lines refer to the bottleneck for the aligned cases.The phonon bottleneck is the smallest of the sum of pore-pore distances among all the cycles in a configuration.",
"The effectiveness of $g$ in describing nanoscale thermal transport in such structures can be estimated by the Spearman correlation rank ($r_s$ ), a quantity that indicates how two variables are monotonically correlated to each other [31].",
"The first step in computing $r_s$ is ranking the values for $\\kappa _{eff}$ and $g$ and collecting the result via the vectors $\\mathbf {K}$ and $\\mathbf {G}$ , respectively.",
"Then, we compute $\\begin{split}r_s = 1-\\frac{6\\sum _i^n \\left(G_i - K_i\\right)}{n(n^2-1)},\\end{split}$ where $n$ = 100 is the number of simulations for each shape.",
"For both DC and DS cases, we obtain a significant Spearman correlation (higher than 0.63), suggesting that $g$ can be used as a good descriptor.",
"We use this knowledge to understand the $\\kappa _{eff}$ distributions for the DC and DS cases in relation to the aligned cases.",
"According to simple geometric considerations, the bottlneck for the aligned cases is simply $g_{AC} = 4L(1-2\\sqrt{\\phi /\\pi })=$ 17.44 nm and $g_{AS}= 4L(1-\\sqrt{\\phi })$ = 20 nm.",
"As shown in Fig.",
"*Fig:30c, for DC, the average $g$ is around $g_{AC}$ ; for DS, almost all the configurations have $g$ smaller than that of AS [as shown in Fig.",
"*Fig:30d], due to the square edges.",
"These results reflect the relative trend in $\\kappa _{eff}$ between the aligned and disordered cases, corroborating the use of $g$ as a valid descriptor for thermal transport.",
"Moreover, we note that most of the bottlenecks have a number of pores ($\\sim $ 6-7) higher than that of their aligned counterparts (4).",
"This result confirms that smaller $\\kappa _{eff}$ , within configurations with the same porosity, can be achieved with anisotropic pore lattices, where the density of pores is higher along the Cartesian direction orthogonal to the applied temperature gradient [18], [19].",
"The introduction of a simple descriptor can be used to estimate the ranking of $\\kappa _{eff}$ among different samples with disordered pores, supporting experiments on realistic materials [17], [8].",
"In summary, by performing calculations of thermal transport in disordered porous materials we have quantified the effect of the randomness in pore arrangement on the thermal conductivity.",
"Furthermore, we have devised a method to identify the set of special pores composing the phonon bottleneck, potentially empowering experimentalists with a simple tool to assess thermal conductivity in disordered porous materials."
]
] | 1612.05669 |
[
[
"Two weighted estimates for generalized fractional maximal operators on\n non homogeneous spaces"
],
[
"Abstract Let $\\mu$ be a non-negative Borel measure on $R^d$ satisfying that the measure of a cube in $R^d$ is smaller than the length of its side raised to the $n$-th power, $0<n\\leq d$.",
"In this article we study the class of weights related to the boundedness of radial fractional type maximal operator associated to a Young function $B$ in the context of non-homogeneous spaces related with the measure $\\mu$.",
"This type of maximal operators are the adequate operators related with commutators of singular and fractional operators.",
"Particularly, we give an improvement of a two weighted result for certain fractional maximal operator proved in [26]."
],
[
"Introduction and statements of the main results",
"Let $\\mu $ be a non-negative upper Ahlfors $n$ -dimensional measure on $\\mathbb {R}^d$ , that is, a Borel measure satisfying $\\mu (Q)\\le l(Q)^n$ for any cube $Q\\subset \\mathbb {R}^n$ with sides parallel to the coordinate axes, where $l(Q)$ stands for the side length of $Q$ and $n$ is a fixed real number such that $0<n\\le d$ .",
"Besides, for $r>0$ , $rQ$ will mean the cube with the same centre as $Q$ and with $l(rQ)=rl(Q)$ .",
"In the last decades, this measure have proved to be adequate for the development of many results in Harmonic Analysis which were known that hold in the context of doubling measures, that is, Borel measures $\\nu $ for which there exists a positive constant $D$ such that $\\nu (2Q)\\le D \\nu (Q)$ for every cube $Q\\subset \\mathbb {R}^d$ .",
"For example, many interesting results related with different operators and spaces of functions with non doubling measures can be found in [15], [16], [13], [24], and [14] between a vast bibliography on this topic.",
"In [26] the authors studied two weighted norm inequalities for a fractional maximal operator associated to a measure $\\mu $ satisfying condition (REF ).",
"Concretely, they considered the following version of the fractional maximal operator defined, for $0\\le \\alpha <1$ , by ${M}_{\\alpha }f(x)= \\sup _{Q\\ni x}\\frac{1}{\\mu (5Q)^{1-\\alpha }}\\int _Q|f(y)|d\\mu (y),$ and proved the following result.",
"Theorem 1.1 Let $1<p<q<\\infty $ and $0\\le \\alpha <1$ .",
"Let $(u,v)$ be a pair of weights such that for every cube $Q$ $l(Q)^{n(1-1/p)}\\mu (Q)^{\\alpha -1}u(3Q)^{\\frac{1}{q}}\\Vert v^{-\\frac{1}{p}}\\Vert _{\\Phi ,Q}\\le C$ where $\\Phi $ is a Young function whose complementary function $\\bar{\\Phi }\\in B_p$ .",
"Then $\\left(\\int _{\\mathbb {R}^d}M_{\\alpha }f(x)^q u(x)\\, d\\mu (x)\\right)^{1/q}\\le C\\left(\\int _{\\mathbb {R}^d}|f(x)|^p v(x)\\, d\\mu (x)\\right)^{1/p}.$ for every $f\\in L^p(v)$ bounded with compact support.",
"The radial Luxemburg type average in theorem above is defined by $\\Vert \\cdot \\Vert _{\\Phi ,Q}=\\inf \\lbrace \\lambda >0:\\frac{1}{l(Q)^n}\\int _{Q}\\Phi \\left(\\frac{|f(x)|}{\\lambda }\\right)\\,d\\mu (x)\\le 1\\rbrace ,$ and a Young function $B$ satisfies the $B_p$ condition, $1<p<\\infty $ , if there is a positive constant $c$ such that $\\int _c^{\\infty } \\frac{B(t)}{t^p}\\frac{dt}{t}<\\infty .$ Let us make some comments about Theorem $\\ref {chinos}$ .",
"When $\\mu $ is the Lebesgue measure and $u=v=1$ , it is easy to note that condition ($\\ref {muwtl}$ ) holds if and only if $1/q=1/p-\\alpha $ for any $\\Phi $ as in the hypothesis.",
"On the other hand, if we consider an upper Ahlfors $n$ -dimensional measure $\\mu $ and if we take $\\Phi (t)=t^{rp^{\\prime }}$ , for $1<r<\\infty $ , $1/q=1/p-\\alpha $ and $u=v=1$ in condition ($\\ref {muwtl}$ ) we have that if the following inequality holds $l(Q)^{n(1-1/p)}\\mu (Q)^{\\alpha -1}\\mu (3Q)^{\\frac{1}{p}-\\alpha } \\left(\\frac{\\mu (Q)}{l(Q)^n}\\right)^{1/rp^{\\prime }}\\le C,$ then $\\left(\\frac{l(Q)^n}{\\mu (Q)}\\right)^{1/(p^{\\prime }r^{\\prime })}\\le C$ which implies that the measure $\\mu $ satisfying the growth condition ($\\ref {mu}$ ) also satisfies the “lower\" case, that is $\\mu (Q)\\ge Cl(Q)^n$ , with a constant independent of $Q$ .",
"So, the weights $u=v=1$ are not allowed in this case unless the measure is Ahlfors, that is $\\mu (Q)\\simeq l(Q)^n$ , for every cube $Q$ .",
"Moreover, let $Mu(x)=\\sup _{Q\\ni x}\\frac{1}{\\mu (Q)}\\int _Q|u(y)|d\\mu (y).$ When $\\mu $ is the Lebesgue measure and $\\Phi (t)=t^{rp^{\\prime }}$ , it is easy to check that the pair of weights $(u,(Mu)^{p/q})$ , with $1/q=1/p-\\alpha $ , satisfies condition ($\\ref {muwtl}$ ).",
"On the other hand, suppose that this pair satisfies the same condition for a measure satisfying ($\\ref {mu}$ ) and let $u\\in A_1(\\mu )$ .",
"Thus, the following chains of inequalities holds $C&\\ge &\\frac{l(Q)^{n/p^{\\prime }}}{\\mu (Q)^{1-\\alpha -1/q}}\\left(\\frac{1}{\\mu (Q)}\\int _Qu\\, d\\mu \\right)^{1/q}\\left(\\frac{1}{l(Q)^n}\\int _Q\\left((Mu)^{p/q}\\right)^{-rp^{\\prime }/p}\\right)^{1/(rp^{\\prime })}\\\\&\\ge &\\frac{l(Q)^{n/p^{\\prime }-n/(rp^{\\prime })}}{\\mu (Q)^{1/p^{\\prime }-1/(rp^{\\prime })}}\\left(\\frac{1}{\\mu (Q)}\\int _Qu^{p/q}\\, d\\mu \\right)^{1/p}\\left(\\frac{1}{\\mu (Q)}\\int _Q\\left(u^{p/q}\\right)^{-rp^{\\prime }/p}\\right)^{1/(rp^{\\prime })}\\\\&\\ge &\\frac{l(Q)^{n/(r^{\\prime }p^{\\prime })}}{\\mu (Q)^{1/(r^{\\prime }p^{\\prime })}}\\left(\\frac{1}{\\mu (Q)}\\int _Qu^{p/q}\\, d\\mu \\right)^{1/p}\\left(\\frac{1}{\\mu (Q)}\\int _Q\\left(u^{p/q}\\right)^{-p^{\\prime }/p}\\right)^{1/(p^{\\prime })}\\\\&\\ge & \\frac{l(Q)^{n/(r^{\\prime }p^{\\prime })}}{\\mu (Q)^{1/(r^{\\prime }p^{\\prime })}},$ which implies again that $\\mu $ must be an Ahlfors measure.",
"In [7] the authors considered the radial fractional maximal function associated to an upper Ahlfors $n$ -dimensional measure $\\mu $ which is defined, for $0\\le \\alpha <n$ , by $\\mathcal {M}_{\\alpha }f(x)= \\sup _{Q\\ni x}\\frac{1}{l(Q)^{n-\\alpha }}\\int _Q |f(y)|d\\mu (y).$ In the same article they study weighted boundedness properties for $\\mathcal {M}_{\\alpha }$ on non homogeneous spaces.",
"In this paper we introduce a generalized version of the radial fractional maximal operator defined in [7], associated to a Young function $B$ .",
"This type of maximal operators are not only a generalization but also they have proved to be the adequate operators related with commutators of singular and fractional integral operators in different settings, (see for example [2], [18], [19], [20], [21], , [12] and [3]).",
"It is important to note that the examples of weights given above satisfy the condition obtained in our theorem when $\\mu $ is an upper Ahlfors $n$ -dimensional measure.",
"In this sense, when $B(t)=t$ , our result is better than the corresponding result in [26].",
"In order to state the main results we introduce some preliminaries.",
"Given a Young function $B$ , we define $L^B_\\mu (\\mathbb {R}^d)$ as the set of all measurable functions $f$ for which there exists a positive number $\\lambda $ such that $\\int _{\\mathbb {R}^d}B\\left(\\frac{|f(x)|}{\\lambda }\\right)\\,d\\mu (x)<\\infty .$ The radial fractional type maximal operator associated to a Young function $B$ is defined by $\\mathcal {M}_{\\alpha ,B}(f)(x)= \\sup _{Q\\ni x} \\; l(Q)^{\\alpha }\\Vert f\\Vert _{B,Q}, \\quad 0\\le \\alpha <n,$ where $\\Vert f\\Vert _{B,Q}=\\inf \\lbrace \\lambda >0:\\frac{1}{l(Q)^n}\\int _{Q}B\\left(\\frac{|f(x)|}{\\lambda }\\right)\\,d\\mu (x)\\le 1\\rbrace $ is the radial Luxemburg average (see §).",
"When $B(t)=t$ then $\\Vert f\\Vert _{B,Q}=\\frac{1}{l(Q)^n}\\int _Q |f|\\, d\\mu .$ When $\\alpha =0$ , we write $\\mathcal {M}_{0,B}=\\mathcal {M}_{B}$ .",
"The following theorem gives sufficient conditions for strong type inequalities for $\\mathcal {M}_{\\alpha ,B}$ on non homogeneous spaces.",
"Theorem 1.2 Let $1<p<q<\\infty $ , $0\\le \\alpha <n$ and let $\\mu $ be an upper Ahlfors $n$ -dimensional measure in $\\mathbb {R}^d$ .",
"Let $B$ be a submultiplicative Young function such that $B^{q_0/p_0}\\in B_{q_0}$ for some $1<p_0\\le n/\\alpha $ and $1/q_0=1/p_0-\\alpha /n$ , and let $\\phi $ and $\\varphi $ be two Young functions such that $C_1\\varphi ^{-1}(t)t^{\\alpha /n}\\le \\;B^{-1}(t)\\le C_2\\phi ^{-1}(t)t^{\\alpha /n}$ for some positive constants $C_1$ and $C_2$ .",
"If $A$ and $C$ are two Young functions such that $A^{-1}C^{-1}\\preceq B^{-1}$ with $C \\in B_p$ and $(u,v)$ is a pair of weights such that for every cube $Q$ $l(Q)^{\\alpha -\\frac{n}{p}}u(3Q)^{\\frac{1}{q}}\\Vert v^{-\\frac{1}{p}}\\Vert _{A,Q}\\le K$ then, for all $f\\in L_\\mu ^p(v)$ .",
"$\\Vert \\mathcal {M}_{\\alpha ,B}(f)\\Vert _{L^q_\\mu (u)}\\le C\\; \\Vert f\\Vert _{L^p_\\mu (v)},$ Remark 1.3 When $u=v=1$ and $1/q=1/p-\\alpha /n$ then condition $(\\ref {good})$ is satisfied for any upper Ahlfors $n$ -dimensional measure $\\mu $ .",
"Thus, this result is an improvement of that given in [26] in the sense that the unweighted boundedness of the operator is obtained for any measure satisfying the growh condition $(\\ref {mu})$ .",
"The same is true for the second example considered above.",
"Remark 1.4 When $B(t)=t \\log (e+t)^k$ it can be easily seen that $B$ is submultiplicative, $B^{q_0/p_0}\\in B_{q_0}$ for every $p_0, q_0>1$ and $B^{-1}(t)\\approx t^{\\alpha /n} \\frac{t^{1-\\alpha /n}}{\\log (e+t)^k}\\approx t^{\\alpha /n}\\phi ^{-1}(t),$ when $\\phi (t)=\\left(t\\log (e+t)^k\\right)^{\\frac{n}{n-\\alpha }}$ .",
"Moreover, the functions $A(t)=t^{rp^{\\prime }}$ and $C(t)=(t\\log (e+t)^k)^{(rp^{\\prime })^{\\prime }}$ satisfy $A^{-1}C^{-1}\\preceq B^{-1}.$ For $\\delta >0$ , other examples are given by $A(t)=t^{p^{\\prime }}\\log (e+t)^{(k+1)p^{\\prime }-1+\\delta }$ and $C(t)=t^{p}\\log (e+t)^{-(1+\\delta (p-1))}$ , (see ).",
"It is also important to note that Theorem 3.1 in is a special case of the previous theorem by considering $A(t)=t^{rp^{\\prime }}$ , $C(t)=t^{(rp^{\\prime })^{\\prime }}$ and $B(t)=t$ .",
"Let us make some comments about the upper Ahlfors $n$ -dimensional measure $\\mu $ satisfying (REF ).",
"It is well known that for such measures the Lebesgue differentiation theorem holds; that is, for every $f\\in L_{loc}^1(\\mathbb {R}^d)$ and a.e.",
"$x$ $\\frac{1}{\\mu (Q)}\\int _Q f(y)\\,d\\mu (y)\\rightarrow f(x),$ when $Q$ decreases to $x$ (see [25]).",
"However, if we take radial averages like those defined in (REF ) this is not longer true.",
"In fact, let us consider $\\mu $ defined by $d\\mu (t)=e^{-t^2}dt$ , which is an upper Ahlfors 1-dimensional measure, and $f(t)=e^{\\theta t^2}$ , $\\theta \\in \\mathbb {R}$ .",
"Let $x \\in \\mathbb {R}$ , then $\\lim _{r\\rightarrow 0}\\frac{1}{2r}\\int _{x-r}^{x+r}f(t)d\\mu (t)=\\lim _{r\\rightarrow 0}\\frac{1}{2r}\\int _{x-r}^{x+r}e^{(\\theta -1)t^2}\\,dt=e^{(\\theta -1)x^2}$ which differs from $f$ in a.e.",
"$x$ .",
"Given a Young function $B$ , let $h_B$ be the function defined by $h_B(s)= \\sup _{t>0}\\frac{B(st)}{B(t)}, \\;\\; 0\\le s<\\infty .$ If $B$ is submultiplicative then $h_B \\approx B$ .",
"More generally, given any $B$ , for every $s$ , $t\\ge 0$ , $B(st)\\le h_B(s)B(t)$ .",
"It is easy to proof (see [4]), that if $B$ is a Young function then $h_B$ is nonnegative, submultiplicative, increasing in $[0,\\infty )$ , strictly increasing in $[0,1]$ and $h_B(1)=1$ .",
"The following theorem gives an modular endpoint estimate for $\\mathcal {M}_{\\alpha ,B}$ on non-homogeneous spaces.",
"When $\\mu $ is the Lebesgue measure, this result was proved in .",
"Theorem 1.5 Let $0\\le \\alpha <n$ and let $\\mu $ an upper Ahlfors $n$ -dimensional measure on $\\mathbb {R}^d$ .",
"Let $B$ be a Young function and suppose that, if $\\alpha >0$ , $B(t)/t^{\\frac{n}{\\alpha }}$ is decreasing for all $t>0$ .",
"Then there exists a constant $C$ depending only on $B$ such that for all $t>0$ , $\\mathcal {M}_{\\alpha ,B}$ satisfies the following modular weak-type inequality $\\phi \\left[\\mu \\left(\\lbrace x \\in \\mathbb {R}^d: \\mathcal {M}_{\\alpha ,B}(f)(x)>t\\rbrace \\right)\\right]\\le \\;C\\;\\int _{\\mathbb {R}^n} B\\left(\\frac{|f(y)|}{t}\\right)d\\mu (y),$ for all $f\\in L_\\mu ^B(\\mathbb {R}^d)$ , where $\\phi $ is any function such that $\\phi (s)\\le C_1 \\phi _1(s)= \\left\\lbrace \\begin{array}{ll}0 & \\text{if } s=0 \\\\ \\\\\\frac{s}{h_B(s^{\\alpha /n})} & \\text{if } s>0\\\\ \\\\\\end{array}\\right.$ Remark 1.6 It is easy to see that the function $B(t)=t\\log (e+t)$ satisfies the hypothesis of the theorem above and thus $\\mu \\left(\\lbrace x \\in \\mathbb {R}^n: \\mathcal {M}_{\\alpha ,B}(f)(x)>t\\rbrace \\right)\\le \\;C\\;\\psi \\left[\\int _{\\mathbb {R}^n} B\\left(\\frac{|f(y)|}{t}\\right)d\\mu (y)\\right],$ for all $f\\in L_\\mu ^B(\\mathbb {R}^d)$ , where $\\psi =[t\\log (e+t^{\\alpha /n})]^{n/(n-\\alpha )}$ .",
"In the context of spaces of homogeneous type this last result was proved in .",
"The following result is a pointwise estimate between the radial fractional type maximal operator associated to a Young function $B$ and the radial maximal operator associated to a Young $\\psi $ related with $B$ on non homogeneous spaces.",
"Theorem 1.7 Let $0\\le \\alpha < n$ and $1<p<n/\\alpha $ .",
"Let $\\mu $ be an upper Ahlfors $n$ -dimensional measure.",
"Let $q$ and $s$ be defined by $1/q=1/p-\\alpha /n$ and $s=1+q/{p^{\\prime }}$ , respectively.",
"Let $B$ and $\\phi $ be Young functions such that $\\phi ^{-1}(t)t^{\\alpha /n}\\ge C\\; B^{-1}(t)$ and $\\psi (t)=\\phi (t^{1-\\alpha /n})$ .",
"Then for every measurable function $f$ , the following inequality $\\mathcal {M}_{\\alpha ,B}(f)(x)\\le C\\;\\mathcal {M}_{\\psi }(|f|^{p/s})(x)^{1-\\alpha /n}\\left(\\int _{\\mathbb {R}^d}|f(y)|^pd\\mu (y)\\right)^{\\alpha /n}$ holds in a.e.",
"$x$ .",
"When $\\mu $ is the Lebesgue measure, the result above was proved in [1] (see also [22] for similar multilinear versions and [9] for the case $B(t)=t$ , both in the euclidean context).",
"The next theorem gives sufficient conditions on the function $B$ in order to obtain the boundedness of $\\mathcal {M}_B$ on $L^p(\\mu )$ .",
"In the euclidean context, this result was proved in and in [23] in the framework of spaces of homogeneous type.",
"Theorem 1.8 Let $\\mu $ be an upper Ahlfors $n$ -dimensional measure.",
"Let $B$ be a Young function such that $B \\in B_p$ , then, $\\mathcal {M}_B :L^p(\\mu )\\rightarrow L^p(\\mu ).$ The following result is a fractional version of Theorem REF and gives a sufficient condition on the function $B$ that guarantees the continuity of the radial fractional type maximal operator $\\mathcal {M}_{\\alpha ,B}$ between Lebesgue spaces with non necessary doubling measures.",
"Theorem 1.9 Let $\\mu $ be an upper Ahlfors $n$ -dimensional measure.",
"Let $0<\\alpha < n$ and $1<p\\le n/\\alpha $ .",
"Let $q$ and $s$ be defined by $1/q=1/p-\\alpha /n$ and $s=1+q/{p^{\\prime }}$ respectively.",
"Let $B$ be a submultiplicative Young function such that $B^{q/p}\\in B_q$ and let $\\phi $ be a Young function such that $\\phi ^{-1}(t)t^{\\alpha /n}\\ge C\\;B^{-1}(t)$ .",
"Then $\\mathcal {M}_{\\alpha ,B}:L^p(\\mu )\\rightarrow L^q(\\mu ).$ The next theorem is very interesting since it allows us to readily find examples of $A_1$ weights on non homogeneous spaces.",
"Theorem 1.10 Given $\\alpha $ , $0<\\alpha <n$ , and a non-negative function $f$ .",
"There exists a constant $C$ such that, $\\mathcal {M}(\\mathcal {M}_\\alpha f)(x)\\le \\; C\\;\\mathcal {M}_\\alpha f(x).$ Proof: Fix a cube $Q$ .",
"We shall see that, $\\frac{1}{l(Q)^n}\\int _Q \\mathcal {M}_\\alpha f(y)\\;d\\mu (y)\\le \\;C\\;\\mathcal {M}_\\alpha f(x)\\;\\;\\;\\;\\; \\text{for a.e.",
"}\\;\\;\\;\\; x\\in Q.$ with $C$ independent of $Q$ .",
"Let $\\widetilde{Q}=Q^3$ , the 3-dilated of $Q$ .",
"We write $f=f_1+f_2$ with $f_1=f\\chi _{\\widetilde{Q}}$ .",
"Then, $\\mathcal {M}_\\alpha f(x)\\le \\mathcal {M}_\\alpha f_1(x)+\\mathcal {M}_\\alpha f_2 (x)$ .",
"$\\frac{1}{l(Q)^n}\\int _Q \\mathcal {M}_\\alpha f_1 (y)d\\mu (y)=\\frac{1}{l(Q)^n}\\int _0^\\infty \\mu \\lbrace x\\in Q: \\mathcal {M}_\\alpha f_1(x)>t\\rbrace dt\\le $ $\\le \\frac{1}{l(Q)^n}\\left(\\mu (Q)R+\\int _R^\\infty \\mu \\lbrace x\\in Q: \\mathcal {M}_\\alpha f_1(x)>t\\rbrace dt\\right)$ By [7], we know that $\\Vert \\mathcal {M}_\\alpha f\\Vert _{L^{\\frac{n}{n-\\alpha },\\infty }(\\mu )}\\le \\Vert f\\Vert _{L^{1}(\\mu )}$ .",
"Then, since $\\mu (Q)\\le l(Q)^n$ $\\frac{1}{l(Q)^n}\\int _Q \\mathcal {M}_\\alpha f_1 (y)d\\mu (y)\\le R+ \\frac{c}{l(Q)^n} \\Vert f_1\\Vert _{L^1(\\mu )}^{n/{n-\\alpha }}\\int _R^\\infty \\frac{dt}{t^{n/{n-\\alpha }}}.$ By taking $R=\\frac{\\Vert f_1\\Vert _{L^1(\\mu )}}{l(Q)^{n-\\alpha }}$ , we get $\\frac{1}{l(Q)^n}\\int _Q \\mathcal {M}_\\alpha f_1 (y)d\\mu (y)\\le \\;C_{\\alpha ,n}\\; \\frac{\\Vert f_1\\Vert _{L^1(\\mu )}}{l(Q)^{n-\\alpha }}= \\frac{C_{\\alpha ,n}}{l(\\widetilde{Q})^{n-\\alpha }}\\int _{\\widetilde{Q}}f(y)d\\mu (y)\\le C_{\\alpha ,n}\\mathcal {M}_\\alpha f(x)$ for every $x \\in Q$ .",
"To deal with $\\mathcal {M}_\\alpha f_2$ is even simpler.",
"It is enough to see that, because of the fact that $f_2$ lives far from $Q$ (outside $\\widetilde{Q}$ ), for any two points $x,y \\in Q$ , we have $\\mathcal {M}_\\alpha f_2 (x)\\le C \\mathcal {M}_\\alpha f_2(y)$ , with $C$ an absolute constant.",
"Indeed if $Q_0$ is a cube containing $x$ and meeting $\\mathbb {R}^{n}\\setminus \\widetilde{Q}$ , then $Q\\subset Q_0^3$ , so that $\\frac{1}{l(Q_0)^{n-\\alpha }}\\int _{Q_0} f_2(t)d\\mu (t)\\le \\frac{3^{n-\\alpha }}{l(Q_0^3)^{n-\\alpha }}\\int _{Q_0^3}f_2(t)d\\mu (t) \\le 3^{n-\\alpha }\\mathcal {M}_\\alpha f_2(y).$ Thus $\\frac{1}{l(Q)^n}\\int _Q \\mathcal {M}_\\alpha f_2 (y)d\\mu (y)\\le \\; C \\frac{\\mu (Q)}{l(Q)^n} \\mathcal {M}_\\alpha f(x)\\le \\; C \\mathcal {M}_\\alpha f(x)$ for every $x\\in Q$ .",
"$\\square $"
],
[
"Preliminaries and auxiliar lemmas",
"A function $B:[0,\\infty )\\rightarrow [0,\\infty )$ is a Young function if it is convex and increasing, if $B(0)=0$ , and if $B(t) \\rightarrow \\infty $ as $t\\rightarrow \\infty $ .",
"We also deal with submultiplicative Young functions, which means that $B(st)\\le B(s)B(t)$ for every $s$ , $t>0$ .",
"If $B$ is a submultiplicative Young function, it follows that $B^{\\prime }(t)\\simeq B(t)/t$ for every $t>0$ Given a Young function $B$ and a cube $Q$ , we define the radial Luxemburg average of $f$ on $Q$ associated to $\\mu $ by $ \\Vert f\\Vert _{B,Q}=\\inf \\left\\lbrace \\lambda >0: \\frac{1}{l(Q)^n}\\int _QB\\left(\\frac{|f(x)|}{\\lambda }\\right)d\\mu (x)\\le 1\\right\\rbrace .$ The radial Luxemburg average has two rescaling properties which we will use repeatedly.",
"Given any Young function $A$ and $r>0$ , $\\Vert f^r\\Vert _{A,Q}=\\Vert f\\Vert _{B,Q}^{r},$ where $B(t)=A(t^r)$ .",
"By convexity, if $\\tau >1$ $\\Vert f\\Vert _{A,Q}\\le \\tau ^n \\Vert f\\Vert _{A,\\tau Q}.$ Given a Young function $B$ , the complementary Young function $\\tilde{B}$ is defined by $\\tilde{B}(t)=\\sup _{s>0}\\lbrace st-B(s)\\rbrace ,\\;\\;\\; t>0.$ It is well known that $B$ and $\\tilde{B}$ satisfy the following inequality $t\\le B^{-1}(t)\\tilde{B}^{-1}\\le 2t.$ It is also easy to check that the following version on the Hölder's inequality $\\frac{1}{l(Q)^n}\\int _{Q} |f(x)g(x)|d\\mu (x)\\le 2\\Vert f\\Vert _{B,Q}\\Vert g\\Vert _{\\tilde{B},Q}$ holds.",
"Moreover, there is a further generalization of the inequality above.",
"If $A$ , $B$ and $C$ are Young functions such that for every $t\\ge t_0>0$ , $B^{-1}(t)C^{-1}(t)\\le c\\, A^{-1}(t),$ then, the inequality $\\Vert fg\\Vert _{A,Q}\\le K \\Vert f\\Vert _{B,Q}\\Vert g\\Vert _{C,Q}$ holds.",
"In this papers we give boundedness results for the maximal operator $\\mathcal {M}_{\\alpha ,B}$ between Lebesgue spaces.",
"We begin with an usefull property related with $B_p$ condition.",
"Proposition 2.1 Let $B$ be a submultiplicative Young function and let $\\phi $ be a Young function such that $\\phi ^{-1}(t)t^{\\alpha /n}\\ge C\\,B^{-1}(t)$ .",
"Let $1<p<n/\\alpha $ , $1/q=1/p-\\alpha /n$ and $s=q(1-\\alpha /n)$ .",
"If $B^{q/p}\\in B_q$ , then the function $\\psi $ defined by $\\psi (t)=\\phi (t^{1-\\alpha /n})$ belongs to $B_s$ .",
"Proof: From the definition of $\\psi $ and by changing variables we obtain that $\\int _1^\\infty \\frac{\\psi (t)}{t^s}\\frac{dt}{t}= \\int _1^{\\infty }\\frac{\\phi (t^{1-\\alpha /n})}{t^s}\\frac{dt}{t}=\\left(\\frac{n}{n-\\alpha }\\right)\\int _1^{\\infty }\\frac{\\phi (r)}{r^{ns/{(n-\\alpha )}}}\\frac{dr}{r}.$ From the relation between $B$ and $\\phi $ it is easy to see that $\\phi $ is a submultiplicative function.",
"Thus, noting that $q=ns/{(n-\\alpha )}$ we obtain $\\int _1^{\\infty } \\frac{\\phi (r)}{r^{ns/{(n-\\alpha )}}}\\frac{dr}{r}&=&\\int _1^{\\infty } \\frac{\\phi (r)}{r^{q}}\\frac{dr}{r}\\\\&\\le & c\\,\\int _c^{\\infty }\\frac{u^{1+q\\alpha /n}}{\\left(\\phi ^{-1}(u)u^{\\alpha /n}\\right)^q}\\frac{du}{u}\\\\&\\le & C\\, \\int _c^{\\infty } \\frac{u^{q/p}}{B^{-1}(u)^q}\\frac{du}{u}=C\\, \\int _c^{\\infty }\\frac{B(t)^{q/p}}{t^q}\\frac{dt}{t}<\\infty .$ $\\square $ The proof of Theorem REF requires any lemmas.",
"The first of them was proved in [5] and the second in [10].",
"Lemma 2.2 Given $0\\le \\alpha <n$ , let $B$ be a Young function such that for $\\alpha > 0$ , $B(t)/t^{n/\\alpha }$ is decreasing for all $t>0$ .",
"Then the function $\\phi _1$ en Theorem REF is increasing, and $\\phi _1(s)/s$ is decreasing.",
"Moreover, there exists $\\phi $ such that $\\phi (s)\\le C_1 \\phi _1(s)$ and $\\phi $ is invertible.",
"Lemma 2.3 If $\\phi (t)/t$ is decreasing, then for any positive sequence $\\lbrace x_j\\rbrace $ , $\\phi \\left(\\sum _j x_j\\right)\\le \\sum _j\\phi (x_j).$ The following lemma is a generalization of Lemma 3.2 in for radial Luxemburg type averages.",
"When $\\mu $ is the Lebesgue measure, it was proved in [5].",
"Lemma 2.4 Suppose that $0\\le \\alpha <n$ , $B$ is a Young function and $f$ is a nonnegative bounded function with compact support.",
"If for $t>0$ and any cube $Q$ $l(Q)^{\\alpha }\\Vert f\\Vert _{B,Q}>t,$ then, there exist a dyadic cube $P$ such that $Q\\subset 3P$ satisfying $l(P)^{\\alpha }\\Vert f\\Vert _{B,P}>\\beta t,$ where $\\beta $ is a nonnegative constant.",
"Proof: Let $Q$ be a cube with $l(Q)^{\\alpha }\\Vert f\\Vert _{B,Q}>t.$ Let $k$ be the unique integer such that $2^{-(k+1)}<l(Q)\\le 2^{-k}$ .",
"There is some dyadic cube with side length $2^{-k}$ , and at most $2^d$ of them, $\\lbrace J_i: i=1,...,N\\rbrace $ , $N\\le 2^d$ , meeting the interior of $Q$ .",
"It is easy to see that for one of these cubes, say $J_1$ , $\\frac{t}{2^d}<l(Q)^\\alpha \\Vert \\chi _{J_1}f\\Vert _{B,Q}.$ This can be seen as follows.",
"If for each $i=1,2,.., N$ we have $l(Q)^{\\alpha }\\Vert \\chi _{J_i}f\\Vert _{B,Q}\\le \\frac{t}{2^d},$ since $Q\\subset \\cup _{i=1}^{N}J_i$ we obtain that $l(Q)^{\\alpha }\\Vert f\\Vert _{B,Q}&=&l(Q)^{\\alpha }\\Vert \\chi _{\\cup _{i=1}^{N}J_i}f\\Vert _{B,Q}\\\\&\\le & l(Q)^{\\alpha } \\sum _{i=1}^{N}\\Vert \\chi _{J_i}f\\Vert _{B,Q}\\le N\\frac{t}{2^d}\\le t,$ contradicting (REF ).",
"Using that $l(Q)\\le l(J_1)<2l(Q)$ we can also show that $\\frac{t}{2^d}<l(Q)^\\alpha \\Vert \\chi _{J_1}f\\Vert _{B,Q}\\le 2^n\\,l(J_1)^\\alpha \\Vert f\\Vert _{B,J_1}$ and $Q\\subset 3J_1$ .",
"$\\square $"
],
[
"Proof of the main results",
"Proof of Theorem (REF ).",
"Fix a non-negative function $f \\in L_\\mu ^B(\\mathbb {R}^d)$ .",
"Fix $t>0$ and define $E_t=\\lbrace x\\in \\mathbb {R}^d:\\mathcal {M}_{\\alpha ,B}f(x)>t\\rbrace .$ If $t$ is such that the set $E_t$ is empty, we have nothing to prove.",
"Otherwise, for each $x \\in E_t$ there exists a cube $Q_x$ containing $x$ such that $l(Q_x)^{\\alpha }\\Vert f\\Vert _{B,Q_x}>t.$ By Lemma REF , there exists a constant $\\beta $ and a dyadic cube $P_x$ with $Q_x\\subset 3 P_x$ such that $l(P_x)^{\\alpha }\\Vert f\\Vert _{B,P_x}>\\beta t.$ Since $f \\in L^B_\\mu (\\mathbb {R}^d)$ , it is not hard to prove that we can replace the collection $\\lbrace P_x\\rbrace $ with a maximal disjoint subcollection $\\lbrace P_j\\rbrace $ .",
"Each $P_j$ satisfies (REF ) and, by our choice of the $Q_x$ 's, $E_t\\subset \\cup _j 3P_j$ .",
"By Lemmas REF and REF , $\\phi _1(\\mu (E_t))\\le \\sum _j \\phi _1(\\mu (3P_j)).$ On the other hand, inequality (REF ) implies that, for each $j$ , $\\frac{1}{l(P_j)^n}\\int _{P_j}B\\left(\\frac{l(P_j)^{\\alpha }|f|}{\\beta t}\\right) \\, d\\mu >1,$ and then by the definition and properties of $h_B$ , $1&<&\\frac{1}{l(P_j)^n}\\int _{P_j} B\\left(\\frac{3^\\alpha l(P_j)^\\alpha |f(x)|}{3^\\alpha \\beta t}\\right)d\\mu (x)\\\\&\\le &\\frac{3^nh_B(3^{-\\alpha }\\beta ^{-1})h_B(l(3P_j)^{\\alpha })}{l(3P_j)^n}\\int _{P_j}B\\left(\\frac{|f(x)|}{t}\\right)d\\mu (x)\\\\&\\le &\\frac{C}{\\phi _1(l(3P_j)^n)}\\int _{P_j}B\\left(\\frac{|f(x)|}{t}\\right)d\\mu (x).$ Hence, since the $P_j$ 's are disjoint, $\\hspace{-28.45274pt}\\sum _j \\phi _1(\\mu (3P_j))&\\le & \\sum _j\\phi _1(l(3P_j)^n)\\\\&\\le & C \\sum _j \\int _{P_j}B\\left(\\frac{|f(x)|}{t}\\right)d\\mu (x)\\\\&\\le & C \\int _{\\mathbb {R}^d}B\\left(\\frac{|f(x)|}{t}\\right)d\\mu (x).$ $\\quad \\hfill \\square $ Proof of Theorem REF .",
"Without loss of generality, we can assume that $f$ is a non-negative bounded function with compact support.",
"This guarantees that $\\mathcal {M}_{\\alpha , B}f$ is finite $\\mu $ -almost everywhere.",
"In fact, $f\\in L_\\mu ^{p_0}(\\mathbb {R}^d)$ where $p_0$ is the exponent of the hypotheses.",
"From Theorem REF we get that $\\mathcal {M}_{\\alpha , B}f\\in L_\\mu ^{q_0}(\\mathbb {R}^d)$ and thus $\\mathcal {M}_{\\alpha , B}f(x)<\\infty \\;\\;\\;\\; \\text{a.e.}",
"\\;\\;x \\in \\mathbb {R}^d.$ For each $k\\in \\mathbb {Z}$ let $\\Omega _k=\\lbrace x\\in \\mathbb {R}^d:2^k<\\mathcal {M}_{\\alpha ,B}f(x)\\le 2^{k+1}\\rbrace $ .",
"Thus $ \\mathbb {R}^d=\\cup _{k\\in \\mathbb {Z}}\\Omega _k.$ Then, for every $k$ and every $x\\in \\Omega _k$ , by the definition of $\\mathcal {M}_{\\alpha , B}f$ , there is a cube $Q_x^k$ containing $x$ , such that $l(Q^k_x)^\\alpha \\Vert f\\Vert _{B,Q^k_x}>2^k,$ and, from Lemma REF there exist a constant $\\beta $ and a dyadic cube $P_x^k$ with $Q_x^k\\subset 3P_x^k$ such that $l(P^k_x)^\\alpha \\Vert f\\Vert _{B,P^k_x}>\\beta 2^k.$ From the fact that $B$ is submultiplicative and $\\mathop {\\rm supp}(f)$ is a compact set, the inequality above allow us to obtain that $\\frac{l(P^k_x)^n}{B(l(P^k_x)^\\alpha )}<\\int _{P^k_x}B\\left(\\frac{|f|}{2^k\\beta }\\right)d\\mu \\le C \\mu ({\\rm supp} (f))\\le C.$ From the hypotheses on $B$ it is easy to check that $C_1\\varphi ^{-1}\\left(\\frac{l(P^k_x)^n}{C}\\right)\\left(\\frac{l(P^k_x)^n}{C}\\right)^{\\alpha /n}\\le B^{-1}\\left(\\frac{l(P^k_x)^n}{C}\\right)\\le l(P^k_x)^\\alpha ,$ which allow us to conclude that, for each $k$ , $l(P^k_x)$ is bounded by a constant independent of $x$ .",
"Then, there is a subcollection of maximal cubes (and so disjoint) $\\lbrace P^k_j\\rbrace _j$ such that every $Q_x^k$ is contained in $3P^k_j$ for some $j$ and, as a consequence, $\\Omega _k\\subset _j 3P^k_j $ .",
"Next, decompose $\\Omega _k$ into the sets $E_1^k=3P_1^k\\cap \\Omega _k, \\; E^k_2 =\\left(3P^k_2\\setminus 3P^k_1\\right)\\cap \\Omega _k, \\; ....,\\; E^k_j=\\left(3P^k_j\\setminus \\cup _{r=1}^{j-1}3P^k_r \\right)\\cap \\Omega _k,....$ Then $\\mathbb {R}^d=\\bigcup _{k\\in \\mathbb {Z}}\\Omega _k=\\bigcup _{j,k}E_j^k$ and these sets are pairwise disjoint.",
"Let $K$ be a fixed positive integer which will go to infinity later, and let $\\Lambda _K=\\lbrace (j,k)\\in \\mathbb {N}\\times \\mathbb {Z}: |k|\\le K\\rbrace $ .",
"By using that $E^k_j\\subset \\Omega _k$ and that the cubes $P^k_j$ satisfy (REF ) we obtain that $\\mathcal {I}_k &= \\int _{\\cup _{-K}^{K}\\Omega _k} \\left(\\mathcal {M}_{\\alpha ,B}f(x)\\right)^qu(x)d\\mu (x)\\\\&= \\sum _{(j,k)\\in \\Lambda _k} \\int _{E^k_j}\\left(\\mathcal {M}_{\\alpha ,B}f(x)\\right)^qu(x)d\\mu (x)\\\\&\\le \\sum _{(j,k)\\in \\Lambda _k} u(E^k_j)2^{(k+1)q}\\\\&\\le C2^q \\sum _{(j,k)\\in \\Lambda _k}u(E^k_j)\\left(l(P^k_j)^\\alpha \\Vert f\\Vert _{B,P^k_j}\\right)^q\\\\& \\le C 2^q \\sum _{(j,k)\\in \\Lambda _k}u(3P^k_j)\\left(l(P^k_j)^\\alpha \\Vert fv^{1/p}\\Vert _{C,P^k_j}\\Vert v^{-1/p}\\Vert _{A,P^k_j}\\right)^q,$ where in the last inequality we have used the generalized Hölder's inequality and the hypothesis on the functions $A$ , $B$ and $C$ .",
"Now, by applying the hypothesis on the weights we obtain that $\\mathcal {I}_k\\le C \\sum _{(j,k)\\in \\Lambda _k}l(3P^k_j)^{nq/p}\\Vert fv^{1/p}\\Vert ^q_{C,P^k_j}=C\\int _{\\mathcal {Y}}T_k(fv^{1/p})^q d\\nu ,$ where $\\mathcal {Y}=\\mathbb {N}\\times \\mathbb {Z}$ , $\\nu $ is de measure in $\\mathcal {Y}$ given by $\\nu (j,k)=l(3P^k_j)^{nq/p}$ and, for every measurable function $h$ , the operator $T_k$ is defined by the expression $T_kh(j,k)=\\Vert \\varphi \\Vert _{C,P^k_j} \\chi _{\\Lambda _k}(j,k).$ Then, if we prove that $T_k:L^p(\\mathbb {R}^d,\\mu )\\rightarrow L^q(\\mathcal {Y},\\nu )$ is bounded independently of $K$ , we shall obtain that $\\mathcal {I}_k\\le C \\int _{\\mathcal {Y}}T_k(fv^{1/p})^q d\\nu \\le C\\left(\\int _{\\mathbb {R}^d}(fv^{1/p})^pd\\mu \\right)^{q/p}=C\\left(\\int _{\\mathbb {R}^d}f^pvd\\mu \\right)^{q/p},$ and we shall get the desired inequality by doing $K\\rightarrow \\infty $ .",
"But the proof of the boundedness of $T_k$ follows the same arguments as in Theorem 5.3 in [7], by using now that the function $C \\in B_p$ , so we omit it.",
"$\\quad \\hfill \\square $ Proof of Theorem REF .",
"Let $g(x)=|f(x)|^{p/s}$ , then $|f(x)|=g(x)^{s/p+\\alpha /n-1}g(x)^{1-\\alpha /n}.$ Let $x\\in \\mathbb {R}^d$ and $Q$ be a fixed cube containing $x$ .",
"By the generalized Hölder's inequality () and the fact that $g(x)^{(s/p+\\alpha /n-1)n/\\alpha }=|f|^p,$ we get $l(Q)^\\alpha \\Vert f\\Vert _{B,Q}&\\le & C \\,l(Q)^\\alpha \\Vert g^{1-\\alpha /n}\\Vert _{\\phi ,Q}\\Vert g^{s/p+\\alpha /n-1}\\Vert _{n/\\alpha ,Q}\\\\&=&C \\;l(Q)^\\alpha \\Vert g\\Vert _{\\psi ,Q}^{1-\\alpha /n}\\left(\\frac{1}{l(Q)^n}\\int _Q |f(y)|^pd\\mu (y)\\right)^{\\alpha /n}\\\\ &\\le & C\\;\\left[\\mathcal {M}_{\\psi }(g)(x)\\right]^{1-\\alpha /n}\\Vert f\\Vert _{L^p(\\mu )}^{{p\\alpha }/n}.\\\\$ $\\quad \\hfill \\square $ Proof of Theorem REF .",
"From Theorem REF applied to the case $\\alpha =0$ it is easy to check that $\\mu (\\lbrace y \\in \\mathbb {R}^d: \\mathcal {M}_B f(y)>2t\\rbrace )\\le C\\int _{|f|>t}B(|f|/t)d\\mu (t).$ Thus, by changing variables and using inequality above we obtain that $\\int _{\\mathbb {R}^d} \\mathcal {M}_B f(y)^p d\\mu (y)&=&C \\int _0^\\infty t^p\\mu (\\lbrace y \\in \\mathbb {R}^d: \\mathcal {M}_Bf(y)>2t\\rbrace )\\frac{dt}{t}\\\\&\\le & C \\int _{\\mathbb {R}^d}\\int _0^{|f(y)|} t^pB\\left(\\frac{|f(y)|}{t}\\right)\\frac{dt}{t}d\\mu (y)\\\\&=& C \\left(\\int _{\\mathbb {R}^d}|f(y)|^pd\\mu (y)\\right)\\left(\\int _{1}^{\\infty }\\frac{B(s)}{s^p}\\frac{ds}{s}\\right).$ Thus, condition $B_p$ allow us to obtain the desired result.",
"$\\quad \\hfill \\square $ Proof of Theorem REF .",
"By Theorem REF , if $1<p<n/\\alpha $ , we have $\\left(\\int _{\\mathbb {R}^d}\\left(\\mathcal {M}_{\\alpha ,B}(f)\\right)^qd\\mu \\right)^{1/q}&\\le &C\\;\\left(\\int _{\\mathbb {R}^d}\\left(\\mathcal {M}_{\\psi }(|f|^{p/s})^{1-\\alpha /n}\\Vert f\\Vert _{L^p(\\mu )}^{p\\alpha /n}\\right)^qd\\mu \\right)^{1/q}\\\\&=& C\\,\\Vert f\\Vert _{L^p(\\mu )}^{p\\alpha /n}\\left(\\int _{\\mathbb {R}^d}\\mathcal {M}_{\\psi }(|f|^{p/s})^{s}d\\mu \\right)^{1/q}.$ From Proposition $\\ref {Bq}$ we have that the function $\\psi \\in B_s$ .",
"Thus Theorem $\\ref {Meta}$ implies that $\\mathcal {M}_{\\psi }:L^s(\\mu )\\rightarrow L^s(\\mu )$ , and thus $\\left(\\int _{\\mathbb {R}^d}\\left(\\mathcal {M}_{\\alpha ,B}(f)\\right)^qd\\mu \\right)^{1/q}\\le C\\, \\Vert f\\Vert _{L^p(\\mu )}^{p\\alpha /n}\\left(\\int _{\\mathbb {R}^d}(|f|^{p/s})^sd\\mu \\right)^{1/q} = C\\;\\Vert f\\Vert _{L^p(\\mu )}.$ On the other hand, if $p=n/\\alpha $ and $Q$ is a cube such that $x\\in Q$ we obtain that $l(Q)^\\alpha \\Vert f\\Vert _{\\eta ,Q}&\\le & C l(Q)^\\alpha \\Vert \\chi _Q\\Vert _{\\phi ,Q}\\Vert f\\Vert _{n/\\alpha ,Q}\\\\&\\le & C \\Vert f\\Vert _{n/\\alpha },$ and thus $\\mathcal {M}_{\\alpha ,B}(f)(x)\\le C \\Vert f\\Vert _{n/\\alpha }$ for a.e.",
"$x$ which leads us with the desired result.",
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"e-mail address: [email protected] Jorgelina Recchi, Instituto de Matemática Bahía Blanca (CONICET-UNS) Departamento De Matemáticas, Universidad Nacional Del Sur, 8000 Bahía Blanca, Argentina.",
"e-mail address: [email protected]"
]
] | 1612.05789 |
[
[
"Delta Networks for Optimized Recurrent Network Computation"
],
[
"Abstract Many neural networks exhibit stability in their activation patterns over time in response to inputs from sensors operating under real-world conditions.",
"By capitalizing on this property of natural signals, we propose a Recurrent Neural Network (RNN) architecture called a delta network in which each neuron transmits its value only when the change in its activation exceeds a threshold.",
"The execution of RNNs as delta networks is attractive because their states must be stored and fetched at every timestep, unlike in convolutional neural networks (CNNs).",
"We show that a naive run-time delta network implementation offers modest improvements on the number of memory accesses and computes, but optimized training techniques confer higher accuracy at higher speedup.",
"With these optimizations, we demonstrate a 9X reduction in cost with negligible loss of accuracy for the TIDIGITS audio digit recognition benchmark.",
"Similarly, on the large Wall Street Journal speech recognition benchmark even existing networks can be greatly accelerated as delta networks, and a 5.7x improvement with negligible loss of accuracy can be obtained through training.",
"Finally, on an end-to-end CNN trained for steering angle prediction in a driving dataset, the RNN cost can be reduced by a substantial 100X."
],
[
"Introduction and Motivation",
"Recurrent Neural Networks (RNNs) have achieved tremendous progress in recent years, with the increased availability of large datasets, more powerful computer resources such as GPUs, and improvements in their training algorithms.",
"These combined factors have enabled breakthroughs in the use of RNNs for processing of temporal sequences.",
"Applications such as natural language processing [1], speech recognition [2], [3], and attention-based models for structured prediction [4], [5] have showcased the advantages of RNNs, as they provide breakthroughs in former stagnating challenges.",
"RNNs are attractive because they equip neural networks with memories, and the introduction of gating units such as long short-term memory (LSTM) units [6] and gated recurrent units (GRU) [7] has greatly improved the training process with these networks.",
"However, RNNs require many matrix-vector multiplications per layer to calculate the updates of neuron activations over time.",
"RNNs also require a large weight memory storage that is expensive to allocate to on-chip static random access memory.",
"In a 45nm technology, the energy cost of an off-chip dynamic 32-bit random access memory (SDRAM) access is about 2nJ and the energy for a 32-bit integer multiply is about 3pJ, so memory access is about 700 times more expensive than arithmetic [8].",
"Architectures can benefit from minimizing this external memory access.",
"Previous work has focused on a variety of algorithmic optimizations for reducing compute and memory access requirements for deep neural networks.",
"These methods include reduced precision for hardware optimization ([9], [10], [11], [12], [13]); weight encoding, pruning, and compression ([14], [15]); and architectural optimizations ([16], [17], [18]).",
"However these studies have not considered temporal properties of the data.",
"It can be observed that natural inputs to a neural network tend to have a high degree of temporal autocorrelation, resulting in slowly-changing network states.",
"Fig.",
"REF demonstrates this property with a standard convolutional network (VGG-S [19]) operating on a standard video dataset.",
"As seen, the neural representation over time is highly redundant.",
"This slow changing activation feature is also seen within the computation of RNNs processing natural inputs, for example, speech (Fig.",
"REF ).",
"Delta networks, as introduced here, exploit the temporal stability of both the input stream and the associated neural representation to reduce memory access and computation without loss of accuracy.",
"By caching neuron activations, computations can be skipped where inputs do not change from the previous update.",
"Because each neuron that is not updated will save fetches of entire columns of several weight matrices, efficiently determining which neurons need to be updated offers significant speedups.",
"The rest of this paper is organized as follows.",
"Sec.",
"introduces the delta network concept in terms of the basic matrix-vector operations.",
"Sec.",
"concretely formulates it for a GRU RNN.",
"Sec.",
"proposes a method using a finite threshold for the deltas that suppresses the accumulation of the transient approximation error.",
"Sec.",
"describes methods for optimally training a delta RNN.",
"Sec.",
"shows accuracy versus speedup for three examples.",
"Finally, Sec.",
"compares this work with other developments and summarizes the results.",
"Figure: Stability of high-level neural representations over time.",
"The first 1000 frames (40s) from a Hollywood-2 scene recognition clip (the introduction to American Beauty) are presented to a standard convolutional network (VGG-S) , with the first 50 (arbitrary) features of the top-level feature vector layer are plotted over time.",
"Note that peaks tend to stay relatively constant over time, showing network output consistency over time rather than random feature activation.Figure: Stability in RNN activations over time.",
"The top figure shows the continually-changing MFCC features for a spoken digit from the TIDIGITS dataset ; the bottom figure shows the corresponding neural network output activations in response to these features.",
"Note the slow evolution of the network states over timesteps."
],
[
"Delta Network Formulation",
"The purpose of a delta network is to transform a dense matrix-vector multiplication (for example, a weight matrix and a state vector) into a sparse matrix-vector multiplication followed by a full addition.",
"This transformation leads to savings on both operations (actual multiplications) and more importantly memory accesses (weight fetches).",
"Fig.",
"REF illustrates the savings due to a sparse multiplicative vector.",
"Zeros are shown with white, while non-zero matrix and vector values are shown in black.",
"Note the multiplicative effect of sparsity in the weight matrix and sparsity in the delta vector.",
"In this example, 20% occupancy of the weight matrix and 20% occupancy of the $\\Delta $ vector requires fetching and computing only 4% of the original operations.",
"To illustrate this methodology, consider a general matrix-vector multiplication of the form c r = W x that uses $n^2$ compute operationsIn this paper, a “compute” operation is either a multiply, an add, or a multiply-accumulate.",
"The costs of these operations are similar, particularly when compared to the cost of an off-chip memory operation.",
"See [8] for a simple comparison of energy costs of compute and memory operations, $n^2 + n$ reads and $n$ writes for a $W$ matrix of size $n \\times n$ and a vector $x$ of size $n$ .",
"Now consider multiple matrix-vector multiplications for a long input vector sequence $x_t$ indexed by $t = 1, 2, \\ldots , n$ .",
"The corresponding result $r_t$ can be calculated recursively with: c rt = W + rt-1, where $\\Delta = x_t - x_{t-1}$ and $r_{t-1}$ is the result obtained from the previous calculation; if stored, the compute cost of $r_{t-1}$ is zero as it can be fetched from the previous timestep.",
"Trivially, $x_0 = 0$ and $r_0 = 0$ .",
"It is clear that cll rt = W (xt - xt-1) + rt-1 = W (xt - xt-1) + W (xt-1 - xt-2) + ...+ r0 = W xt Thus this formulation, which uses the difference between two subsequent steps and referred to as the delta network formulation, can be seen to produce exactly the same result as the original matrix-vector multiplication.",
"Figure: Illustration of saved matrix-vector computation using delta networks with sparse delta vectors and weight matrices."
],
[
"Theoretical Cost Calculation",
"The $\\Delta $ from () results in a reduction of computes and memory accesses of the weight matrix if $\\Delta $ is a sparse vector.",
"To illustrate this, begin by defining $o_c$ to be the occupancy of a vector if a ratio $o_c$ of the vector elements are nonzero.",
"Consider the compute cost for $r_t$ ; it consists of the sum of the cost of calculating $\\Delta $ (requiring $n$ operations for a vector of size $n$ ), the cost of adding in the stored previous result $r_{t-1}$ ($n$ operations), and the cost of the sparse matrix multiply $W \\Delta $ ($o_c \\cdot n^2$ operations for an $n \\times n$ weight matrix and a sparse $\\Delta $ vector of occupancy ratio $o_c$ ).",
"Similarly, the memory cost for calculating $r_t$ requires fetching $o_c \\cdot n^2$ weights for $W$ , $2n$ values for $\\Delta $ , $n$ values for $r_{t-1}$ and writing out the $n$ values of the result.",
"Overall, the compute cost for the standard formulation ($C_{\\rm comp, dense}$ ) and the new delta formulation ($C_{\\rm comp, sparse}$ ) will be: rl Ccomp, dense = n2 Ccomp, sparse = oc n2 + 2n while the memory access costs for both the standard ($C_{\\rm mem, dense}$ ) and delta networks ($C_{\\rm mem, sparse}$ ) can be seen from inspection as: rl Cmem, dense = n2 + n Cmem, sparse = oc n2 + 4n Thus, the arithmetic intensity (ratio of arithmetic to memory access costs) as $n\\rightarrow \\infty $ is 1 for both the standard and delta network methods.",
"This means that for both methods of calculating $r_t$ , every arithmetic operation requires a memory access, unfortunately placing computational accelerators at a disadvantage.",
"However, if a sparse occupancy $o_c$ of $\\Delta $ is assumed, then the decrease in computes and memory accesses due to storing the previous state will result in a speedup of: rl Cdense/Csparse n2 / (n2 oc) = (1/oc) For example, if $o_c=10\\%$ , then the theoretical speedup will be 10X.",
"Note that this speedup is determined by the occupancy in each computed $\\Delta = x_t - x_{t-1}$ , implying that this sparsity is determined by the data stream.",
"Specifically, the regularity with which values stay exactly the same between $x_t$ and $x_{t-1}$ , or as demonstrated later, within a certain absolute value called the threshold, determines the speedup.",
"In a neural network, the vector $x$ can represents inputs, intermediate activation values, or outputs of RNNs.",
"If $x$ changes slowly between subsequent timesteps then the input values $x_t$ and $x_{t-1}$ will be highly redundant, leading to a low occupancy $o_c$ and a correspondingly increased speedup."
],
[
"delta network GRUs",
"In GRUs, the matrix-vector multiplication operation that can be replaced with a delta network operation appears several times, shown in bold below.",
"This GRU formulation is from [22]: cl rt = r(Wxr xt + Whr ht - 1 + br) ut = u(Wxu xt + Whu ht - 1 + bu) ct = c(Wxc xt + rt (Whc ht - 1) + bc) ht = (1 - ut) ht - 1 + ut ct Here $r$ , $u$ , $c$ and $h$ are reset and update gates, candidate activation, and activation vectors, respectively, typically a few hundred elements long.",
"The $\\sigma $ functions are nonlinear logistic sigmoids that saturate at 0 and 1.",
"The $\\odot $ signifies element-wise multiplication.",
"Each term in bold can be replaced with the delta update defined in (), forming: rCl x = xt - xt-1 h = ht - 1 - ht - 2 rt = r(Wxr x + zxr + Whr h + zhr + br) ut = u(Wxu x + zxu + Whu h + zhu + bu) ct = c(Wxc x + zxc + rt (Whc h + zhc) + bc) ht = (1 - ut) ht - 1 + ut ct where the values $z_{xr}$ , $z_{xu}$ , $z_{xc}, z_{hr}$ , $z_{hu}$ , $z_{hc}$ are recursively defined as the the stored result of the previous computation for the input or hidden state, i.e.",
": rCCCl zxr := zxr, t-1 = Wxr (xt-1 - xt-2) + zxr, t-2 The above operation can be applied for the other five values $z_{xu}$ , $z_{xc}, z_{hr}$ , $z_{hu}$ , $z_{hc}$ .",
"The initial condition at time $x_0$ is $z_0 := 0$ .",
"Also, as can be seen from the equations above, many of the additive terms, including the stored full-rank pre-activation states as well as the biases, can be merged into single values resulting into four stored memory values ($M_{r}$ , $M_{u}$ , $M_{xc}$ , and $M_{hr}$ ) for the three gates: rCCCl Mt-1 := zx, t-1 + zh, t-1 + b Finally, in accordance with the above definitions of the initial state, the memories $M$ are initialized at their corresponding biases, i.e., $M_{r, 0}=b_r$ , $M_{u, 0}=b_u$ , $M_{xc, 0}=b_c$ , and $M_{hr, 0}=0$ , resulting in the following full formulation of the delta network GRU: rCl x = xt - xt-1 h = ht - 1 - ht - 2 Mr, t := Wxr x + Whr h + Mr, t-1 Mu, t := Wxu x + Whu h + Mu, t-1 Mxc, t := Wxc x + Mxc, t-1 Mhc, t := Whc h + Mhc, t-1 rt = r(Mr,t) ut = u(Mu,t) ct = c(Mxc,t + rt Mhc,t ) ht = (1 - ut) ht - 1 + ut ct"
],
[
"Approximate Calculations in Delta Networks",
"Note that the formulations described in Secs.",
"and are designed to give precisely the same answer as the original computation in the network.",
"However, a more aggressive approach can be taken in the update, inspired by recent studies that have shown the possibility of greatly reducing weight precision in neural networks without giving up accuracy [10], [23].",
"Instead of skipping a vector-multiplication computation if a change in the activation $\\Delta = 0$ , a vector-multiplication can be skipped if a value of $\\Delta $ is smaller than the threshold (i.e $|\\Delta _{i,t}| < \\Theta $ , where $\\Theta $ is a chosen threshold value for a state $i$ at time $t$ ).",
"That is, if a neuron's hidden-state $M$ activation has changed by less than $\\Theta $ since it was last memorized, the neuron output will not be propagated, i.e., its $\\Delta $ value is set to zero for that update.",
"Using this threshold, the network will not produce precisely the same result at each update, but will produce a result which is approximately correct.",
"Moreover, the using a threshold substantially increases activation sparsity.",
"Importantly, if a non-zero threshold is used with a naive delta change propagation, errors can accumulate over multiple time steps through state drift.",
"For example, if the input value $x_t$ increases by nearly $\\Theta $ on every time step, no change will ever be triggered despite an accumulated significant change in activation, causing a large drift in error.",
"Therefore, in our implementation, the memory records the last value causing an above-threshold change, not the difference since the last time step.",
"More formally, we introduce the states $\\hat{x}_{i, t-1}$ and $\\hat{h}_{j, t-1}$ .",
"These states store the $i-$ th input and the hidden state of the $j-$ th neurons, respectively, at their last change.",
"The current input $x_{i, t}$ and state $h_{j, t}$ will be compared against these values to determine the $\\Delta $ .",
"Then the $\\hat{x}_{i, t-1}$ and $\\hat{h}_{j, t-1}$ values will only be updated if the threshold is crossed: rll xi, t-1 = {ll xi,t-1 if |xi, t - xi, t-1 | > xi, t-2 otherwise .",
"xi,t = {ll xi, t - xi, t-1 if |xi, t - xi, t-1 | > 0 otherwise .",
"hj, t-1 = {ll hj, t-1 if |hj, t - hj, t-1 | > hj, t-2 otherwise .",
"hj, t = {ll hj, t - hj, t-1 if |hj, t - hj, t-1 | > 0 otherwise .",
"That is, when calculating the input delta vector $\\Delta {x_{i, t}}$ comprised of each element $i$ at time $t$ , the difference between two values are used: the current value of the input $x_{i,t}$ , and the value the last time the delta vector was nonzero $\\hat{x}_{i, t-1}$ .",
"Furthermore, if the delta change is under the threshold $\\Theta $ , then the delta change is set to zero, producing a small approximation error that will be corrected when a sufficiently large change produces a nonzero update.",
"The same formulation is used for the hidden state delta vector $\\Delta {h_{j, t}}$ ."
],
[
"Methods for Reducing Approximation Error and Increasing Speedup",
"This section presents training methods, constraints, and optimization schemes that yield faster and more accurate delta networks."
],
[
"Rounding Network Activations",
"The thresholded delta network computation described in Sec.",
"performs a rounding function similar to a rounding of the partially-computed state, since small changes are rounded to zero while large changes are propagated.",
"Since many previous investigations have demonstrated methods to train networks to be robust against small rounding errors by rounding during training, one method that could increase accuracy is to perform activation rounding.",
"Then, using the techniques outlined in [23], [10], a network can be successfully trained so that it is robust to these small rounding errors.",
"Furthermore, low-precision computation and low-precision parameters can further reduce power consumption and improve the efficiency of the network for dedicated hardware implementations.",
"As explored in previous studies, a low-resolution activation $\\theta _L$ in signed fixed-point format $Qm.f$ with $m$ integer bits and $f$ fractional bits can be produced from a high-resolution activation $\\theta $ by using a deterministic and gradient-preserving rounding: cl L = round(2f ) 2-f with $2^f \\cdot \\theta $ clipped to a range $[-2^{m+f-1}, 2^{m+f-1}]$ .",
"Thus, the output error cost will incorporate the errors due to small rounding approximations, and the process of stochastic gradient descent used to increase the accuracy will learn to avoid these errors through exposure during training."
],
[
"Adding Gaussian Noise to Network Activations",
"Once thresholding has been introduced, the network must be robust to the non-propagation of small changes, while large changes should be considered important.",
"Another way to provide robustness against small changes is to add Gaussian noise $\\eta $ to terms that will have a thresholded delta activation: lll rt = r((xt + x) Wxr + (ht - 1+h) Whr + br) ut = u((xt + x) Wxu + (ht - 1+h) Whu + bu) ct = c((xt + x) Wxc + rt ((ht - 1+h) Whc) + bc) ht = (1 - ut) ht - 1 + ut ct where $\\eta \\sim \\mathcal {N}(\\mu , \\sigma )$ .",
"That is, $\\eta $ is a vector of samples drawn from the Gaussian distribution with mean $\\mu $ and variance $\\sigma $ , and $\\eta \\in \\lbrace \\eta _{x}, \\eta _{h}\\rbrace $ .",
"Each element of these vectors is drawn independently.",
"Typically, the value $\\mu $ is set to 0 so that the expectation is unbiased, e.g., $\\mathbf {E}[x_t + \\eta _{x}] = \\mathbf {E}[x_t]$ .",
"As a result, the Gaussian noise should prevent the network from being sensitive to minor fluctuations, and increase its robustness to truncation errors."
],
[
"Training Directly on Delta Networks",
"However, injecting Gaussian noise at many points in the network computation is still not the same as the truncation operation performed by a thresholded delta network.",
"To best model that truncation, the network should be trained directly on the errors that arise from a delta network.",
"The resulting network will then be robust against the types of errors that a thresholding delta network typically makes.",
"More accurately, instead of training on the original GRU equations Eq.",
"–, the state is updated using the delta network model described in Eq. –.",
"This change should incur no accuracy loss between train accuracy and test accuracy, but the model may yet have more difficulty during the training if the model proves harder to optimize and possibly result in an overall lower accuracy level."
],
[
"Considering Additional Speedup from Weight Sparsity",
"Furthermore, the speedup from using a delta network so far has been considered to only arise from the sparse delta vectors that allow skipping columns of the weight matrices.",
"However, the amount of sparsity in the weight matrices of deep networks after training also can affect the savings in the computational cost and the speedup.",
"Studies such as in [24] show that in trained low-precision networks, the weight matrices can be quite sparse.",
"For example, in a ternary or 3-bit weight network the weight matrix sparsity can exceed 80% for small RNNs.",
"Since every nonzero input vector element is multiplied by a column of the weight matrix, this computation can be skipped if the weight value is zero.",
"That is, the zeros in the weight matrix act multiplicatively with the delta vector to produce even fewer necessary multiply-accumulates, as illustrated above in Fig.",
"REF .",
"The calculation of the matrix-vector product then costs: rl Ccomp, sparse = om oc n2 + 2n Cmem, sparse = om oc n2 + 4n for a weight matrix with occupancy $o_m$ .",
"By comparison to Eq.",
"REF , the system can achieve a theoretical speedup of $1/(o_m \\cdot o_c)$ .",
"That is, by compressing the weight matrix and only fetching nonzero weight elements that combine with the nonzero state vector, a higher speedup can be obtained without degrading the accuracy."
],
[
"Incurring Sparsity Cost on Changes in Activation",
"Finally, if the network is trained using the delta network model, a cost can be associated with the delta terms and added into the overall cost.",
"In a batch of input samples, the $L_1$ norm for $\\Delta _h$ can be calculated as the mean absolute delta changes, and this norm can be scaled by a weighting factor $\\beta $ .",
"This $L_{\\rm sparse}$ cost can then be additively incorporated into the standard loss function.",
"That is: cl Lsparse = ||h||1 Here the $L_1$ norm is used to encourage sparse values in $\\Delta {h}$ , so that fewer delta updates are required."
],
[
"Results",
"This section presents the results showing the trade-off between compute savings and accuracy loss from RNNs trained on the TIDIGITS digit recognition benchmark.",
"Furthermore, it also demonstrates that the results found on small datasets also appear in the much larger Wall Street Journal speech recognition benchmark.",
"The final example is for a CNN-RNN stack trained on end-to-end steering control using a recent driving dataset.",
"The fixed-point $Q3.4$ (i.e $m=3$ and $f=4$ ) format was used for network activation values in all speech experiments except the “Original” RNN line for TIDIGITS in Fig.",
"4, which was trained in floating-point representation.",
"The driving dataset in REF used Q2.5 activation.",
"The networks were trained with Lasagne [25] powered by Theano [26].",
"The training time on an Nvidia GTX980 Ti GPU is reported to indicate training difficulty, per discussions in the deep learning symposium at NIPS 2016."
],
[
"TIDIGITS dataset",
"The TIDIGITS dataset [21] was used as an initial evaluation task for the methods introduced in Sec. .",
"Single digits (“oh” and zero through nine) from this database, with a total of 2464 digits in the training set and 2486 digits in the test set, were transformed in the standard way [27] to produce a 39-dimensional Mel-Frequency Cepstral Coefficient (MFCC) feature vector using a 25 ms window, 10 ms frame shift, and 20 filter bank channels.",
"The labels for “oh” and “zero\" were collapsed to a single label.",
"Training time is approximately 8 minutes for 150 epochs of training per experiment.",
"Figure: Test accuracy results from standard GRUs run as delta networks after training (curves 1, 1a, and 1ab) and those trained as delta networks (curves 2, 2a, and 2ab) under different constraints on the TIDIGITS dataset.",
"The delta networks are trained for Θ=0.5\\Theta =0.5.",
"Note that the methods are combined, hence the naming scheme.",
"Additionally, the accuracy curve for 2 is hidden by the curve 2a, since both achieve the same accuracy and only differ in speedup metric.Figure: Accuracy-speedup tradeoff by adjusting threshold for TIDIGITS dataset.",
"By increasing the threshold (indicated by sample point size), greater speedups can be obtained at greater losses of accuracy.",
"For networks trained as delta networks, the training threshold is the first (leftmost) point in the line point sequence.The results of applying the methods introduced in Sec.",
"can be found in Fig.",
"REF .",
"There are two quantities measured: the change in the number of memory fetches, and the accuracy as a function of the threshold $\\Theta $ .",
"Fig.",
"REF shows the same results, but removes the threshold axis to allow easier comparison among the different training methods.",
"First, a standard GRU RNN was trained, achieving 96.59% accuracy on the TIDIGITS task without data augmentation and regularization.",
"This network consists of a layer of 200 GRU units connected to a layer of 200 fully-connected units and finally to a classification layer for the 10 digits.",
"This network was then subsequently tested using the delta network GRU formulation given in Sec. .",
"The standard RNN run as a delta network (“Original”) achieves 95% accuracy (a drop from zero delta threshold accuracy of 96%) with a speedup factor of about 2.2X.",
"That is, only approximately 45% of the computes or fetches are needed in achieving this accuracy.",
"By adding the rounding constraint during training (“+ Rounding during Training”), the accuracy is nearly 97% with an increase to a 3.9X speedup.",
"By incorporating Gaussian noise (“+ Noise”), the accuracy can be boosted even further to about 97% with a 4.2X speedup.",
"Essentially, these methods added generalization robustness to the original GRU, while preventing small changes from influencing the output of the network.",
"These techniques allow a higher threshold to be used while maintaining the same accuracy, which results in a decrease of memory fetches and a corresponding speedup.",
"Furthermore, training on the delta network itself (“Train on DN”) allows a considerable speedup, achieving 98% accuracy with a 8X speedup.",
"Accounting for the effect of pre-existing sparsity in the weight matrix (“+ Account for Sparse Weights”) increases the speedup to 10X, without affecting the accuracy (as it is the same network).",
"Finally, incorporating an L1 cost on network changes in addition to training on the delta network model (“+ L1 cost”) achieves 97% accuracy while boosting speedup to 11.9X.",
"Adding in the sparseness cost on network changes decreases the accuracy slightly, since the loss minimization must find a tradeoff between both error and delta activation instead of considering error alone.",
"However, using the L1 loss can offer a significant additional speedup while retaining an accuracy increase over the original GRU network.",
"Finally, Fig.",
"REF also demonstrates the primary advantage given by each algorithm; an increase in generalization robustness manifests as an overall upward shift in accuracy, while an increase in sparsity manifests as a rightward shift in speedup.",
"Methods 1, 1a, and 1b generally increase generalization robustness while only modestly influencing the sparsity.",
"Method 2 greatly increases both, while method 2a only increases sparsity, and finally method 2ab slightly decreases accuracy but offers the highest speedup."
],
[
"Wall Street Journal dataset",
"While the gains seen on TIDIGITs are significant, the delta network methodology was applied to an RNN trained on a larger dataset to determine whether it could produce the same gains.",
"Here, the Wall Street Journal dataset comprised of 81 hours of transcribed speech, as described in [28].",
"Similar to that study, the first 4 layers of the network consisted of bidirectional GRU units with 320 units in each direction.",
"Training time for each experiment was about 120h.",
"Fig.",
"REF presents results on the achieved word error rate (WER) and speedup on this dataset for two cases: First, running an existing speech transcription RNN as a delta network (results shown as solid curves labeled “RNN used as a DN”), and second, a network trained as a delta network with results shown as the dashed curves “Trained Delta Network”.",
"The speedup here accounts for weight matrix sparsity as described in Sec.",
"REF .",
"Surprisingly, the existing highly trained network already shows significant speedup without loss of accuracy as the threshold, $\\Theta $ , is increased: At $\\Theta =0.2$ , the speedup is about 5.5 with a WER of 10.8% compared with the WER of 10.2% at $\\Theta =0$ .",
"However, training the RNN to run as a delta network yields a network that achieves a slightly higher 5.7X speedup with the same WER.",
"For this large, multilayer RNN that processes complex and constantly-changing speech data, even the conventionally-trained RNN run as a delta network can provide greater than 5X speedup with only a 5% increase in the WER.",
"Figure: Accuracy and speedup tradeoffs on the Wall Street Journal dataset.",
"The solid lines show results from an existing deep RNN trained on this dataset butrun as a delta network.",
"The dashed lines show results from a network trained as a delta network using Θ=0.2\\Theta =0.2 during training.The horizontal lines indicate the non-delta network accuracy level; similarly, the solid and dashed horizontal lines indicate the accuracy of the normal network and the DN network prior to rounding, respectively."
],
[
"Comma.ai Driving DataSet",
"While speech applications are a common area of exploration for RNNs, driving scenarios are rapidly emerging as another area of focused research.",
"Here, the delta network model was applied to determine the gains of exploiting the redundancy of real-time video input.",
"The open driving dataset from comma.ai [29] with 7.25 hours of driving data was used, with video data recorded at 20 FPS from a camera mounted on the windshield.",
"The network is trained to predict the steering angle from the visual scene similar to [30], [31].",
"We followed the approach in [30] by using an RNN on top of the CNN feature detector as shown in Fig.",
"REF .",
"The CNN feature detector has three convolution layers without pooling layers and a fully-connected layer with 512 units.",
"During training, the CNN feature detector was pre-trained with an analog output unit to learn the recorded steering angle from randomly selected single frame images.",
"Afterwards, the delta network RNN was added, and was trained by feeding sequences of the visual features from the CNN feature detector to learn sequences of the steering angle.",
"Since Q2.5 format was used for the GRU layer activations, the GRU input and output vectors were normalized to match this range.",
"However, this raw dataset results in a few practical difficulties and requires data preprocessing.",
"In particular, the driver's intention of changing lanes or taking a specific route at the fork of a road cannot be learned by training on this dataset, as no intent or route goal is provided (though could be addressed in the future using precise route planning and localization techniques).",
"Additionally, the ground truth of the steering angle is noisy due to the sometimes unpredictable behavior of the driver; for example, the driver occasionally idly adjusts the steering wheel when the car is at a stop.",
"As a a result, the recorded steering angle occasionally becomes uncorrelated with the direction of movement of the car.",
"However, this issue can be addressed in a straightforward way by excluding the frames recorded during periods of low speed driving.",
"Training time of the CNN feature detector was about 8 hours for 10k updates with the batch size of 200.",
"Training of the RNN part took about 3 hours for 5k updates with the batch size of 32 samples consisting of 48 frames/sample.",
"Fig.",
"REF shows the compute cost of the delta network GRU layer in comparison with a conventional GRU, in the steering angle prediction task on 2000 consecutive frames (100s) from the validation set.",
"While the number of operations per frame remains constant for the conventional GRU layer, those for the delta network GRU layer varies dynamically depending on the change of visual features.",
"Since the output of the CNN feature detector is very stable over time as shown in the top figure, a huge speedup of about 100X (see Fig.",
"REF ) is obtained by removing the large temporal redundancy of the visual features in driving data.",
"However, in this steering network, the computational cost of the CNN (about 37 MOp/frame) dominates the RNN cost (about 1.58 MOp/frame).",
"Thus the overall, system-level computational savings for this example is only about 4.2%.",
"However, future applications will likely have efficient dedicated vision hardware or require a greater role for RNNs in processing numerous and complex data streams, which result in RNN models that consume a greater percentage of the overall energy/compute cost.",
"Even now, steering angle prediction already benefits from a delta network approach.",
"Figure: Network architecture for steering angle prediction.",
"The CNN feature detector consists of three convolution layers (Conv) and a fully-connected layer (FC) with 512 units.",
"Conv(64K5S2) represents a convolution layer with 64 feature maps and 5x5 kernel with stride 2.",
"Visual features of each image frame are fed into the GRU-based RNN to predict steering angle.Figure: Reduction of RNN compute cost in the steering angle prediction task on the comma.ai driving dataset.The top figure shows the output of CNN feature detector.",
"The middle figure shows the required # of ops per frame for the delta network GRU layer (trained with Θ=0.1\\Theta =0.1) in comparison with the conventional GRU case.A huge speedup is obtained because of the large temporal redundancy of the driving visual scenes.",
"The bottom figure compares the prediction errors of CNN predictor and CNN+RNN predictor.Note that the RNN slightly improves the steering angle prediction by using multiple frames.",
"(See Fig.",
")Figure: Tradeoffs between prediction error and speedup of the GRU layer on the steering angle prediction of the comma.ai driving dataset.",
"The result was obtained from 1000 samples with 48 consecutive frames sampled from the validation set.",
"Speedup here does not include weight matrix sparsity.",
"The network was trained with Θ=0.1\\Theta =0.1.",
"A speedup of approximately 100X can be obtained without increasing the prediction error, using Θ\\Theta between 0.1 and 0.25."
],
[
"Discussion and Conclusion",
"Although the delta network concept can be applied to other network architectures, as was shown in similar concurrent work for CNNs [32], in practice a larger benefit is seen in RNNs because we already need to store all the intermediate activation values for the delta networks.",
"For example, the widely-used VGG19 CNN has 16M neuron states [19].",
"Employing the delta network approach for CNNs requires doubled memory access and a lot of additional memory space for neuron states.",
"Because the cost of external memory access is hundreds of times larger than that of arithmetic operations, delta network CNNs seem impractical without new memory technology to address this issue.",
"In contrast, for RNNs, the number of weight parameters is much bigger than the number of activations.",
"The sparsity of the deltas allows large savings in power consumption by reducing the number of memory access for weight parameters.",
"CNNs do not have this advantage since the weight parameters are shared by many units and their number is much smaller than the number of activations.",
"Whereas the work in [32] focuses on an optimization method for converting a pre-trained CNN into a Sigma-Delta network to reduce the compute cost, our work shows that the delta networks can be optimized in terms of accuracy and speedup by directly training the original network to run as a delta network.",
"Recurrent neural networks can be highly optimized due to the redundancy of their activations over time.",
"When the use of this temporal redundancy is combined with robust training algorithms, this work demonstrates that speedups of 6X to 9X can be obtained with negligible accuracy loss in speech RNNs, and speedups of over 100X are possible in steering angle prediction RNNs.",
"Acknowledgements: We thank S. Braun for helping with the WSJ speech transcription pipeline.",
"This work was funded by Samsung Institute of Advanced Technology, the University of Zurich and ETH Zurich."
]
] | 1612.05571 |
[
[
"Pre-seismic ionospheric anomalies detected before the 2016 Kumamoto\n earthquake"
],
[
"Abstract On April 15, 2016, the Kumamoto earthquake (Mw 7.3) occurred in Japan with no warning signals.",
"Global Navigation Satellite System (GNSS) receivers provide useful information on disturbances in ionosphere by calculating the changes in Total Electron Content (TEC), which is the number of electrons in ionosphere.",
"Here we show our recently proposed correlation analysis of TEC data which can detect the pre-seismic ionospheric anomalies from the public GNSS data.",
"Our method detected the ionospheric anomaly several tens of minutes before the 2016 Kumamoto earthquake near its epicenter.",
"Furthermore, we gave an indicator to distinguish between the pre-seismic TEC anomalies and the medium scale traveling ionospheric disturbances (MSTIDs) by calculating the anomalous area rates.",
"These results support the hypothesis for existence of the preceding phenomena before large earthquakes."
],
[
"Introduction",
"Ionosphere, a shell of a large amount of electrons spreading above us, is disturbed by various natural phenomena such as volcanic eruptions [6], [4], solar flares [5], earthquakes [3], [1], [2], and so on.",
"Observing the ionospheric disturbances provides miscellaneous information on the condition of the upper atmosphere [10].",
"Japan has a dense GNSS observation network [21] (GNSS earth observation network, GEONET) and the GNSS data from this network are available freely via the Internet (terras.gsi.go.jp).",
"Daily TEC values are estimated by calculating the phase differences between the two carrier waves from the GNSS satellites [22], [18].",
"TEC data are useful for monitoring the conditions of the ionosphere, and are used by many researchers.",
"Hypotheses for preceding phenomena of earthquakes include the ionospheric disturbances before large earthquakes [7], [9].",
"As the hypothesis is still under discussion, we have to verify whether the hypothesis is right or not [13], [23], [8], [17].",
"We can get typically two TEC time series from a pair of a satellite and a GNSS station on the ground because each GPS satellite is seen twice per day at each GPS receiver.",
"Since ~30 satellites are available in orbit, we can get over 60,000 time series in a day from GEONET, composed of about 1300 GNSS stations."
],
[
"Methods",
"First of all, we choose a GNSS station as a “central station”.",
"Next, we calculate the distances between the central station and the other GNSS stations in Japan to pick up the 30 stations which are the nearest to the central station and call the 30 stations “surrounding stations”.",
"We number the central station and each surrounding stations from 0 to 30, where the number 0 means the central station and the numbers 1 to 30 are allotted to the surrounding stations.",
"Since the data length of the TEC data in a day varies by station, we extract the data from the TEC data at the center station and the surrounding stations that have common time information.",
"We set up parameters $t_{sample}$ and $t_{test}$ .",
"In this research, we set up $t_{sample} = 2.0$ [hours] and $t_{test} = 0.25$ [hours].",
"At each station $i$ and each time epoch $t$ , let Sample Data be the TEC data from time $t$ to $t + t_{sample}$ and Test Data be the TEC data from time $t + t_{sample}$ to $t + t_{sample} + t_{test}$ .",
"That is, the time length of the Sample Data is $t_{sample}$ and the time length of the Test Data is $t_{test}$ .",
"At each station, we fit a reference curve to Sample Data by the least square method.",
"We can choose some kinds of reference curves here, but the result does not highly depend on the choice of reference curve [12].",
"In this research, we choose the 7th polynomial functions as the reference curves.",
"In this way, we get the reference curves from the Sample Data.",
"Next, we calculate a deviation of the Test Data from the reference curve at each station.",
"The deviation at station $i$ at time $t$ is denoted as $x_{i, t}$ .",
"This value $x_{i, t}$ means the abnormality of TEC at each station.",
"Finally, we calculate a summation of correlations $C(T)$ between the deviations at the central station and the surrounding stations as follows: $C(T) = 1/(N * 30) \\sum _{i=1}^{30} \\sum _{j=0}^{N-1} x_{i,t + t_{sample} + j\\Delta t} x_{0,t + t_{sample} + j\\Delta t}$ $T = t + t_{sample} + t_{test}$ Here, $N$ is the number of data in Test Data, $\\Delta t$ is a sampling interval in Test Data, which means $\\Delta t = t_{test}/(N-1)$ .",
"Note that we do not use the TEC data after time epoch $T$ to calculate $C(T)$ ."
],
[
"Correlation values all over Japan on 15th April, 2016",
"The 2016 April 15 (16:25 UT) Kumamoto earthquake occurred in southwest Japan.",
"Its magnitude is Mw 7.3 and this earthquake severely damaged to around Kumamoto Prefecture.",
"We analyzed the TEC data on the day and found that our correlation analysis captured the TEC anomalies near the epicenter about one hour before the main shock (Fig.",
"REF ).",
"Our correlation analysis is an analysis method to calculate the correlation of TEC anomalies between one GNSS station and the other GNSS stations surrounding it [12].",
"This method is effective because the noises on the data can be effectively suppressed by the statistical superposition effect and the genuine anomalies are emphasized by calculating the correlation.",
"Here, we used GPS (Global Positioning System, American GNSS) satellite 17 and all the GNSS stations in Japan.",
"The elevation mask of the line-of-sight is $15^\\circ $ .",
"The red points in Fig.",
"REF mean high TEC correlation values (anomalous) and the yellow points mean calm condition (normal).",
"The location of each point on this Japan map indicates the intersection of ionosphere with the line of sight between the satellite and each GNSS station (Sub Ionospheric Point, SIP).",
"For simplification, we make an assumption that ionosphere is a thin layer about 300 kilometers above us.",
"Therefore, the points in Fig.",
"REF slightly move every minute as the satellite goes around the earth.",
"Since the ionosphere is subject to the natural phenomena, the average TEC values in Japan are continuously changing as a function of the time.",
"For example, it is known that enhanced ultraviolet flux by solar flares causes sudden increase of electron densities in ionosphere in a wide area [16].",
"Figure: Correlation values at all GNSS stations in Japan before the 2016 Kumamoto earthquake.",
"We used every GNSS station as a central station and mapped the results into the Japan map.",
"The GPS satellite 17 is used here.",
"Red points in the figures show high correlation values (anomalous).",
"The black x marks represent the epicenter of the earthquake.",
"The earthquake occurrence time is 16:25 [UT] and each time in the figures corresponds to about one hour, 40 minutes, 25 minutes and 5 minutes before the main shock."
],
[
"Correlation values near the epicenter",
"Figure REF shows the result of the correlation analysis on the earthquake day near the epicenter.",
"We chose the GPS satellite 17 and the 0087 (Koga, Fukuoka Prefecture) GNSS station as a central station.",
"The x-axis is universal time and the y-axis represents an accumulated correlation, here, briefly written $C(T)$ defined in Eq.",
"(1).",
"The black line represents T=16:25 [UT], i.e.",
"the time when the 2016 Kumamoto earthquake occured.",
"A typical example for a single GPS (PRN 17) satellite pass seen from a few representative receivers, including the central GNSS station (ID:0087) used in Fig.",
"REF , is shown in Fig.",
"REF .",
"The maximum distance between the 30 receivers and the central receiver (ID:0087) is about 78.8 km on the ground and the maximum distance of associated SIPs of PRN 17 is about 72.7 km.",
"Figure: Correlation values before the 2016 Kumamoto earthquake.",
"The vertical axis shows the correlation C(T)C(T) and the horizontal one the time tt [UTC].",
"The black line indicates the exact time 16:25 [UTC] when the 2016 Kumamoto earthquake occured.",
"We used the pair of the 0087 (Koga, Fukuoka Prefecture) GNSS station as a central station and GPS satellite 17.Figure: The geographical position relations of a single GPS (PRN 17) satellite pass (UT 14:00-17:00) seen from a few representative receivers, including the central GNSS station (ID:0087) used in Fig.",
".",
"The colored lines represent the SIP tracks and the black points on them are the SIPs when the earthquake occured."
],
[
"Correlation values on the days before the earthquake",
"Generally, when we think of anomality detection, it is important to confirm that how often the method makes a wrong detection in a normal condition (type I error, false positive) as well as the method does not make a right detection in an abnormal condition (type II error, false negative).",
"To confirm them, we calculated the TEC correlation on non-earthquake days as well (Fig.",
"REF ).",
"This figure shows the correlation values at the GNSS station near the epicenter (ID:0087, Koga, Fukuoka Prefecture) with the satellite 17 from 2016/04/08 to 2016/04/15.",
"2016/04/15 is the day the Kumamoto earthquake occurred at 16:25 [UT].",
"In comparison to the correlation values on the earthquake day, the correlation values on the other days except 2016/04/13 are quite small and actually quiet.",
"The most correlation values on these days are less than 10, whereas the correlation values are more than 30 about 20 minutes before the main shock.",
"However, the correlation values on 2016/04/13 are excessively large in spite that no large earthquakes occurred on that day.",
"Such anomalous correlation values should be caused by the medium scale traveling ionospheric disturbances (MSTIDs).",
"Figure REF shows the result of correlation analysis on 2016/04/13.",
"We can see the ionospheric disturbances in a wide range on the day.",
"These TEC anomalies move from northeast to southwest at a speed of 100-200 m/s, which is consistent with the previous study of MSTIDs [19].",
"Distinguishing between the pre-seismic TEC anomalies and the MSTIDs is one major obstacle to establishing an earthquake prediction method from ionospheric conditions.",
"One of the methods to solve this problem is shown in the next subsection.",
"These abnormal correlation values also show the characteristic wave-like patterns, which can be seen also in the 2011 Tohoku-Oki earthquake [12].",
"Figure: Correlation values near the epicenter on the non-earthquake days and the earthquake day.",
"We used the pair of GPS satellite 17 and the GNSS station 0087 (Koga, Fukuoka Prefecture), which is near the epicenter when the earthquake occurred.",
"The x-axis shows time [UT] and the y-axis shows a correlation value at each time.",
"The black vertical line on 2016/04/15 represents the earthquake occurrence time 16:25 [UT].Figure: Correlation values at all GNSS stations in Japan on 2016/04/13.",
"We used every GNSS station as a central station and mapped the results into the Japan map.",
"The GPS satellite 17 is used here."
],
[
"Seasonal MSTIDs and the 2016 Kumamoto earthquake",
"The ionospheric disturbances before the Kumamoto earthquake look like the seasonal MSTIDs at first, which can be caused by atmospheric gravity waves that propagate upward from the lower atmosphere, or created in conjunction with auroral activity [19].",
"The statistical study of MSTIDs shows that the MSTIDs occurrence rate in Japan strongly depends on the season and local time [19].",
"According to their research, a high occurrence rate can be seen at four regions: dawn (05:00-07:00 LT) in summer (May-August), daytime (08:00-12:00 LT) in winter (November-February), dusk (17:00-20:00 LT) in summer, and nighttime (21:00-03:00 LT) in summer.",
"Figure REF shows the correlation values of PRN 17 over Japan with smaller time steps (3 minutes).",
"3 minutes time step is much smaller than the typical MSTID periods of 1000 seconds.",
"As seen in the figure, a part of anomalous area seems to propagate Southwestward, which is the typical MSTID behavior at nighttime and spring-summer seasons in several places of the world [11].",
"As a whole, however, the anomalous area stays near the epicenter.",
"This moveless behavior is unlike typical night-time MSTID.",
"Figure REF shows the correlation values of PRN 17 after the earthquake.",
"The anomalous area near the epicenter is getting vanished as time elapses.",
"In the case of MSTIDs typically generated right after the earthquake, an ionospheric perturbation should propagates from the epicenter region as a circular MSTID [3], [1], [2].",
"The TEC anomalies before large earthquakes, however, do not propagate as a circular MSTID because the mechanism to be considered is radically different from the mechanism of MSTIDs after earthquakes [15], [20].",
"Figure REF shows the typical MSTIDs which are observed on 2016/01/01 and 2016/01/02.",
"During this period, the TEC anomalies can be seen sporadically in Japan and propagate southwestward as time elapses.",
"In the rest of this paper, we examine the differences of natures between MSTIDs (on 2016/01/01, 2016/01/02 and 2016/04/13) and TEC anomalies before the Kumamoto earthquake (2016/04/15)."
],
[
"The anomalous area rates",
"As seen in Fig.",
"REF and Fig.",
"REF , the ionospheric disturbences are captured on the non-earthquake day (2016/04/13).",
"We have to show the evidences which can prove clear differences between the TEC anomalies on the earthquake day (2016/04/15) and the non-earthquake day (2016/04/13).",
"In order to show that, we calculated the anomalous area rates in Japan at each time.",
"\"The anomalous area rates\" mean the rate of GNSS stations whose correlation values $C(T)$ exceed a certain threshold $\\theta $ , which is set up in advance.",
"The anomalous rate $r(T)$ is given by the following equation.",
"$r(T) = num(\\theta , T)/total\\_num(T)$ Here, $num(\\theta , T)$ is the number of GNSS stations whose $C(T)$ exceeds $\\theta $ and $total\\_num(T)$ is the total number of GNSS stations in Japan at time $T$ .",
"In this paper, we set up $\\theta = 20$ .",
"Figure REF shows the comparison of $r(T)$ from 2016/04/12 to 2016/04/15.",
"It is clear that $r(T)$ on 2016/04/13 is very large in comparison to the other days, including the earthquake day.",
"This is because MSTIDs cause TEC anomalies in a relatively wide range, whereas the TEC anomalies before large earthquakes can occur in a relatively narrow range.",
"We also examined the anomalous area rates from 2016/01/01 to 2016/01/05.",
"2016/01/01 and 2016/01/02 are the days that MSTIDs are observed by correlation analysis as seen in Fig.",
"REF and MSTIDs are not observed on the other days.",
"Figure REF shows the comparison of $r(T)$ from 2016/01/01 to 2016/01/05.",
"$r(T)$ on 2016/01/01 and 2016/01/02 are slightly larger than those on the other days.",
"Figure: Correlation values of PRN 5 over Japan on 2016/01/01 (above 2 rows) and 2016/01/02 (bottom 2 rows).",
"The time period is 02:40-03:30 in both cases.",
"Seasonal MSTIDs are captured by correlation analysis.",
"No large earthquakes occured in this period.Figure: Correlation values of PRN 17 over Japan with smaller time step (3 minutes) from 16:00UT to 16:24UT on 2016/04/15.",
"The earthquake occurrence time is 16:25 [UT].",
"The red area in each map means TEC anomalous area.",
"The x mark represents the epicenter.Figure: Correlation values of PRN 17 over Japan after the main shock (16:25 UT).",
"The anomalous (red) area is getting vanished as time elapses.",
"The x mark represents the epicenter.Figure: The anomalous area rates from 2016/04/12 to 2016/04/15 in Japan.",
"The anomalous area rates mean the rates of GNSS stations whose correlation values C(T)C(T) exceed the threshold, which is set up in advance.",
"The thereshold is 20 here.",
"We used the GPS satellite 17 in this figure.Figure: The anomalous area rates from 2016/01/01 to 2016/01/05 in Japan.",
"The anomalous area rates mean the rates of GNSS stations whose correlation values C(T)C(T) exceed the threshold, which is set up in advance.",
"The thereshold is 20 here.",
"We used the GPS satellite 5 in this figure."
],
[
"The propagation velocities of the clustered TEC anomalies",
"In order to show other differences of natures between MSTIDs and pre-seismic ionospheric anomalies, we investigated the propagation velocities of these TEC anomalies.",
"First of all, we classify the TEC anomalies into some clusters based on the geographical position.",
"Here, we defined the GNSS stations with $C(T) \\ge 20$ as anomaly at each time $T$ .",
"Next, we track and calculate the propagation velocities of the clustered TEC anomalies.",
"In this way, we investigated the propagation velocities of TEC anomalies on 2016/01/01 02:40-03:30, 2016/01/02 02:40-03:30, 2016/04/13 15:40-16:25 and 2016/04/15 15:40-16:25.",
"Figure REF shows the propagation velocities of TEC anomalies on 2016/01/01 02:40-03:30.",
"During this period, the TEC anomalies which are detected by correlation analysis are classified into 8 groups and each propagation velocity is calculated.",
"Though the TEC anomaly which is observed in 03:01-03:12 is relatively slow, other TEC anomalies show typical speeds of seasonal MSTIDs.",
"Figure REF shows the propagation velocities of TEC anomalies on 2016/01/02 02:40-03:30.",
"During this period, the TEC anomalies are classified into 11 groups and each propagation velocity is calculated.",
"Figure REF shows the propagation velocities of TEC anomalies on 2016/04/13 15:40-16:25.",
"During this period, the TEC anomalies are classified into 6 groups and each propagation velocity is calculated.",
"Figure REF shows the propagation velocities of TEC anomalies on 2016/04/15 15:40-16:25.",
"During this period, the TEC anomalies are classified into 5 groups and each propagation velocity is calculated.",
"We can confirm that some clustered TEC anomalies move slower than the typical seasonal MSTIDs observed on other days.",
"These slowly propagating TEC anomalies can be considered as pre-seismic TEC anomalies rather than seasonal MSTIDs.",
"Figure REF shows the summary of Fig.",
"REF - REF .",
"As seen in this figure, the mean velocity on 2016/04/15 is lower than those on the other days.",
"In addition, more points which show the low velocity (less than 100 m/s) exist on 2016/04/15 (3 points of the 5 points).",
"These results suggest that some, if not all, TEC anomalies on 2016/04/15 behave differently from other MSTIDs in terms of propagation velocities.",
"Figure: The propagation velocities of the clustered TEC anomalies detected by correlation analysis on 2016/01/01 02:40 - 03:30.",
"The colored track in each window represents the track of clustered TEC anomaly's center.",
"The time period in which each clustered TEC anomaly is observed is described at the top left.",
"The propagation velocity of each clustered TEC anomaly is described at the underneath.Figure: The propagation velocities of the clustered TEC anomalies detected by correlation analysis on 2016/01/02 02:40 - 03:30.",
"The colored track in each window represents the track of clustered TEC anomaly's center.",
"The time period in which each clustered TEC anomaly is observed is described at the top left.",
"The propagation velocity of each clustered TEC anomaly is described at the underneath.Figure: The propagation velocities of the clustered TEC anomalies detected by correlation analysis on 2016/04/13 15:40 - 16:25.",
"The colored track in each window represents the track of clustered TEC anomaly's center.",
"The time period in which each clustered TEC anomaly is observed is described at the top left.",
"The propagation velocity of each clustered TEC anomaly is described at the underneath.Figure: The propagation velocities of the clustered TEC anomalies detected by correlation analysis on 2016/04/15 15:40 - 16:25.",
"The colored track in each window represents the track of clustered TEC anomaly's center.",
"The time period in which each clustered TEC anomaly is observed is described at the top left.",
"The propagation velocity of each clustered TEC anomaly is described at the underneath.Figure: The summary of Fig.",
"- .",
"The propagation velocities in each figure are dotted in this figure.",
"The x-axis represents the date when the TEC anomalies are observed by correlation analysis.",
"The y-axis represents the velocities of the clustered TEC anomalies shown in Fig.",
"- .",
"The colored dots and bar represent the mean values and the error bar, respectively.As a further research, we need to determine whether the TEC anomalies detected before the Kumamoto earthquake are caused by (i) MSTID or (ii) compound of MSTID and the earthquake or (iii) the earthquake.",
"In order to detect the TEC anomalies before large earthquakes during the seasons in which MSTIDs appear frequently, the more detailed analysis supposing the case of the compound of MSTID and the preseismic TEC anomalies (case (ii)) is needed."
],
[
"Discussion",
"In order to give a reasonable explanation to TEC anomalies before the 2016 Kumamoto earthquake, some more evidences in any on-shore earthquakes of Mw $\\ge $ 7.0 are required (e.g.",
"the 1995 Mw7.3 Kobe earthquake).",
"In this research, however, we could not get the GNSS data in such cases from GEONET because GEONET was not established but just planned in 1995.",
"In this research, we aimed only the TEC analysis method.",
"Our method showed TEC anomalies before the 2016 Kumamoto earthquake and this is one of the required condition to imply the causation between the pre-seismic activity and the TEC anomaly.",
"In order to examine the various aspects of the phenomena and lead more convincing conclusion, other geo-magnetic deflections need to be investigated in like manner with Heki and Enomoto [8].",
"The physical mechanism responsible for the TEC anomalies before large earthquakes is still unclear, but it has been researched so far and the physical models such as the coupling model for the lithosphere-atmosphere-ionosphere system which explain the preseismic TEC anomalies have been shown [14], [15], [20].",
"Kuo model assumed unusually large electric field, which was not observed in the 2016 Kumamoto earthquake, however.",
"Pulinets and Ouzounov model's approach is based on the most fundamental principles of tectonics giving understanding that earthquake is an ultimate result of relative movement of tectonic plates and blocks of different sizes.",
"The TEC anomalies detected by the correlation analysis before the earthquakes have some peculiar characteristics.",
"First, the TEC anomalies can be seen near the epicenter, but not just above the epicenter.",
"Second, as seen in Fig.",
"REF , the abnormality time series shows a wave-like pattern.",
"This characteristic pattern also appears in the 2011 Tohoku-oki case [12].",
"At this stage, there is no physical model which can explain these characteristics clearly.",
"Our research, however, revealed some remarkable traits (anomalous area rates and propagation velocities) of MSTIDs and pre-seismic TEC anomalies.",
"These data should be helpful for more understanding of the physical models for both of MSTIDs and pre-seismic ionospheric anomalies.",
"Furthermore, these characteristics are very interesting because they might be a clue to reveal the relation between pre-seismic ionosphere behavior and large earthquakes with Mw $\\ge 7.0$ in the future."
],
[
"Conclusion",
"In conclusion, the clear ionospheric anomalies are detected about one hour before the 2016 Kumamoto earthquake by the correlation analysis of TEC data.",
"As far as we know, this is the first pre-seismic observation of anomalies for the 2016 Kumamoto earthquake of Mw 7.3.",
"The pre-seismic TEC anomalies before the earthquake showed charasteristic patterns.",
"We also showed a method to distinguish the pre-seismic ionospheric disturbences and MSTIDs by considering the anomalous area rates.",
"Although we still do not know the physical mechanism which cause the pre-seismic ionospheric disturbences, these analysis results of TEC may help to understand the relationship between the ionosphere and earthquakes.",
"Further investigation of other large earthquakes and understanding of the physical mechanism of the pre-seismic ionospheric disturbances should be proceeded."
]
] | 1612.05667 |
[
[
"$\\theta$ and the $\\eta^\\prime$ in Large $N$ Supersymmetric QCD"
],
[
"Abstract We study the large $N$ $\\theta$ dependence and the $\\eta^\\prime$ potential in supersymmetric QCD with small soft SUSY-breaking terms.",
"Known exact results in SUSY QCD are found to reflect a variety of expectations from large $N$ perturbation theory, including the presence of branches and the behavior of theories with matter (both with $N_f \\ll N$ and $N_f \\sim N$).",
"However, there are also striking departures from ordinary QCD and the conventional large $N$ description: instanton effects, when under control, are not exponentially suppressed at large $N$, and branched structure in supersymmetric QCD is always associated with approximate discrete symmetries.",
"We suggest that these differences motivate further study of large $N$ QCD on the lattice."
],
[
"Conjectured Behaviors of QCD at large $N$",
"In [1], Witten suggested that instantons fail to provide even a qualitative picture of the $\\theta $ dependence of QCD and the solution of the $U(1)$ problem.",
"Instead, he advanced strong arguments that the large $N$ approximation was a much more useful tool.",
"Particularly remarkable was his observation that in large $N$ , the anomaly can be treated as a perturbation and the $\\eta ^\\prime $ understood as a pseudogoldstone boson.",
"The large $N$ picture for the physics of $\\theta $ and the $\\eta ^\\prime $ rests on the assumption that correlation functions of $F \\tilde{F}$ at zero momentum behave with $N$ as similar correlation functions at non-zero momentum in perturbation theory.",
"In particular, a Green's function with $n$ insertions of $F \\tilde{F}$ behaves as $\\langle \\left(\\int F\\tilde{F}\\right)^n\\rangle \\sim N^{-n+2}\\;.$ With this assumption, and the requirement of $2 \\pi $ periodicity in $\\theta $ , the vacuum energy must behave, to leading order in $1/N$ , as $E(\\theta )={\\rm min}_k c \\left( \\theta + 2 \\pi k \\right)^2$ The minimization over $k$ reflects a branched structure in the theory, and ensures that $\\theta $ is a periodic variable [1], [4], [2].",
"The branches are characterized by a constant background topological charge density, $\\langle F\\tilde{F}\\rangle _k \\propto (\\theta +2\\pi k)\\;,$ and are smoothly traversed under $\\theta \\rightarrow \\theta +2\\pi $ .",
"A dual description of the branches in a higher dimensional gravity theory was analyzed in [2].",
"With $N_f \\ll N$ , the fermions are expected to be a small perturbation of the large $N$ pure gauge theory.",
"In particular, the axial anomaly can be treated as a perturbation [3], [4].",
"The mass of the $\\eta ^\\prime $ is an ${\\cal O}\\left({1 \\over N}\\right)$ effect, and a pseudo-Goldstone boson, the $\\eta ^\\prime $ , should be included in chiral perturbation theory in order to nonlinearly realize the approximate axial symmetry.",
"To leading order in $1/N$ and in the chiral limit, its potential is obtained by the replacement $\\theta \\rightarrow \\theta + {N_f \\eta ^\\prime \\over f_\\pi }$ in the vacuum energy.",
"This form is fixed by the axial anomaly.Here $\\eta ^\\prime /f_\\pi $ is normalized as an ordinary angle, valued on $[0,2\\pi )$ .",
"In chiral perturbation theory, it is included at leading order in large $N$ by the substitution $\\Sigma \\rightarrow \\Sigma e^{i\\eta ^\\prime /f_\\pi }$ , where $\\Sigma $ are the $SU(3)$ $\\sigma $ -model fields.",
"The axial symmetry can be realized as $\\eta ^\\prime \\rightarrow \\eta ^\\prime +\\beta f_\\pi $ , and the anomaly coefficient is $N_f$ , constraining the potential to have the form (REF ).",
"A different periodicity and anomaly are obtained if the $\\eta ^\\prime $ is instead introduced with canonically normalized kinetic term.",
"Including the branch label, $V_k(\\eta ^\\prime )=c\\cdot \\Lambda ^4 \\left(\\theta +2\\pi k+\\frac{N_f\\eta ^\\prime }{f_\\pi }\\right)^2\\;.$ Taking $N_f=1$ as an example, under $\\eta ^\\prime \\rightarrow \\eta ^\\prime +2\\pi f_\\pi $ , the state passes from one branch to another.",
"Because $f_\\pi ^2 \\propto N$ , the $\\eta ^\\prime $ (mass)$^2$ is a $1/N$ effect.",
"Higher order interactions of the $\\eta ^\\prime $ are suppressed by powers of $N$ , behaving as $V_n \\sim \\Lambda ^4 N^2 \\left({\\eta ^\\prime \\over N f_{\\pi }} \\right)^n\\;.$ In other words, the $\\eta ^\\prime $ a true Goldstone boson in the large $N$ limit, in the sense that its interactions vanish rapidly as $N \\rightarrow \\infty $ .",
"Note that $\\theta $ can be absorbed into the $\\eta ^\\prime $ .",
"With at least one massless quark, and ignoring terms in the chiral lagrangian associated with high scale (weak or above) physics, the $\\eta ^\\prime $ potential has a minimum at the CP conserving point.",
"These expressions for $\\theta $ dependence and the $\\eta ^\\prime $ potential are in stark contrast with qualitative expectations from instantons, assumed to be cut off in the infrared in some manner.",
"In this case, one would expect a convergent Fourier series, for example, for $E(\\theta )$ in the pure gauge theory: $E(\\theta ) = \\Lambda ^4 \\sum _q c_q \\cos (q\\theta ).$ Correlators of $n$ insertions of $F \\tilde{F}$ at zero momentum would scale with $N$ in a manner independent of $n$ , i.e.",
"the extra powers of $1/N$ expected from perturbation theory counting at non-zero momentum would be absent.",
"Likewise this picture makes a distinctive physical prediction for the couplings of the $\\eta ^\\prime $ : the extra powers of $1/N$ in equation REF should be absent.",
"We refer to behavior of the type of Eq.",
"(REF ) as “monodromy\" or “branched\" behavior, while that of Eq.",
"(REF ) as “instanton\" behavior.",
"Lattice gauge theory is the only framework available in which the conjectured $\\theta $ dependence of large $N$ QCD can be tested.",
"However, such questions are technically extremely challenging.",
"Some recent progress in testing Eq.",
"(REF ) was recently reported in [5], but concrete tests of the predicted cuspy behavior near $\\theta =\\pi $ , or the existence, lifetime, and other properties of the tower of $k$ branches, remains elusive.See discussion in [6].",
"On the other hand, there are a variety of known theories that are similar but more tractable than QCD, including supersymmetric QCD (SQCD), deformed Yang-Mills, and QCD at large `t Hooft coupling, in which the $\\theta $ dependence and existence of branches can be studied analytically [7], [8], [9], [2], [10], [11].",
"While differing in the details, these theories largely reflect the behaviors in Eqs.",
"(REF ,REF ).",
"The case of SQCD will be analyzed in detail in this work.",
"More generally, progress in the understanding of the dynamics of strongly coupled supersymmetric gauge theories [12], [13], [14], [15] led to new studies of ordinary QCD, considering it as a limit of Softly Broken Supersymmetric QCD (SBQCD), or SUSY QCD with $N_f$ vectorlike flavors and soft SUSY-breaking masses [16], [17], [7], [18], [19].",
"We will study aspects of $\\theta $ dependence in large $N$ SBQCD, including the existence of branches, $N$ scalings, the physics of the $\\eta ^\\prime $ , the role of instantons, and the sense in which adding matter can be thought of as a perturbation.",
"In Secs.",
"and , we observe a number of properties consistent with the large $N$ conjectures for ordinary QCD, including, as noted previously in [7], [17], branched behavior (associated with the gaugino condensate in the SUSY limit, as well as $F\\tilde{F}$ in the presence of soft breakings), and, as noted in [16], [19], a supersymmetric version of the $\\eta ^\\prime $ with mass of order $1/N$ in certain regions of parameter space.",
"We also make several new observations.",
"The behavior of ordinary QCD is different if the quark mass dominates over the effects of the $U(1)_A$ anomaly and when the anomaly dominates.",
"In the former case, there are $N$ branches, while in the latter limit there are $N_f$ branches.",
"Phase transitions are expected in passing between these regimes.",
"In Sec.",
"we show that the same phenomenon arises in SBQCD and we exhibit the phase structure.",
"In Sec.",
"we demonstrate that small changes in the number of flavors $\\Delta N_f\\ll N$ leads to small changes in the physics of different vacua at large $N$ : this provides a concrete realization of “matter as a perturbation.\"",
"Finally, we point out two ways in which the properties of SBQCD differ from the conjectured properties of QCD.",
"First, in Sec.",
"we return to the fate of instantons in large $N$ : the conjectured exponential suppression of instanton effects in QCD is critical to the large $N$ scaling properties described above.",
"A simple heuristic argument suggests that if IR divergences associated with QCD instantons are cut off at a scale of order $\\Lambda _{QCD}^{-1}$ , there is no exponential suppression.",
"As a counterargument, Ref.",
"[1] emphasized that because of the extreme nature of the power law divergences, the result is extremely sensitive to how the cutoff is chosen, and the notion that such a cutoff computation makes sense, even at a qualitative level, is hard to support.",
"But in SQCD with $N_f = N-1$ , where a systematic instanton computation of holomorphic quantities is possible, we show that the results are not suppressed by $e^{-N}$ , and that the gauge boson mass acts as an infrared cutoff approaching $\\Lambda $ at precisely the required rate.",
"On the other hand, the $N$ -scalings are, in fact, exactly as predicted by perturbative arguments, and the $\\theta $ -dependence reflects the branched structure!",
"We provide other evidence, in less controlled situations, that a notion of cut-off instantons may survive in supersymmetric theories in large $N$ .",
"Secondly, in Sec.",
", we comment on the role of discrete symmetries.",
"Unlike QCD, branched structure in SBQCD is associated with an approximate $Z_N$ symmetry, and a corresponding set of $N$ quasi-degenerate, metastable ground states.",
"What happens to these states in the limit of large soft breakings, where the discrete symmetry is lost and QCD is recovered?",
"A priori, one possibility is that these states, and the associated branch structure, disappears.",
"The possibility of phase transitions as parameters are varied is already realized in supersymmetric QCD in the controlled approximation of small soft breakings.",
"Against this possibility is the usual large $N$ scaling of perturbative correlation functions, suggesting that the branches should remain.",
"As we briefly review, a possible microscopic realization of the branches in real QCD is provided by 't Hooft's proposal of oblique confinement [20] (particularly as realized in deformed $N=2$ theories).We thank Ed Witten and Davide Gaiotto for stressing this possibility to us.",
"On the other hand, the fact that instantons are not suppressed as $e^{-N}$ in controlled situations raises questions about these arguments.",
"Whether the states disappear or survive cannot be conclusively established without non-perturbative computations.",
"In Sec.",
", we summarize and conclude.",
"We argue that while the traditional large $N$ branched picture of [1], [3], [4], [2] remains likely, only lattice calculations can ultimately settle the issues."
],
[
"Large N Scaling of the Gaugino Condensate",
"Much is understood about the dynamics of supersymmetric gauge theories.",
"For a pure supersymmetric gauge theory, for example, the value of the gaugino condensate is known, from arguments which resemble neither perturbation theory nor a straightforward instanton computation [21], [22], [12], [23], [14], [24].",
"It is interesting that, as we now show, the $N$ dependence agrees with that expected from the usual diagrammatic counting.",
"Let us recall the Coleman-Witten argument [25] for the $N$ -scaling of the chiral condensate in QCD and apply it to supersymmetric QCD.",
"By ordinary $N$ counting, an effective potential for ${\\cal M}=\\langle \\bar{\\psi }\\psi \\rangle $ (with $\\psi , ~\\bar{\\psi }$ two-component fermions) would take the form $V({\\cal M}) = N f\\left({{\\cal M}^\\dagger {\\cal M} \\over N^2\\Lambda _{QCD}^6}\\right)\\;,$ in the fermion normalization where $1/g^2$ sits in front of the whole action.",
"Thus ${\\cal M} \\propto N \\Lambda ^3.$ For supersymmetric gauge theories, the corresponding analysis for the $\\langle \\lambda \\lambda \\rangle $ effective potential gives $V(\\langle \\lambda \\lambda \\rangle ) = N^2 f\\left({\\langle \\lambda \\lambda \\rangle \\langle \\lambda \\lambda \\rangle ^*\\over N^2\\Lambda ^6}\\right)$ again in the gaugino normalization where $1/g^2$ sits in front of the whole action.",
"So, we expect $\\langle \\lambda \\lambda \\rangle = N\\Lambda ^3$ .",
"The exact result in pure gauge theory is $\\langle \\lambda \\lambda \\rangle = 32 \\pi ^2 \\Lambda _{hol}^3e^{\\frac{2\\pi i k}{N}}\\;.$ (For a review, see [26].)",
"Here $\\Lambda _{hol}$ is the holomorphic $\\Lambda $ parameter, proportional to $e^{\\frac{i\\theta }{3N}}$ .",
"In general, as discussed in [27], the holomorphic $\\Lambda $ parameter differs from the more conventional $\\Lambda $ parameter, as defined in [28], by an $N$ -dependent factor: $\\Lambda _{hol} = \\Lambda \\left({b_0 \\over 16 \\pi ^2} \\right)^{b_1/b_0^2}\\;.$ We review this connection in Appendix A. Eq.",
"(REF ) reflects the fact that $\\Lambda $ is fixed as $N \\rightarrow \\infty $ with $g^2 N$ fixed, while $\\Lambda _{hol}^3\\propto N \\Lambda ^3$ .",
"It is striking that the $N$ scaling of $\\langle \\lambda \\lambda \\rangle $ agrees with the diagrammatic expectation, although the physics leading to the exact computation appears quite different."
],
[
"$\\theta $ and the {{formula:624001b5-11cd-40a0-bf1b-e1595eb1d279}} Potential in SQCD",
"In this section, we will see that with small soft breakings, both without matter and with $N_f \\ll N$ , supersymmetric theories exhibit precisely the branched behavior anticipated by Witten, with the branches being associated with the breaking of an approximate discrete symmetry."
],
[
"Supersymmetric $SU(N)$ Gauge Theory Without Matter",
"For vanishing gaugino mass, the gaugino condensate is given by Eq.",
"(REF ).",
"In the presence of a small holomorphic soft-breaking mass, $m_\\lambda $ , the vacuum energy is $V(\\theta ,k) \\simeq {m_\\lambda } \\vert \\Lambda _{hol}\\vert ^3 \\cos \\left({\\theta + 2 \\pi k \\over N}\\right).$ In terms of physical quantities, $m_\\lambda \\Lambda _{hol}^3 = N^2 m_{phys} \\Lambda ^3,$ where $m_{phys} = g^2 m_\\lambda $ .",
"Therefore, for very large $N$ with $\\theta $ and $k$ fixed $V(\\theta ,k)\\simeq N^2 m_{phys} \\vert \\Lambda \\vert ^3 \\left({\\theta + 2 \\pi k \\over N}\\right)^2.$ This is compatible with the $N$ -scaling and $\\theta $ dependence of [2].",
"For small $m_\\lambda $ , the separate branches are long-lived.",
"As $m_\\lambda $ increases, approaching real QCD, the fate of the branches is not clear; we will comment on this further in Sec.",
"."
],
[
"$N_f \\ll N$ in supersymmetric QCD: A model for the {{formula:92e39a2c-6aba-474d-b38f-5b53abead13a}}",
"Supersymmetric QCD with $N_f<N$ flavors possesses an $SU(N_f)_L \\times SU(N_f)_R \\times U(1)_B\\times U(1)_R$ symmetry.",
"Dynamically, a non-perturbative superpotential is generated [12], $W_{np} = (N-N_f){ \\Lambda _{hol}^{3N - N_f \\over N-N_f} \\over (\\det \\bar{Q} Q)^{1 \\over N-N_f}}.$ Including supersymmetric mass terms for the quarks, the system has $N$ supersymmetric vacua.",
"Turning on general soft breakings gives a set of theories which, in certain limits, should reduce to $SU(N)$ QCD with $N_f$ flavors of fermionic quarks.",
"For small values of the supersymmetric mass terms and the soft breaking terms, the system can be studied in a systematic perturbative/semiclassical approximation [16], [19].",
"Consider first adding only soft squark and gaugino masses: $\\delta V = \\tilde{m}^2 \\sum _f \\left( \\vert Q_f \\vert ^2 + \\vert \\bar{Q}_f \\vert ^2 \\right)+ m_\\lambda \\lambda \\lambda .$ With universal soft scalar mass terms, the first terms respect the full $SU(N_f)_L \\times SU(N_f)_R \\times U(1)_B\\times U(1)_R$ symmetry of the supersymmetric theory.",
"The gaugino mass term breaks the $U(1)_R$ .",
"Ignoring the gaugino mass, the potential $V = \\sum _f \\left(\\vert {\\partial W \\over \\partial Q_f }\\vert ^2 + \\vert {\\partial W \\over \\partial \\bar{Q}_f }\\vert ^2 \\right)+ \\delta V$ (along with the $\\sum (D^{a})2$ terms) yields a minimum at $Q^a_f = v \\delta ^a_f~~~\\bar{Q}^a_f = Q^a_{f^\\prime } U_{f^\\prime f}\\;,$ where $U$ is a unitary matrix describing the Goldstone fields.",
"If $U=1$ , the symmetry is broken to the diagonal subgroup.",
"$v$ is given, in the large $N$ limit, by: $v =\\Lambda _{hol} \\left({\\Lambda _{hol}^2 \\over \\tilde{m}^2} \\right)^{1/4}.$ (If we take $\\tilde{m}^2 \\sim \\Lambda ^2$ , and recall that $\\Lambda _{hol}^3 \\sim N \\Lambda ^3$ , then $v = f_{\\eta ^\\prime } \\sim \\sqrt{N}$ , as expected by standard large $N$ arguments.",
"The same result is obtained if the moduli are stabilized by a small quark mass, $v^2\\sim \\Lambda _{hol}^3/m\\Rightarrow v\\sim \\sqrt{N}$ .)",
"The gaugino bilinear $\\lambda \\lambda $ has an expectation value in this theory, which is essentially the derivative with respect to $\\tau $ of the expectation value of the non-perturbative superpotential [14], [26], $\\langle \\lambda \\lambda \\rangle = {32 \\pi ^2} \\left\\langle { \\Lambda _{hol}^{3N - N_f \\over N-N_f} \\over (\\det \\bar{Q} Q)^{1 \\over N-N_f} } \\right\\rangle .$ To leading order, the expectation value is obtained simply using the value of $v$ in Eq.",
"(REF ).",
"For large $N$ , the condensate behaves as $\\langle \\lambda \\lambda \\rangle = \\Lambda _{hol}^3 e^{{2 \\pi i k \\over N}+ i\\,{\\rm arg} \\det U^{1/N}},$ where $U$ is the unitary matrix in Eq.",
"(REF ).",
"Now consider turning on a small $m_\\lambda $ .",
"The gaugino mass breaks the classical, anomalous $U(1)_R$ as well as the quantum, non-anomalous $U(1)_R$ .",
"It also breaks the quantum $Z_N$ symmetry.",
"Through a field redefinition, we can take $m_\\lambda = \\vert m_\\lambda \\vert e^{i\\theta /N}$ .",
"Gaugino condensation then generates a potential for the fields $U$ , which at large $N$ takes the form: $V(\\theta ,\\eta ^\\prime ) = \\vert m_\\lambda \\vert \\Lambda _{hol}^3 \\cos \\left({\\theta + 2 \\pi k + {\\eta ^\\prime \\over v} \\over N} \\right),$ where we have written $\\arg \\det U = {\\eta ^\\prime \\over v}$ .",
"Recall that in conventional large $N$ scaling, $m_\\lambda \\propto N m_{\\lambda }^{phys}$ , where $m_{\\lambda }^{phys}$ is the physical gaugino mass.",
"Therefore, expanding for very large $N$ and taking $m_{\\lambda }^{phys} \\sim \\Lambda $ gives the potential for the $\\eta ^\\prime $ proposed in [4].",
"The scaling with $N$ is exactly as predicted.",
"For zero supersymmetric quark mass, $\\theta $ and $k$ can be removed by a redefinition of the $\\eta ^\\prime $ field.",
"In the presence of a quark mass term, this is no longer the case.",
"The $\\eta ^\\prime $ potential contains an additional term, which at large $N$ takes the form $V(\\theta ,\\eta ^\\prime ) = \\vert m_\\lambda \\vert \\Lambda _{hol}^3 \\cos \\left({\\theta + 2 \\pi k + {\\eta ^\\prime \\over v} \\over N} \\right) + \\vert m_q \\vert \\Lambda _{hol}^3\\cos \\left(\\frac{\\eta ^\\prime }{v} + \\beta \\right),$ where $\\beta $ is the phase of the quark mass.",
"We comment on the properties of this potential in Sec.",
"."
],
[
"Phases with General $N_f<N$",
"In QCD, the realization of branched structure is thought to vary with $m_q$ [4].",
"At zero $m_q$ , a field redefinition can eliminate $\\theta $ -dependence.",
"At large $N$ , this corresponds to the fact that $\\theta $ can be eliminated by a shift of the $\\eta ^\\prime $ .",
"On the other hand, at sufficiently large $m_q$ , the quarks can be integrated out and $\\theta $ -dependence should reappear, along with any branched structure.",
"In SBQCD, already in the limit of soft breakings, an intricate phase structure arises by varying the soft breaking parameters and the quark masses.",
"This can be anticipated because in the theory of Eq.",
"(REF ), before including the quark masses $m_q$ , the discrete symmetry is $Z_{N_f}$ , a preserved subgroup of the anomalous $U(1)_A$ axial symmetry acting on $Q,\\bar{Q}$ .",
"If we set $m_\\lambda $ to zero, with non-zero $m_q$ , the discrete symmetry is $Z_N$ , a preserved subgroup of the anomalous $U(1)_R$ symmetry acting only on $\\lambda $ .",
"It is easy to check that varying the parameter $x = {m_\\lambda \\over m_q}\\;,$ the number of local minima of the potential changes from $N$ at small $x$ to $N_f$ at large $x$ .",
"To see this explicitly, take the simplified case $\\vert m_q \\vert ^2, \\vert m_\\lambda \\vert ^2 \\ll \\tilde{m}^2$ , and $\\tilde{m}^2$ , $m_q$ proportional to the unit matrix in flavor space.",
"We can then take $\\bar{Q} Q = v_0^2 e^{i \\eta ^\\prime }$ (note here we are working with a dimensionless $\\eta ^\\prime $ ).",
"The potential for the $\\eta ^\\prime $ then has the form: $V(\\eta ^\\prime ) = m_q \\Lambda _{hol}^{3N-N_f \\over N-N_f} v_0^{-{2N_f \\over N-N_f}} \\cos \\left({N \\over N-N_f} \\eta ^\\prime \\right)+ N m_\\lambda {\\Lambda _{hol}^{3N-N_f \\over N-N_f} \\over v_0^{2N_f \\over N-N_f} } \\cos \\left( \\eta ^\\prime {N_f \\over N-N_f}\\right),$ or, for $N \\gg N_F$ , $V(\\eta ^\\prime ) = m_q \\Lambda _{hol}^{3} v_0^{-{2N_f \\over N}} \\cos ( \\eta ^\\prime )+ N m_\\lambda \\Lambda _{hol}^{3} \\cos \\left( \\eta ^\\prime {N_f \\over N}\\right).$ This potential is similar in structure to that for the ordinary $\\eta ^\\prime $ proposed in [4].",
"It exhibits $N$ vacua in the limit of small $x$ , and $N_f$ in the limit of large $x$ .",
"Analogously, in ordinary QCD, the large-$N$ $\\eta ^\\prime $ potential has $N_f$ vacua in the limit $m_q\\ll \\Lambda /N$ , and $N$ vacua in the opposite limit.",
"In SQCD, the transitions between these phases occur for $x$ of order one.",
"As the vacua disappear, they become increasingly unstable.",
"In the limit of large $x$ , correlation functions with successively more insertions of $\\int d^4 x F \\tilde{F}$ are suppressed by $N_f$ , not $N$ .",
"The potential can also be analyzed in the case of $N_f = N-1$ , where a reliable instanton computation is possible.",
"In this case, there are of order $N$ branches in either limit, but one can still observe transitions between different phases, increasing confidence in the small $N_f$ analysis.",
"The phase structure also offers some insight into the lifetimes of states of a system as one approaches the critical values $x_0$ where they disappear.",
"The bounce action vanishes as a power of $x-x_0$ (of course, the semiclassical analysis breaks down once the lifetime becomes short)."
],
[
"Matter as a Perturbation",
"In the large $N$ limit, we might expect that small changes in the number of flavors only affect the properties of the theory at order $1/N$ : in this sense, matter is a perturbation.",
"There are two classes of quantities we might study.",
"In actual QCD, we might ask about the $N_f$ dependence of the glueball mass or $F \\tilde{F}$ correlation functions, expecting weak sensitivity of these quantities to ${\\cal O}(1)$ changes in $N_f$ at large $N$ .",
"Alternatively, we can consider the structure of the quark sector.",
"Here we expect the features of the effective action for the $\\eta ^\\prime $ , for example, to be determined by the large $N$ pure gauge theory.",
"In the supersymmetric theories, the gluino condensate is in the first class, and we expect small changes in the number of flavors to yield only small changes in the condensate.",
"To test this idea, we must be precise about what is perturbed.",
"As we vary $N_f$ , we hold the ultraviolet cutoff $M$ and the gauge coupling $g^2(M)$ fixed.",
"For simplicity, we take all quarks to have mass $m_q$ , with $m_q \\gg \\Lambda $ , and we study the Wilsonian effective action at a scale $\\mu $ such that $m_q \\gg \\mu \\gg \\Lambda $ .",
"Integrating out the quarks generates a term ${\\cal L} = -{1 \\over 32 \\pi ^2} \\int d^2 \\theta \\left({8 \\pi ^2 \\over g^2} + 3N \\log (\\mu /M) - N_f \\log (m_q/M) \\right)W_{\\alpha }^2 .$ From it, we can compute the holomorphic low energy scale, $\\Lambda _{LE}$ , which in turn determines $\\langle \\lambda \\lambda \\rangle $ , $\\langle \\lambda \\lambda \\rangle = \\Lambda _{LE}^3 = \\Lambda ^3\\left( {m_q \\over \\Lambda }\\right)^{N_f \\over N}.$ This expression is clearly smooth with respect to changes in $N_f$ .",
"Indeed, we can treat an additional flavor as a perturbation, computing first the change in the effective action, and from that the change in $\\Lambda _{LE}$ .",
"Alternatively, we can consider a quantity involving the quark superfields, for small number of flavors.",
"As before, we can think of a fixed cutoff scale and coupling, and take universal quark masses $m_q \\gg \\Lambda $ .",
"Then integrating out the heavy fermions yields $\\langle \\bar{Q} Q \\rangle = {1 \\over 16 \\pi ^2 m_q} \\langle \\lambda \\lambda \\rangle \\;.$ (this is an example of the Konishi anomaly [29]).",
"This agrees with the exact result, and by holomorphy, it holds for all $m_q$ .",
"Thus for small $N_f$ the quark condensate is determined in large $N$ by the pure gauge theory.",
"One can provide a heuristic derivation of this result at small $m_q$ as well.",
"For larger values of $N_f$ , small changes $\\Delta N_f\\ll N$ should also produce only small changes in the theory, for appropriate choices of ground states.",
"This is particularly interesting for $N_f=N-2, N-1, N, N+1, N+2$ , where the dynamics, when the quarks are light, is substantially different in each case (described via gaugino condensation, instantons, the deformed moduli space, s-confinement, and Seiberg duality, respectively [12], [13], [15].)",
"Yet, in large $N$ , all descriptions must in some sense converge, up to $1/N$ corrections!",
"Let us understand a few simple reflections of this fact, again taking $m_{f\\bar{f}}\\rightarrow m\\delta _{f\\bar{f}}$ and $Q\\bar{Q}_{f\\bar{f}}\\rightarrow v^2\\delta _{f\\bar{f}}$ .",
"For $N_f=N-1$ , there is a Wilsonian effective superpotential [12], $W_{Wilsonian}=\\frac{\\Lambda ^{2N+1}}{v^{N-1}}+N_f m v^2$ and the vacuum is $v=\\Lambda \\left(\\frac{\\Lambda }{m}\\right)^\\frac{1}{2N}$ which approaches $v\\rightarrow \\Lambda $ in the large $N$ limit, losing its $m$ -dependence.",
"In contrast, the case $N_f=N-1$ has a deformed moduli space [13], described by a 1PI effective superpotential with Lagrange multiplier $X$ , $W_{1PI}=X\\left(v^{2N}-B\\bar{B}-\\Lambda ^{2N}\\right)+N_f mv^2\\;.$ Since there are no baryonic operators in $N_f=N-1$ , vacua on baryonic branches are not connected to vacua in $N_f=N-1$ .",
"The meson vacuum, however, is: in the large $N$ limit, the $N_f=N-1$ vacuum becomes the $B=\\bar{B}=0$ vacuum $v=\\Lambda $ of $N_f=N$ .",
"The gaugino condensates likewise match in large $N$ , and vanish in the massless limit.",
"A similar result is obtained for $N_f=N+1$ with small quark mass: the meson vev takes the form $v^{2N+1}=m\\Lambda ^{2N}$ , so $v\\rightarrow \\Lambda $ in large $N$ .",
"The new feature of the $N_f=N+1$ theory, the chiral preserving vacuum, is obtained in the limit $m\\rightarrow 0$ , which does not commute with $N\\rightarrow \\infty $ ."
],
[
"Instantons at Large $N$",
"We see that approximately supersymmetric theories exhibit many of the features anticipated for real QCD, within controlled approximations.",
"Much of our understanding of supersymmetric dynamics, on the other hand, involves instantons in an essential way.",
"This suggests that instanton effects are not necessarily suppressed at large $N$ , and can have controlled large $N$ limits, at least in SQCD."
],
[
"Heuristic treatment of instantons: the infrared cutoff",
"In the introduction, we discussed two potential behaviors for large $N$ QCD as a function of $\\theta $ , referred to as branched and instanton behaviors, respectively.",
"We have seen that supersymmetric SQCD with small gaugino mass exhibits the former behavior.",
"Ref.",
"[1] offered a simple argument against the latter, suggesting that instanton effects are exponentially suppressed in large $N$ .",
"Let us recapitulate the argument.",
"Consider QCD without flavors.",
"The one-instanton contribution to $V(\\theta )$ has the structure: $V(\\theta ) = \\int d\\rho \\rho ^{-5 + {11 N \\over 3}} M^{11 N \\over 3} N e^{-{8 \\pi ^2 \\over g(M)^2}} \\cos (\\theta )$ where $M$ is a renormalization scale.",
"Since $g^2(M) \\sim 1/N$ , this is formally exponentially suppressed, but the expression is also infrared divergent.",
"Suppose that the integral is cut off at $\\rho \\approx \\Lambda ^{-1}$ .",
"The result would then be simply $V(\\theta ) = {\\rm C} \\Lambda ^4 \\cos (\\theta ).$ which is of order one in large $N$ .",
"Of course, this argument is handwaving at best.",
"If the cutoff is $c~ \\Lambda $ , with $c$ an order one constant, then the result can be exponentially suppressed or enhanced by $c^N$ .",
"Ref.",
"[1] suggested that the most likely smooth limit for instanton effects in large $N$ is zero.",
"Imagine, however, that $c$ approaches 1 as $e^{1/N}$ : in this case, the limit of the single instanton term would be smooth and finite.",
"In QCD, such a picture could only be qualitative; perturbative corrections and instanton-antiinstanton corrections are all be nominally of the same order, and a reliable semiclassical calculation is not possible.",
"The only statement one could make, in general, is that $\\theta $ dependence would be described by a series of the form of Eq.",
"(REF ).",
"One could speculate on the convergence of the series, for example whether cusps arise in the potential.",
"This appears to occur in the $CP^N$ models, where finite temperature provides an infrared cut-off on instanton size [30], [31], [32], and the series (REF ) exhibits cusps in the limit $T \\rightarrow 0$ (the Fourier expansion for $dE \\over d\\theta $ does not converge).",
"This will be discussed more fully in a subsequent publication."
],
[
"Scaling of Reliable Instanton Computations with $N$",
"In SQCD with $N_f=N-1$ , the role of instantons in large $N$ can be assessed sharply, exploiting the existence of a pseudomoduli space.",
"The effective superpotential can be computed systematically, and infrared divergences are cut off by $Q \\bar{Q}_{f\\bar{f}} \\equiv v^2\\delta _{f\\bar{f}}$ .",
"The $\\rho $ integrals take the form $W \\;\\sim \\; \\int d\\rho \\,{(\\Lambda \\rho )}^{2N+1} {(v^*)}^{2N-2} \\rho ^{4N-5} e^{-c ^2\\rho ^2 \\vert v \\vert ^2}\\;\\sim \\; {\\Lambda ^{2N+1} \\over v^{2N-2}}.$ A careful analysis yields [14] $W = {\\Lambda _{hol}^{2N + 1} \\over \\det {\\bar{Q} Q} }\\;,$ which is naïvely of order $e^{-N}$ .",
"However, $v^2$ also depends on $\\Lambda $ .",
"For simplicity, taking all of the quarks to have equal mass, $v^N = \\Lambda _{hol}^{N} \\left( {\\Lambda _{hol} \\over m_q} \\right)^{1 \\over N}.$ At the stationary point, $\\langle W \\rangle = a \\Lambda _{hol}^2 m_q \\left[ {\\Lambda _{hol} \\over m_q} \\right]^{1/N}.$ This structure is dictated by symmetries and holomorphy.",
"In particular, there is a non-anomalous, spurious $R$ symmetry under which $m_q \\rightarrow e^{2 i \\alpha {N \\over N_f} }m_q.$ Similarly, there is a non-anomalous $R$ symmetry under which $m_q$ (and $Q,\\bar{Q}$ ) are neutral, and $\\Lambda \\rightarrow e^{i \\alpha 2N/(2N+1) } \\Lambda $ .",
"Eq.",
"(REF ) is notable.",
"First, there is no exponential suppression with $N$ : $\\Lambda _{hol}^{2 + {1 \\over N}} = M^{2+ {1 \\over N}} e^{-{8 \\pi ^2 \\over g^2 N}+ i {\\theta \\over N}}.$ Not only do the $e^{-{8 \\pi ^2 \\over g^2}}$ factors appear with a suitable power to avoid $e^{-N}$ suppressions, but there are no factors like $\\pi ^N$ or $2^N$ which might have obstructed a suitable large $N$ limit.",
"At the same time, the result exhibits monodromy, arising from the $N$ roots of Eq.",
"(REF ).",
"It is also important to stress that, unless $m_q$ is exponentially small, the stationary point lies in a region of strong coupling.",
"So a reliable calculation is possible taking $m_q = \\epsilon ^N \\Lambda $ , for small $\\epsilon $ , and then using holomorphy and symmetries to extend the result to $m_q=\\Lambda $ .",
"For $m_q \\sim \\Lambda $ , the instanton result is not reliable in the sense that non-holomorphic quantities like the scalar potential are not properly computed.",
"But the result for $\\langle W\\rangle $ qualitatively has the instanton structure, and it is equivalent to say that it is saturated by the single instanton.",
"We also note that in presence of a gaugino mass, we again find the usual formula for the vacuum energy, $E(\\theta ) = m_\\lambda \\langle W \\rangle = m_\\lambda \\Lambda _{hol}^3 \\cos \\left({\\theta +2 \\pi k \\over N}\\right).$ So in this case, we have complete agreement with expectations based on $N$ counting of perturbative Feynman diagrams, yet the result arises entirely from an instanton!",
"In particular, correlators of $n$ $F \\tilde{F}$ operators at zero momentum behave as $N^{2-n}$ , precisely as expected.",
"We have already noted how a cutoff might approach $\\Lambda $ in large $N$ so that instanton amplitudes are unsuppressed.",
"Here we see that, in the nearly supersymmetric case, the $\\Lambda $ which appears in the argument is the holomorphic $\\Lambda $ , yielding $\\cos (\\theta /N)$ .",
"To summarize, on the one hand, we see evidence for a branched structure, a structure originally suggested by a presumed suppression of instanton effects.",
"On the other hand, we see that instantons are not suppressed, and the branches are associated with an approximate discrete symmetry.",
"We cannot draw conclusions about the fate of the branched structure as SUSY breaking is increased, but the instanton argument for the branched structure, by itself, is at least misleading in the nearly-SUSY limit."
],
[
"Further circumstantial evidence for the role of instantons",
"Also instructive are instanton computations in the pure supersymmetric gauge theory.",
"This subject was pioneered in [33], [34].",
"In pure $SU(N)$ supersymmetric QCD, one can attempt to calculate the correlation function $G^{2N}=\\langle \\lambda \\lambda (x_1) \\dots \\lambda \\lambda (x_N) \\rangle \\;.$ A single instanton makes an infrared finite contribution to this correlator, $G^{2N} \\sim \\Lambda ^{3N}$ , which is formally of order $e^{-N}$ .",
"This paper argued that this correlation function, as the correlator of the lowest component of a set of chiral fields, was independent of coordinates, and in addition advanced arguments that it was not renormalized.",
"The authors of [35] argued, invoking cluster decomposition, that the $N^{\\rm th}$ root of this expression is $G = \\langle \\lambda \\lambda \\rangle $ .",
"It is known that the single instanton computation makes an order one error in these quantities.",
"The corrections can be understood as dilute gas corrections (in the sense that they can be shown to arise from the sector with topological number one [36]).",
"If the naive reasoning were correct, these effects would be suppressed by further powers of $e^{-N}$ , but this is not the case.",
"This is consistent with the infrared cutoff computations suggested in [37]."
],
[
"Speculations on Real QCD",
"We have seen that instantons and large $N$ behavior are not necessarily incompatible, and emphasized that the appearance of branches in supersymmetric QCD is associated with the spontaneous breaking of a discrete symmetry.",
"As we take the soft breakings large, most of the $N$ vacua might disappear, leading to what we have called “instanton\" behavior.",
"On the other hand, given that the lifetimes of the states scale as $e^{-N^4}$ (in the region over which we have control), they might survive."
],
[
"Spontaneous breaking of an explicitly broken discrete symmetry",
"In this brief section, we describe the possible behaviors in terms of the realization of a spurious symmetry.",
"At the level of the classical action, the softly broken supersymmetric theory exhibits a symmetry with $m_{\\lambda }$ viewed as a spurion: $\\lambda \\lambda \\rightarrow e^{2 \\pi i k \\over N} \\lambda \\lambda ~~~~m_\\lambda \\rightarrow e^{-{2 \\pi i k \\over N} }m_\\lambda .$ If $E(m_\\lambda ) = E(\\vert m_\\lambda \\vert ,m_\\lambda ^N)$ , this spurious symmetry is not spontaneously broken.",
"If $E(m_{\\lambda })$ is not invariant under $m_\\lambda \\rightarrow e^{-{2 \\pi i k \\over N}} m_\\lambda $ , however, spontaneous symmetry breaking has occurred.",
"This is the option realized in SBQCD, and is associated with $N$ stationary points of the vacuum energy.",
"$E$ has an imaginary part outside a finite range of $\\alpha =\\arg m_\\lambda $ .",
"The existence of branches in real QCD can be mapped to the question of whether the spurious symmetry is broken or unbroken as $m_\\lambda $ becomes much larger than $\\Lambda $ .",
"As $m_\\lambda \\rightarrow \\infty $ and $\\lambda $ is integrated out, we generate $\\theta = \\arg (m_\\lambda ) N$ .",
"The question is: does $E$ behave (in the pure gauge theory) as a function of $\\arg (m_\\lambda ) $ or $\\arg (m_\\lambda ) N$ ?",
"Needless to say, analytic tools to address this question are not available, but we can look to toy models to gain some understanding of the possibilities.",
"We can illustrate these possible behaviors of the pure gauge theory in a field theory of scalars, treating the system classically and including certain non-renormalizable couplings.",
"With a complex field, $\\phi $ , the potential $V(\\phi ) = -\\mu ^2 \\vert \\phi \\vert ^2 + {\\lambda \\over 2} \\vert \\phi \\vert ^4 - \\Gamma (\\phi ^N + \\phi ^{*N})$ respects a $Z_N$ symmetry.",
"If $\\Gamma $ is small, we can write: $\\phi =f e^{ia/f}\\,, ~~~f = \\sqrt{\\mu ^2 \\over \\lambda }$ The field $a$ acquires a potential $V(a) =- \\Gamma f^N\\cos \\left(N{a \\over f}\\right).$ The system has $N$ degenerate minima, at ${a \\over f} = {2 \\pi k \\over N}$ , reflecting the spontaneous breaking of the discrete symmetry.",
"Adding a coupling $\\delta V = m_\\lambda \\Lambda ^2 \\phi + \\rm c.c.$ breaks the $Z_N$ symmetry explicitly, and the parameter $m_\\lambda $ is a spurion analogous to $m_\\lambda $ in SUSY QCD.",
"For small $m_\\lambda = \\vert m_\\lambda \\vert e^{i\\alpha }$ , $\\phi $ does not shift significantly, and the classical vacuum energy has a contribution $E(\\alpha ,k) = \\vert m_\\lambda \\vert \\Lambda ^2 f \\cos \\left(\\alpha + {2 \\pi k \\over N}\\right).$ The potential reflects the spontaneous breaking of the spurious symmetry.",
"Quantum mechanically, $E$ has a small imaginary part except for $k$ such that $\\vert \\alpha + {2 \\pi k \\over N}\\vert < \\pi .$ Elsewhere in the parameter space, however, the branches disappear.",
"For example, for $\\mu ^2$ negative, the potential has a unique minimum, and this is not altered by the addition of the $m_\\lambda $ term.",
"Instead, $\\langle \\phi \\rangle = {m_\\lambda ^* \\Lambda ^2\\over \\mu ^2},$ and $E(\\alpha ,k) = { |m_\\lambda |^2 \\Lambda ^4 \\over \\mu ^2}.$ Thinking of this as a toy model of supersymmetric QCD, the parameters $\\mu ^2 \\rightarrow \\mu ^2(m_\\lambda )$ , $\\Gamma \\rightarrow \\Gamma (m_\\lambda )$ .",
"If, for example, $\\mu ^2(m_\\lambda )$ becomes negative and $\\Gamma $ does not grow too rapidly for large $m_\\lambda $ , the branched structure disappears.",
"Alternatively, if for large $m_\\lambda $ , $\\mu ^2 >0$ and if $\\Gamma $ grows rapidly with $m_\\lambda $ , then the branched structure survives.",
"In this toy model, the $N$ vacua reflect an approximate $Z_N$ symmetry which survives in the limit."
],
[
"Stability of Branches",
"In SBQCD, both with and without matter, we can ask about the stability of different branches.",
"Take $k=0$ and $0<\\theta <2 \\pi $ and consider what happens as $\\eta ^\\prime $ increases.",
"At some point, the state with $k= -1$ has lower energy, and the system can tunnel.",
"For small $m_\\lambda $ , the tunneling rate is highly suppressed, roughly asThis estimate appears also in [38], which notes that due to numerical factors, even for $m_\\lambda \\Lambda _{QCD}$ , the states may be short-lived unless $ N> 100$ or so.",
"If true, it would be challenging to understand how the large $N$ limit could be valid for $N \\approx 3$ ., $\\Gamma = C e^{-a N^4{\\Lambda ^3 \\over m_\\lambda ^3}}.$ We can repeat this for larger $k$ , producing a large set of metastable states.",
"Increasing $m_\\lambda $ , eventually we can no longer perform a reliable computation, but based on (REF ) it is possible that tunneling rates remain exponentially suppressed with $N$ .",
"The presence of metastable states in QCD is an interesting target for the lattice [6], and could conceivably have implications for physics in the early universe."
],
[
"'t Hooft's Picture of Confinement: A Candidate Setting for Branched Structure",
"Nambu, Mandelstam, and 't Hooft suggested that condensation of magnetically charged objects in a non-abelian theory could account for confinement of color charge [39], [40], [41].",
"Subsequently, 't Hooft studied the adjoint-valued composite field $\\Phi =F_{\\mu \\nu } \\tilde{F}^{\\mu \\nu }$ , choosing a gauge in which $\\Phi $ is everywhere diagonal and leaving unfixed a $U(1)^{N-1}$ symmetry [20].",
"He speculated that singular points with respect to the gauge choice correspond to massless, condensing monopoles of the $U(1)^{N-1}$ theory, and noted that in the presence of $\\theta $ , the monopoles acquire a charge through the Witten effect.",
"When $\\theta \\rightarrow \\theta +2 \\pi $ , the spectrum is the same, but “rearranged\": what were monopoles with one charge at $\\theta =0$ become monopoles of a different charge at $\\theta = 2\\pi $ .",
"This picture of confinement thus gives rise to an explicit realization of branched structure with $\\theta $ .",
"The details, including whether there are $N$ vacua of a spurious $Z_N$ symmetry, depend on unknown features of the monopole/dyon spectrum.",
"Such dynamical features are also suggested by consideration of the algebra of Wilson and 't Hooft lines [42], [43].",
"$N=2$ supersymmetric Yang-Mills, with a small mass $m_A$ for the adjoint chiral multiplet, exhibits many of these features explicitly, including a $U(1)^{N-1}$ symmetry in the small $m_A$ limit.",
"Seiberg and Witten showed that the theory possesses massless monopoles at points in the moduli space [44].",
"In the case of $SU(N)$ , there are $N$ such points, related by a discrete $Z_N$ symmetry.",
"Turning on $m_A$ , the massless monopoles condense, and the theory confines.",
"The condensate is proportional to $m_A$ , and, for small $m_A$ , the monopole and $U(1)$ gauge field masses are also suppressed by $m_A$ .",
"The theory possesses precisely the sort of branched structure anticipated by 't Hooft, with $\\tau = {8 \\pi ^2 \\over g^2} + ia$ , and the branches are associated with the $Z_N$ symmetry of the theory.",
"As $m_A$ becomes larger than $\\Lambda $ , it is not clear what becomes of the monopole picture; the $U(1)$ gauge bosons are no longer light relative to other states in the spectrum, nor are the monopoles.",
"But we know that the $N=1$ theory exhibits a branched structure.",
"If 't Hooft's picture for confinement is qualitatively correct for real QCD, it can account for a branched structure.",
"However, the applicability of the monopole condensation picture to real QCD remains unclear.",
"For example, one does not expect that the theory exhibits light states corresponding to $U(1)^{N-1}$ gauge bosons.",
"Starting from $N=2$ , it is also not clear that the monopole picture is instructive for large $m_A$ , let alone after adding a soft breaking gaugino mass."
],
[
"Summary",
"We have studied the large $N$ $\\theta $ dependence of supersymmetric QCD, using small soft breakings as a probe of the nonsupersymmetric limit.",
"We have seen that certain aspects of the usual large $N$ picture, including the presence of branches and the behavior in theories with matter (both with $N_f \\ll N$ and $N_f \\sim N$ ), are reflected in SBQCD.",
"However, there are also striking departures from ordinary QCD and the conventional large $N$ description.",
"First, in supersymmetric theories, instanton effects are sometimes calculable and do not fall off exponentially with $N$ .",
"Second, branched structure in SBQCD is always associated with approximate discrete symmetries, which are badly broken in the nonsupersymmetric limit.",
"In light of these differences, and to advance our understanding of nonperturbative phenomena in QCD, it would be of great interest to have additional lattice probes of the branched structure of large $N$ QCD.",
"In future work we will explore aspects of lattice tests, particularly the possibility of searching directly for the tower of metastable states at $\\theta =0$ .",
"Acknowledgements: This work was supported in part by the U.S. Department of Energy grant number DE-FG02-04ER41286.",
"We are grateful for conversations with and critical comments from Tom Banks, Nathan Seiberg, Steve Shenker, and Edward Witten."
],
[
"Quantities in supersymmetric gauge theories are readily derived in terms of an object referred to as the holomorphic scale, $\\Lambda _{hol}$ .",
"In the case of $SU(N)$ SUSY QCD without chiral fields, we can make this notion precise in a very simple way, embedding the theory in an ${\\cal N}=4$ theory, with masses for the adjoint fields providing a cutoff for the SQCD theory [45], [37].",
"In a presentation in which the $SU(4)$ symmetry is (almost) manifest, the action is ${\\cal L} = -{1 \\over 32 \\pi ^2} \\int d^2 \\theta \\tau W_\\alpha ^2 + {1 \\over g^2} \\int d^4 \\theta \\Phi _i^\\dagger e^V \\Phi ^i+ \\int d^2 \\theta {1 \\over g^2} f_{abc} \\epsilon ^{ijk} \\Phi ^a_i \\Phi ^b_j \\Phi ^c_k.$ Here $\\tau $ is $\\tau = {8 \\pi ^2 \\over g^2} + i \\theta .$ In order that the superpotential be a holomorphic function of $\\tau $ , we rescale the $\\Phi ^a$ fields.",
"We can also add holomorphic mass terms: ${\\cal L} = -{1 \\over 32 \\pi ^2} \\int d^2 \\theta \\tau W_\\alpha ^2 + {1 \\over g^{2/3} }\\int d^4 \\theta \\Phi _i^\\dagger e^V \\Phi ^i+ \\int d^2 \\theta ( f_{abc} \\epsilon ^{ijk} \\Phi ^a_i \\Phi ^b_j \\Phi ^c_k + M \\Phi ^a_i \\Phi ^a_i).$ Holomorphy of the gauge coupling function gives, for the renormalized coupling, ${8 \\pi ^2 \\over g^2(m)} = {8 \\pi ^2 \\over g^2(M)} + b_0 \\log (m/M) .$ Here $m$ and $M$ are holomorphic parameters (this is discussed further in [37]).",
"The physical masses are related to these by a factor of $g^{2/3}(m), g^{2/3}(M)$ ; substituting yields the standard $\\beta $ function through two loops (issues involving the exact $\\beta $ function are discussed, again, in [37]).",
"$\\Lambda _{hol}$ is then defined through: $\\Lambda _{hol} = M e^{-\\tau /b_0} = g^{-2/3} M_{phys} e^{-\\tau /b_0} .$ This is almost the conventionally defined $\\Lambda $ parameter, but in large $N$ it differs by a power of $N$ , as noted in [27] and we now review.",
"The Particle Data Group presents the strong coupling as (with slight redefinition of $b_0$ and $b_1$ to agree with our conventions above): $\\alpha _s(\\mu ) = {4 \\pi \\over b_0 t} \\left(1 - {b_1 \\over b_0^2} {\\log t \\over t } \\right),~~~t = \\log \\left({\\mu ^2 \\over \\Lambda ^2}\\right).$ Comparing with the solution of the RGE, ${8 \\pi ^2 \\over g^2(\\mu )} = {8 \\pi ^2 \\over g^2(M_{phys})} + b_0 \\log (\\mu /M_{phys}) - {b_1 \\over b_0} \\log (g(\\mu )/g(M_{phys})),$ we see that inserting $\\Lambda = M_{phys} e^{-{8 \\pi ^2 \\over b_0 g^2(M_{phys})} } \\left( \\sqrt{b_0 \\over 8 \\pi ^2} g(M_{phys}) \\right)^{-b_1/b_0^2},$ and $\\log t \\approx \\log \\left({8 \\pi ^2 \\over b_0 g^2(\\mu )}\\right)$ into Eq.",
"(REF ), we recover Eq.",
"(REF ).",
"Using $b_1^2/b_0=2/3$ in pure SYM, one obtains $(\\Lambda _{hol}/\\Lambda )^3\\sim N$ in large $N$ ."
]
] | 1612.05770 |
[
[
"Measuring the core rotation of red giant stars"
],
[
"Abstract Red giant stars present mixed modes, which behave as pressure modes in the convective envelope and as gravity modes in the radiative interior.",
"This mixed character allows to probe the physical conditions in their core.",
"With the advent of long-duration time series from space-borne missions such as CoRoT and Kepler, it becomes possible to study the red giant core rotation.",
"As more than 15 000 red giant light curves have been recorded, it is crucial to develop a robust and efficient method to measure this rotation.",
"Such measurements of thousands of mean core rotation would open the way to a deeper understanding of the physical mechanisms that are able to transport angular momentum from the core to the envelope in red giants.",
"In this work, we detail the principle of the method we developed to obtain automatic measurements of the red giant mean core rotation.",
"This method is based on the stretching of the oscillation spectra and on the use of the so-called Hough transform.",
"We finally validate this method for stars on the red giant branch, where overlapping rotational splittings and mixed-mode spacings produce complicated frequency spectra."
],
[
"Introduction",
"Red giant stars present mixed modes, which behave as pressure modes in the convective envelope and as gravity modes in the radiative interior (Beck et al.",
"[2]).",
"This mixed character allows to study the core of red giants.",
"This is not the case for main-sequence stars, where gravity modes are confined in the inner radiative zone and evanescent in the convective envelope.",
"Red giants differ from main sequence stars because they have a very dense radiative core, where the Brunt-Väisälä frequency reaches high values (Montalbán et al.",
"[12]).",
"Moreover, the evanescent region between gravity and pressure mode resonant cavities is much narrower than in the case of main-sequence stars (Goupil et al.",
"[13]).",
"Thus physical conditions in red giants are met to cause a coupling between pressure modes in the convective envelope and gravity modes in the radiative interior, giving birth to non-radial mixed modes.",
"The study of red giant oscillation spectra led to the automatic measurement of many seismic parameters such as the large separation $\\Delta \\nu $ (Mosser et al.",
"[5]), the frequency of maximum power oscillation $\\nu _{\\mathrm {max}}$ (Kallinger et al.",
"[4]), and the gravity mode period spacing $\\Delta \\Pi _{\\mathrm {1}}$ (Vrard et al.",
"[11]).",
"Many information can be retrieved from these parameters.",
"The measurement of $\\Delta \\nu $ and $\\nu _{\\mathrm {max}}$ allows to determine stellar masses and radii with a precision between 4 to 8 % for the radius and 8 to 16 % for the mass (Kallinger et al.",
"[4]).",
"The measurement of $\\Delta \\Pi _{\\mathrm {1}}$ provides the determination of stars evolutionary status: we can now distinguish between the subgiant phase, the red giant branch (RGB), the red clump where stars burn their helium in their core, and the beginning of the asymptotic giant branch (Mosser et al.",
"[6]).",
"Although the measurement of all these parameters is now entirely automated, this is not the case for the measurement of rotational splittings.",
"The difficulty comes from the fact that mixed modes are not evenly spaced in frequency: pressure-dominated mixed modes are nearly equally spaced in frequency with a spacing close to the large separation $\\Delta \\nu $ , while gravity-dominated mixed modes are nearly equally spaced in period with a spacing close to the gravity mode period spacing $\\Delta \\Pi _{\\mathrm {1}}$ (Mosser et al.",
"[7]).",
"Red giant spectra exhibit numerous dipole mixed modes, which apparently form an inextricable mixed mode forest.",
"The asymptotic expansion of mixed modes is used to identify the mixed mode pattern.",
"In the case of rapid core rotation, it is however difficult to disentangle the frequency spacings between two consecutive mixed modes from the rotational splittings in frequency oscillation spectra.",
"Measurements of the mean core rotation were manually obtained for about 300 red giants (Mosser et al.",
"[8]).",
"They showed that a very efficient angular momentum transport from the core to the envelope is at work in red giants.",
"The physical mechanism responsible for this transport is still not yet fully understood.",
"Mean core rotation measurements for a red giant set as large as possible are therefore needed to constrain the nature of this mechanism.",
"In this work, we detail the principle of the method developed in order to obtain automated measurements of the mean core rotation of red giants.",
"We aim at validating it for stars on the RGB, for which automated measurements are fully consistent with manual measurements (Mosser et al.",
"[8]).",
"Such automated measurements are essential to pave the way for the future analysis of PLATO mission data, with a potential as high as half a million red giants."
],
[
"Stretching the spectra",
"The first step of the method consists in stretching the frequency spectra, which provides spectra where mixed modes are now regularly spaced with a spacing close to $\\Delta \\Pi _{\\mathrm {1}}$ .",
"Therefore, frequencies are changed into stretched periods $\\tau $ through the differential equation (Mosser et al.",
"[9]) $\\mathrm {d}\\tau = \\frac{\\mathrm {d}\\nu }{\\zeta \\nu ^2}.$ The $\\zeta $ function is defined by $\\zeta = \\left[1 + \\frac{1}{q} \\frac{\\nu ^2 \\Delta \\Pi _{\\mathrm {1}}}{\\Delta \\nu } \\frac{\\cos ^2 \\left[\\pi \\frac{1}{\\Delta \\Pi _{\\mathrm {1}}} \\displaystyle {\\left(\\frac{1}{\\nu } - \\frac{1}{\\nu _{\\mathrm {g}}} \\right)}\\right]}{\\cos ^2 \\displaystyle {\\left(\\pi \\frac{\\nu - \\nu _{\\mathrm {p}}}{\\Delta \\nu }\\right)}} \\right]^{-1},$ where $q$ is the coupling parameter of mixed modes, $\\Delta \\Pi _{\\mathrm {1}}$ is the gravity mode period spacing, $\\Delta \\nu $ is the large separation, $\\nu _{\\mathrm {g}}$ are the pure dipole gravity mode frequencies, $\\nu _{\\mathrm {p}}$ are the pure pressure mode frequencies.",
"For the pure dipole gravity mode frequencies $\\nu _{\\mathrm {g}}$ we can use the first-order asymptotic expansion (Tassoul [10]) $\\frac{1}{\\nu _{\\mathrm {g}}} = -({n_{\\mathrm {g}}}+ \\varepsilon _{\\mathrm {g}}) \\, \\Delta \\Pi _{\\mathrm {1}},$ where ${n_{\\mathrm {g}}}$ is the gravity radial order usually defined as a negative value, and $\\varepsilon _{\\mathrm {g}}$ is a small but complicated function sensitive to the stratification near the boundary between the radiative core and the convective envelope.",
"For the pure dipole pressure mode frequencies $\\nu _{\\mathrm {p}}$ we use the universal red giant oscillation pattern (Mosser et al.",
"[7]) $\\nu _{\\mathrm {p}}= \\left({n_{\\mathrm {p}}}+ \\frac{1}{2} + \\varepsilon _{\\mathrm {p}}+ d_{01}+ \\frac{\\alpha }{2} [{n_{\\mathrm {p}}}- n_{\\mathrm {max}}]^2 \\right) \\Delta \\nu ,$ where ${n_{\\mathrm {p}}}$ is the pressure radial order, $\\varepsilon _{\\mathrm {p}}$ is the pressure mode offset, $d_{01}$ is the small separation, $\\alpha $ represents the curvature of the oscillation pattern, and $n_{\\mathrm {max}}= \\nu _{\\mathrm {max}}/ \\Delta \\nu - \\varepsilon _{\\mathrm {p}}$ is the non-integer order at the frequency $\\nu _{\\mathrm {max}}$ of maximum oscillation signal."
],
[
"Rotation signature in stretched spectra",
"In the stretched spectra, rotation induces a small departure from an evenly spaced pattern.",
"The stretched period spacing between rotational multiplet components with the same azimuthal order $m$ expresses (Mosser et al.",
"[9]) $\\Delta \\tau _{\\mathrm {m}}= \\Delta \\Pi _{\\mathrm {1}}(1 + m x_{\\mathrm {rot}}).$ It presents a small departure from $\\Delta \\Pi _{\\mathrm {1}}$ , expressed by $x_{\\mathrm {rot}}= 2 \\, \\zeta \\, \\frac{\\delta \\nu _{\\mathrm {rot, core}}}{\\nu },$ where $\\delta \\nu _{\\mathrm {rot, core}}$ is the rotational splitting induced by the mean rotation of the radiative core.",
"In the case of rapid core rotation, this equation can be used to characterize the crossing of the multiplet components and to infer $\\delta \\nu _{\\mathrm {rot, core}}$ (Gehan et al.",
"[3]).",
"When $x_{\\mathrm {rot}}\\ll 1$ for slow rotation, a relevant approximation of $x_{\\mathrm {rot}}$ is given by (Mosser et al.",
"[9]) $x_{\\mathrm {rot}}\\simeq 2 \\, \\frac{N}{N+1} \\frac{\\delta \\nu _{\\mathrm {rot, core}}}{\\nu _{\\mathrm {max}}},$ where $N$ is the number of gravity modes per $\\Delta \\nu $ -wide frequency range, defined as $N = \\frac{\\Delta \\nu }{\\Delta \\Pi _{\\mathrm {1}}\\, \\nu _{\\mathrm {max}}^2}.$ It is then possible to build échelle diagrams based on this stretched period $\\tau $ , where the different components of the rotational multiplet draw ridges and are disentangled (Figure REF ).",
"Hence, it is easy to obtain measurements of the mean core rotation.",
"The number of ridges that are visible depends on the stellar inclination.",
"Only the ridge associated to the azimuthal order $m = 0$ is visible if the star is seen pole-on, whereas the two components $m = \\pm \\, 1$ are visible when the star is seen equator-on, and finally all the three components $m = \\lbrace -1, 0, 1 \\rbrace $ are visible for intermediate inclinations.",
"In the case of rapid core rotation, the ridges overlap in the échelle diagram, and the frequency where the crossing occurs together with the knowledge of the $\\Delta \\Pi _{\\mathrm {1}}$ parameter provide a precise estimate of the mean core rotation value (Gehan et al.",
"[3]).",
"Figure: Échelle diagram representing τ\\tau as a function of τ\\tau modulo ΔΠ 1 \\Delta \\Pi _{\\mathrm {1}} for the star KIC 6144777.",
"The rotational multiplet components corresponding to the azimuthal orders m={-1,0,1}m=\\lbrace -1, 0, 1\\rbrace are represented by blue dots.",
"Red lines represent the automatic identification of each component through the Hough transform."
],
[
"Measuring rotational splittings: the Hough transform",
"We aim at identifying the ridges corresponding to the different rotational multiplet components (Figure REF ).",
"According to Equation REF , these ridges are nearly vertical in a stretched échelle diagram.",
"The measurement of the rotation can be derived from the identification of these lines, in the form of $y=ax+b$ , where $x$ and $y$ are the cartesian coordinates of the points, $a$ is the slope and $b$ is the intercept of the line.",
"Detecting lines using the previous equation proves to be problematic for lines which are almost vertical.",
"In such cases, the slope tends towards infinity and the determination of the slope and the intercept is highly imprecise.",
"The Hough transformThe classical Hough transform is a feature extraction technique commonly used to identify lines in an image, but it can also be used to identify arbitrary shapes as circles or ellipses (Ballester [1]).",
"allows to get around this problem through the use of the polar coordinates ($\\rho $ , $\\theta $ ) that characterize the position and orientation of the line (Figure REF ): $\\theta $ corresponds to the angle between the line which is perpendicular to the alignment that we consider and the x-axis, $\\rho $ represents the distance between the alignment that we consider and the origin.",
"The Hough transform is particularly useful to detect the different ridges drawn by red giant core rotation in échelle diagrams, which are quasi vertical.",
"The relation between $\\rho $ and $\\theta $ is: $\\rho = x \\cos \\theta + y \\sin \\theta .$ For an isolated point, the $\\theta $ angle may take all possible values between $-\\pi /2$ and $\\pi /2$ .",
"A point have fixed ($x$ , $y$ ) values and is thus represented by a sinusoid in the Hough space ($\\rho $ , $\\theta $ ) (Figure REF top).",
"If several points are aligned on the same line, the corresponding sinusoids all intersect at the same ($\\rho $ , $\\theta $ ) coordinates in the Hough space (Figure REF middle and bottom).",
"Therefore, detecting lines in the ($x,y$ ) space is equivalent to detect intersections in the Hough space.",
"In practice, the algorithm looks for points which are almost aligned and almost vertical in échelle diagrams to identify in an automated way the different ridges.",
"From the Hough parameters, we can derive the mean core rotation, since the angle $\\theta $ used in the Hough transform is linked to $x_{\\mathrm {rot}}$ by $\\tan \\theta = m x_{\\mathrm {rot}}.$ The correction to the $\\Delta \\Pi _{\\mathrm {1}}$ gravity mode spacing induced by rotation in stretched spectra is small.",
"In practice, $x_{\\mathrm {rot}}\\le 0.02$ , so that $\\theta \\le 0.01$ .",
"Hence, we have the direct relation $\\theta \\simeq 2 \\, m\\, \\frac{N}{N+1} \\frac{\\delta \\nu _{\\mathrm {rot, core}}}{\\nu _{\\mathrm {max}}},$ from which we can derive $\\delta \\nu _{\\mathrm {rot, core}}$ .",
"Figure: ΔΠ 1 \\Delta \\Pi _{\\mathrm {1}}-Δν\\Delta \\nu diagram.",
"Blue symbols represent the sample analysed by Mosser et al.",
", with blue crosses accounting for RGB stars and blue triangles accounting for red clump stars.",
"Green triangles represent the stars on which we applied the automatic method developed in this study, orange diamonds represent the rapidly rotating red giants analysed by Gehan et al.",
"."
],
[
"Validation of the method on RGB stars",
"We applied this method to eight red giants.",
"The obtained measurements are resumed in Table 1.",
"With the use of the Hough transform, we could obtain core rotational splitting measurements with a relative uncertainty on the order of a few percents.",
"The sample represents seven RGB stars and a red giant which is transiting to the clump phase (Figure REF ).",
"RGB stars are located very close to the confusion limit in frequency between the rotational splitting $\\delta \\nu _{\\mathrm {rot}}$ and the spacing between consecutive mixed modes (Mosser et al.",
"[8]).",
"That means that the rotational splitting and the mixed-mode frequency spacing sometimes overlap for RGB stars.",
"Thus measuring the mean core rotation in RGB stars is particularly difficult.",
"Nevertheless, we could obtain automatic measurements for stars in this evolutionary stage.",
"In these conditons, obtaining automatic measurements of the mean core rotation of red clump stars will be easier providing that we develop a method which would be more appropriate to their lower core rotation values.",
"Table: Core rotational splittings δν rot , core \\delta \\nu _{\\mathrm {rot, core}}, uncertainties δ(δν rot , core )\\delta (\\delta \\nu _{\\mathrm {rot, core}}) and relative uncertainties δ(δν rot , core )/δν rot , core \\delta (\\delta \\nu _{\\mathrm {rot, core}}) / \\delta \\nu _{\\mathrm {rot, core}} measured with the Hough transform"
],
[
"Conclusions",
"Disentangling rotational splittings from mixed modes is now possible with the use of stretched spectra.",
"We developed a largely automated method to measure the mean core rotation and validated it on RGB stars, where large rotational splittings lead to complicated frequency oscillation spectra.",
"The entire automation of red giant core rotation measurement is in progress, and we aim at obtaining mean core rotation measurements for thousands of red giants observed by Kepler in the near future.",
"These measurements will be essential to pave the way for the future analysis of PLATO data, with potentially 500 000 red giants.",
"Such measurements will also allow us to get more information on the physical mechanisms responsible for angular momentum transport in these stars, thus improving our understanding of stellar physics in deep stellar interiors."
]
] | 1612.05414 |
[
[
"Charted Metropolis Light Transport"
],
[
"Abstract In this manuscript, inspired by a simpler reformulation of primary sample space Metropolis light transport, we derive a novel family of general Markov chain Monte Carlo algorithms called charted Metropolis-Hastings, that introduces the notion of sampling charts to extend a given sampling domain and making it easier to sample the desired target distribution and escape from local maxima through coordinate changes.",
"We further apply the novel algorithms to light transport simulation, obtaining a new type of algorithm called charted Metropolis light transport, that can be seen as a bridge between primary sample space and path space Metropolis light transport.",
"The new algorithms require to provide only right inverses of the sampling functions, a property that we believe crucial to make them practical in the context of light transport simulation.",
"We further propose a method to integrate density estimation into this framework through a novel scheme that uses it as an independence sampler."
],
[
"Introduction",
"Light transport simulation can be notoriously hard.",
"The main problem is that forming an image requires evaluating millions of infinite dimensional integrals, whose integrands, while correlated, may contain an infinity of singularities and different modes at disparate frequencies.",
"Many approaches have been proposed to solve the rendering equation, though most of them rely on variants of Monte Carlo integration.",
"One of the most robust algorithms, Metropolis light transport (MLT), has been proposed by Veach and Guibas in 1997 [18] and has been later extended in many different ways.",
"One of the most commonly used variants is primary sample space MLT [10], partly because in some scenarios it is more efficient (though not always), partly because it is generally considered simpler to implement.",
"However, both variants are still considered relatively complex compared to other algorithms that are not based on Markov chain Monte Carlo (MCMC) methods, or that employ a simplified target distribution [4].",
"In this paper we show that the original primary sample space MLT uses a suboptimal target distribution, and that fixing the problem makes the algorithm more efficient while also greatly simplifying it at the same time.",
"Inspired by this simpler formulation, we then propose a novel family of general Markov chain Monte Carlo algorithms called charted Metropolis-Hastings (CMH).",
"The core idea is to extend the concept of primary sample spaces into that of sampling charts of the target space, extending the domain of the desired target distribution and introducing novel mutation types that swap charts and perform coordinate changes (analogous to those found in regular tensor calculus) in order to craft better proposals.",
"We then apply the new MCMC algorithm to light transport simulation, obtaining a type of algorithms called charted Metropolis light transport (CMLT), that considers all local path sampling methods as parameterizations of the path space manifold, and employs stochastic path inversion as a way to perform coordinate transformations between charts.",
"Our algorithm is made practical by avoiding the requirement to use fully invertible path sampling methods - a property we believe fundamental - and only requiring stochastic right inverses.",
"This new type of algorithms can be seen as fundamentally bridging the difference between the original formulation of path space MLT and the primary sample space version, allowing to easily combine both.",
"Finally, we briefly propose a novel scheme to integrate density estimation inside MCMC frameworks that exploits its robustness with respect to sampling near-singular and singular paths while mantaining overall simplicity and efficiency of implementation."
],
[
"Main contribution",
"The main contribution of our paper is extending primary sample space MLT [10] by introducing mutations that allow to swap bidirectional sampling techniques at any time while preserving the underlying path.",
"Alternatively, adopting a different viewpoint, we could say our main contribution is allowing to freely apply all types of primary space mutations to any given path.",
"The key strength, missing from the original primary space formulation, is allowing to break the path in the middle at any arbitrary point along it and mutate the two resulting subpaths using the corresponding primary space perturbations, bringing back the flexibility of path space MLT, combined with primary space BSDF importance sampling.",
"This is achieved in two ways: the first is realizing that the single primary sample space defined in the original work of Kelemen et al Kelemen:2002 can be more flexibly thought of as a collection of different primary sample spaces stitched together through Russian Roulette, with each space corresponding to a specific bidirectional sampling technique.",
"The second is realizing that each primary sample space is nothing more than a parameterization of path space, and that if we could invert them we could effectively transform this set of parameterizations into a proper atlas, where each primary space is a chart.",
"Once this is achieved, crafting mutations that jump between the charts while not changing the represented path is just a matter of applying proper transformations and following the rules for mantaining detailed balance.",
"However, this second step is made complicated by the fact that the parameterizations typically used in bidirectional path tracers are not always classically invertible, making it impossible to unambiguosly recover the primary space coordinates of a given path.",
"In fact, in the presence of layered materials, sampling the BSDF, which is at the core of any local path sampling technique, is often based on the use of non-injective maps from primary coordinates to the sphere of outgoing directions: for example, if a diffuse and a glossy layer are present, each outgoing direction might be sampled by both layers.",
"In these cases the local primary sample space corresponding to each scattering event is typically divided in two or more strata, each of which maps to the entire sphere (or hemisphere) of directions.",
"As this means we cannot employ the notion of charts used in standard manifold geometry, which requires the parameterizations to be invertible, we hence introduce the notion of sampling charts, that unlike the deterministic counterpart doesn't rely on classical inverses, but rather requires to only provide stochastic right inverses.",
"This new definition allows to move freely between different primary sample spaces even in cases of ambiguity, employing the probability densities associated to these stochastic inverses to compute the transition probabilities needed to satisfy detailed balance.",
"The rest of the paper is dedicated to explaining our framework in detail.",
"In particular, the following sections are organized as follows: section 3 introduces some preliminaries required to properly frame the problem, as well as a simpler reformulation of primary sample space MLT in which all the primary spaces are kept explicitly separate; section 4 introduces our new framework in a very abstract and general mathematical setting; finally section 5 details its application to light transport simulation, and section 6 and 7 are dedicated to describing our massively parallel implementation of the algorithms, and providing test results.",
"This paper is a preprint of a SIGGRAPH publication [14].",
"Concurrent to our work Otsu et al Anon:0462 have developed a novel set of mutations relying on an inverse mapping from path space to primary sample space: while proposing different solutions and mathematical methods, our algorithms share a similar underlying idea."
],
[
"Preliminaries",
"Veach Veach:PHD showed that light transport simulation can be expressed as the solution of per-pixel integrals of the form: $I_j = \\int _{\\Omega } f_j({\\bf x}) d\\mu ({\\bf x})$ where $\\Omega = \\bigcup _{k=1}^{\\infty } \\Omega _k$ represents the space of light paths of all finite lengths $k$ and $\\mu $ is the area measure, and $j$ is the pixel index.",
"For a path ${\\bf x} = x_0 \\rightarrow x_1 \\dots \\rightarrow x_k$ , the integrand is defined by the measurement contribution function: $f_j({\\bf x}) &=&L_e(x_0 \\rightarrow x_1) \\nonumber \\\\&\\cdot & \\prod _{i=0}^{k-1} \\big [ f_s(x_{i-1} \\rightarrow x_i \\rightarrow x_{i+1}) G(x_i \\leftrightarrow x_{i+1}) \\big ] \\nonumber \\\\&\\cdot & W_e^j(x_{k-1} \\rightarrow x_{k})$ where $L_e$ is the surface emission, $W_e^j$ is the pixel response (or emitted importance), $f_s$ denotes the local BSDF and $G$ is the geometric term.",
"To simplify notation, in the following we will simply omit the pixel index and consider the positions $f = f_j$ and $I = I_j$ .",
"Veach further showed that if one employs a family $\\mathcal {F}_k = \\lbrace s,t : s+t-1 = k\\rbrace $ of local path sampling techniques to sample subpaths ${\\bf y} = {y_0 \\dots y_{s-1}}$ and ${\\bf z} = {z_0 \\dots z_{t-1}}$ from the light and the eye respectively, and build the joined path ${\\bf x} = y_0 \\dots y_{s-1} z_{t-1} \\dots z_0$ , an unbiased estimator of $I$ can be obtained as a multiple importance sampling combination: $F = \\sum _{s,t} C_{s,t}({\\bf x})$ with the following definitions: $C_{s,t}({\\bf x}) = w_{s,t}C^*_{s,t}$ $C^*_{s,t}({\\bf x}) = \\frac{f({\\bf x})}{p_{s,t}({\\bf x})} $ $p_{s,t}({\\bf x}) = p_s({\\bf x}) p_t({\\bf x})$ $w_{s,t} = \\frac{p_{s,t}({\\bf x})}{ \\sum _{(i,j) \\in \\mathcal {F}_k} p_{i,j}({\\bf x})}$ While a complete analysis of the above formulas is beyond the scope of this paper (we refer the reader to [19]), we feel it is important to make the following:"
],
[
"Remark:",
"if importance sampling is used, the connection term $C^*_{s,t}$ effectively contains only the parts of $f$ which have not been importance sampled; particularly, if $p_s$ and $p_t$ importance sample all terms of the measurement contribution function up to the $s$ -th and $t$ -th light and eye vertex respectively, $C^*_{s,t}$ will be proportional to the BSDFs at the connecting vertices times the geometric term $G(y_{s-1},z_{t-1})$ .",
"This is the only remaining singularity, which gets eventually suppressed in $C_{s,t}$ by the multiple importance sampling weight $w_{s,t}$ .",
"In fact, simplifying equation (4), one gets: $C_{s,t}({\\bf x}) = \\frac{f({\\bf x})}{ \\sum _{(i,j) \\in \\mathcal {F}_k} p_{i,j}({\\bf x}) } \\nonumber $ The Metropolis-Hastings algorithm is a Markov-Chain Monte Carlo method that, given an arbitrary target distribution $\\pi (x)$ , builds a chain of samples $X_1, X_2, \\dots $ that have $\\pi $ as the stationary distribution, i.e.",
"$\\lim _{n \\rightarrow \\infty } p(X_n) = \\pi (X_n)$ .",
"The algorithm is based on two simple steps:"
],
[
"proposal:",
"a new sample $Y$ is obtained from $X=X_i$ by means of a transition kernel $K(Y|X)$"
],
[
"acceptance-rejection:",
"$X_{i+1}$ is set to $Y$ with probability: $A(Y|X) = \\min \\left( 1, \\frac{\\pi (Y)K(X|Y)}{\\pi (X)K(Y|X)} \\right)$ and to $X_i$ otherwise.",
"Importantly, note that $\\pi $ can be defined up to a constant.",
"In other words, if $\\int \\pi (x) dx = c$ , the algorithm will simply admit $\\pi /c$ as its stationary distribution.",
"Finally, it is also possible to use mutations in which the proposal $K(Y|X) = K(Y)$ depends only on $Y$ : in this case, the mutation type is called an independence sampler [17]."
],
[
"Primary sample space Metropolis light transport, revisited",
"Kelemen et al Kelemen:2002 showed that if one considers the transformation $T : U \\rightarrow \\Omega $ that is typically used to map random numbers to paths when performing forward and backward path tracing (i.e.",
"when sampling eye and light subpaths), one can apply the Metropolis-Hastings algorithm on the unit hypercube $U$ instead of working in the more complex path space.",
"The advantage is that crafting mutations in $U$ is much easier to implement - a simple Gaussian kernel will do - and will often lead to better mutations, since they will naturally follow the local BSDFs.This can, however, be detrimental in cases of complex occlusion, where the original path space MLT is generally superior.",
"The reason is that the BSDF parameterizations might squeeze unoccluded, off-specular directions into vanishingly small regions of the primary sample space.",
"The only requirement is pulling back the desired measure from $\\Omega $ to $U$ , which is easily achieved by multiplying by the Jacobian of the transformation $T$ , which is nothing more than the reciprocal of path probability: $I = \\int _U f(T(u)) \\left|\\frac{dT(u)}{du}\\right| du = \\int _U \\frac{ f(T(u)) }{p(T(u))} du$ We now provide a novel formulation that improves the choice of mapping and target distributions compared to the ones employed by Kelemen et al Kelemen:2002.",
"In fact, what was done in the original work was to consider a mapping from the product of two infinite-dimensional unit hypercubes,A formulation which, technically, poses some definition challenges, as infinite dimensional spaces do not possess a Lebesgue measure.",
"to the product space of light and eye subpaths sampled using Russian Roulette terminated path tracing.",
"Furthermore, instead of simply considering the single path obtained by joining the two endpoints of the respective subpaths, and using the measurement contribution function as the target distribution, they considered the sum of the MIS weighted contributions from all paths obtained joining any two vertices of the light and eye subpaths.",
"The reason why this was done can be understood: this was the historical way to perform bidirectional path tracing.",
"In order not to waste any vertex, one would reuse all of them at the expense of some added correlation and some added shadow rays.",
"However, this is undesirable for several reasons:"
],
[
"1.",
"by joining all vertices in the generated subpaths, and summing up all the weighted contributions from the obtained paths (which are in fact truly different paths, except for the fact they share their light and eye prefixes), they were using a target distribution which was no longer proportional to path throughput (or, more precisely, the measurement contribution function we are finally interested in).",
"In other words, the obtained paths have a skewed distribution which is not necessarily optimal.One can consider their technique to generate path bundles and in this sense their target distribution is optimal for the constructed bundles, relative to the overall bundle contribution, but not for the individual paths."
],
[
"2.",
"dealing with the infinite dimensional unit hypercubes introduces some unnecessary algorithmic complications, including the need for lazy coordinate evaluations."
],
[
"3.",
"by joining all vertices in the generated subpaths, we are introducing some additional sample correlation that might not necessarily improve the per-sample efficiency.",
"In some situations, for example in the presence of incoherent transport or complex occlusion, it will in fact reduce it.",
"In light of these problems, we now propose a much simpler variant.",
"Let's for the moment consider the space of paths of length $k$ , and a single technique $i \\in \\mathcal {F}_k$ to generate them, where $i$ defines the number of light vertices and the number of eye vertices is given as $j = k+1-i$ .",
"If sampling $n$ vertices through path tracing requires $m \\cdot n$ random numbers, we will consider the following definition of the primary sample space: $U_i = [0,1]^{m \\cdot i} \\times [0,1]^{m \\cdot (k+1-i)}.$ The transformation $T = T_i: U \\rightarrow \\Omega _k$ will have the following Jacobian: $\\left| \\frac{dT(u)}{du} \\right| = \\frac{1}{ p_{i}(T(u)) }.$ We now have two options for the choice of our target distribution.",
"The simplest is to set: Definition: Importance sampled distributions $\\pi _i(u) = \\frac{ f(T(u)) }{ p_{i}(T(u)) }.$ This choice keeps the corresponding path space distribution invariant relative to the area measure $\\mu $ , as we have: $\\pi _i(u) du&=& \\pi _i(u) p_i(T(u)) |d\\mu (T(u))/du|du \\nonumber \\\\&=& \\pi _i(u) p_i(T(u)) d\\mu (T(u)) \\nonumber \\\\&=& f(T(u)) d\\mu (T(u)) \\nonumber \\\\&=& \\bar{\\pi }(T(u)) d\\mu (T(u)).$ In other words, it ensures that all our distributions $\\pi _i(u)$ are designed to have a distribution in their primary space $U_i$ that becomes the same distribution $\\bar{\\pi }({\\bf x}) = f({\\bf x})$ in path space.",
"The second choice is to use the following: Definition: Weighted distributions $\\pi _i(u) = w_i(T(u)) \\frac{ f(T(u)) }{ p_i(T(u)) },$ exploiting the fact that, while now the corresponding path space distributions $\\bar{\\pi }_i({\\bf x}) = w_i({\\bf x})f({\\bf x})$ are biased,In practice instead of sampling $f$ , they are sampling a version downscaled locally according to the efficiency of $p_i$ our desired path space distribution $f$ is obtained as their sum: $\\sum _{i\\in \\mathcal {F}_k}\\bar{\\pi }_i({\\bf x})= \\sum _{i\\in \\mathcal {F}_k} w_i({\\bf x}) f({\\bf x})= f({\\bf x}).$ This definition leads to some interesting properties.",
"First and foremost, we have the following simplifications: $\\pi _i(u) = \\frac{ f(T(u)) }{\\sum _{j \\in \\mathcal {F}_k} p_j(T(u))}$ Second, in each primary sample space the target distribution depends only on the path ${\\bf x} = T(u)$ , but not on the particular choice of technique $i$ used to generate it.",
"In other words, if $u^i \\in U_i$ and $u^j \\in U_j$ map to the same path ${\\bf x} = T_i(u^i) = T_j(u^j)$ , we have: $\\pi _i(u^i) = \\pi _j(u^j)$ In particular, the target distribution depends only on how well the sumEquivalently, their average, since $\\pi $ is here defined up to a constant.",
"of the individual pdfs $p_i$ approximate $f$ .",
"This is an interesting result, as we will see later on.",
"Third, notice that if all bidirectional techniques are included in $\\mathcal {F}_k$ , the target distribution does not contain any of the weak singularities induced by the geometric terms.",
"This is the case because each pdf includes all but one of the geometric terms: thus their sum will contain all of them, and counterbalance those in the numerator of (REF ).",
"In particular, this means there will be no singular concentration of paths near geometric corners.The only sources of singularie Diracs in unsampled specular BSDFs in SDS paths (not containing any DD edge).",
"Notice that this would have not been the case if we simply adopted $\\pi = f / p_i$ , omitting the multiple importance sampling weight."
],
[
"Auxiliary Distributions",
"Šik et al Sik:2016 proposed using an auxiliary distribution in conjunction with replica exchange [16] to help the primary MLT chain escape from local maxima.",
"The auxiliary distribution is designed to be easier to sample, and hence favor exploration.",
"Given they were working in the context of the original PSSMLT formulation where all connections are performed, they proposed using an auxiliary distribution with a target defined as 1 if any of the paths formed provides a non-zero contribution, and 0 otherwise.",
"With our new primary sample space formulation, a similar but even easier objective can be achieved by simply dropping all connection terms except for visibility, i.e.",
"the only terms which are not sampled by the $i$ -th local path sampling technique, giving: $\\pi _i^{\\prime }(u) = V(x_{i-1} \\leftrightarrow x_{i})$ which in path space becomes: $\\bar{\\pi _i}^{\\prime }(x) = V(x_{i-1} \\leftrightarrow x_{i}) p_i(x)$ Notice that due to our use of primary sample space mutations, this function is very easy to sample, as our base sampling technique already generates samples distributed according to $p_i$ .",
"Importantly, we might not even need Metropolis at all, as we could simply use our path generation technique as an independence sampler, akin to the large steps in the original work of Kelemen et al Kelemen:2002.",
"However, using Metropolis with local perturbations might still help in regions of difficult visibility."
],
[
"Handling color",
"In the above we treated $f$ as a scalar, though in practice it is actually a color represented either in RGB or with some other spectral sampling.",
"While handling spectral rendering in all generality can require custom techniques [21] and is beyond the scope of this paper, for RGB (and even in many cases of spectral transport) it is sufficient to use the maximum of the components $f^* = \\max _i\\lbrace (f)_i\\rbrace $ when constructing the target distribution, and weighting the resulting color samples accordingly before final image accumulation.",
"Before introducing our light transport algorithm, we introduce a novel family of general Markov chain Monte Carlo algorithms inspired by the primary sample space MLT formulation we just described.",
"The idea is that we want to allow jumping between different primary sample spaces, as this will allow to more freely escape from local maxima in situations in which the current parameterization is not the best fit for the target distribution.",
"Suppose in all generality that we have an arbitrary target space $(\\Omega ,\\mu )$ , a function $f:\\Omega \\rightarrow \\mathbb {R}$ we are interested in sampling, and a parametric family $\\mathcal {F} = (U_i,T_i,R_i) _{i = 0,\\dots ,n-1}$ , such that: $U_i$ is a measured primary sample space; $T_i$ , the forward map, is a function $T_i:U_i \\rightarrow \\Omega $ ; $R_i$ , the reverse map, is a right-inverse of $T_i$ , i.e.",
"$R_i:\\Omega \\rightarrow U_i$ with: $T_i(R_i(x)) = x \\quad \\forall x \\in \\Omega ;$ Let's also consider the density $p_i:\\Omega \\rightarrow \\mathbb {R}$ defined as the pdf of the transformation $T_i(U)$ of a uniform random variableMore precisely, $p_i$ is uniquely defined almost everywhere as the function that satisfies the equation: $P(T_i(U) \\in A) = \\int _{A} p_i(x) d\\mu (x)$ , for any measurable subset $A \\subseteq \\Omega $ and $U \\sim Uniform(U_i)$ ., and the function $r_i:U_i \\rightarrow \\mathbb {R}$ defined as its reciprocal: $r_i(u) = \\frac{1}{p_i(T_i(u))}.",
"\\nonumber $ Now, consider again the weighted distributions defined by: $\\pi _i(u) = \\frac{ f(T_i(u)) }{ \\sum _i p_i(T_i(u)) }$ The idea is that we could use the reverse maps $R_i$ , which can be interpreted as inverse sampling functions, to perform the desired jumps between primary sample spaces, e.g performing swaps in the context of a replica exchange framework where we run $n$ chains, each sampled according to a different $\\pi _i$ .",
"We now show how to achieve it.",
"Figure: Charted Metropolis-Hastings allows performing coordinate changes between the target space Ω\\Omega and its sampling charts.",
"When multiple points of a given sampling domain map to a single point in Ω\\Omega , it's sufficient for the right inversion mappings to return one of them (as for the case of u 0 u_0), or return one picked at random inside the set (as for the case of u 3 u_3) with the help of an additional sampling domain (V 3 V_3, light violet box).Given two states, $u_1^i$ , generated by the $i$ -chain, and $u_2^j$ , generated by the $j$ -chain, consider their target space mappings: ${x_1} := T_i(u_1^i) \\nonumber \\\\{x_2} := T_j(u_2^j) \\nonumber $ and their reverse mappings: ${u_1^j} := R_j(x_1) \\nonumber \\\\{u_2^i} := R_i(x_2) \\nonumber $ if we wanted to perform a swap, preserving detailed balance between the chains requires accepting the swap with probability: $A =\\min \\left( 1,\\frac{\\pi _i(u_2^i)\\pi _j(u_1^j)r_i(u_1^i)r_j(u_2^j)}{\\pi _i(u_1^i)\\pi _j(u_2^j)r_i(u_2^i)r_j(u_1^j)}\\right)$ This can be proven by looking at the two chains as an ensemble in the space $U_i \\times U_j$ , with target distribution $\\pi _i \\cdot \\pi _j$ .",
"Equation (REF ) is then obtained from equation (8) following the usual Metropolis-Hastings rule described in section 3.1, viewing $(u_1^i,u_2^j)$ as the current state and $(u_2^i,u_1^j)$ as the proposal.",
"In the previous section we saw that our target distributions $\\pi _i$ assume the same value on the same points of $\\Omega $ , independently of the underlying technique $i$ used to generate it.",
"Now since $R_i$ has been defined as a right inverse of $T_i$ , if $u^j = R_j(T_i(u^i))$ , we would again have: $\\pi _j(u^j) = \\pi _i(u^i).$ This property is essentially stating that our target distribution is invariant under a change of charts of the target space.",
"Hence, equation (REF ) simplifies to: $A =\\min \\left( 1,\\frac{r_i(u_1^i)r_j(u_2^j)}{r_i(u_2^i)r_j(u_1^j)}\\right)$ without requiring any evaluation of the target distributions.",
"Notice that we didn't require the transformations $T_i$ to be fully invertible: if the fiber of $x$ under $T_i$ , i.e.",
"the set $T^\\leftarrow _i(x) = \\lbrace u | T_i(u) = x\\rbrace $ , contains several points, it's sufficient that $R_i$ returns one of them.",
"This approach is very general, as such a function can always be constructed.",
"However, it can be made even more general by randomizing the selection of the point in the fiber.",
"We do so by extending the domains in which the functions $R_i$ operate."
],
[
"Sampling Atlas.",
"We call sampling atlas a family $\\mathcal {F} = (U_i,V_i,T_i,R_i)_{i=0,...,n-1}$ where $U_i$ and $T_i$ are defined as before, but: $V_i$ is a measured reverse sampling space, and $R_i$ is an extended right-inversion map, $R_i:\\Omega \\times V_i \\rightarrow U_i$ , such that: $T_i(R_i(x,v)) = x \\quad \\forall x \\in \\Omega \\quad \\textrm {and} \\quad \\forall v \\in V_i.",
"\\nonumber $ Each tuple $(U_i,V_i,T_i,R_i)$ is called a sampling chart.",
"With these definitions, we can draw two uniform random variables $v_1 \\in V_i$ and $v_2 \\in V_j$ , and replace the reverse mappings $u_1^j$ and $u_2^i$ with: ${u_1^j} := R_j(x_1,v_1) \\nonumber \\\\{u_2^i} := R_i(x_2,v_2) \\nonumber $ which can now be tested for acceptance with the same acceptance ratio: $A =\\min \\left( 1,\\frac{r_i(u_1^i)r_j(u_2^j)}{r_i(u_2^i)r_j(u_1^j)}\\right).",
"\\nonumber $ This construction is depicted in Figure REF , where: a. the chart $U_0$ contains two points, $u_0$ and $u^{\\prime }_0$ , that map to the same point ${\\bf x} \\in \\Omega $ , but $R_{0}({\\bf x})$ selects just one of them, in this case $u_0$ ; b. the chart $U_3$ contains an entire set that maps to $x$ , but its points are identified by means of points of the reverse sampling domain $V_3$ .",
"A similar mathematical framework can be used in the context of serial (or simulated) tempering [12].",
"In this context, one could run a single chain $u^i = (u,i)$ in an extended state space $U \\times \\mathcal {F}$ , where $i$ denotes the technique used to map the chain to target space.",
"Drawing a uniform random variable $v \\in V_i$ and swapping from $i$ to $j$ through the transformation: $u^j = R_j(u^i,v) \\nonumber $ would then require accepting the swap with probability: $\\min \\left( 1,\\frac{r_i(u^i)}{r_j(u^j)}\\right)$ and rejecting it otherwise.",
"Once again, no evaluation of the target distributions is required.",
"We call both this and the above mutations chart swaps or coordinate changes.",
"Notice that if there is a way to craft mutations in the target space itself, it is always possible to add the identity chart to $\\mathcal {F}$ : $U_n = \\Omega $ , $V_n = \\emptyset $ $T_n(x) = R_n(x) = x$ ; care must only be taken in adding the probability $p_n = 1$ to the denominator of all the distributions $\\pi _i$ in equation (REF ).",
"Finally, we consider another type of mutation, inverse primary space perturbations, which can be in a sense considered the dual of the above.",
"Suppose we are now running a chain in the target space $\\Omega $ , distributed according to $\\pi ({\\bf x})$ .",
"We can then use inversion to momentarily parameterize the target space through a given technique $i$ and take a detour or move down from $\\Omega $ to $U_i$ to perform a symmetric primary sample space perturbation there, before finally getting back to $\\Omega $ .",
"With this scheme, given a state ${\\bf x}$ and a uniform random variable $v \\in V_i$ , applying the transformation $R_i$ to obtain $u = R_i({\\bf x},v)$ and the perturbation kernel $K$ to obtain the proposal $u^{\\prime } = K(u)$ and ${\\bf y} = T_i(u^{\\prime })$ , would result in the following acceptance ratio: $A({\\bf y}|{\\bf x}) =\\min \\left( 1,\\frac{\\pi ({\\bf y})K(u|u^{\\prime })r_i(u^{\\prime })}{\\pi ({\\bf x})K(u^{\\prime }|u)r_i(u)}\\right)$ which simplifies to the standard primary sample space formula if $K$ is symmetric: $A({\\bf y}|{\\bf x}) =\\min \\left( 1,\\frac{\\pi ({\\bf y})}{p_i({\\bf y})}\\cdot \\frac{p_i({\\bf x})}{\\pi ({\\bf x})}\\right).$ We call this family of MCMC algorithms that jump between charts of the target space charted Metropolis-Hastings, or CMH.",
"Figure: A visualization of two path space charts,where one of the bidirectional sampling techniques, in this case T 3,2 T_{3,2}, maps multiple points to the same selected path, while T 2,3 T_{2,3} is locally invertible.Notice how a naive transfer of coordinates such as that employed in MMLT (dashed gray lines) could result in a very different path."
],
[
"Charted Metropolis Light Transport",
"It should now be clear how the above algorithms can be applied to light transport simulation.",
"If we consider the framework for primary sample space MLT outlined in section 3.2, it is sufficient to add functions for path sampling inversion to be able to apply our new charted Metropolis-Hastings replica exchange or serial tempering mutations in conjunction with the standard set of primary sample space perturbations.",
"The advantage of these mutations is that they will allow to more easily escape from local maxima when the current sampling technique is not locally the best fit for $f$ .",
"The mutations are relatively cheap, as they don't require any expensive evaluations of the target distribution.",
"Moreover, and very importantly, the algorithm is made practical by not requiring the path sampling functions $T_i$ to be classically invertible.",
"In the context of light transport simulation this property is crucial, as BSDF sampling is seldom invertible: in fact, with layered materials often a random decision is taken to decide which layer to sample, but the resulting output directions could be equally sampled (with different probabilities) by more than one layer.",
"Our framework requires to return just one of them, but it also allows selecting which one at random with a proper probability.",
"All is needed is the ability to compute the density of the resulting transformation.",
"This construction is illustrated in Figure REF , which shows how the same path ${\\bf x}$ can be represented both in the chart corresponding to the bidirectinal technique $(2,3)$ and the one coresponding to the technique $(3,2)$ , where the latter contains two distinct points, $u_{3,2}$ and $u^{\\prime }_{3,2}$ , that map to ${\\bf x}$ .",
"In the picture $R_{3,2}({\\bf x})$ selects just one of them, in this case $u^{\\prime }_{3,2}$ .",
"Further on, by adding the identity target space chart, we can also add the original path space mutations proposed by Veach and Guibas Veach:1997:MLT, potentially coupled with the new inverse primary space perturbations.",
"We call the family of such algorithms charted Metropolis light transport, or CMLT."
],
[
"Connection to path space MLT",
"The new algorithms can be considered as a bridge between primary sample space MLT and the original path space MLT proposed by Veach and Guibas Veach:1997:MLT.",
"In fact, one of the advantages of the original formulation over Kelemen's variant Kelemen:2002 was its ability to break the path in the middle and resample the given path segment with any arbitrary bidirectional technique.",
"This ability was entirely lost in primary sample space, as the bidirectional sampling technique was implicitly determined by the sample coordinates (or needed to be chosen ahead of time in the version we outlined in section 3.2).",
"While Multiplexed Metropolis Light Transport (MMLT) [6] added the ability to change technique over time, as the coordinates $u$ were kept fixed such a scheme was leading to swap proposals that sample unrelated paths: in fact, two techniques $i$ and $j$ map the same coordinates $u$ to different paths $T_i(u) \\ne T_j(u)$ that share only a portion of their prefixes (in other words, the two resulting paths are spuriously correlated by the algorithm, whereas in fact there is no reason for them to be - see Fig.REF ).",
"Our coordinate changes, in contrast, preserve the path while changing its parameterization, thus allowing to simply perturb it later on with a different bidirectional sampler.",
"Adding the identity path space chart and inverse primary space perturbations makes the connection even tighter, allowing to smoothly integrate the original bidirectional mutations and perturbations with an entirely new set of primary sample space perturbations.",
"Notice that while inverse primary space perturbations could also be applied to a single path space chain, the advantage of also incorporating primary space chains in a replica exchange or serial tempering context is that the target distributions (defined by equation REF ) become generally smoother due to the implicit use of the multiple importance sampling weight, raising the acceptance rate."
],
[
"Alternative parameterizations",
"While the original primary sample space Metropolis used path space parameterizations based on plain BSDF sampling, it is also possible to use other parameterizations that can provide further advantages: for example the half vector space parameterizations that have been recently explored [9], [7]."
],
[
"Density estimation",
"So far, we have concentrated on standard bidirectional path tracing with vertex connections.",
"However, all the above extends naturally to density estimation methods, using the framework outlined in [5].",
"The only major difference is the computation of the subpath probabilities.",
"However, we here suggest an alternative approach.",
"Instead of using density estimation as an additional technique, applying multiple importance sampling to combine it into a unique estimator, we can use it only to craft additional proposals.",
"In other words, we can use density estimation as another independence sampler.",
"Suppose we are running an MCMC simulation in $\\Omega _k$ , and at some point in time our chain is in the state $u^i$ , with $s = i$ , and $t = k - s + 1$ .",
"We can then try to build a candidate path through density estimation with the $(s+1,t)$ -technique and, if the resulting path has non-zero contribution, we can drop one light vertex (and the corresponding primary sample space coordinates) and consider it as a new proposal $u^i_{de}$ .",
"Notice that in doing so, we have to adjust the acceptance ratio for the actual proposal distribution.",
"For clarity, we will now omit the superscripts $i$ , and obtain: $A(u_{de}|u) = \\min \\left( 1,\\frac{\\pi (u_{de})p_{de}(T(u))}{\\pi (u)p_{de}(T(u_{de}))} \\right)$ where $p_{de}(x)$ is the probability of sampling the path $x$ by density estimation (which can be approximated at the cost of some bias as described in [5] or estimated unbiasedly as in [15]).",
"If we want to further raise the acceptance rate, we can also mix this proposal scheme with an independence sampler based on bidirectional connections and combine the two, calculating the total expected probability to make both more robust: $A(u^{\\prime }|u) = \\min \\left( 1,\\frac{\\pi (u^{\\prime })(p_{de}(T(u)) + p_{bc}(T(u)))}{\\pi (u)(p_{de}(T(u^{\\prime })) + p_{bc}(T(u^{\\prime })))}\\right).$ Notice that this formula is now agnostic of how the samples were generated in the first place, i.e.",
"whether the candidate $u^{\\prime }$ was proposed by density estimation or bidirectional connections: this is a positive side-effect of using expectations.While this looks similar to multiple importance sampling, it is not quite the same: multiple importance sampling is a more general technique used to combine estimators, whereas here we are just interested in computing an expected probability density, using so called state-independent mixing [3].",
"However, multiple importance sampling using the balance heuristic is equivalent to using an estimator based on the average of the probabilities, which is exactly the expected probability we need: hence the reason of the similarity.",
"This approach is the same used in the original MLT to compute the expected probability of bidirectional mutations.",
"Figure: A schematic visualization of the basic bidirectional path tracing pipeline, showing the different shading and tracing kernels.",
"Notice that while they are shown here side by side, light path tracing and eye path tracing happen in subsequent phases of the algorithm."
],
[
"Designing a complete algorithm",
"So far we have only constructed a theoretical background to build novel algorithms, but we didn't prescribe practical recipes.",
"The way we combine all the above techniques into an actual algorithm is described here.",
"First of all, we start by estimating the total image brightness with a simplified version of bidirectional path tracing.",
"The algorithm first traces $N_{init}$ light subpaths in parallel and stores all generated light vertices.",
"Then it proceeds tracing $N_{init}$ eye subpaths, and connects each eye vertex to a single light vertex chosen at random among the ones we previously stored.",
"At the same time, the emission distribution function at each eye vertex is considered, forming pure path tracing estimators with light subpaths with zero vertices.",
"All evaluated connections (both implicit and explicit) with non-zero contribution (which represent entire paths, each with a different number of light and eye vertices $s$ and $t$ ) are stored in an unordered list.",
"Second, in order to remove startup bias, we resample a population of $N$ seed paths for a corresponding amount of chains.",
"In order to do this, we build the cumulative distribution over the scalar contributions of the previously stored paths, and resample $N$ of them randomly.",
"Notice that the $N$ seed paths will be distributed according to their contribution to the image.",
"Particularly, the number of paths sampled with technique $i$ will be proportional to the overall contribution of that technique, and similarly for path length.",
"At this point, though not crucial for the algorithm, we sort the seeds by path length $k$ so as to improve execution coherence in the next stages.",
"In practice, sorting divides the $N$ seeds into groups of $N_k$ paths each, such that $\\sum _k N_k = N$ .",
"Finally, we run the $N$ Markov chains in parallel using both classic primary sample space perturbations and the novel simulated tempering or replica exchange mutations described in sections 3 and 4.",
"As the new mutations have a low cost compared to performing actual perturbations, they can be mixed in rather frequently (with very low overhead up to once every four iterations).",
"boxruled x, $\\omega _i$ , $\\omega _o$ u (primary space coordinates) probs[] $\\leftarrow $ layerSamplingProbabilities(x,$\\omega _i$ ) prob_sum $\\leftarrow $ probs[diffuse] + probs[glossy] v $\\leftarrow $ random() * prob_sum v $<$ probs[diffuse] u $\\leftarrow $ (v, invertLambert(x,$\\omega _i$ ,$\\omega _o$ )) u $\\leftarrow $ (v, invertGGX(x,$\\omega _i$ ,$\\omega _o$ )) inversion of a composite BSDF containing a diffuse and a glossy layer"
],
[
"Implementation",
"We implemented our algorithm, together with MMLT, PSSMLT and bidirectional path tracing (BPT) in CUDA C++, exposing massive parallelism at every single stage, including ray tracing, shading, cdf construction (prefix sum), resampling and sorting (radix sort).",
"The basic bidirectional path tracing algorithm is constructed as a pipeline of kernels (also known as wavefront tracing [11]), and relies on the OptiX Prime library for ray tracing.",
"We ran all tests on an NVIDIA Maxwell Titan X GPU.",
"The basic bidirectional path tracing pipeline, composed of seven shading and tracing stages, is shown schematically in Figure REF .",
"This pipeline is further extended in all the MCMC rendering algorithms by additional stages performing primary sample space coordinates generation (applying both perturbations and chart swaps in the case of CMLT), and the final acceptance-rejection step.",
"All the pipeline stages communicate through global memory work queues.",
"In order to keep storage and bandwidth consumption to a minimum, only minimal information is stored for each path vertex (including instance id, primitive id and uv coordinates), using on the fly vertex attributes interpolation where needed (such as during path inversion).",
"For $256K$ paths, of a maximum of 10 vertices each, this requires about 64MB of storage.",
"Both our CMLT and MMLT implementations run several thousand chains in parallel, using the seeding algorithm described in section 5.4.",
"Besides being strictly necessary to scale to massively parallel hardware, we found this to produce some additional image stratification, as discussed in the Results section.",
"The CMLT implementation is based on the serial tempering formulation.",
"Our framework employs a layered material system that combines a diffuse BSDF (Lambertian) and rough glossy reflection and transmission BSDFs (GGX) using a Fresnel weighting.",
"Sampling of the glossy component is implemented using the distribution of visible normals [8], and selection between the diffuse and glossy components is performed based on Fresnel weights.",
"Clearly, this path sampling scheme is not invertible, as both the diffuse and glossy components can map different primary sample space values to the same outgoing directions.",
"Hence, we used the machinery described in section 4 to enable randomized inversion."
],
[
"Chart swaps and path inversion",
"Given a bidirectional path generated by the technique $(s,t)$ using coordinates $u$ , in order to perform a chart swap we propose a new pair $(s^{\\prime },t^{\\prime })$ with $s^{\\prime } + t^{\\prime } = s + t$ distributed according to the total energy of the techniques (i.e.",
"the normalization constants of the target distributions).",
"After the candidate is sampled, path inversion needs to be performed using the transformation $u^{\\prime } = R_{s^{\\prime },t^{\\prime }}(T_{s,t}(u))$ .",
"This transformation can be widely optimized noticing that there are only two cases: $s^{\\prime } > s$ : in this case it is only necessary to invert the coordinates of the light subpath vertices $\\lbrace y_s, ..., y_{s^{\\prime } - 1}\\rbrace $ .",
"$t^{\\prime } > t$ : in this case it is only necessary to invert the coordinates of the eye subpath vertices $\\lbrace z_t, ..., z_{t^{\\prime } - 1}\\rbrace $ .",
"Computing the inverse pdf $r_{s,t}$ can be optimized analogously.",
"In each of these cases, we start the stochastic inversion from the end of the selected subpath, and proceed backwards.",
"At each vertex, we consider the local composite BSDF, and compute the forward probabilities originally used to select which layer to sample (for example, based on their Fresnel weighted albedos).",
"Using these, we draw a single random number $v$ to select which of the layers to use for inversion.",
"Pseudocode for a material with a diffuse and glossy layer is provided in Algorithm 1.",
"Pseudocode for a serial version of the overall CMLT algorithm is given in Algorithm 2.",
"The Appendix provides further details and pseudocode for the inversion of typical BSDFs."
],
[
"Results",
"We performed two sets of tests.",
"The first is aimed at testing the many possible algorithmic variations of CMLT on a simplified light transport problem.",
"The second, using full light transport simulation, compares a single CMLT variant against MMLT, which could be currently considered state-of-the-art in primary sample space MLT."
],
[
"Simplified light transport tests",
"This test consists of rendering an orthographic projection of the $XY$ plane directly lit by two area light sources.",
"The first light is a unit square on the plane $Y = 0$ , with a spatially varying emission distribution function changing color and increasing in intensity along the $X$ axis.",
"The light source is partially blocked by a thin black vertical strip near its area of strongest emission.",
"The second light is another unit square on the plane $Y = 1$ , with uniform green emission properties.",
"This light is completely blocked except for a tiny hole.",
"In this case, our path space consists of two three-dimensional points: the first on the ground plane, the second on the light source.",
"As charts, we used two different parameterizations:"
],
[
"generating a point uniformly on the visible portion of the ground plane and a point on the light sources distributed according to their spatial emission kernels - corresponding to path tracing with next-event estimation, i.e.",
"the bidirectional path tracing technique $(s,t) = (1,1)$ ;"
],
[
"generating a point uniformly on the visible portion of the ground plane, sampling a cosine distributed direction, and intersecting the resulting ray with the scene geometry to obtain the second point - corresponding to pure path tracing, i.e.",
"the bidirectional path tracing technique $(s,t) = (0,2)$ .",
"Both charts have a four dimensional domain, and in both cases we used exact inverses of the sampling functions.",
"We tested six different MCMC algorithms: PSSMLT-1: a single PSSMLT chain using the first parameterization; PSSMLT-2: a single PSSMLT chain using the second parameterization; PSSMLT-AVG: two PSSMLT chains using both the first and the second parameterization, both using the importance sampled distribitions (equation REF ), where the accumulated image samples are weighted (i.e.",
"averaged) through multiple importance sampling with the balance heuristic; PSSMLT-MIX: two PSSMLT chains using both the first and the second parameterization, with the weighted distributions (equation REF ); CMLT-IPSM: a single CMLT chain in path space alternating inverse primary space mutations using the first and the second parameterizations; CMLT: two CMLT chains using both the first and the second parameterization as charts, coupled with replica-exchange swaps performed every four iterations; Results are shown in Figure REF , while their root mean square error (RMSE) is reported in Table 1.",
"All images except for the reference were produced using the same total number of samples $n = 16 \\cdot 10^6$ : PSSMLT-1, PSSMLT-2 and CMLT-IPSM running a single chain of length $n$ , whereas PSSMLT-AVG, PSSMLT-MIX and CMLT running two chains of length $n/2$ .",
"In table 1 we further report RMSE values for $n = 128 \\cdot 10^6$ .",
"The reference image has been generated by plain Monte Carlo sampling.",
"As can be noticed, our PSSMLT-MIX formulation using the distributions defined by equation (REF ) is superior to simply averaging two PSSMLT chains using multiple importance sampling (PSSMLT-AVG), which is in fact worse than PSSMLT using a single chain according to the second distribution (PSSMLT-2).",
"Table: Root mean square error of the images computed by the various algorithms we tested in figure .CMLT-IPSM produces results that are just slightly worse than PSSMLT-MIX, but still superior to all other PSSMLT variants.",
"The reason why CMLT-IPSM is inferior to PSSMLT-MIX is that while the target distribution for CMLT-IPSM is proportional to $f$ , the target distributions of the chains in PSSMLT-MIX are smoother due to the embedded multiple importance sampling weights, and contain no singularities.",
"Finally, CMLT produces the best results among all algorithms."
],
[
"Full light transport tests",
"For these tests we compared the CMLT implementation described in section 5 against our own implementation of MMLT.",
"We provide five test scenes representative of different transport phenomena: The Gray & White Room: a scene from Bitterli's repository resources16.",
"Escher's Room: an M.C.",
"Escher themed adaptation of the above scene, featuring multi-layer materials with variable surface properties.",
"This scene contains many light sources of different size: the large back wall, with a variable Lambertian emission distribution displaying a famous painting by the artist, a smaller area light on the ceiling, and the external lighting coming from the windows.",
"The smaller light is partially blocked by a rough glossy reflector, which causes a blurry caustic on the partially glossy ceiling.",
"Notice how all the above elements contribute to forming an all-frequency lighting scenario.",
"Escher's Glossy Room: a variation of the above scene in which all surfaces are glossy (with no diffuse component), with variable roughness (with GGX exponents ranging between 5 and 100).",
"Notice that this scene contains a variety of caustics of all frequencies (in a sense, all lighting is due to caustics).",
"This scene stresses the advantages of chart swaps in the presence of near-specular transport, where there are many narrow modes and there is often no single best sampling technique.",
"Wall Ajar: another variation of the above scene mimicking Eric Veach's famous scene the door ajar.",
"Most of the lighting in the scene comes from a narrow opening in the sliding back wall, which covers an equally large but completely hidden emissive wall.",
"Hence, the room is almost entirely indirectly lit, except for the blue light coming from the windows.",
"The ceiling area light source is also considerably smaller, casting a sharper caustic, and most surfaces are now about half diffuse half glossy.",
"The sofa also features some rough transmission.",
"Salle de bain: another scene from Bitterli's repository resources16.",
"While in terms of light transport this scene is considerably simpler than any of the others, we chose it as representative of some typical architectural lighting situations.",
"It is important to note that while the first four scenes look superficially similar, they stress entirely different transport phenomena.",
"Moreover, all of them, while relatively simple in terms of geometric complexity, are very hard in terms of pure light transport, requiring between $16 \\cdot 10^3$ and $64 \\cdot 10^3$ samples per pixel (spp) for bidirectional path tracing to converge.",
"Figure REF shows equal-time comparisons of MMLT and CMLT on all scenes.",
"Except for the last row, both the MMLT and CMLT renders were generated using 256 spp, taking roughly the same computation time, whereas the reference images have been rendered with bidirectional path tracing using $16 \\cdot 10^3$ spp.",
"The images in the last row used 512 spp for MMLT and CMLT, and $64 \\cdot 10^3$ spp for the reference image.",
"CMLT produces considerably less noise on all test scenes.",
"In particular, it is very effective in cases of complex glossy reflections and reflections of caustics, where there is no clear winner among all bidirectional sampling techniques.",
"Figure REF shows the convergence of MMLT and CMLT on the Salle de bain scene.",
"Notice how MMLT needs more than twice as many samples as CMLT to get approximately the same RMSE.",
"In the early stages, MMLT is not capable of finding many important light paths, leading to an apparently darker image (due to energy being concentrated on a subset of the pixels); the difference vanishes at higher sampling rates.",
"Figure REF shows a similar graph comparing also to PSSMLT.",
"Since each PSSMLT sample requires both more shadow rays and BSDF evaluations, in our implementations PSSMLT can perform roughly one half the mutations as CMLT in the same time.",
"Finally, Figure REF shows the effect of varying the number of chains run in parallel, trading it against chain length to keep the total number of samples fixed.",
"The images in the top row are obtained running $32 \\cdot 10^3$ chains in parallel, whereas the ones in the bottom row are obtained using $256 \\cdot 10^3$ chains.",
"It can be seen that using more, shorter chains generally improves stratification and widely reduces the spotty appearance typical of Metropolis autocorrelation.",
"The exception is the caustic on the ceiling that benefits from the higher adaptation of the longer chains.",
"Note that the additional stratification is similar to the one obtained by ERPT [2], which however was using a different, per-pixel chain distribution strategy (as opposed to our global resampling stage), and was not specifically targeted at introducing massive parallelism.",
"While Cline et al Cline:2005 discussed only the stratification benefits, we believe it is worth documenting what seems an intrinsic tradeoff between local exploration and better stratification: running more, shorter chains generally helps image stratification, while necessarily losing some exploration capabilities in narrow regions of path space.",
"In all cases, for CMLT we used one chart swap proposal every 16 mutations, resulting in negligible overhead.",
"Figure: Parallel CMLT convergence using respectively 32K (top row) and 256K chains (bottom row).",
"From left to right: 32, 128 and 512 samples per pixel.Figure: Equal-time comparisons of our CMLT and MMLT on four scenes testing different transport phenomena.Figure: RMSE comparison of CMLT (bottom) and MMLT (top) at 4, 8, 16, 64 and 1024 spp."
],
[
"Performance analysis",
"On our system, the 1024 spp CMLT and MMLT images take roughly 80s to render at a resolution of $1600 \\times 900$ using $256 \\cdot 10^3$ chains.",
"Figure REF shows a performance breakdown on Salle de bain: roughly 50% of the time is spent in ray tracing, with shading taking 45%, and the initial path sampling and path inversion taking roughly 2.5% each.",
"If we substantially reduce the number of chains we start to notice a slowdown due to underutilization of the hardware resources, mostly caused by insufficient parallelism in the late stages of the bidirectional path tracing pipeline needed to process longer than average paths.",
"This could likely be mitigated by better scheduling policies, for example not requiring all chains to be processed in sync (currently we finish applying a mutation to all paths before starting to process the next)."
],
[
"Discussion",
"We proposed a novel family of MCMC algorithms that use sampling charts to extend the sampling domain and allow better exploration.",
"We applied the new scheme to propose a new type of light transport simulation algorithms that bridge primary sample space and path space MLT.",
"We also showed that the new algorithms arising from this framework require to implement only a new set of relatively cheap mutations that can be constructed using simple, stochastic right inverses of the path sampling functions: particularly, the fact our framework requires only such type of probabilistic inversion is what makes the algorithm practical, as classical BSDF inversion with layered material models is generally impossible.",
"We believe this to be a major strength of our work.",
"We implemented both the old and new methods exposing massive parallelism at all levels, and showed how increasing the number of chains that run in parallel can increase stratification.",
"Finally, we suggested a novel, simpler method to integrate path density estimation into MCMC light transport algorithms as a mechanism to craft independent proposals."
],
[
"Future work",
"There are multiple avenues in which this work could be extended.",
"The first is testing all possible variants of our new algorithmic family more thoroughly.",
"In such a context, it will be particularly interesting to test the combination with the original path space MLT mutations, which might provide some advantages in regions with complex visibility.",
"Similarly, it would be interesting to test the new technique for including path density estimation as an independence sampler.",
"Another potential venue is considering dimension jumps to switch between the charts underlying different path spaces $\\Omega _k$ and $\\Omega _{k^{\\prime }}$ .",
"This could be achieved using the Metropolis-Hastings-Green with Jacobians algorithm as described by Geyer Geyer:2011.",
"Finally, it would be interesting to integrate half vector space light transport [9], [7] as yet another path space chart."
],
[
"We would like to thank Cem Cebenoyan at NVIDIA for constantly supporting our work; Luca Fascione and Marc Droske at Weta Digital for early reviews and continuous feedback; Matthias Raab at NVIDIA for helping us with modern layered material sampling methods; Nicholas Hull and Nir Benty at NVIDIA for their precious help with the setup and import of the original Gray & White Room and Salle de bain scenes and Thomas Iuliano for providing beautiful artwork that ought to be included in this paper, and was not for mere lack of time.",
"Finally, we would like to thank the anonymous SIGGRAPH reviewers, particularly #30, for their detailed comments which led to significant improvements in the exposition of the paper."
],
[
"Appendix",
"We here describe how to invert the sampling functions for typical BSDF layers as needed to implement chart swaps.",
"The key insight is that most common BSDF sampling methods can be seen as bijective functions $S(\\omega _i)$ from the unit square to the hemisphere of directions: $S(\\omega _i):[0,1]^2 &\\rightarrow & H \\\\(u,v) &\\mapsto & \\omega _o \\nonumber $ where the notation $S(\\omega _i)$ denotes the potential dependence on the incident direction $\\omega _i$ .",
"Hence, in order to perform BSDF inversion, we need to simply compute the inverse $S^\\leftarrow (\\omega _i): H \\rightarrow [0,1]^2$ ."
],
[
"Lambertian distribution",
"Lambertian BSDFs are typically importance sampled using the mapping: $S: (u,v) \\mapsto (\\theta ,\\phi ) = (acos(\\sqrt{v}), u \\cdot 2 \\pi )$ where $(\\theta , \\phi )$ represent spherical coordinates relative to the surface normal.",
"Inverting this mapping can hence be done very easily: $S^\\leftarrow : (\\theta ,\\phi ) \\mapsto (u,v) = \\left(\\frac{\\phi }{2 \\pi },cos^2(\\theta )\\right)$"
],
[
"GGX distribution",
"Sampling the GGX distribution is slightly more involved as it is the composition of two functions: $S(\\omega _i) = R(\\omega _i) \\circ F_m$ , where the function $F_m:[0,1]^2 \\rightarrow H$ samples a microfacet according to the roughness parameter $m$ , and $R(\\omega _i) : H \\rightarrow H$ returns the input direction $\\omega _i$ reflected about the sampled microfacet normal.",
"Its inverse can hence be obtained as $S^\\leftarrow (\\omega _i) = F_m^\\leftarrow \\circ R^\\leftarrow (\\omega _i)$ .",
"Finding the microfacet normal given the incident and outgoing directions $\\omega _i$ and $\\omega _o$ is trivial, as the normal can be simply computed using the half vector formula: $R^\\leftarrow (\\omega _i) : H &\\rightarrow & H \\\\\\omega _o &\\mapsto & \\frac{\\omega _i + \\omega _o}{|\\omega _i + \\omega _o|}.",
"\\nonumber $ The forward mapping for sampling a microfacet is instead given by the following expression: $F_m: (u,v) \\mapsto (\\theta ,\\phi ) = \\left( acos\\left(\\frac{1}{\\sqrt{1 + t(v)}}\\right), u \\cdot 2 \\pi \\right)$ with: $t(v) = \\frac{v}{ (1 - v) \\cdot m^2 }.$ The inverse can hence be computed as: $F_m^\\leftarrow : (\\theta ,\\phi ) \\mapsto (u,v) = \\left(\\frac{\\phi }{2 \\pi },\\frac{q(\\theta ) }{1 + q(\\theta ) }\\right)$ with: $q(\\theta ) = m^2 \\cdot (1 / cos^2(\\theta ) - 1).$ The composition of the two can now be obtained considering the polar coordinates $(\\theta _h, \\phi _h)$ of the vector: ${\\bf h} = R^\\leftarrow (\\omega _i,\\omega _o),$ and finally computing: $(u,v) = F_m^\\leftarrow (\\theta _h,\\phi _h).$"
],
[
"Specular scattering",
"Specular scattering introduces singularities in the transformations $T_i$ , which manifest as Dirac deltas in the respective pdfs $p_i$ .",
"While we did not explicitly study how to handle these in this work, we believe it would be possible to include them in our chart swaps, as long as the scattering mode at specular vertices is not altered.",
"In fact, altering the mode from specular to diffuse would simply result in a zero acceptance rate: this can be verified looking at equation (REF ), and considering the fact that the numerator, equal to the reciprocal of the density of the current (specular) pdf $p_i$ , would be zero.",
"Conversely, if the mode was not altered, the implicit Dirac deltas in the numerator and denominator would cancel out.",
"boxruled // fill the three arrays: // u_init[] : primary sample space path coordinates // st_init[] : technique number of each path // C_init[] : contribution of each path (u_init,st_init,C_init) $\\leftarrow $ bptSamplePaths($N_{init}$ ) // build a cdf over the $N_{init}$ path contributions cdf[] $\\leftarrow $ prefixSum(C_init) // resample N paths based on their contribution i=1 ... N seed $\\leftarrow $ sampleCdf(cdf,(i + random())/N) u[i] $\\leftarrow $ u_init[seed] st[i] $\\leftarrow $ st_init[seed] C[i] $\\leftarrow $ C_init[seed] // retrace the bidirectional path path[i] $\\leftarrow $ bptTracePath(u[i], st[i]) // loop across the number of mutations L l=1 ... L i=1 ... N selectMutation(l) == ChartSwap // propose a chart swap (s, t) $\\leftarrow $ st[i] k $\\leftarrow $ s + t - 1 (s',t') $\\leftarrow $ chartProposal(k) s' > s (u',r') $\\leftarrow $ invertLightSubpath(path[i],s,s') r $\\leftarrow $ eyeSubpathInversionPdf(path[i], t', t) (u',r') $\\leftarrow $ invertEyeSubpath(path[i],t,t') r $\\leftarrow $ lightSubpathInversionPdf(path[i], s', s) // compute the acceptance-ratio according to eq (REF ) a $\\leftarrow $ r / r' random() < a u[i] $\\leftarrow $ u' st[i] $\\leftarrow $ (s',t') // apply a standard primary sample space mutation u' $\\leftarrow $ perturb(u[i]) (path',C') $\\leftarrow $ bptTracePath(u', st[i]) a $\\leftarrow $ min(1, C' / C[i]) random() < a u[i] $\\leftarrow $ u' C[i] $\\leftarrow $ C' path[i] $\\leftarrow $ path' // accumulate the new sample accumulate(path[i], C[i]); pseudo-code for our CMLT algorithm Figure: RMSE comparison of PSSMLT (top), MMLT (middle) and CMLT (bottom) at equal computation time.Figure: Performance breakdown for running 256K chains of length 350 (equivalent to about 64 spp at a resolution of 1600×9001600 \\times 900).",
"All timings are in milliseconds."
]
] | 1612.05395 |
[
[
"Invariant subspaces for commuting operators in a real Banach space"
],
[
"Abstract It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\\in A$ satisfies the condition $$\\|1- \\varepsilon T^2\\|_e \\le 1 + o(\\varepsilon) \\text{ when } \\varepsilon\\searrow 0,$$ where $\\|\\cdot\\|_e$ is the essential norm.",
"This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators in a real Hilbert space."
],
[
"Introduction ",
"One of the most known unsolved problems in the invariant subspace theory is the question of existence of a (non-trivial, closed) invariant subspace for an operator $T$ with compact imaginary part (= essentially selfadjoint operator = compact perturbation of a selfadjoint operator).",
"There is a lot of papers concerning this subject; we only mention that the answer is affirmative for perturbations by operators from Shatten - von Neumann class $\\mathfrak {S}_p$ (Livshits [1] for $p=1$ , Sahnovich [2] for $p=2$ , Gohberg and Krein [3], Macaev [4], Schwartz [5] — for arbitrary $p$ ), or, more generally, from the Macaev ideal (Macaev [6]).",
"But the general question is still open.",
"It was proved in [7] that an essentially self-adjoint operator in a complex Hilbert space has an invariant real subspace.",
"Then in [8] the following general theorem of Burnside type was proved: Theorem 1.1 Suppose that an algebra $A$ of bounded operators in a (real or complex) Banach space $X$ is not dense in the algebra $\\mathcal {B}(X)$ of all bounded operators on $X$ with respect to the weak operator topology (WOT).",
"Then there are non-zero $x\\in X^{**}, y\\in X^*$ , such that $|(x,T^*y)| \\le \\Vert T\\Vert _e \\text{ for all } T\\in A,$ where $\\Vert T\\Vert _e$ is the essential norm of $~T$ , that is $\\Vert T\\Vert _e = \\inf \\lbrace \\Vert T+K\\Vert : K\\in \\mathcal {K}(X)\\rbrace $ .",
"Here $\\mathcal {K}(X)$ is the ideal of all compact operators in $X$ .",
"Using this result and developing a special variational techniques, Simonic [9] has obtained a significant progress in the topic: he proved that each essentially selfadjoint operator in a real Hilbert space has invariant subspace.",
"Deep results based on Theorem REF were proved then by Atzmon [10], Atzmon, Godefroy and Kalton [11], Grivaux [12] and other mathematicians.",
"Here we show that every commutative family of essentially selfadjoint operators in a real Hilbert space has an invariant subspace, and consider some analogs of this result for operators in Banach spaces.",
"Our proof is very simple and short but it heavily depends on Theorem REF .",
"More precisely, we use the following improvement of Theorem REF obtained in [13]: Theorem 1.2 If an algebra $A\\subset \\mathcal {B}(X)$ is not WOT-dense in $\\mathcal {B}(X)$ , then there are non-zero vectors $x\\in X^{**}, y\\in X^*$ , such that $(x,y) \\ge 0$ and $|(x,T^*y)| \\le \\Vert T\\Vert _e (x,y), \\text{ for all } T\\in A.$ Let us mention that if $A$ contains a non-zero compact operator then by [14] $A$ has a non-trivial invariant subspace $L$ in $X$ (so $x$ can be chosen from $L$ and $y$ can be chosen from $L^\\bot $ ).",
"The original proof of Theorem REF was essentially simplified by Lindstrom and Shluchtermann [15].",
"For completeness, we give at the end of the paper a short proof of Theorem REF , unifying arguments from [15] and [13] (and correcting some inaccuracy in [13]) — in this form it was not published before."
],
[
"Main results ",
"In this section $X$ is a real Banach space (complex spaces are considered as real ones).",
"The standard epimorphism from $\\mathcal {B}(X)$ to the Calkin algebra $\\mathcal {C}(X) = \\mathcal {B}(X)/\\mathcal {K}(X)$ is denoted by $\\pi $ .",
"Let us say that an element $a$ of a unital normed algebra is positive, if $\\Vert 1-\\varepsilon a\\Vert \\le 1+o(\\varepsilon )$ for $\\varepsilon > 0$ , $\\varepsilon \\rightarrow 0$ .",
"And let us say that an element $a$ is real, if $a^2$ is positive.",
"An operator $T\\in \\mathcal {B}(X)$ is essentially real, if $\\pi (T)$ is a real element of $\\mathcal {C}(X)$ .",
"Clearly all selfadjoint operators in a Hilbert space are real.",
"It is not difficult to check that Hermitian operators in a complex Banach space (defined by the condition $\\Vert \\exp (itT)\\Vert = 1$ for $t\\in \\mathbb {R}$ ) are real.",
"Many other operators, for instance, all involutions and all nilpotents of index two are also real.",
"So we can see that the class of essentially real operators is very wide.",
"Theorem 2.1 If $A\\subset \\mathcal {B}(X)$ is a commutative algebra of essentially real operators, then there exists a non-trivial closed subspace of $X^*$ , invariant for the algebra $A^*= \\lbrace T^*: T \\in A\\rbrace $ .",
"Proof.",
"Note that the set of all positive elements of a Banach algebra is a convex cone.",
"Moreover this cone is norm-closed.",
"Indeed let $a = \\lim _{n\\rightarrow \\infty }a_n$ where all $a_n$ are positive.",
"If $a$ is not positive then there is a sequence $\\varepsilon _n \\rightarrow 0$ and a number $C> 0$ , such that $\\Vert 1 - \\varepsilon _n a\\Vert > 1 + C\\varepsilon _n$ for all $n$ .",
"Taking $k$ with $\\Vert a-a_k\\Vert < C/2$ , we get that $\\Vert 1 - \\varepsilon _na_k\\Vert > 1 + C\\varepsilon _n - \\Vert a-a_k\\Vert \\varepsilon _n > 1 + (C/2)\\varepsilon _n$ , which is a contradiction to positivity of $a_k$ .",
"Hence the set of real elements is also norm closed.",
"Applying this to $\\mathcal {C}(X)$ we see that the set of essentially real operators is norm-closed as well.",
"This allows us to assume that the algebra $A$ is norm-closed.",
"Obviously we may assume also that $A$ is unital.",
"Therefore $\\exp (T)\\in A$ , for each $T\\in A$ .",
"Since $A$ is commutative it is not WOT-dense in $\\mathcal {B}(X)$ .",
"Applying Theorem REF , we find non-zero vectors $x\\in X^{**}, y\\in X^*$ , such that the condition (REF ) holds.",
"Therefore, for $T\\in A$ and $\\varepsilon \\searrow 0$ , we have $(x,(1-\\varepsilon (T^2)^*)y) \\le \\Vert 1-\\varepsilon T^2\\Vert _e(x,y) \\le (1+o(\\varepsilon ))(x,y),$ hence $-\\varepsilon (x, (T^2)^*y) \\le o(\\varepsilon )(x,y)$ and $(x,(T^2)^*y)\\ge 0$ , because $(x,y)\\ge 0$ .",
"Since $\\exp (T) = (\\exp (T/2))^2$ , we get that $(x,\\exp (T^*)y) \\ge 0 \\text{ for }T\\in A.$ Let $K$ be the closed convex envelope of the set $M = \\lbrace \\exp (T^*)y: T\\in A\\rbrace \\subset X^*$ .",
"Since $M$ is invariant under all operators $\\exp (T^*)$ , the same is true for $K$ .",
"Since $K\\ne X^*$ (by (REF )) and $K$ is not a singleton (otherwise we trivially have an invariant subspace $\\mathbb {C}y$ ), the boundary $\\partial K$ of $K$ is not a singleton.",
"By the Bishop-Phelps theorem [17], the set of support points is dense in $\\partial K$ , so there is a non-zero functional $x_0\\in X^{**}$ supporting $K$ in some point $0\\ne y_0\\in K$ .",
"That is $(x_0,y_0) \\le (x_0,z) \\text{ for all } z\\in K.$ By the above arguments, $\\exp (T^*)y_0\\in K$ for $T\\in A$ , therefore $ (x_0,y_0) \\le (x_0,\\exp (T^*)y_0) \\text{ for all } T\\in A.$ It follows that for each $T\\in A$ , the function $\\phi (t) = (x_0,\\exp (tT^*)y_0)$ has a minimum at $t = 0$ .",
"For this reason $\\phi ^{\\prime }(0) = 0$ and $(x_0,T^*y_0) = 0$ .",
"Hence the subspace $A^*y_0$ is not dense in $H$ , and its norm-closure $\\overline{A^*y}$ is a non-trivial invariant subspace.",
"Corollary 2.2 A commutative algebra of essentially real operators in a reflexive real Banach space has a non-trivial invariant subspace.",
"Since the algebra generated by a commutative family of essentially selfadjoint operators consists of essentially selfadjoint operators we get the following result: Corollary 2.3 Any commutative family of essentially selfadjoint operators in a real Hilbert space has a non-trivial invariant subspace.",
"Returning to individual criteria of continuity let us consider the class $(E)$ of operators $T$ such that all polynomials $p(T)$ of $T$ are essentially real.",
"Corollary 2.4 Each operator $T\\in (E)$ in a reflexive real Banach space has a non-trivial invariant subspace.",
"Atzmon, Godefroy and Kalton [11] introduced the class $(S)$ of all operators, satisfying the condition $\\Vert p(T)\\Vert _e \\le \\sup \\lbrace |p(t)|: t\\in L\\rbrace , \\text{ for each polynomial } p,$ where $L$ is a compact subset of $\\mathbb {R}$ .",
"It was proved in [11] that all operators in $(S)$ have invariant subspaces.",
"It is not difficult to see that $(S)\\subset (E)$ (indeed if $T\\in (S)$ then $\\Vert 1- \\varepsilon p(T)^2\\Vert _e \\le \\sup \\lbrace |1-\\varepsilon p(t)^2|: t\\in L\\rbrace \\le 1$ if $\\varepsilon $ is sufficiently small) , so this result follows from Corollary REF ."
],
[
"Proof of Theorem ",
"Without loss of generality one can assume that $A$ is norm-closed.",
"Since the algebra $A$ is not WOT-dense in $\\mathcal {B}(X)$ , the algebra $A^*$ is not WOT-dense in $\\mathcal {B}(X^*)$ .",
"Suppose, aiming at the contrary, that $A^*$ is transitive (has no invariant subspaces).",
"Set $F = \\lbrace T^*\\in A^*: \\Vert T\\Vert _e < 1\\rbrace $ and fix $\\varepsilon \\in (0, \\frac{1}{10})$ .",
"Choose $y_0\\in X^*$ with $\\Vert y_0\\Vert = 3$ and let $S$ be the ball $\\lbrace y\\in X^*: \\Vert y-y_0\\Vert \\le 2\\rbrace $ .",
"Let us suppose firstly that $Fy$ is dense in $X^*$ for each non-zero $y\\in X^*$ .",
"Then the same is true for $\\varepsilon Fy$ .",
"It follows that for every $y\\in S$ there exists $T^*_y\\in \\varepsilon F$ with $\\Vert T^*_yy - y_0\\Vert <1$ .",
"By the definition of $F$ , $T^*_y = R^*_y + K^*_y$ , where $\\Vert R_y\\Vert < \\varepsilon $ , $K_y \\in \\mathcal {K}(X)$ .",
"Thus $\\Vert K^*_yy - y_0\\Vert \\le \\Vert T^*_yy-y_0\\Vert + \\Vert R^*_yy\\Vert < 1 + 5\\varepsilon $ (since $\\Vert y\\Vert \\le 5$ , for each $y\\in S$ ).",
"Let $\\tau $ denote the (relative) *-weak topology of $S$ , then compactness of $K_y$ implies that $K_y^*$ continuously maps $(S,\\tau )$ to $(X^*, \\Vert \\cdot \\Vert )$ .",
"Therefore, for each $y\\in S$ , there is a $\\tau $ -neighborhood $V_y$ of $y$ such that $\\Vert K^*_yz - y_0\\Vert < 1 +5\\varepsilon $ , for each $z\\in V_y$ , and $\\Vert T^*_yz-y_0\\Vert < 1+5\\varepsilon + 5\\varepsilon < 2$ .",
"In other words $T^*_y$ maps $V_y$ to $S$ .",
"The sets $V_y$ , $y\\in S$ , form an open covering of $S$ .",
"Since $S$ is $\\tau $ -compact there is a finite subcovering $\\lbrace V_{y_i}: 1\\le i\\le n\\rbrace $ .",
"Let $\\lbrace \\varphi _i: 1\\le i\\le n\\rbrace $ be a partition of unity related to this subcovering.",
"We define a $\\tau $ -continuous map $\\Phi : S \\rightarrow S$ by $\\Phi (y) = \\sum _{i=1}^n\\varphi _i(y)T^*_{y_i}(y)$ .",
"By Tichonov's Theorem, $\\Phi $ has a fixed point $z\\in S$ .",
"This means that $T^*z = z$ , where $T^* = \\sum _{i=1}^n\\varphi _i(z)T^*_{y_i}$ .",
"Since the set $\\varepsilon F$ is convex we get that $T^* \\in \\varepsilon F$ and $\\Vert T^*\\Vert _e \\le \\Vert T\\Vert _e < \\frac{1}{10}$ .",
"For this reason 1 is an eigenvalue of $T^*$ exceeding $\\Vert T^*\\Vert _e$ and hence it is an isolated point in $\\sigma (T^*)$ .",
"The corresponding Riesz projection is of finite rank and belongs to $A^*$ .",
"But it is well known (see e.g.",
"[16]), that a transitive algebra containing a non-zero finite rank operator is $WOT$ -dense in the algebra of all operators.",
"Since $A^*$ is not $WOT$ -dense in $\\mathcal {B}(X^*)$ , we obtain a contradiction.",
"Hence there exists $y_0\\in X^*$ such that $Fy_0$ is not norm-dense in $X^*$ .",
"Let $V$ be the norm-closure of $Fy_0$ .",
"If $V = \\lbrace 0\\rbrace $ then we have the inequality (REF ) with $y=y_0$ and any non-zero $x$ .",
"If $V \\ne \\lbrace 0\\rbrace $ then $V$ is a norm-closed convex proper subset of $X^*$ containing more then one point.",
"By the Bishop - Phelps Theorem [17], there are $0\\ne y \\in V$ and $0\\ne x \\in X^{**}$ such that $Re(x,y) = \\sup \\lbrace Re(x,w): w\\in V\\rbrace $ .",
"Since the set $V$ is invariant under multiplication by any number $t$ with $|t| \\le 1$ , we have $(x,y) \\ge 0$ and $(x,y) \\ge |(x,w)|$ for all $ w\\in V$ .",
"Since $F^2\\subset F$ , we have that $Fy\\subset V$ and $|(x,T^*y)| \\le (x,y)$ , for any $T^*\\in F$ .",
"Therefore the inequality $|(x,T^*y)|\\le \\Vert T\\Vert _e(x,y)$ holds for all $T\\in A$ .",
"$~~\\blacksquare $ Dept of Math.",
"Kent State University Kent OH 44242, USA [email protected] Dept of Math.",
"Vologda State University Vologda 160000, Russia [email protected]"
]
] | 1612.05821 |
[
[
"Evidence of reverse and intermediate size segregation in dry granular\n flows down a rough incline"
],
[
"Abstract In a dry granular flow, size segregation behave differently for a mixture containing a few large beads with a size ratio (S) above 5 (Thomas, Phys.Rev.E 62,96(2000)).",
"For moderate large S, large beads migrate to an intermediate depth in the bed: this is called intermediate segregation.",
"For the largest S, large beads migrate to the bottom: this is called reverse segregation (in contrast with surface segregation).",
"As the reversal and intermediate depth values depend on the bead fraction, this numerical study mainly uses a single large tracer.",
"Small fractions are also computed showing the link between a tracer behavior and segregation process.",
"For half-filled rotating drum and for rough incline, two and three (3D) dimensional cases are studied.",
"In the tumbler, trajectories of a large tracer show that it reaches a constant depth during the flow.",
"For large S, this depth is intermediate with a progressive sinking when S increases.",
"Largest S correspond to tracers at the bottom of the flow.",
"All 3D simulation are in quantitative agreement with the experiments.",
"In the flow down an incline, a large tracer reaches an equilibrium depth during flow.",
"For large S, its depth is intermediate, inside the bed.",
"For the largest S, its depth is reverse, near the bottom.",
"Results are slightly different for thin or thick flow.",
"For 3D thick flows, the reversal between surface and bottom positions occurs within a short range of S: no tracer stabilizes near mid-height and two reachable intermediate depth layers exist, below the surface and above the bottom.",
"For 3D thin flows, all intermediate depths are reachable, depending on S. The numerical study of larger tracer fractions (5-10%) shows the 3 segregation patterns (surface, intermediate, reverse) corresponding to the 3 types of equilibrium depth.",
"The reversal is smoother than for a single tracer.",
"It happens around S=4.5, in agreement with experiments."
],
[
"Introduction",
"Size segregation in dry granular flow has been extensively studied as it is an important phenomenon occurring in natural flows or in industrial applications [1], [2], [3], [4], [5], [6], [7], [8].",
"Recently, there have been significant advances in the modeling of segregation in dense granular flows.",
"Models based on kinetic theory have been established for segregation in rapid flows, in the case of particles of different sizes and/or densities [9], [10], [11].",
"These models, based on particle properties and with no adjustable parameter, are able to predict the evolution of the volume fraction of two types of particles that do not differ much in size or mass [9].",
"Alternative models based on mixture theory have been proposed in which unequal stress partitioning reflects the mechanisms that are responsible for the segregation: kinetic sieving and squeeze expulsion [12].",
"In this continuum framework, particle segregation results from the lithostatic pressure gradient induced by gravity [13], [7].",
"Several groups have proposed improvements to take into account the effects of shear rate [14], [15], [16], [17], kinetic stress gradients (derived from vertical chutes) [18], [19], or the polydispersity of flows with particles of different sizes and densities [20], leading to further developments for flows on inclines [21].",
"Quantitative agreement with experiments has been obtained for the stationary concentration profile of a mixture with a size ratio of 2 [22].",
"A comparative review can be found in [23].",
"Most of these studies are concerned with small size ratios, the large particles being generally 1.5 to 2 times the size of the small ones.",
"In some studies, size ratio is varied up to 3 [16], 3.5 [24], or 4 [25].",
"This variation remains small compared with the size ratio range in our present study.",
"Even so, it already induces a non-monotonic variation of some parameters, e.g.",
"the segregation rate [25].",
"One of the studies concerning the measurement of the force acting on an object plunged into a granular flow [26], [27], [28] provides interesting information on the segregation phenomenon because the intruder is free to move with the flow [29], instead of being an obstacle exerting drag.",
"In these 2D simulations, the authors also noticed an extremum for the normalized segregation force obtained at the size ratio 2.",
"Some segregation theory has been extended to large size ratios (up to 10) [20] and predicts a monotonic decrease in the segregation time with the size ratio and without any change in the segregation pattern.",
"Most of the models do not explicitly depend on the size ratio, but rather on a segregating velocity determined for each species [23].",
"In the few models where the size ratio is explicitly mentioned [20], [15], the segregating velocity cannot change sign when the size ratio is increased, for any particle fraction.",
"In these models, only a difference in density between particles could induce a reversal of the segregating velocity direction [15].",
"Nevertheless, the segregation phenomenon is observed to be different when increasing the size ratio above 4 or 5.",
"It has been shown experimentally that large particles do not reach the surface, as they usually do in surface segregation, but move downwards and stabilize either at an intermediate depth or at the bottom of the flow for the largest size ratios [30].",
"Particle stabilization at an intermediate depth has been named “intermediate segregation\".",
"Large particle segregation at the bottom of the flow has been named “reverse segregation\" by analogy with the “reverse Brazil-nut effect\" [31], [32], [33] observed in vibrated granular systems.",
"The origin of this vibrating effect [34] is due to an inertia driven segregation process induced by high amplitude vibrations [31], [34] as well as to the absence of convective motion [35].",
"The reverse and intermediate segregations of particles of different sizes (and having the same density) have been observed experimentally in various sheared flows: channel flow, half-filled cylindrical rotating tumbler and 3D heap flow [30], [36].",
"The corresponding segregation patterns take different forms.",
"In a rotating tumbler, if large particles are close to the tumbler center in the static part, reverse segregation occurs because particles move to the bottom of the flowing layer during flow.",
"By contrast, tracers having a small size ratio (from 1.5 to 3) end up at the periphery on the solid part, undergoing surface segregation during flow.",
"For a flow down an incline, the reverse-segregating large particles disappear from the surface during flow, and are present near the bottom of the deposit, while the surface-segregating large particles cover the flow and deposit surface.",
"For a flow feeding a heap, very large beads form a vertical core (reverse segregation).",
"For a small size ratio, a ring of large beads forms at the bottom periphery of the heap (surface segregation).",
"Intermediate segregation has been precisely observed in the tumbler: all large particles are found at the same intermediate radial position in the static part (Fig.",
"REF ) [30], [36], forming a segregation half-ring pattern.",
"An intermediate ring corresponds to an intermediate depth in the flowing layer (Fig.",
"REF (a)).",
"This was measured for size ratios ranging from 4 to approximately 15, for small fractions of large particles (3%) [30].",
"In the experiments, the ring mean radius decreased continuously with increasing size ratios, corresponding in the flowing layer to a mean depth passing continuously from surface to bottom.",
"Figure: A D=4.85D=4.85 cm rotating cylindrical tumbler with d=0.3d= 0.3 mmsmall particles and d t =d_t= 3 mm large particles (tracers): (a) cross-section of an experiment with 3% blue tracers, slowly impregnated with water after the flow has stopped, then sliced , (b) 2D simulation with one tracer.The reversal of the segregation from the surface toward the bottom depends both on the size ratio and on the relative fraction of particles [30]: a limit between surface and reverse segregations can be defined and it corresponds to a size ratio between 4 and 5 for small fractions (1 to 10%) of large particles; around 14 for a 30% fraction; and no reversal has been observed for a 50% fraction, for size ratios below 45.",
"As most of the size segregation studies were done for equivalent fractions of both species, the reversal was not observed.",
"Moreover, surface and reverse segregations give opposite, although not symmetrical, patterns between the two species.",
"This asymmetry is partly due to the use of a smaller fraction of large particles.",
"However, when the tracer fraction is increased in an attempt to reduce the pattern difference, this asymmetry is enhanced: surface segregation leads to a bi-layered (or two concentric zones) system made up of pure components, although reverse segregation progressively leads to an apparent mixing, except near the surface (near the tumbler periphery) [30].",
"Reverse segregation is not another kind of surface segregation process, by which large particles are placed at the bottom: it is a different phenomenon with a different behavior.",
"Note that the reversal of the segregation pattern is not due to percolation effects, as suggested by some authors [37], because they happen for a large fraction of large particle [38].",
"For these reasons, we limit our present study to one single large particle or to small fractions (5% and 10%).",
"Another series of experiments involves particles of different densities and sizes in tumbler flow [36].",
"Similarly, reverse and intermediate segregations of large particles are observed.",
"The mean segregation depths are shifted toward the surface for less dense large particles, and shifted toward the bottom for denser large particles.",
"For each tracer particle material, the reversal of the segregation is therefore enhanced (resp.",
"reduced) by an increase (resp.",
"a decrease) in the density of large tracers.",
"Only large beads made of very light material always segregate to the surface, and only very dense beads always segregate to the bottom (reverse segregation), whatever their size.",
"These observations suggest that for particles of the same density, the reversal of the size segregation is due to the increase in their mass ratio.",
"Heavy (because large) particles push light (because small) particles around to make their way down.",
"Moreover, the fact that large particles locate themselves at a precise intermediate depth shows the existence of vertical gradients of force acting on them.",
"Further studies are needed to extent these results to the case of flow down a solid rough incline.",
"In fact, for an incline flow, we have the “intermediate segregation\" if large particles are found at intermediate depths inside the deposit.",
"Our previous experiments have shown that the mean depth for the large beads in the deposit varies continuously with the size ratio from top to bottom [30].",
"However, these experiments were not precise enough to assess the occurrence of intermediate segregation in channel flow: there was a large spread of the individual positions for these intermediate mean depths.",
"This may be due to the use of 10% of large beads, but it could also be related to a non-stationary state of the flow, and/or to the modification of the tracer positions during the deposit aggradation.",
"For these reasons, the existence or non-existence of intermediate equilibrium depths for a single large tracer in a granular flow down an incline is the main focus of this article.",
"The case of several tracers (5% and 10% volume fraction) will also be considered for a comparison with a single tracer behavior and with previous experiments of reverse segregation [30].",
"This article is organized as follows.",
"In section 2, the numerical method is presented.",
"Section 3 studies the tracer trajectory and equilibrium radial position in a rotating cylindrical tumbler in two (2D) and three dimensions (3D).",
"The method is validated through quantitative comparison with previous 3D experimental results.",
"In section 4, the displacement of tracers in a granular flow down an incline is studied in 2D, and in 3D.",
"Similarities and discrepancies between 2D and 3D, as well as the comparison between incline and tumbler flow are discussed.",
"Then, a study of multiple-tracer flows and a comparison with previous experiments are presented."
],
[
"The numerical method",
"The numerical method used is the distinct element method (DEM).",
"A linear-spring and viscous damper force model [39], [40] is used to calculate the normal force between contacting particles: $ \\mathbf {F}_{ij}^n=[k_n\\,\\delta -2\\gamma _n m_{\\textrm {eff}}(\\mathbf {V}_{ij}\\cdot \\mathbf {\\hat{r}}_{ij} )] \\mathbf {\\hat{r}}_{ij} $ where $\\delta $ and $\\mathbf {V}_{ij}=\\mathbf {V}_{i}-\\mathbf {V}_{j}$ are the particle overlap and the relative velocity of contacting particles, respectively, $\\mathbf {\\hat{r}}_{ij}$ is the unit vector in the direction between two particles $i$ and $j$ , $m_{\\textrm {eff}}=m_i m_j/(m_i+m_j) $ is the reduced mass of the two particles, $ k_n=m_{\\textrm {eff}}[(\\frac{\\pi }{\\Delta t})^2+\\gamma _n^2]$ is the normal stiffness and $\\gamma _n=\\ln e\\,/\\Delta t$ is the normal damping with $\\Delta t$ the collision time and $e$ the restitution coefficient.",
"A standard tangential force with elasticity is implemented: $\\mathbf {F}_{ij}^t=-\\min (| \\mu \\mathbf {F}_{ij}^n|,|k_s\\zeta |){\\rm sign}(\\mathbf {V}^s_{ij}) $ where $\\mathbf {V}^s_{ij}$ is the relative tangential velocity of the two particles, $k_s$ is the tangential stiffness, and $\\mathbf {\\zeta }(t)=\\int _{t_0}^{t}\\mathbf {V}^s_{ij}(t^{\\prime })\\, {\\rm d}t^{\\prime }$ is the net tangential displacement after contact is first established at time $t=t_0$ .",
"The gravitational acceleration is $g=9.81$ m s$^{-2}$ .",
"The particle properties correspond to those of cellulose acetate: density $\\rho =1308$ kg m$^{-3}$ , restitution coefficient $e=0.87$ and friction coefficient $\\mu =0.7$ [39], [41], [42], [43].",
"To prevent the formation of a close-packed structure, the particles have a uniform size distribution ranging from 0.95$d$ to 1.05$d$ , with $d$ the particle diameter.",
"The collision time is $\\Delta t$ =10$^{-4}$ seconds, consistent with previous simulations [44], [45], [43] and sufficient for modeling hard spheres [46], [47], [48].",
"These parameters correspond to a stiffness coefficient $k_n = 7.32\\times 10^4$ N m$^{-1}$ [39] and a damping coefficient $\\gamma _n = 0.206$ kg s$^{-1}$ .",
"The integration time step is $\\Delta t/50 = 2\\times 10^{-6}$ seconds to meet the requirement of numerical stability [46].",
"The rough inclined plane and the tumbler walls are modeled as a monolayer of bonded particles of the same size.",
"The tumbler walls are composed of small particles in solid body rotation.",
"In the incline simulations, small beads (or disks) are placed randomly in the simulation domain and, as gravity is set, they fall on a sticky plane (or line).",
"All small beads touching the bottom of the domain ($z=0$ ) stop moving and form the rough bottom of the inclined plane.",
"The other beads constitute the flowing granular material.",
"With this procedure, rough planes whose compacity is around 0.57 are obtained in 3D.",
"A large tracer bead (or disk) is placed usually at the top of the free surface and at time zero gravity is tilted from 0 to 23$^\\circ $ in 3D (or to $\\theta =20$$^\\circ $ in 2D), except where otherwise stated, and the flow starts.",
"For tumblers, the large tracer is placed first randomly inside the drum, or at a defined location if needed.",
"The other flowing particles are then placed randomly inside the tumbler.",
"At time zero, gravity is switched on, the flowing particles fall and the wall particles start a rotational movement.",
"In tumblers and inclined planes, wall particles are assumed to have an infinite mass for calculation of the collision force between flowing and fixed particles.",
"The velocity-Verlet algorithm is used to update the position, orientation, and linear and angular momentum of each particle.",
"Periodic boundary conditions are applied in the directions $x$ or $x$ -$y$ of the box (flow direction or flow - horizontal directions) in the case of an incline, and along the tumbler horizontal axis $y$ in the case of a 3D cylinder.",
"In the tumbler case, velocity maps are obtained by binning particles into boxes.",
"The simulation domain is divided into 60$\\times $ 60 boxes in the $x-z$ directions.",
"The tumbler having a diameter of 4.85 cm (plus 2 small bead diameters), each box is a square of size around 0.8 mm.",
"From these maps, streamlines and velocity profiles are extracted.",
"Velocity maps are obtained, the tracer being either included or excluded in the binning, or by generating a monodisperse flow where the tracer is replaced by exactly the same volume of small particles.",
"All the velocity maps obtained are identical."
],
[
"Rotating cylindrical tumblers",
"In this part, the aim is to obtain numerical results in 2D and 3D rotating cylindrical tumblers, in order to compare them precisely with previous 3D experimental results.",
"This will provide a validation of the numerical method and some insights into the processes happening during flow.",
"The previous experiments used glass beads of different diameters and of the same density [30], [36].",
"In those experiments, the rotating cylindrical tumbler (4.2 cm long and 4.85 cm in diameter) was half-filled with small beads and a few large beads (typically 50) named tracers, initially placed so that they barely interacted.",
"The volume fraction of tracers was 3%.",
"The diameter of the tracers ($d_t=$ 3 mm) was kept constant while the size of the small particles (diameter $d$ ) was decreased from $d=$ 2.5 mm to $d=$ 90 $\\mu $ m to explore size ratios ranging from $d_t/d=1.2$ to 33.",
"The cylinder was rotated around its horizontal axis at about 3.6 rpm, so that a continuous flow with a flat free surface developed.",
"After three revolutions, a stationary state was reached, with tracers at nearly identical radial positions, leading to a half-ring segregation pattern (Fig.",
"REF (a)).",
"Since each radial position $R_{ti}$ in the solid rotating part corresponds to a depth $h_i$ during flow, we interpreted the ring by the fact that all the tracers located themselves at the same preferential depth within the flowing layer (Fig.",
"REF (a)).",
"The radial segregated position $R_t$ was defined as the mean of all radial positions $R_{ti}$ .",
"It is important to choose the same experimental dimensions for the simulations, so that experimental and numerical results can be compared, because the link between the radial position and the depth within the flowing layer is mainly a function of the tumbler and particle diameters.",
"For instance, equivalent size or density ratios give different radial positions in different tumbler diameters [36], [49].",
"From a numerical point of view, this experimental protocol is not easy to reproduce since the number of small particles increases strongly with increasing size ratio, already reaching 10$^5$ for 90 $\\mu $ m small particles in 2D.",
"First, we will use dimensions as close as possible to those used in the experiments.",
"Then, the tracer size will be increased carefully to reach larger size ratios."
],
[
"Direct comparison with experiments",
"The 2D numerical tumbler of inner diameter ($D=~2R$ ) 4.85 cm is half-filled with monodisperse small disks and one large tracer (disk) of the same density.",
"The diameter of the small particles varies from 2.5 mm to 90 $\\mu $ m and that of the large tracer is 3 mm.",
"The tumbler rotates at 15 rpm to ensure a continuous flow with a flat free “1D surface\".",
"Figure REF (b) shows the trajectory of a large tracer ($d_t/d=16$ ) passing successively through the flowing layer and the solid rotating zone.",
"After a few revolutions (4 to 5, not shown here), the trajectory converges to and fluctuates around an equilibrium radial position: a stationary state is reached.",
"Figure: (a) Each depth h i h_i in the flowing layer corresponds to a radial position R ti R_{ti} in the static part, (b) A 4.85 cm diameter rotating tumbler with 187 μ\\mu m red small disks and a 3 mm white tracer, d t /d=d_t/d= 16.",
"The blue curve is the tracer trajectory.",
"The tracer stabilizes at an intermediate depth and radial position.",
"Green thin lines are the streamlines of the small disks.Each time $i$ the tracer passes through the vertical plane $x=0$ in the static rotating zone, the distance from the tracer center to the cylinder center $R_{ti}$ is measured.",
"A mean position $R_t$ and a standard deviation are computed.",
"Small standard deviations indicate strong localization at the same radial position from turn to turn.",
"This corresponds to stabilization at a constant depth $h$ in the flowing layer.",
"This also corresponds to segregation rather than to mixing since several non-interacting tracers would all stabilize at these well-defined depth and radial position.",
"Consequently, they would regroup, i.e., segregate on this ring, exhibiting this small deviation.",
"We choose to call the equilibrium radial position $R_t$ a radius of segregation, because it corresponds to the experimental segregation half-ring radius obtained with 3% of tracers (for a comparison between experiment and simulation see Fig.",
"REF ).",
"Averaging and deviation calculation are done on one tracer during several turns for numerical data, or on several tracers at a given moment for experimental data, thus including trajectory fluctuations, but also potential tracer interactions and experimental errors.",
"Figure: Relative tracer radial positions in the cylindrical tumbler versus sizeratio d t /dd_t/d, for : several tracers in 3D experiments (green ▪\\blacksquare ) and several passages of one tracer in 2D simulations (blue ▴\\blacktriangle ).",
"Error bars show the standard deviation.In the simulations, a tracer with a size ratio from 1.2 to 3 is at the periphery, which corresponds to a surface position during flow.",
"Each larger tracer nearly remains at an intermediate radial position $R_t$ inside the drum (Fig.",
"REF (b)), which corresponds to an intermediate depth during flow.",
"As $R_t$ decreases toward zero with increasing size ratio, we deduce that the tracer position is progressively deeper in the flowing layer.",
"Fig.",
"REF represents the evolution of $R_t$ with the diameter ratio $d_t/d$ showing the reversal of the tracer position with increasing size ratio.",
"Each standard deviation value indicates whether there is a well-defined position or a dispersed trajectory within the tumbler.",
"In the event that several tracers are used a well-defined position leads to segregation and the dispersed trajectory leads to mixing.",
"In the tumbler, the spatial organization passes from a spread of the instantaneous positions (for size ratio near 1) to well-defined equilibrium mean positions: at the surface (maximum of $R_t$ is $R-d_t/2$ ), then at intermediate depths when $R_t$ decreases, and toward reverse depths for the lowest values of $R_t$ .",
"We compare the successive numerical positions of one tracer, and the experimental positions of several tracers, both giving a value for $R_t$ and a standard deviation.",
"The agreement between experiments and simulations is good, but only qualitative, with a similar evolution of the curve.",
"Both simulations and experiments show the reversal of either the equilibrium position or the segregation location (Fig.",
"REF ).",
"There are differences between 3D experiments and 2D simulations: (1) In 3D experiments, the decrease of the curve $R_t/R$ versus $d_t/d$ is more rapid than in 2D simulations.",
"(2) In 2D simulations, the asymptotic value of the curve is close to 0.55, a larger value than the asymptotic value in the 3D experiments, around $R_t/R= 0.35$ .",
"This 2D asymptotic value will barely be reduced for larger size ratios (Fig.",
"REF ).",
"(3) Another difference is observed regarding the maximum of the curve (surface segregation) which occurs for $d_t/d=$ 1.5 or 1.8 in experiments, instead of $d_t/d=$ 2.5 in the 2D simulations (Fig.",
"REF ).",
"We will see that these differences are due to the 2D nature of these simulations rather than to an experiment-simulation discrepancy.",
"A longer discussion on that point is presented with the 3D simulations."
],
[
"Higher size ratios",
"To explore the asymptotic value, we need to reach larger size ratios, which would require the use of a high number of small particles in the simulations.",
"To overcome this disadvantage several larger tracer sizes are tested ($d_t=3$ , 4.85, 6 and 9.7 mm) in the tumbler $D=48.5$ mm, and their equilibrium positions $R_t$ are compared (for size ratios 25 and 40).",
"Up to a diameter of $d_t=6$ mm, $R_t$ are almost identical.",
"For the largest tracer ($d_t=9.7$ mm, whose size is to be compared with the drum diameter $D=5d_t$ ), a small discrepancy (relative error of 4%) is observed.",
"We choose to keep the size of the tracer under $d_t=D/10$ to be sure that there would be no effect of the tracer size.",
"For the particles and tumbler studied here, the thickness of the flowing layer is always larger than one tracer diameter.",
"A tracer diameter $d_t=4.85$ mm is adopted to reach large size ratios up to 60.",
"Figure: Relative tracer positions in the tumbler versus size ratio d t /dd_t/d, for 3Dexperiments with 3% of tracers and 2D simulations with 2 tracer sizes, 3 and 4.85 mm(no standard deviation here).Fig.",
"REF shows the relative positions $R_t/R$ of the 3 mm and 4.85 mm tracers, compared to 3D experimental results.",
"When the 2 different tracers have the same size ratio $d_t/d$ , the resulting positions coincide.",
"For the largest size ratios used in these 2D simulations, the radial position slowly decreases but remains close to 0.5, and does not reach the experimental value of 0.35.",
"2D simulation and 3D experiment asymptotic $R_t$ values are different.",
"Even if there is a qualitative agreement, 3D simulations are needed for an accurate comparison."
],
[
"Comparison with experiments",
"To obtain a quantitative agreement, 3D simulations are conducted (Fig.",
"REF ).",
"Figure: 3D simulation of a rotating cylinder (diameter 48.5 mm) with a 3 mm tracer in 0.5 mm small beads.The tumbler inner diameter is equal to $D= 48.5$ mm and it rotates around the $y$ axis at 15 rpm.",
"In a first series (size ratio up to 8), the tracer diameter is set to $d_t = 3$ mm as in experiments, then for larger size ratios (from 5 to 25) it is set to $d_t=4.8$ mm to reduce the number of small simulated beads.",
"For size ratios $d_t/d=5$ and 8, both tracer sizes are tested.",
"Larger tracers ($d_t=6$ and 9 mm) are also used respectively from size ratios 5 to 25 and 12 to 25 to check the sensitivity to the tracer size.",
"As in 2D, no differences are observed for the 6 mm tracer, and very small discrepancies are observed for the 9 mm tracer.",
"Figure: Relative radial positions of the tracers versus size ratio, in 3D experiments (3% of tracers) and 3D simulations (one tracer, 3 or 4.8 mm).",
"The numerical standard deviation is very small (error bars).",
"The dashed line is the position of a tracer touching the bottom of the flow.The 3D numerical results show the evolution of the tracer radial position $R_t$ from the periphery to intermediate positions, toward the reverse position when the size ratio is increased (Fig.",
"REF ).",
"The standard deviation is very small, indicating a strong localization on the same radial position from turn to turn.",
"The 3D numerical radial position quantitatively matches the 3D experimental radial segregated position of several tracers.",
"Agreement is very good, even on precise points like: 1) the slope of the curve, 2) the asymptotic value of $R_t/R$ for large size ratios, and 3) the diameter ratio which corresponds to the maximum of the curve.",
"The agreement confirms our hypothesis that a few tracers locate themselves on a ring which has the same radius as the equilibrium radial position of one single tracer.",
"The segregation of several non-interacting tracers can be seen as the regrouping at an identical position because the equilibrium radial position of each tracer depends only on its size ratio.",
"The tracers do not interact much at this small fraction (3%), nevertheless their interaction leads to a slight increase in the standard deviation, with no observable change in the mean value.",
"We can speak interchangeably of segregation radius or of equilibrium radial position.",
"Moreover, as the agreement is really quantitative, we are confident in our simulation method to be used to study other systems, such as flows on rough inclines."
],
[
"Trajectories in 3D tumbler",
"To gain a better understanding of the segregation phenomenon, tracer trajectories are studied in details.",
"Fig.",
"REF (a) shows the trajectory of a large particle with a size ratio of 4 and the streamlines of small beads in a plane $x-z$ .",
"Two phases are distinguished: first, the unsteady stage, second a stationary trajectory when the equilibrium depth is reached.",
"The tracer initially falls after the tumbler has been filled (vertical line), then the rotation starts with the tracer relatively close to the stagnation point.",
"During the first, second, third passages, and the first part of the fourth passage in the flowing layer, the tracer exhibits an upward motion when compared to the small bead streamlines.",
"It migrates towards its equilibrium position.",
"Accordingly, in the static zone, from one passage to the next, the radial position $R_{ti}$ increases.",
"Then, after these 4 passages, the trajectory is stationary: the tracer flows along the streamlines at each passage, and presents a nearly constant radial position $R_{ti}$ with some fluctuations from turn to turn.",
"This confirms the experimental observation that after 3 rotations ($\\simeq 6$ passages through the flowing layer for a half-filled drum) the whole segregation process is over [30].",
"The convergence to an equilibrium depth, and consequently the segregation process, happens mainly during flow, and is not due to processes happening during the entrance into and/or the exit from the flowing layer.",
"Here, the tracer starts from a central position, and moves upwards to reach its equilibrium depth.",
"It could have been downwards if the tracer had been released from the surface of the flow (or periphery in the static part).",
"An equivalent upward motion is observed for a tracer with a size ratio 8 (Fig.",
"REF (b)), but its amplitude is smaller, as the starting position is closer to the equilibrium $R_t/R$ corresponding to this size ratio.",
"A more rapid downward motion toward the same equilibrium radial position is observed when the tracer is released at the periphery, probably because of the longer distance traveled in the flowing layer.",
"Figure: Trajectories of the tracer center (black curves), and small bead streamlines (red curves) (D=48.5D=48.5 mm, d t =3d_t=3 mm).",
"The thick green curve is the freesurface.",
"The first rotations concern the convergence, the following rotations concern the stationary phase.",
"(a) d t /d=4d_t/d=4, the tracer is at the limit between surface and intermediate positions, its top touching the free surface, (b) d t /d=8d_t/d=8 either starting from the periphery (dashed line) or from the tumbler center (solid line), the tracer is at an intermediate position.Once in the steady phase, the tracer trajectory and small bead streamlines are parallel in the flowing zone.",
"There is no relative motion any longer, either up or down.",
"Plotting a circle 3 mm on the trajectory shows that the tracer with size ratio 4 is just below the surface, and that the tracer with size ratio 8 is on a mid-height intermediate depth.",
"Each depth in the flowing layer corresponds to one radial position in the rotating part of the tumbler.",
"However, the tracer trajectory does not match exactly the same small bead streamline in the rotating zone and in the flowing zone.",
"There are two small shifts between the tracer trajectory and the small bead streamlines when going in and out of the flowing layer.",
"At the entrance (Fig.",
"REF (b)), the tracer starts to move after the small beads on the same streamline (despite the shift occurring at the previous exit), probably because its bottom is still surrounded with non-moving small beads.",
"At the exit of the flow, the tracer stops before the small beads on its corresponding streamline because its lower part is touching the static curved bottom (note that these entrance and exit shifts are enhanced in 2D (Fig.",
"REF )).",
"In conclusion, these shifts are not responsible for the segregation from turn to turn.",
"But they exist, they probably vary with $R_t$ and might be one cause of the discrepancy between 2D and 3D.",
"For that reason, it is not possible to easily deduce the flowing depth positions from both data of small particle streamlines and $R_t/R$ .",
"Nevertheless, the shifts are very small and the $R_t/R$ variation mainly reflects a variation in depth within the flowing layer.",
"A more accurate examination of the trajectory reveals that the entrance in the flow induces a starting point slightly above the equilibrium depth that the tracer will reach (Fig.",
"REF (a)).",
"Each time it passes through the flowing layer, the tracer exhibits a tiny descent towards its equilibrium depth, then remains at a constant depth to the end of the flow, parallel to streamlines.",
"The length at which the constant depth is reached seems to decrease with the tracer size ratio, approximately at mid-length for ratio 4, almost immediately for ratio 8 (Fig.",
"REF ).",
"Segregation is so fast that slight destabilization can be rebalanced in less than one passage in the flow.",
"In conclusion, the study of trajectory in the 3D cylindrical tumbler shows that the process responsible for the segregated radial positions of tracers is a vertical migration and stabilization of the tracer at various depths, occurring during flow.",
"We then expect a similar phenomenon to happen during flow on an incline."
],
[
"Radial position and depth within the flowing layer",
"One may wonder how the different values of $R_t/R$ should be interpreted in terms of equilibrium depth within the flowing layer of the tumbler, to anticipate conclusions across the tumbler study and the following incline study.",
"In particular, the question arises whether the asymptotic small values of $R_t/R$ do correspond or not to a reverse segregation within the flow.",
"A tracer touching the bottom of the flow undergoes a small decrease in $R_t/R$ with the size ratio, because with our protocol ($d$ decreasing) the thickness of the flowing layer slightly decreases [50].",
"Numerical thickness measurements, added with a tracer radius to obtain the tracer center position, are shown as a dashed line on Fig.",
"REF : it would correspond to reverse positions, turning around the stagnation point.",
"We deduce that size ratios 20 and 25 are in reverse position, and that the small decrease between them is explained by the choice of the protocol.",
"In addition, in some simulations, we measure the depth of the tracer directly on its trajectory.",
"For a ratio 8, (Fig.",
"REF (b)) the tracer is at an intermediate depth.",
"For the largest size ratios (for example, ratio 20), the tracer is just touching the bottom of the flowing layer, i.e., the bottom of the tracer passes where the streamlines are reduced to the stagnation point, which is nowhere else than the middle point of the bottom.",
"But this method is not precise: it is difficult to define the bottom of a flowing layer near the tracer.",
"The bottom of the flowing layer is defined by an averaging of small bead streamlines.",
"The tracer passage has almost no effect on the averaging, although it probably deforms locally and during a short time the granular material below and around it when it passes “at the bottom\".",
"Consequently, the bottom of the averaged flowing layer may not be the same as the local bottom of the flow around the tracer.",
"Nevertheless, we choose to call the positions of these tracers with the largest ratios \"reversed\", keeping in mind that this is somehow arbitrary.",
"In fact, denser tracers may be found at lower $R_t$ than the asymptotic value, probably because they more strongly deform the bottom [36].",
"With that choice, all the asymptotic $R_t/R$ positions correspond to a tracer in a reverse bottom position within the flow.",
"In conclusion, reverse depth is reached for tracers with a size ratio$\\geqslant $ 20 in 3D.",
"The same measurements on trajectories are made in 2D: tracers are found at intermediate depth for size ratio 10, 16 and 20, and in reverse position for size ratios above 40.",
"The reverse position can be reached both in 2D and 3D, but for greater size ratios in 2D."
],
[
"Differences between 2D and 3D tumblers",
"Compared with 3D results, 2D results are shifted, as if the size ratio had a reduced effect: the maximum of the $R_t/R$ curve occurs for a higher size ratio, the dependency is smaller, and the asymptotic value is higher (Figs.",
"REF and REF ).",
"We first check that the difference between 2D and 3D is not due to a variation of the thickness of the flowing zone.",
"Indeed, the thicknesses have been measured nearly identical for a given small bead size in 2D and 3D tumblers.",
"Secondly, for a given size ratio ($d_t/d=$ 20) we compare the depth of each tracer on its trajectory within the flow: in 3D, the tracer is touching the bottom, while in 2D, 6 small beads are under the tracer (this flow thickness is 25$d$ ).",
"The shift between 2D and 3D $R_t/R$ curves does correspond to a real difference in depth positions within the flowing layer.",
"Nevertheless, a radial position difference in 2D and 3D can also be seen for tracers at the same depth.",
"For example, for the asymptotic values, the largest tracers in 2D (size ratios above 40) and in 3D (size ratios 20 and 25) are all measured touching the bottom of the flowing layer.",
"The size of the tracer is fixed ($d_t=4.8$ mm or 4.85) and the thickness of the flowing layer is almost unchanged for these small bead sizes $d$ (a slight decrease indicated by the dashed line in Fig.",
"REF ).",
"Nevertheless, there is a gap between the 2D and 3D values of $R_t/R$ corresponding to these identical reverse depths (Fig.",
"REF ).",
"Note that to take into account the slight variation in the flowing layer thickness with the small bead size, one can simply extrapolate $R_t/R$ values in 3D up to 40 (Fig.",
"REF ), and compare radial positions exactly at the same flowing thickness: the conclusion is unchanged.",
"Considering tracers at a same depth, the $R_t/R$ difference is mainly due to the larger trajectory fluctuations in 2D which shift $R_t$ to larger values in 2D when approaching the bottom (and to smaller values when nearing the surface).",
"It is also due to a difference in the entrance and exit of the flowing layer, which gives smaller $R_t$ values in 2D.",
"This latter effect pushes in the opposite direction, but is small compared to the former one.",
"We have seen that there is a difference between the 2D and 3D equilibrium depths within the flowing layer all along their evolution with the size ratio.",
"To understand the cause of this difference, one should compare the effective densities of the medium made up of small particles.",
"If we note $\\rho $ the density of the small (or large) particles, the effective density of the granular medium made up of small particles is equal to $c\\,\\rho $ , where $c$ is the compacity.",
"A large tracer is denser than a sphere/disk of the same diameter filled with a random close packing of small particles.",
"In 2D, such a packing gives a compacity close to $c_{2d}\\simeq 0.8$ , while in 3D, $c_{3d}\\simeq 0.6$ .",
"Thus, the density ratio between the tracer and the medium is larger in 3D than in 2D, leading to deeper intermediate segregation and advanced reverse segregation.",
"This result was confirmed using tracers of decreasing densities [36].",
"For tracers less dense than a random packing of small particles, only surface segregation of the tracer is observed.",
"Even if the names and limits of the equilibrium depths are arguable, there are similarities but also discrepancies between 2D and 3D cases.",
"In 3D, the evolution of the position shows a shift of the curve maximum toward smaller size ratios and a stronger dependency with size ratio.",
"If the enhancement of the effect of the size ratio is due to the compacity around 0.6 (in 3D) instead of 0.8 (in 2D), we expect that it will always be present in all types of flow.",
"Thus, care should be taken when extrapolating these 2D studies to the 3D case.",
"The experimental study of granular segregation occurring during flow down an incline is a difficult task to achieve in wide and thick channels (3D).",
"Indeed, in our previous experiments [30] only the surface of the flow was visible.",
"A fraction of 10% of large particles was used.",
"For large size ratios ($d_t/d\\geqslant 6$ ), no tracers were visible at the surface during flow, although for $d_t/d\\leqslant 3$ , large particles were at the surface.",
"The volume of the deposit could be accessed after the flow had stopped due to a slope change or a vertical end wall.",
"But the aggradation of the deposit may have modified the particle depths.",
"Size ratio had been varied and the segregation pattern in the deposit changed according to the size ratio: for small size ratios, the large particles covered the surface of the deposit, while for larger size ratios, the large particles were found inside the deposit.",
"The individual positions were moderately spread inside the deposit.",
"Nevertheless, their mean position was at an intermediate depth, which was deeper and deeper with increasing size ratios.",
"Because of this spread and because of the aggradation, it was not possible to conclude on whether these tracers were located at a well-defined intermediate depth during flow (corresponding to an intermediate segregation).",
"Simulations will allow measurements during flow, in an established steady-state regime, with a single tracer.",
"The main advantage of the incline geometry is that the measurements of tracer depths within the flows are direct, while measurements of radial positions in the tumbler involve entrance in the flowing layer, acceleration and exit from the flowing layer.",
"Moreover, for a solid rough incline, the bottom depth can be accurately determined, which is not the case in a partially filled tumbler where flow passes on loose granular matter having a curved bottom shape.",
"Another difference is that the flow thickness in the tumbler is mainly imposed by the dimensions (tumbler diameter and small particle size).",
"For the chosen protocol (decreasing small bead size), this layer thickness decreases with the size ratio (around $8d$ for $d_t/d= 2$ ; $21d$ for $d_t/d=10$ ; 34$d$ for $d_t/d=25$ , respectively: 4, 2.1 and 1.4$d_t$ ), while for confined flows in an inclined channel the thickness of the flow can be varied independently.",
"In the present study, the smallest thickness chosen is comparable to those encountered in non-confined flows on an incline (around $10d$ ) [51], and, for this reason, the results on such thin flows are not without interest.",
"The thickness will be increased ($37d$ ), to explore thickness effects, and will reach the values for experimental channel flows, for comparison [30].",
"As experimental results were obtained without following any protocol, we choose to keep the small bead size constant ($d=6$ mm), and vary the tracer size ($d_t$ ).",
"Indeed, decreasing the small bead size would have resulted in increased flow velocity and increased calculation time for a constant flow thickness.",
"Nevertheless, we may expect some deviations between experimental and numerical results if the tracer becomes too large compared with the flow thickness."
],
[
"Intermediate segregation",
"Even though quantitative agreement cannot be taken for granted, we first perform 2D simulations.",
"The simulation domains are 160$d$ long, or 300$d$ long for the larger tracers ($d_t/d\\geqslant 8$ ).",
"Fig.",
"REF shows a 48 mm diameter tracer (disk) in a granular flow made up of 6 mm small disks flowing down an incline.",
"The plane slope is 20$^\\circ $ and the thickness of the flow $h_{max}$ is around 36 cm.",
"Figure: A 2D granular flow down a rough incline with 6 mm smalldisks and a 48 mm large tracer, moving from left to right.",
"The slope angle is 20 ∘ ^\\circ , the flow thickness is 36 cm.The tracer with this size ratio ($d_t/d=8$ ) is not far from the free “1D surface\" but remains below it, fluctuating around an intermediate depth.",
"This does not correspond to the behavior of a large particle during the classical granular surface segregation of large particles, but to that of a particle flowing inside the bed, at an equilibrium intermediate depth: this would lead to intermediate segregation if several non-interacting tracers of the same size were present.",
"Figure: Trajectories of the center of 3 tracers versus time.",
"The horizontal dashed line is the free 2D “surface\".",
"Thick circles show the sizes of the three tracers.Figure REF shows the depth $z(t)$ of each tracer center for three different tracer sizes versus time $t$ (each simulation involving one single tracer).",
"For each tracer size, several initial positions at the bottom or at the surface are tested, but only one is plotted here.",
"The steady state tracer depth $h$ does not depend on the initial location ($h$ is the mean of $z(t)$ , the initial convergence time being removed).",
"For the size ratio $d_t/d= 8$ , the tracer almost never reaches the free “surface\" and stays at an intermediate depth.",
"It is the noisiest trajectory.",
"At intermediate depths, the trajectory is not stabilized by the existence of the free “surface\" or the bottom nearby.",
"For the size ratio $d_t/d=20$ , the tracer reaches an equilibrium depth located near the center of the flow with a layer of around 22 small particles below it.",
"The $d_t/d= 3$ tracer, initially placed at the bottom, reaches the surface, as in a surface segregation phenomenon.",
"Stationary positions are reached after horizontal displacements of 10000$d$ , 25000$d$ and 30000$d$ for tracers of size ratio 20, 8, and 3 respectively.",
"The distance is mainly due to the gap between the initial vertical position of each tracer and its corresponding stationary position.",
"Large values are explained by the use of a thick flow (60$d$ ), and by the poor efficiency of the 2D segregation.",
"Figure: Trajectories of a tracer released at the top (solid lines)and the bottom (dashed lines) of the flow.",
"Both trajectories converge to the sameequilibrium depth (intermediate for d t /d=16d_t/d=16, at surface for d t /d=3d_t/d=3).",
"Circles show the tracer sizes.",
"The horizontal black line is the mean position of the free 2D “surface\".For a size ratio of 16, trajectories starting from the top and from the bottom reach the same equilibrium depth in about 12-15 seconds (around 4000-5000$d$ ) (Fig.",
"REF ).",
"The initial gap to stationary position is the main parameter which determines the time or distance to travel along.",
"It takes a longer time (and distance), around 55 seconds, (22000$d$ ) to the tracer with size ratio 3 starting from the bottom to reach the surface: it has to move across the whole flow thickness, on a trajectory showing larger fluctuations.",
"Figure: Equilibrium depths of the tracer center in a flow down a 2D inclineversus size ratio, for 3 flow thicknesses: (blue ▴\\blacktriangle ) 0.12 m, (black •\\bullet ) 0.24 m, (red ▪\\blacksquare ) 0.36 m. Error bars show the standard deviation.",
"Horizontal lines show the free “surfaces\".",
"The oblique dashed line corresponds to the position of a tracer whose top is at the surface of the thinnest flow.In order to study where the tracer stabilizes, the mean depth of one tracer in the stationary regime $h$ is reported for several size ratios $d_t/d$ (with $d$ = 6 mm) and for several thicknesses of the flow $h_{max}$ (Fig.",
"REF ).",
"$h$ is calculated from the flow bottom to the tracer center.",
"Moderately large tracers ($2\\leqslant d_t/d\\leqslant 6$ ) are found at or near the surface, and $h$ is maximum for a size ratio between 2 and 3.",
"For these low size ratios, the values of $h$ seem related to the distance to the free “surface\", independently of the thickness of the flow, showing the same curve shape relatively to the flow surface.",
"For larger size ratios ($d_t/d \\geqslant 7$ ), the tracer position gets deeper and with increasing size ratios.",
"It is compatible with the $R_t/R$ vs $d_t/d$ decrease in the tumbler.",
"The $h$ asymptotic value for very large size ratios is close to $h_{max}/2$ , and thus scales with the thickness of the flow (Fig.",
"REF ).",
"The interesting result is that there are some tracers which stabilize at intermediate depths inside the flow.",
"This shows the occurrence of intermediate segregation in a 2D flow on a rough incline, at least for non-interacting tracers.",
"We can assume that small fractions of large disks would undergo intermediate segregation for these size ratios.",
"The small standard deviations represented as error bars indicate that each tracer does not explore the whole thickness of the flow, but remains at an intermediate well-defined depth, with little randomness in its trajectory.",
"These small fluctuations would correspond to a small standard deviation in the segregation of several non-interacting tracers.",
"Note that for thin flows, tracers with size ratios above 10 are very large compared to the flow thickness and they are close to appearing at the “surface\", although they interact with the bottom at the same time.",
"The oblique dashed line shows the position of a tracer such that its top is flush with the free “surface\" of the flow (Fig.",
"REF ).",
"It defines a boundary between surface and intermediate positions (for small size ratios, here around 5), and also shows a reasonable size limit for a tracer in such thin flow (here, $d_t/d=12$ ).",
"Figure: Relative equilibrium depths of the tracer center in the 2D flowversus size ratio, for three flow thicknesses: (blue ▴\\blacktriangle ) 0.12 m, (black •\\bullet ) 0.24 m, (red ▪\\blacksquare ) 0.36 m.If the vertical position is renormalized by the thickness of the flow (Fig.",
"REF ), the three previous curves collapse reasonably well.",
"Note also that for $1.5\\leqslant d_t/d \\leqslant 6$ , rescaling like $h_{max}-h$ is a better choice as curves match well in their upper part, but they will no longer collapse for large size ratios.",
"Thus the behavior in a 2D flow shows two regimes: a first one (tracer near or at the surface) where the equilibrium position depends on the distance to the “surface\", independently of $h_{max}$ value, and a second one where the equilibrium depth is intermediate and tends towards $h_{max}/2$ , and thus scales with the flow thickness.",
"The positions $h$ show that only surface and intermediate depths are obtained in 2D granular flows on an incline.",
"We conclude that reverse segregation is not obtained in 2D, at least in this parameter range.",
"The large tracer does not reach positions below mid-height of the flow, even for very large size ratios.",
"For the thinnest flow, the depths are compatible both with a reverse and an intermediate pattern, considering the small number of small particles below the tracer, but for thicker flows both types of depths can be differentiated.",
"Equilibrium positions end up really at mid-flow for the largest ratios.",
"There are 14 small particles below the largest tracer in the thickest flow, significantly above a reverse position.",
"Since in tumblers the dependency of the position $R_t$ on $d_t/d$ is stronger in 3D than in 2D, we expect different results for a 3D incline flow.",
"Another point worth noting is that the dependency of the position ($h$ or $R_t/R$ ) on $d_t/d$ is also greater for a 2D incline than for a 2D tumbler and consequently, the asymptotic value is approached for smaller size ratios on an incline than in a tumbler flow."
],
[
"Slope angle",
"In a granular flow down an incline, the easiest way to increase the shear rate, without changing the thickness of the flow, is to increase the slope.",
"Fig.",
"REF shows the relative position of four tracers, with size ratios $d_t/d=6$ , 8, 10 and 30, for several angles of the plane.",
"Even if small evolutions are measurable, the relative vertical position of tracers ($d_t/d\\leqslant 10$ ) is almost unchanged for a slope change from 17 to 23$^\\circ $ although this change induces an increase in the mean velocity of the flow, and thus in the shear rate, by a factor of 4.",
"In the case of a $d_t/d=30$ tracer, a slight monotonic increase in the tracer depth with the slope is observed.",
"For size ratio 10, the same increase is obtained but only for slopes larger than 20$^\\circ $ .",
"A series of simulations is conducted on the 3D incline, first in a thin flow, then in thicker flows.",
"Even though very large size ratios are not reachable with our computational facilities, this captures most of the phenomena and allows a comparison with the 2D case and with previous experiments in a 3D channel.",
"Figure: A 3D incline granular flow, with a tracer (d t /d=6d_t/d=6), moving from left to right (slope is 23 ∘ ^\\circ , h max =0.112h_{max}=0.112 m).",
"Side beads have been removed to show the tracer."
],
[
"Equilibrium positions",
"Figure REF shows a 3D flow, with a tracer having a size ratio $d_t/d=6$ (small beads are $d=6$ mm).",
"The horizontal dimensions of the simulation domain are 20$d$ $\\times $ 20$d$ (0.12 m $\\times $ 0.12 m) or (40$d$ $\\times $ 40$d$ ) for the largest size ratios.",
"Both domain sizes are used for several size ratios to be sure that the simulated domain is large enough (Fig.",
"REF ).",
"The flow thickness ($h_{max}=0.112$ m $\\simeq $ 18$d$ ) corresponds to a relatively thin flow, comparable to the flows encountered in our 3D tumbler for size ratios around 8.",
"The tilt angle is 23$^\\circ $ .",
"Figure: Equilibrium depths of the tracer center in the 3D flow down an incline versus size ratio d t /dd_t/d.",
"Error bars show the standard deviation.The horizontal line is the free surface (h max =0.112h_{max}=0.112 m, the slope is 23 ∘ ^\\circ ).",
"Two numerical domain sizes are used: 20dd ×\\times 20dd (red □\\square ) and 40dd ×\\times 40dd(green •\\bullet ).For each size ratio (from 1.2 to 12), the large tracer depth $z(t)$ evolves rapidly during flow (see below Figs.",
"REF and REF ) to stabilize finally at a constant depth $h$ .",
"Some tracers have been initially placed at the bottom of the flow, and some at the surface without any final difference.",
"Once in steady state, trajectory fluctuations are small and give small standard deviation associated with each $h$ .",
"The equilibrium depth $h$ depends on the size ratio between tracer and small beads.",
"Fig.",
"REF plots the tracer depths (from 7.2 to 72 mm in size) for size ratios ranging from $d_t/d= 1.2$ to 12.",
"For moderately large size ratios (below 4), $h$ is near or at the surface, in accordance with the surface segregation of large beads.",
"As in the 3D rotating tumbler, the maximum of the curve (i.e., tracer at the free surface) is obtained for size ratios between 1.5 and 1.8 (Fig.",
"REF ).",
"For size ratios approximately between 4 and 6, the tracer reaches an equilibrium depth inside the flow, suggesting the occurrence of intermediate segregation in 3D flow down an incline, for non-interacting tracers.",
"For larger size ratios, $d_t/d>6$ , the equilibrium depths reach a saturation value near the bottom, in a reverse position.",
"We note that the equilibrium positions are independent of the size of the simulation domain.",
"The slight increase of the curve for the largest size ratios (10 and 12) is due to the increase in the tracer size itself, showing that the tracer is in strong interaction with the bottom.",
"There are only about 4 small beads below the tracer.",
"The three types of equilibrium positions (surface-intermediate-reverse) are thus found in this thin 3D flow, suggesting that the three segregation patterns would exist for a small fraction of non-interacting tracers.",
"Comparing 2D and 3D cases (Figs.",
"REF and REF respectively), the overall behavior is the same but some differences are present.",
"In the 3D case, the equilibrium depth decreases more rapidly and reaches a smaller saturation value earlier, at a size ratio close to $d_t/d= 6$ in 3D, instead of $d_t/d=$ 10 or 15 in 2D.",
"We also note that the standard deviations are much smaller in 3D."
],
[
"Thickness of the flow",
"Figure REF shows the trajectories for the first 50 seconds of two tracers, $d_t/d= 2$ and 8, immersed in granular flows having three different thicknesses.",
"The horizontal lines show the positions of the free surfaces of the flows: $h_{max}= 0.112$ m, 0.167 m and 0.223 m. For the three thicknesses, the tracer with a size ratio of 2 remains at or goes to the surface of the flow showing the same final position as in a surface segregation process (only the case of the thinnest flow is shown for the ratio 2 tracer placed at the bottom).",
"When crossing the whole thickness, the convergence is longer for this small size ratio ($d_t/d=2$ ) than for a larger one ($d_t/d=8$ ), and the trajectory presents more fluctuations.",
"The large tracer ($d_t/d= 8$ ) sinks to reach a depth near the bottom of the flow.",
"Its stationary depth is close to 0.05 m, independently of the thickness of the flow.",
"As the tracer radius is $r_t= 0.024$ m, it does not touch the rough inclined plane made up of small glued beads, but about 4 small beads remain between the tracer and the plane.",
"We consider this position close enough to the bottom to be called “reverse\".",
"When using a $t-z$ representation, parallel trajectories on Fig.",
"REF (b) show that the sinking velocity is constant ($v_{sink}=-0.0105$ m/s).",
"For a given size ratio, the time of convergence is mainly related to the thickness of material to travel through.",
"A constant sinking velocity is an interesting feature since it can be used in theoretical models to describe granular segregation.",
"From an experimental point of view, an $x-z$ representation (Fig.",
"REF (b) insert) is more interesting since it gives the incline length required for an experiment.",
"For a size ratio of $d_t/d=8$ , changing the thickness of the flow from $h_{max}=19d$ to 37$d$ , decreases the slope of the trajectories (compared to the rough incline) and increases the horizontal settling distance from $\\simeq 400d$ to $\\simeq 2500d$ .",
"This distance increase comes from the flow thickness increase but also from the induced increase of the mean velocity which is about a factor 3 here.",
"In the case of a larger tracer $d_t/d=12$ (Fig.",
"REF ), the sinking is more rapid ($v_{sink}=-0.021$ m/s), and the slope of the trajectories also decreases with the increase in thickness (not represented).",
"Consequently, the settling distance increases from $\\simeq 200d$ to $\\simeq 1200d$ , for $h_{max}=0.112$ and 0.233 m respectively.",
"For a downward motion, convergence is more rapid for high size ratios (comparing Figs.",
"REF (b), REF and REF ).",
"Downward forces acting on tracers are stronger when tracers are larger, and consequently heavier.",
"Figure: Equilibrium depths of tracers versus size ratio in the 3D flow for 3 thicknesses: (blue ▴\\blacktriangle ) 0.112 m, (black •\\bullet ) 0.167 m, (red ▪\\blacksquare ) 0.223 m. Error bars show the standard deviation.Figure REF shows the equilibrium depth $h$ for a large tracer for three different thicknesses $h_{max}$ .",
"For size ratios up to 4, the large tracer remains at or near the surface, independently of $h_{max}$ .",
"For size ratios larger than 5, the tracer sinks close to the bottom of the flow, and $h$ is independent of $h_{max}$ (as in Figs.",
"REF (b) and REF ).",
"The slight increase with the tracer size shows the strong interaction with the bottom when in reverse position.",
"For the two thickest flows, a sharp transition between the surface position range and the reverse position range appears for size ratios $d_t/d$ between 4.2 and 4.5, while a relatively progressive variation is observed for the thinnest flow.",
"The tracer depth $h$ depends on the flow thickness only during the transition.",
"Both parts of the curves, $h-h_{max}$ for small (below 4) or $h$ for large size (above 5.5) ratios are independent of $h_{max}$ .",
"Figure: Equilibrium distances from the tracers to the surface h-h max h-h_{max}, in the 3D flow on an incline for 3 thicknesses: (blue ▴\\blacktriangle ) 0.112 m, (black •\\bullet ) 0.167 m, (red ▪\\blacksquare ) 0.223 m.Plotting $h$ (Fig.",
"REF ) shows that the distance to the bottom controls the equilibrium position for very large tracers, independently of the thickness of the flow.",
"In the same way, plotting $h-h_{max}$ (Fig.",
"REF ) shows that the distance to the surface is independent of the flow thickness for moderately large tracers ($1.5 \\leqslant d_t/d \\leqslant 4.2$ ), when positions near surface are reached.",
"It seems that two independent phenomena, one influenced by the presence of the surface and one by the bottom, determine the equilibrium of the tracer in each case.",
"For a thin flow, the free surface and the bottom are close enough so that the two phenomena interact, and the result is a progressive transition between the two influences, creating a larger range of intermediate segregation positions.",
"In the case of a thick flow, both influences are almost separated and could be studied independently.",
"It could be tempting to associate the three zones coming from these curves (Figs.",
"REF and REF ) with the three types of equilibrium positions: surface, intermediate and reverse (or the three segregation types).",
"But they do not exactly match.",
"For example, tracers just below the surface (as for a ratio 3.5) are not visible at the surface, whatever the flow thickness is: they should be considered in intermediate position.",
"Symmetrically, the tracer with a size ratio 5 is floating above the larger ones, and is in intermediate position.",
"As for the largest tracers ($d_t/d\\geqslant 8$ ), which show a slight increase in their center depth due to the increase in their size, they are in strong interaction with the bottom plane and are thus in reverse position.",
"The separation into three types of equilibrium depths (surface-intermediate-reverse) is convenient but may not be representative of the phenomena happening in the granular matter.",
"Only two mechanisms may be the cause for equilibrium depths: one due to the influence of the surface and one due to the influence of the bottom.",
"Their potential combination appears or does not appear at around mi-height of the flow, depending on the flow thickness.",
"Nevertheless, in the present study we keep the separation in the three types (surface-intermediate-reverse) that correspond to particular positions of the tracers (and not to mechanisms).",
"In this view, we have to split the surface zone of the $h$ curve in two layers: one layer with surface positions (visible tracers), and one layer with intermediate depths.",
"In the same way, we split the bottom zone of the $h$ curve in two layers: a second layer with intermediate depths and one layer with reverse depths (Fig REF ).",
"In this view, thick flows have two intermediate depth layers which are separated by an empty central zone where there is no equilibrium depth for a tracer.",
"Thin flows have their two intermediate depth layers continuously connected, forming a “thick\" central layer of intermediate equilibrium depths.",
"Figure: The upper part of the hh curve defines two layers, with surface and intermediate equilibrium depths, and the lower part, two layers, with intermediate and reverse equilibrium depths (red hh curve from Fig.",
").",
"In a thick flow (drawn here), there are no equilibrium depths in a layer around mid-height.",
"The bottom is never reached, partly due to the tracer size (h⩾d t /2h\\geqslant d_t/2) and partly due to the presence of some small beads (around 4) under the tracer."
],
[
"Comparison between the 2D and 3D flows on an incline",
"The main difference between 2D and 3D is the equilibrium depth of very large tracers (Figs.",
"REF and REF ).",
"The large tracers sink near the bottom, exhibiting a reverse position in the 3D case while they locate themselves at intermediate depths in 2D, near mid-height of the flow.",
"For $h_{max}=0.112$ in 3D, the asymptotic equilibrium depth is also close to $h_{max}/2$ (Fig.",
"REF ), but this is just a coincidence: other flows with different thicknesses show the same constant asymptotic value.",
"The fact that the stabilization of a large tracer in 2D is $h_{max}$ dependent, while it is independent of $h_{max}$ in 3D, shows that 2D and 3D segregations of a few non-interacting large tracers may be different processes.",
"Moreover, the transition between the surface and the deepest positions is steeper in 3D than in 2D (see Figs.",
"REF and REF ).",
"The transition occurs between size ratios $d_t/d = 4$ and 6 in 3D, while in 2D the whole transition occurs between $d_t/d = 5$ and 15.",
"This stronger dependency in the 3D case is also observed in the tumbler.",
"Moreover, the maximum does not occur for exactly the same size ratio in 2D and 3D.",
"Nevertheless, similar behaviors are also noticed: for small size ratios in 2D and 3D the tracers positions are both related to the distance to the surface, independently of the $h_{max}$ value.",
"As for the tumbler system, the 2D incline case should not be carelessly extrapolated in 3D: evolutions with the size ratio are different, even though some strong similarities are observed.",
"These differences are probably linked to the compacity difference between 2D and 3D."
],
[
"Comparison between tumbler and incline",
"On an incline, granular matter flows on a solid rough surface whereas in tumblers, it flows on loose curved granular material.",
"The comparison of the equilibrium position ($h$ or $R_t/R$ ) versus $d_t/d$ in both types of flows (incline or tumbler) gives information on the influence of the structure of the flow.",
"Fig.",
"REF shows normalized depth in the incline flow $h^*$ and radial positions $R_t/R$ in the tumbler at the same scale.",
"We choose to adjust the minimal and maximal positions of $h$ to the asymptotic and maximum values of $R_t/R$ , which corresponds to bottom and surface tracer position, respectively, within the tumbler flowing layer.",
"The curves match relatively well for $d_t/d\\leqslant 3.5$ , indicating that for these small size ratios the process is mainly controlled by surface phenomena, which are quite insensitive to the substratum.",
"For larger ratios $d_t/d\\geqslant 4$ , curves shift with a stronger dependency in the case of rough inclines.",
"The difference may come from the substratum.",
"As the conclusions drawn from Figs.",
"REF and REF , these data suggest that the equilibrium at a given depth comes from one phenomenon influenced by the surface, or/and one influenced by the bottom.",
"Figure: Equilibrium positions of the tracer versus size ratio, in the 3D tumbler (green ▾\\blacktriangledown ) and equivalent rescaled (see text) positions in the 3D incline flow, for 3 different thicknesses.Note that for 3D inclines, the tracer vertical position increases for size ratio starting from 1, reaches a maximum for size ratios between 1.5 and 1.8, and decreases for larger values.",
"For 3D tumblers, the maximum is obtained for size ratios between 1.5 and 1.8 in experiments, and between 1.5 and 2 in simulations.",
"These non-monotonic variations are analogous to those observed experimentally in an annular shear cell where the segregation time and the segregation rate both present an extremum for a size ratio of 1.6 [25].",
"This is also related to the variation of the segregation Péclet number, defined as a segregation rate on a diffusive remixing, which shows a slight maximum at 1.7 [37], or the variation of the force acting on a tracer in 2D, which shows a maximum at 2 [29]."
],
[
"Slope angle",
"Several simulations are done for different slope angles of the plane.",
"For thin flows, positions $h$ for tracers with small and large size ratios show no dependency on the slope, i.e., on the velocity of the flow (Fig.",
"REF (a)).",
"On the contrary, when getting close to the transition between surface and reverse depths (size ratio between 4 and 6), $h$ depends on the slope.",
"The greater the angle, the deeper the tracer stabilizes.",
"This can be interpreted by the fact that the flow being more rapid, it loses cohesion and is less able to carry large and consequently heavy tracer.",
"Figure: Equilibrium depths of tracers in a 3D flow versus slope (size ratios d t /dd_t/d from 2 to 10).",
"Flow thickness h max h_{max}= (a) 0.112 m (b) 0.223 m.Error bars show the standard deviation.Figure: Tracer trajectories (x-z plane) measured in small bead diameters.",
"d t /d=10d_t/d= 10, h max =0.223h_{max}=0.223 m (37dd) and slope angles 22 to 25 ∘ ^\\circ .",
"Dashed lines show the free surface.",
"A set of noisier trajectories are plotted for comparison (size ratio d t /d=5.5d_t/d=5.5).",
"Insert: time evolution of the tracer depth (z-t).In the case of a thicker flow, the equilibrium depth of the tracer shows almost no dependency on the slope (Fig.",
"REF (b)).",
"But no size ratios between $d_t/d= 4$ and 5 are presented here: they do not present the usual rapid convergence to an equilibrium depth.",
"Further investigations are needed (ongoing study on [52]).",
"The time evolution of a tracer depth $z$ plotted for different angles (22 to 25$^\\circ $ ), shows that the tracer sinks more rapidly when the slope is larger (Fig.",
"REF insert).",
"However, trajectories (depth $z$ versus displacement along the flow $x$ ) for different angles all superimpose (Fig.",
"REF ).",
"This shows that the sinking of a large tracer is due to successive geometrical reorganizations between particles.",
"At higher slope, flow velocity and shear rate are increased and reorganizations are more frequent: the tracer sinks more rapidly ($z$ vs $t$ ).",
"The trajectories considered (in a $z-x$ space) all coincide independently of the flow rate (Fig.",
"REF ): only the number of reorganizations plays a role.",
"The slope of the trajectories (compared to the rough incline) are constant with the incline angle.",
"The horizontal settling distances are $\\simeq 1800d$ for a size ratio $d_t/d=10$ and $\\simeq 10000d$ for $d_t/d=5.5$ .",
"This implies that the sinking velocities increase with the incline slope.",
"Note that in 2D, the trajectory slope is also found constant for rough incline slopes from 17$^\\circ $ to 23$^\\circ $ and for $d_t/d=30$ .",
"By contrast, if the shear rate is increased due to an increase in flow thickness (Figs.",
"REF (b) and REF ), the downward tracer velocities are nearly identical (giving parallel trajectories in a $z-t$ space) and the spatial trajectories do not match in a $z-x$ space (Fig.",
"REF (b) insert).",
"In this case, the increase of the flow thickness induces an increase in the shear rate and in the frequency of reorganizations, but also an increase in the normal stress, which reduces the downward velocity of the tracer.",
"Both mechanisms compensate to induce a constant downward velocity.",
"Note, that the constant downward velocity is also found in 2D for $h_{max}=$ 36 and 24$d$ and size ratios 20 and 30, even though fluctuations are large.",
"To conclude, an increase in the flow velocity has a different effect on the downward motion of the tracer if coming from a slope or from a thickness increase.",
"Note that neither the constant velocity, nor this type of dependence with the traveled distance has been observed when the trajectory variation is due to a change in tracer size ratio.",
"Choosing $x$ instead of $t$ in Figs.",
"REF and REF does not have give any additional information."
],
[
"Multiple tracers flows on a 3D incline",
"In previous experiments, 10% volume fraction of tracers was used [30].",
"To compare simulations and experiments, the tracer fraction is numerically varied.",
"This will also allow comparison between the segregation process and the stabilization of one single tracer.",
"The segregated position (also labelled $h$ ) is the mean of the tracer positions once the flow has reached the stationary regime.",
"The mean flow velocity $v$ is measured for $d_t/d= 8$ and $h_{max}= 0.223$ m: it decreases by a factor 2 while the fraction increases from one tracer ($\\simeq 0.8$ %) up to a 5% (or to 10%) volume fraction.",
"As pointed out (Figs.",
"REF and REF ), tracer trajectory depths $z$ vs time $t$ cannot be compared for flows having non-equal velocities, only stationary depths $h$ can be compared.",
"For a full comparison of trajectories, $z$ vs $x$ displacements should be used.",
"Figure: Tracer trajectories z-x (in color) measured in small beaddiameters (dd) in a 3D flow (h max =0.223h_{max}= 0.223 m (37dd)) with: (a) 10%, (b) 5% of tracers (d t /d=8d_t/d=8).Trajectory (thick black) of an identical single tracer released at the surface.Figure REF compares the trajectories of a single tracer and of several tracers (5% and 10%) in the case of a thick flow ($h_{max}= 0.223$ m $\\simeq 37d$ ).",
"Tracers are initially randomly placed.",
"The tracer trajectories reveal a succession of displacements: horizontal displacements (the tracers cannot move downwards due to the steric exclusion effect) alternated with downward displacements (with a slope less steep than the case of a single tracer).",
"In the case of multiple tracers, the overall downward displacement is slower than that of a single tracer.",
"The depth equilibrium position of the lowest layer in the case of multiple tracers is rapidly identical to the depth of one single tracer.",
"For a 10% volume fraction, three, then two, layers of tracers form in the lower part the flow (Fig.",
"REF (a)).",
"Successive down cascading from one layer to another corresponds to an increase in the local fraction of the lowest layers.",
"Very rare up-motions of tracers are observed.",
"For 5% volume fraction (Fig.",
"REF (b)), the downward slopes of a single tracer trajectory and of multiple tracer trajectories can even be comparable.",
"Only one layer of tracers is present at the end, whose depth is identical to that of a single tracer.",
"First, the tracer fraction has no influence on the depth of the lowest layer.",
"Consequently, it is possible to compare experimental and numerical data using the lowest numerical trajectories, and the lowest experimental tracer depths.",
"The main effect of the increased fraction (5 or 10%) is the persistence of a second, possibly a third layer above the basal layer of tracers which is full and cannot include any more tracers.",
"For this reverse segregation, there is an asymmetric upwards spread of the tracer positions, with a position distribution maximum at the lowest layer depth.",
"Secondly, the convergence to the final state of segregation is longer to establish for 5 or 10% of tracers than for one single tracer.",
"Even though some individual downward velocities are locally the same, it takes time for tracers to move from one layer to another.",
"Even though the global segregation pattern is rapidly obtained (around $2000-4000d$ ), the distance of convergence is so long (around $10000d$ ) that it is not reachable in usual laboratory conditions.",
"These values are to be compared with experiments in channel with thicknesses from $28d$ to $45d$ , and a surface pattern obtained at 70 cm [22].",
"Nevertheless, in our simulations, the depth of the lowest trajectory is rapidly defined for thick flows (Fig.",
"REF ).",
"In our previous experiments, flows and deposits sometimes presented a thickness larger than 37$d$ .",
"To see how such a thickness could affect the previous results, one simulation is performed with $h_{max}=100d$ , 10% of tracers and $d_t/d=8$ corresponding to an equilibrium reverse depth (Fig.",
"REF ).",
"The number of basal layers increases, because for a constant volume fraction the number of tracers increases with the flow thickness.",
"As several layers of tracers develop (instead of 2 or 3) and as tracers cascade between layers, the time and the distance needed for convergence strongly increase (Fig.",
"REF ).",
"For experiments done with $d=300-400$ $\\mu $ m particles, a distance of convergence of $100 000d$ requires a plane of 35 m. Nevertheless, the results for $h_{max}=100d$ are similar to those for $37d$ (Fig.",
"REF ): reverse segregation is obtained, the bottom layer depth at $9d$ (equal to the single tracer depth), and the formation of several layers of tracers.",
"For larger size ratios, we may expect shorter convergence times and distances, since a single larger tracer reaches its equilibrium depth faster (Figs.",
"REF (b) and REF , or REF ).",
"Figure: Trajectories (z-x) of a few tracers (10%, d t /d=8d_t/d=8) in a 3D very thick flow (h max =h_{max}=100dd) measured in small bead diameters (dd).Measurement of the segregated positions is made for 5% of tracers, in a thick flow $h_{max}=0.223$ m and for several size ratios in an interval around the value 4.3, i.e., the reversal transition of a single tracer position from surface to bottom (Fig.",
"REF ).",
"The segregated position of several tracers at a given moment also presents a reversal, evolving from the surface to intermediate depths, and then to reverse depths.",
"The standard deviation is small enough to consider that segregation occurs: tracers are not spread all through the bed, but regrouped near the mean position, especially for surface and reverse segregations.",
"We then quantitatively compare the results for a single tracer ($h$ is the mean on the trajectory) and for several segregated tracers, and further down with experimental results on several tracers.",
"Except minor differences, the two curves are very close.",
"For small size ratios ($\\leqslant 4$ ), mean depths are the same, but there is an increase in standard deviation for several tracers.",
"The deviation includes both the trajectory fluctuations and the interaction between tracers.",
"For size ratios $4.2\\leqslant $ $d_t/d$ $\\leqslant 4.7$ , larger standard deviations and an upshift of mean depths are observed for several tracers, giving a smoother transition.",
"For larger size ratios ($\\geqslant 5$ ), the upshift disappears, and only larger standard deviations are observed for several tracers (note that for ratios above 8, all beads fit in the lowest layer, giving the same standard deviations as a single tracer).",
"As the mean depth of a single tracer and those of several tracers are almost identical, it confirms the hypothesis that the segregation process for this low fraction is a regrouping of near non-interacting tracers at the same equilibrium depth, because this depth depends only on the size ratio.",
"Studying a single tracer is valuable for understanding the segregation phenomena for a low fraction of tracers.",
"With this low fraction, we observe successively surface segregation, intermediate segregation, and reverse segregation when increasing the size ratio.",
"The reversal from surface to bottom happens for a size ratio (around 4.5) similar to the reversal size ratio for a single tracer(around 4.3).",
"One consequence of the smoother transition than for a single tracer is the disappearance of the empty central region where no single tracer stabilizes in a thick flow (Fig.",
"REF ).",
"For these fractions, there is a thick central layer of intermediate segregation.",
"In the case of multiple tracers, the segregation pattern organizes in three layers (surface, intermediate and reverse), very much like the equilibrium depths of a single tracer in a thin flow.",
"Figure: Equilibrium depths versus size ratio in a 3D flow (h max =0.223h_{max}= 0.223 m) for: a single tracer (blue □\\square ), 5% of tracers (red •\\bullet ).",
"Error bars show the standard deviation, but are only partly representative of the vertical spread of individual tracer positions which can be asymmetric, never higher than the surface or lower than the reverse position (see Fig.",
")."
],
[
"Comparison with experiments in channel",
"Experiments were performed by the sudden release of 1 kg of an initially homogeneous mixture of glass beads with 10% of large tracers, in a 6 cm wide, 1 m long rough channel inclined with a slope about 26.5$^\\circ $ (more details in [30]).",
"Flows were observed to be 1 to 2 cm thick, with deposits aggraded over 2 to 5 cm thick after the flow had been stopped by a perpendicular wall, or by the change of slope to horizontal.",
"On cross-sections of the deposit, the segregation pattern could be separated in three main cases, depending on the size ratio: (1) the small size ratios (1.75, 2, 2.14, 3.5) (resp.",
"for tracer diameter 0.35, 0.7 or 3, 1.5, 0.7 mm) for which all tracers were at the surface with a small standard deviation, (2) the 4.3 ratio (for 3 mm tracers) for which tracers were rather everywhere (surface and inside), and (3) the large ratios (5.9, 8.6, 10.4, 10.7, 15, 21.4, 44) (resp.",
"for tracer diameter 3, 3, 0.7, 7.5, 3, 7.5, 3 mm) for which tracers were found inside, with a small layer free of tracers near the surface.",
"Decrease in the mean position of tracers and in their standard deviation was observed when increasing the size ratio (for these $d_t/d\\geqslant 4.3$ ).",
"The three patterns agree with the three types of tracer mean depth found in the simulations for several or for a single tracer (Fig.",
"REF ).",
"The upper limit of the transition between surface and reverse segregations is experimentally found for 4.3.",
"As this transition numerically occurs between 4.2 and 4.3 for one tracer and between 4.2 to 4.7 for 5% of tracers, the agreement is very good.",
"But experimental standard deviations are large, not only for ratio 4.3, but for all larger size ratios, which is not observed in simulations at high size ratios.",
"Nevertheless, standard deviations decrease with the size ratio in both experiments and simulations.",
"Three experiments were done with a wall to stop the flow at various distances from the start (30, 60 and 90 cm).",
"Tracers were 3 mm, and small beads were 300-400 $\\mu $ m (size ratio 8.6).",
"Tracers were found inside the deposit, with no major differences in the segregation pattern.",
"But the mean depth of tracers in a cross-section taken at the same distance from the end wall (for example at 10 cm), showed a slight decrease passing from the 30 cm to the 60 cm, and to the 90 cm long experiment.",
"The convergence to a final mean position was still developing at the time when the flow stopped.",
"We conclude that all our experimental data, established for a 90 cm traveling distance, do not concern a perfect stationary state.",
"This convergence distance is compatible with the simulations, where a stationary state is not reached at $2500d$ (equivalent to 90 cm) for flows of $37d$ (equivalent to 1.3 cm) (Fig.",
"REF ) or of $100d$ (equivalent to 3.5 cm) (Fig.",
"REF ).",
"The fact that experimental standard deviations are larger than numerical ones can be explained by this non-fully converged state.",
"It can also be explained by the use of 10% of tracers instead of 5%.",
"The decreases in the experimental mean depth and standard deviation with increasing size ratios are compatible with a better convergence towards the reverse position.",
"This better convergence is compatible with the faster migration of one single tracer when increasing the size ratio (comparing Figs.",
"REF (b) and REF , or Fig.",
"REF ).",
"This also explains why reverse segregation is experimentally nearly reached for the ratio 44, despite a quite short channel.",
"Figure: Cross-section of the deposit in a 1 m long and 6 cm widechute flow experiment.",
"The flow is composed of glass beads: 90% of300-400 μ\\mu m and 10% of 3 mm.In the simulations (Figs.",
"REF (a) and REF ), for traveling lengths corresponding to experiments (90 cm=2500$d$ ), tracers above the first layer do not organize in superimposed layers like those obtained at the end of the simulations.",
"For such a short flowing distance, the first bottom layer is well defined and the second layer is in formation.",
"The other layers are still emerging and have not reached their final depth yet.",
"Indeed, in experiments, most sections presented tracers organized in one bottom layer (Fig.",
"REF ), and sometimes in a blurred second layer.",
"Experimental measurements of this bottom layer depth were done at a distance between 15 and 30 cm from the end wall.",
"We tried to avoid perturbations due to the collision with the end wall and the possible local variations of the tracer fraction near the flow front.",
"For size ratios from 4.3 to 8.6, the bottom layer depth was experimentally measured between 10 and $11.5d$ with randomness (and $13d$ for size ratio 15, above our simulation range).",
"These values are close to the 9 to $10.2d$ values numerically found for size ratios from 5 to 12 (Fig.",
"REF ).",
"Even though the stationary stage has not been reached in our experiments, experimental data reproduce well the existence of the bottom layer and its depth, the reversal between surface and reverse segregations, the variation of the associated standard deviations, and the exact size ratio ($d_t/d=4.3$ ) for which the reversal occurs.",
"Moreover, the numerical study has shown that experimental tracers settled inside ($d\\geqslant 5.9$ ) correspond to non-converged states of reverse segregation at different degrees of convergence.",
"Simulation and experiment results both show that the size ratio 4.3 induces intermediate segregation of the tracers, for which the spread of the experimental positions is maximum (in addition, it is experimentally obtained for a non-fully converged system).",
"The result is a nearly homogeneous mixture.",
"This experimental spread of tracers all through the deposit is compatible with the numerical results obtained for a converged state: a quite large standard deviation combined with a mean position at mid-height.",
"But longer times of convergence, and effects coming from the increase in the tracer fraction up to 10% might also be involved in the experimental process for explaining the tracer spread.",
"For that reason, the range of size ratios around 4.3 needs further investigations.",
"Nevertheless, combining an intermediate segregation and an appropriate tracer fraction could be a means to prevent any segregation during a flow."
],
[
"Conclusion",
"In 3D granular flows, the selection of an equilibrium depth of a large tracer depends mainly on the size ratio between the tracer and the small beads, and to a lesser extent on the nature of the flow.",
"Comparison between depths of single tracers and mean depths of several tracers (3 to 10%) shows that the stabilization of one tracer and the segregation process select identical equilibrium depths.",
"In that case, the segregation is the regrouping of non-interacting large tracers at the same equilibrium depth.",
"In a tumbler, a precise study of trajectories has shown that the depth is established during the flowing phase and is recorded in the static rotating part.",
"The flow substratum is a loose granular material whose boundary with the flowing layer is difficult to define, but trajectories of the largest size ratios seem to place tracers at the bottom of the flow.",
"Thus surface positions, intermediate positions with deeper and deeper depths toward reverse positions are observed when increasing the size ratio between the tracer and the small particles.",
"The transition between surface and reverse depths is progressive, and a large range of tracer size ratios is found to be at intermediate depths.",
"For all 3D flows down a rough incline, the reversal also happens when increasing the size ratio.",
"Two cases are clear: positions near the surface, which corresponds to tracers at the surface, or to non visible tracers, just under the surface (surface and intermediate depths), and positions of tracers floating at or very close to the bottom (intermediate and reverse depths).",
"The existence of intermediate positions near half-height depends on the flow thickness.",
"For thick flows, the reversal between surface and bottom positions is sharp, with no tracers stabilized around mid-height.",
"For thin flows, the reversal is progressive and tracers stabilize at every intermediate depth inside the flow.",
"We conclude that, in 3D, the tracer position is also determined by the type of substratum (solid or loose) and the flow thickness.",
"For multiple tracer flows on 3D incline (5 to 10%), the three segregation patterns (surface, intermediate, reverse) are observed when increasing the size ratio, corresponding to the three types of depth stabilization of a single tracer.",
"The transition is smoother and happens at the same size ratio (around 4.3) for simulations and experiments, corresponding to single tracer reversal.",
"But reverse segregation is long to establish, and a large spread in the positions remains over a long traveling distance.",
"During this travel, the flows (with size ratio above 5) can be considered as nearly homogeneous, except near their surface where only small particles are present.",
"The intermediate case (size ratio 4.3) remains almost homogeneous.",
"The choice of reverse and especially intermediate positions could be an opportunity to maintain a homogeneous mixture for usual industrial transfers.",
"Further studies are needed to set the precise parameter range where these processes can be used.",
"The case of 2D flows has been studied in tumblers and inclines.",
"For small size ratios, the position of tracers relative to the free surface behaves similarly in 2D and 3D, although deeper positions are found for large size ratios in 3D.",
"The dependency of the stabilized depth on the size ratio is similar but weaker in 2D both in tumbler and on incline.",
"Moreover for the largest size ratio tracers on 2D inclines, the reverse positions do not exist.",
"Tracers stabilize at intermediate positions near mid-height and their position scales with the flow thickness contrarily to the 3D case.",
"In 2D tumblers, the equilibrium position evolution with size ratio is also weaker, with a shifted maximum, and leads to an intermediate radial position of equilibrium for the largest tracers.",
"However, the position of these largest tracers does correspond to a reverse depth comparable to that of the 3D case, but this depth is obtained for larger size ratios than in 3D.",
"The difference between 2D and 3D, probably due to a granular packing compacity difference, does not emerge in all processes in the same manner.",
"The highest care should be taken before extrapolating results of studies between 2D and 3D cases.",
"The authors thank F. Schwander, F. Smith, J. Favier and D. Martinand for carefully rereading this manuscript.",
"This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program “Investissements d'Avenir\" supervised by the Agence Nationale de la Recherche."
]
] | 1612.05415 |
[
[
"Partition-free approach to open quantum systems in harmonic\n environments: an exact stochastic Liouville equation"
],
[
"Abstract We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment.",
"The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the Extended Stochastic Liouville-von Neumann equation.",
"Our approach generalises previous work based on Caldeira-Leggett models and a partitioned initial density matrix.",
"This provides a simple, yet exact, closed-form description for the evolution of open systems from equilibriated initial conditions.",
"The applicability of this model and the potential for numerical implementations are also discussed."
],
[
"Introduction",
"Much of the work in the canon of physics has been derived under an assumption of isolation, where the system of interest has no interaction with its environment.",
"Often, particularly in the classical regime, this approximation has been successful in generating accurate predictions.",
"There are however numerous systems whose behaviour cannot be explained by their actions in a vacuum [1].",
"In these cases stochastic terms are used, often as an a priori part of the model (and without proper justification), to capture the effect of the environment.",
"Brownian motion is the most famous case of this technique in classical physics, but quantum physics and its applications have many examples where a similarly careful treatment of external effects is required [2], [3], [4].",
"These systems can collectively be termed open dissipative quantum systems, and the problem of how to most accurately model them remains an active field of research.",
"Approaches to these systems can be split into two broad categories.",
"The first method uses the paradigmatic example of a damped system, where the damping is an effective loss-mechanism that approximates the environment's effect and fluctuations are neglected.",
"A typical example of this is the early work of Kerner and Stevens on sets of damped harmonic oscillators [5], [6].",
"The basis of this method in classical, phenomological equations means that it is capable of providing exact solutions for some simple systems, such as the damped harmonic oscillator.",
"These solutions are however undermined by being fundamentally incompatible with quantum mechanics.",
"This can be illustrated by the fact that there are no time-independent Hamiltonians that can replicate the equation of motion for a damped oscillator, $m\\ddot{x}+\\alpha \\dot{x}+m\\omega ^{2}x=0$ which has frequency $\\omega $ and friction $\\alpha $ .",
"While there exists a time-dependent Hamiltonian that leads to this equation of motion [7], after quantisation the fundamental commutation relation becomes time-dependent [8].",
"This unphysical result means that another approach to dissipative systems, to be detailed below, is the method of choice.",
"In this approach, pioneered by Callen, Welton, Senitzky and Lax, dissipative systems are modelled as a primary system (the “open system”) of interest coupled to an explicit secondary system (the “environment” or “heat bath”) which together describe the overall system being modeled (the “total system”) [9], [10], [8].",
"In comparison to the first method, this model is lossless when considering the total system, and incorporates both the dissipation and fluctuations experienced by the open system as a consequence of its explicit coupling to the environment.",
"Combining this model with appropriate approximations (e.g.",
"weak coupling between the open system and environment) allows quantum master equations to be derived, which retain the correct behaviour in the classical limit [11], [12], [13], [14], [15].",
"The general scheme then is to treat the coupled systems as a single closed sytem which can be straightforwardly quantised.",
"The environmental coordinates can then be eliminated in order to obtain an equation of motion for the primary system.",
"In practice the functional form of the environment (secondary system) and its coupling must be chosen subject to several conditions.",
"For example in the high-temperature classical limit we expect to recover a classical Brownian motion.",
"In addition, if the summation over environmental coordinates is to be exact, yet analytically tractable, the choice of environment is largely restricted to a set of harmonic oscillators, with a bilinear coupling to the open system.",
"A particularly popular model is the Caldeira-Leggett (CL) Hamiltonian [16]: $H=H_{q}(q)+\\frac{1}{2}\\sum _{i}\\left(m_{i}\\dot{x}_{i}^{2}+m_{i}\\omega _{i}^{2}x_{i}^{2}\\right)-q\\sum _{i}c_{i}x_{i}+\\frac{q^{2}}{2}\\sum _{i}\\frac{c_{i}^{2}}{m_{i}\\omega _{i}^{2}}$ This model couples the open system (described by the coordinate $q)$ to an environment of independent harmonic oscillators (masses $m_{i}$ , frequencies $\\omega _{i}$ , and displacement coordinates $x_{i}$ ) with each oscillator being coupled to the open system with a strength $c_{i}$ .",
"The final term is a counter-term included to enforce translational invariance on the system and eliminate quasi-static effects [17].",
"Recently, a more general Hamiltonian of the combined system (the open system and harmonic environment) was introduced [18] which is only linear with respect to the environmental variables, but remains arbitrary with respect to the positions of atoms in the open system (this model is detailed in section ).",
"In this Hamiltonian interactions within the environment are not diagonalised.",
"This is convenient because all parameters of the environment and its interaction with the open system can then be extracted by expanding the Hamiltonian of the combined system in atomic displacements in the bath and keeping only harmonic terms, i.e.",
"the open system can be considered as a part of the expansion of the total system.",
"This rather general choice of total system Hamiltonian enables one to derive classical equations of motion [in the form of the Generalised Langevin Equation (GLE)] for the atoms in the open system [18] and propose an efficient numerical scheme for solving them [19], [20], [21].",
"This method has been recently generalised to the fully quantum case [22] where it was shown, using the method based on directly solving the Liouville equation, that equations of motion for the observable positions of atoms in the open system have the GLE form with friction memory and non-Gaussian random force terms.",
"Although this method enables one to develop the general structure of the equations to be expected for the open system, this method lacks an exact mechanism for establishing the necessary expressions for the random force correlation functions.",
"In the study of quantum Brownian motion, the path integral representation has been perhaps the most fruitful.",
"Some specific successful applications include tunnelling and decay rate calculations (Kramer's problem) [23], [24], [4], [25], [26], [3] as well as recent first-principle derivations for the rate of processes in instanton theory [27], [28].",
"In particular the Feynman-Vernon influence functional formalism [29] can be used to exactly calculate the effect of the environment on the open system using path integrals.",
"Approximations such as weak coupling between the primary system and environment are no longer necessary.",
"Path integrals also remove the need for an explicit quantisation of the system Hamiltonian, as in this formalism quantum-mechanical propagators are represented as phase-weighted sums over trajectories, where the phase associated to each trajectory is proportional to the action of that path in the classical system [30].",
"A useful consequence of this is that the classical limit is easily obtained [31], and the quantisation of the system is automatic when choosing this representation.",
"Finally, and probably most importantly, bath degrees of freedom can be integrated out exactly if the environment is harmonic and interacts with the open system via an expression that is at most up to the second order in its displacements.",
"The key simplification of the Feynman-Vernon approach is that initially the density matrix of the total system $\\hat{\\rho }_{0}^{\\mbox{tot}}$ can be partitioned, $\\hat{\\rho }_{0}^{\\mbox{tot}}=\\hat{\\rho }_{0}\\otimes \\hat{\\rho }_{0}^{X}$ i.e.",
"it can be expressed as a direct product of the initial density matrices of the open system $\\hat{\\rho }_{0}$ and the environment $\\hat{\\rho }_{0}^{X}$ , where each subsystem has equilibriated separately.",
"In the context of open, dissipative quantum systems, much work has been done using this formalism, expanding the methodology of the Feynman-Vernon influence functional for both exact and approximate results [32], [33], [34].",
"Using this model, quantum Langevin equations for the reduced density matrix have been rigorously derived using path integrals [16], [35], [36], [37], [38], [39].",
"In special cases, further analytical results have also been obtained by Kleinert [40], [41] and Tsusaka [42].",
"Generalisations of these results to anharmonic baths produce approximate but more realistic models [43], [44], while time-dependent heat exchange can also be exactly included [45].",
"Parallel to this is the work of Stockburger, exactly deriving a stochastic Liouville-von Neumann (SLN) equation, and applying it to two-level systems [46].",
"Approaches based on influence functionals have also found use in the real time numerical simulations of dissipative systems [47], [48], [49], [50], [51], [52], [53].",
"With this corpus of techniques, path integrals (and specifically influence functionals) represent a powerful and flexible formalism that can be used to attack the problem of open quantum systems.",
"So far, we have been discussing methods based on initially partitioning the total system.",
"The initial condition of Eq.",
"(REF ) is however unphysical, as it is impossible in a real experiment to “prepare” a quantum system with the interaction between the open system and environment switched off, prior to any perturbation being applied.",
"As a result, the transient behaviour we predict for perturbations away from a partitioned initial condition will always be spurious due to the artificial equilibriation of each system seperately.",
"If we wish to extract the exact transient dynamics of an open system we must therefore use a more realistic, non-partitioned initial condition.",
"Fortunately, the influence functional formalism has the capacity to naturally generalise the initial conditions of the overall system and environment, rendering the assumption of a partitioned initial state unnecessary.",
"This possibility was first noted by Smith and Caldeira [32], before being properly explored by Grabert, Ingold and Schramm [54], who derived the time dependent expression for the reduced density matrix of an open system where all path integrals associated with the environment are fully eliminated.",
"In this partition-free case, the limits on our ability to describe the reduced dynamics via a Liouville operator have been derived by Karrlein and Grabert [55].",
"In this work however, no differential equation for the reduced density matrix was derived, and the authors still used a simplified CL Hamiltonian.",
"We also note that a differential equation for the equilibrium reduced density matrix for the CL Hamiltonian was obtained using path integrals in Ref.",
"[56] and is consistent with our results.",
"In this paper, we derive, using the path integral formalism, a set of stochastic differential equations for the reduced density matrix of an open system which describe its dynamics exactly.",
"The derived equation does not have the GLE form obtained previously in Ref.",
"[22].",
"Indeed, it does not have a clearly defined friction term and the stochastic fields it contains are Gaussian.",
"Nevertheless, our Hamiltonian is identical to the one used in Ref.",
"[22], which is more general than the CL Hamiltonian.",
"Using it, we obtain a system of first order stochastic differential equations over real and imaginary time that exactly describe the evolution of the state of a dissipative quantum system for partition-free initial conditions.",
"These equations, which we term the Extended Stochastic Liouville Equation (ESLN), represent both a synthesis and extension of the work outlined above, allowing for a simple and exact closed form description of an arbitrary open system evolving from realistic initial conditions.",
"The derivation of the ESLN, (and therefore the paper itself) will be organised as follows: Section details the model employed, and the class of applicable initial conditions.",
"In section the path integral representation for the density matrix of the primary system will be introduced, along with the influence functional and its explicit evaluation.",
"In section the two-time Hubbard-Stratonovich transformation is applied to the influence functional found in the previous section, introducing the corresponding complex Gaussian stochastic fields.",
"Section presents the path integral describing the reduced density matrix of the primary system and the operator ESLN equations of motion that it implies, which represents the central result of this work.",
"These equations account for both the generalised Hamiltonian and partition-free initial conditions.",
"Finally, section concludes the paper with a discussion of the ESLN, its connection to previous results and the potential for numerical implementations."
],
[
"Model",
"Consider a many-body phonon system of the type shown in Figure REF .",
"It consists of a general central system (the open system), described by coordinates $q$ , acting under an arbitrary Hamiltonian $H_{q}\\left(q\\right)$ .",
"The secondary system (the environment) is composed of $M$ harmonic oscillators (with masses $m_{i}$ ) coupled both internally and with the open system.",
"The open system may be subjected to time-dependent external fields.",
"The environment uses displacement coordinates $\\xi _{i}$ and the interaction between the two systems is linear in $\\xi \\equiv \\left\\lbrace \\xi _{i}\\right\\rbrace $ but arbitrary in $q$ : $H_{\\textrm {tot}}(q,\\xi )=H_{q}(q)+\\frac{1}{2}\\sum _{i=1}^{M}m_{i}\\dot{\\xi }_{i}^{2}+\\frac{1}{2}\\sum _{i,j=1}^{M}_{ij}\\xi _{i}\\xi _{j}-\\sum _{i}^{M}f_{i}(q)\\xi _{i}$ This Hamiltonian differs from the standard CL Hamiltonian in Eq.",
"(REF ) in two important respects.",
"First, the interaction between the primary and secondary systems is no longer strictly bilinear, but can depend arbitrarily on $q$ .",
"In addition, the atomic displacements that form the environment are now coupled to each other as well as the system, with the coupling described by the force-constant matrix $_{ij}$ .",
"These alterations will have a material effect on our results.",
"We also note the counter-term found in Eq.",
"(REF ) has been dropped as it is no longer needed, since when the Hamiltonian of an arbitrary combined system is expanded in the power series in terms of atomic displacements $\\xi _{i}$ , this kind of term does not appear.",
"In this sense our model Hamiltonian is the second-order expansion of any conceivable system-bath Hamiltonian.",
"The density matrix evolves in the usual manner according to the Liouville equation: $\\hat{\\rho }^{\\textrm {tot}}(t)=\\widehat{U}(t;t_{0})\\hat{\\rho }^{\\textrm {tot}}(t_{0})\\widehat{U}^{\\dagger }(t;t_{0})$ where $\\widehat{U}(t;t_{0})=\\exp \\left[-\\frac{i}{\\hbar }\\int _{t_{0}}^{t}\\mathrm {d}t^{\\prime }\\ \\widehat{H}_{\\textrm {tot}}(t^{\\prime })\\right]$ is the corresponding evolution operator.",
"Importantly we need not assume that the system Hamiltonian $H_{q}(q)$ is time-independent.",
"i.e.",
"$H_{q}\\left(q\\right)\\equiv H_{q}\\left(q,t\\right)$ .",
"The dynamics of the open system are found by tracing the full density matrix over the $\\xi $ coordinates: $\\hat{\\rho }(t)=\\textrm {Tr}_{\\xi }\\left[\\hat{\\rho }^{\\textrm {tot}}(t)\\right]$ while the total and reduced density matrices in coordinate space are, respectively: $\\rho _{t}^{\\textrm {tot}}\\left(q,\\xi ;q^{\\prime },\\xi ^{\\prime }\\right)=\\left\\langle q,\\xi \\right|\\hat{\\rho }^{\\textrm {tot}}(t)\\left|q^{\\prime },\\xi ^{\\prime }\\right\\rangle $ $\\rho _{t}(q,q^{\\prime })=\\left\\langle q\\right|\\hat{\\rho }(t)\\left|q^{\\prime }\\right\\rangle $ The propagators in this space are given by: $U\\left(q,\\xi ,t;\\bar{q},\\bar{\\xi },t_{0}\\right)=\\left\\langle q,\\xi \\left|\\widehat{U}\\left(t;t_{0}\\right)\\right|\\bar{q},\\bar{\\xi }\\right\\rangle $ $\\left\\langle \\bar{q},\\bar{\\xi }\\left|\\widehat{U}^{\\dagger }\\left(t;t_{0}\\right)\\right|q,\\xi \\right\\rangle =\\left\\langle \\bar{q},\\bar{\\xi }\\left|\\widehat{U}\\left(t_{0};t\\right)\\right|q,\\xi \\right\\rangle =U\\left(\\bar{q},\\bar{\\xi },t_{0};q,\\xi ,t\\right)$ The second equality has been constructed to demonstrate that in coordinates, $U^{\\dagger }$ has the form of a backward propagation in time.",
"Setting $t_{0}=0$ for convenience, the open system density matrix in the coordinate representation is: $\\rho _{t}(q,q^{\\prime })=\\int \\mathrm {d}\\overline{\\xi }\\ \\mathrm {d}\\overline{\\xi }^{\\prime }\\ \\mathrm {d}\\bar{q}\\ \\mathrm {d}\\bar{q}^{\\prime }\\ \\mathrm {d}\\xi \\mathrm {\\ d}\\xi ^{\\prime }\\ \\delta \\left(\\xi -\\xi ^{\\prime }\\right)U(q,\\xi ,t;\\bar{q},\\bar{\\xi },0)\\rho _{0}^{\\textrm {tot}}(\\bar{q,}\\bar{\\xi };\\bar{q}^{\\prime },\\bar{\\xi }^{\\prime })U(\\bar{q}^{\\prime },\\overline{\\xi }^{\\prime },0;q^{\\prime },\\xi ^{\\prime },t)$ At this point we transform to a normal mode representation $\\xi \\rightarrow x=\\left\\lbrace x_{\\lambda }\\right\\rbrace $ , where $x_{\\lambda }=\\sum _{i}^{M}\\sqrt{m_{i}}e_{\\lambda i}\\xi _{i}\\ \\ ,\\ \\ \\ \\xi _{i}=\\sum _{\\lambda }^{M}\\frac{1}{\\sqrt{m_{i}}}e_{i\\lambda }x_{\\lambda }$ and $e_{\\lambda }=\\left\\lbrace e_{\\lambda i}\\right\\rbrace $ are eigenvectors of the dynamical matrix $D=\\left\\lbrace D_{ij}\\right\\rbrace $ , where $D_{ij}=\\Lambda _{ij}/\\sqrt{m_{i}m_{j}}$ , with eigenvalues $\\omega _{\\lambda }^{2}$ .",
"The eigenvectors satisfy the usual orthogonality, $e_{\\lambda }^{T}e_{\\lambda ^{\\prime }}=\\delta _{\\lambda \\lambda ^{\\prime }}$ , and completeness, $\\sum _{\\lambda }e_{\\lambda }e_{\\lambda }^{T}=1$ , conditions (the superscript $T$ stands for transpose).",
"Applying these transformations, the Hamiltonian can be expressed as: $H_{\\textrm {tot}}(q,x)=H_{q}(q)+\\frac{1}{2}\\sum _{\\lambda =1}^{M}\\left(\\dot{x}_{\\lambda }^{2}+\\omega _{\\lambda }^{2}x_{\\lambda }^{2}\\right)-\\sum _{\\lambda }g_{\\lambda }(q)x_{\\lambda }$ where $g_{\\lambda }(q)=\\sum _{i}^{M}\\frac{1}{\\sqrt{m_{i}}}e_{\\lambda i}f_{i}(q)\\;,\\quad f_{i}(q)=\\sqrt{m_{i}}\\sum _{\\lambda }^{M}e_{i\\lambda }g_{\\lambda }(q)$ The reduced density matrix is now given by: $\\rho _{t}(q,q^{\\prime })=\\int \\mathrm {d}\\bar{x}\\mathrm {d}\\bar{x}^{\\prime }\\mathrm {d}x\\ \\mathrm {d}\\bar{q}\\mathrm {d}\\bar{q}^{\\prime }\\,U\\left(q,x,t;\\bar{q},\\bar{x},0\\right)\\rho _{0}^{\\textrm {tot}}\\left(\\bar{q,}\\bar{x};\\bar{q}^{\\prime },\\bar{x}^{\\prime }\\right)U\\left(\\bar{q}^{\\prime },\\bar{x}^{\\prime },0;q^{\\prime },x,t\\right)$ Before Eq.",
"(REF ) can be solved, we must specify the form of the initial density matrix $\\rho _{0}^{\\textrm {tot}}$ .",
"As was explained in the Introduction, in most systems of interest the interaction between the primary system and its environment is an integral part of the system and hence one cannot assume the two systems are initially partitioned.",
"One solution employed by Grabert et al.",
"[54] is to consider the full interacting system as being allowed to equilibrate with some time-independent Hamiltonian $H_{0}$ before applying any time-dependent perturbation.",
"In this case the initial state would then be described by the canonical density matrix: $\\hat{\\rho }_{0}^{\\textrm {tot}}\\equiv \\hat{\\rho }_{\\beta }=\\frac{1}{Z_{\\beta }}\\mathrm {e}^{-\\beta H_{0}}$ where $\\beta =1/k_{B}T$ is the inverse temperature and $Z_{\\beta }=\\mbox{Tr}\\left(e^{-\\beta H_{0}}\\right)$ is the corresponding partition function of the entire system.",
"Note that a class of more general initial density matrices can be considered [54], however, here we shall limit ourselves only to the canonical density matrix.",
"Having specified the initial conditions, the goal is now to derive an equation of motion that will describe the exact evolution of the reduced density matrix $\\rho _{t}\\left(q,q^{\\prime }\\right)$ as given by Eq.",
"(REF ).",
"To do this we will utilise the influence functional to eliminate the environmental degrees of freedom in Eq.",
"(REF )."
],
[
"The Path Integral Representation and Influence Functional ",
"To proceed we will insert the path integral representation of both propagators and the initial density matrix into Eq.",
"(REF ).",
"The expression for the forward propagator $U\\left(q,x,t_{f};\\bar{q},\\bar{x},0\\right)$ as a path integral up to a time $t_{f}$ is given by $U\\left(q,x,t_{f};\\bar{q},\\bar{x},0\\right)=\\int _{q\\left(0\\right)=\\bar{q}}^{q\\left(t_{f}\\right)=q}\\mathcal {D}q\\left(t\\right)\\int _{x\\left(0\\right)=\\bar{x}}^{x\\left(t_{f}\\right)=x}\\mathcal {D}x\\left(t\\right)\\,\\exp \\left(\\frac{i}{\\hbar }S\\left[q\\left(t\\right),x\\left(t\\right)\\right]\\right)$ with a similar definition for the backward propagator $U\\left(\\bar{q}^{\\prime },\\bar{x}^{\\prime },0;q^{\\prime },x,t_{f}\\right)=\\int _{q^{\\prime }\\left(t_{f}\\right)=q^{\\prime }}^{q^{\\prime }\\left(0\\right)=\\overline{q}^{\\prime }}\\mathcal {D}q^{\\prime }\\left(t\\right)\\int _{x^{\\prime }\\left(t_{f}\\right)=x}^{x^{\\prime }\\left(0\\right)=\\bar{x}^{\\prime }}\\mathcal {D}x^{\\prime }\\left(t\\right)\\,\\exp \\left(-\\frac{i}{\\hbar }S\\left[q^{\\prime }\\left(t\\right),x^{\\prime }\\left(t\\right)\\right]\\right)$ The limits of the path integral in the second propagator are reversed as compared to the first one to emphasize its backward nature, as in Eq.",
"(REF ).",
"In both expressions the integration is performed with respect to both the open system ($q,q^{\\prime }$ ) and environment ($x,x^{\\prime }$ ) variables between the boundaries indicated.",
"Here $S$ is the action corresponding to the Hamiltonian in Eq.",
"(REF ) describing the total system.",
"It is defined in both propagators in the usual manner (i.e.",
"the time integral of the Langrangian from 0 to $t_{f}$ ), hence the extra negative in the exponent of the backwards propagator.",
"Integration over the environmental variables can be performed exactly as the environment and interaction Hamiltonians added together have the form of a set of displaced harmonic oscillators in the environment variables.",
"This means the path integral over environmental trajectories is Gaussian, and can be evaluated (see, e.g., [30], [29], [54]).",
"The propagator therefore becomes a path integral over the trajectories of the open system only: $U\\left(q,x,t_{f};\\bar{q},\\bar{x},0\\right)=A\\int _{q\\left(0\\right)=\\bar{q}}^{q\\left(t_{f}\\right)=q}\\mathcal {D}q\\left(t\\right)\\,\\exp \\left(\\frac{i}{\\hbar }S_{\\textrm {tot }}\\left[q\\left(t\\right);x,\\overline{x};t_{f}\\right]\\right)$ Here $A$ is a fluctuating factor that corresponds to a closed loop path integral: $A=\\prod _{\\lambda }A_{\\lambda }=\\prod _{\\lambda }\\sqrt{\\frac{\\omega _{\\lambda }}{2\\pi i\\hbar \\sin \\left(\\omega _{\\lambda }t_{f}\\right)}}$ while the action $S_{\\textrm {tot}}$ is the composition of the action of two systems, which is functionally dependent only on $q\\left(t\\right)$ .",
"Explicitly: $S_{\\textrm {tot }}\\left[q\\left(t\\right);x,\\overline{x};t\\right]=S_{q}\\left[q\\left(t\\right)\\right]+S_{x}\\left[q\\left(t\\right);x,\\overline{x};t_{f}\\right]$ where $S_{q}$ is the open system action $S_{q}\\left[q\\left(t\\right)\\right]=\\int _{0}^{t_{f}}\\textrm {d}t\\ \\,L_{q}\\left[q\\left(t\\right)\\right]=\\int _{0}^{t_{f}}\\textrm {d}t\\ \\,\\left[\\frac{1}{2}m\\dot{q}^{2}\\left(t\\right)-V\\left(q\\left(t\\right)\\right)\\right]$ and $S_{x}$ is the classical action of a set of displaced harmonic oscillators for an external “force” given by $g\\left(q\\left(t\\right)\\right)$ .",
"This has no functional dependence on the $x$ coordinates; $S_{x}$ only depends on the limits of the path integral over the environment: $S_{x}\\left[q(t);x,\\overline{x};t_{f}\\right]=\\sum _{\\lambda }\\left\\lbrace \\frac{\\omega _{\\lambda }}{\\sin \\left(\\omega _{\\lambda }t_{f}\\right)}\\left[\\frac{1}{2}\\left(x_{\\lambda }^{2}+\\bar{x}_{\\lambda }^{2}\\right)\\cos \\left(\\omega _{\\lambda }t_{f}\\right)-x_{\\lambda }\\bar{x}_{\\lambda }\\right.\\right.$ $+\\frac{x_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{t_{f}}\\textrm {d}t\\,g_{\\lambda }\\left(t\\right)\\sin \\left(\\omega _{\\lambda }t\\right)+\\frac{\\bar{x}_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{t_{f}}\\textrm {d}t\\,g_{\\lambda }\\left(t\\right)\\sin \\left(\\omega _{\\lambda }\\left(t_{f}-t\\right)\\right)$ $\\left.\\left.-\\frac{1}{\\omega _{\\lambda }^{2}}\\int _{0}^{t_{f}}\\int _{0}^{t}\\textrm {d}t\\textrm {d}t^{\\prime }\\,g_{\\lambda }\\left(t\\right)g_{\\lambda }\\left(t^{\\prime }\\right)\\sin \\left(\\omega _{\\lambda }\\left(t_{f}-t\\right)\\right)\\sin \\left(\\omega _{\\lambda }t^{\\prime }\\right)\\right]\\right\\rbrace $ In the final equation above, we have abbreviated by setting $g\\left(q\\left(t\\right)\\right)=g\\left(t\\right)\\equiv \\left\\lbrace g_{\\lambda }(t)\\right\\rbrace $ , in addition to the limits $x\\left(t_{f}\\right)=x\\equiv \\left\\lbrace x_{\\lambda }\\right\\rbrace $ and $x\\left(0\\right)=\\bar{x}\\equiv \\left\\lbrace \\overline{x}_{\\lambda }\\right\\rbrace $ .",
"The backward propagator has a similar expression as compared to the forward propagator in Eq.",
"(REF ): $U\\left(\\bar{q}^{\\prime },\\bar{x}^{\\prime },0;q^{\\prime },x,t_{f}\\right)=A^{*}\\int _{q^{\\prime }\\left(t_{f}\\right)=q^{\\prime }}^{q^{\\prime }\\left(0\\right)=\\overline{q}^{\\prime }}\\mathcal {D}q^{\\prime }\\left(t\\right)\\,\\exp \\left(-\\frac{i}{\\hbar }S_{\\textrm {tot }}\\left[q^{\\prime }\\left(t\\right);x,\\overline{x}^{\\prime };t_{f}\\right]\\right)$ with the same expression (REF ) for the action, but using the substitution $\\overline{x}\\rightarrow \\overline{x}^{\\prime }$ .",
"The abbreviation $g\\left(q^{\\prime }\\left(t\\right)\\right)=g^{\\prime }\\left(t\\right)\\equiv \\left\\lbrace g_{\\lambda }^{\\prime }(t)\\right\\rbrace $ will also be used when referring to the backward propagator.",
"As well as the propagators, the initial density matrix may also be expressed as a path integral over both the open system and environmental coordinates.",
"After performing the same integration over the environment as for the propagators, we obtain: $\\rho _{\\beta }\\left(\\bar{q},\\bar{x};\\bar{q}^{\\prime },\\bar{x}^{\\prime }\\right)=\\frac{A^{E}}{Z_{\\beta }}\\int _{\\bar{q}(0)=\\bar{q}^{\\prime }}^{\\bar{q}(\\hbar \\beta )=\\bar{q}}\\mathcal {D}\\bar{q}\\left(\\tau \\right)\\,\\exp \\left(-\\frac{1}{\\hbar }S_{\\textrm {tot}}^{E}\\left[\\bar{q}(\\tau );\\bar{x},\\bar{x}^{\\prime };\\hbar \\beta \\right]\\right)$ $A^{E}=\\prod _{\\lambda }A_{\\lambda }^{E}=\\prod _{\\lambda }\\sqrt{\\frac{\\omega _{\\lambda }}{2\\pi \\hbar \\sinh \\left(\\omega _{\\lambda }\\hbar \\beta \\right)}}$ Here $Z_{\\beta }$ is the partition function for the total system, while $S_{\\textrm {tot}}^{E}$ is the Euclidean action, defined as the Wick rotation of $S_{\\textrm {tot }}\\left[\\bar{q}(\\tau );\\bar{x},\\bar{x}^{\\prime };\\hbar \\beta \\right]$ .",
"Using the notation $g(\\bar{q}(t))=\\bar{g}(t)\\equiv \\left\\lbrace \\overline{g}_{\\lambda }(t)\\right\\rbrace $ , $\\bar{x}(\\hbar \\beta )=\\bar{x}\\equiv \\left\\lbrace \\overline{x}_{\\lambda }\\right\\rbrace $ and $\\bar{x}(0)=\\bar{x}^{\\prime }\\equiv \\left\\lbrace \\overline{x}_{\\lambda }^{\\prime }\\right\\rbrace $ , we obtain a familiar (albeit Wick rotated) definition for $S_{\\textrm {tot}}^{E}$ (see, e.g., [54]): $S_{\\textrm {tot}}^{E}\\left[\\bar{q}(\\tau );\\bar{x},\\bar{x}^{\\prime };\\hbar \\beta \\right]=S_{q}^{E}\\left[\\bar{q}(\\tau )\\right]+S_{x}^{E}\\left[\\bar{q}(\\tau );\\bar{x},\\overline{x}^{\\prime };\\hbar \\beta \\right]$ where the system and bath contributions are given as follows: $S_{q}^{E}\\left[\\bar{q}(\\tau )\\right]=\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\,L_{q}^{E}\\left[\\bar{q}(\\tau )\\right]=\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\,\\left[\\frac{1}{2}m\\dot{\\bar{q}}^{2}(\\tau )+V(\\bar{q}(\\tau ))\\right]$ and $S_{x}^{E}\\left[\\bar{q}(\\tau );\\bar{x},\\overline{x}^{\\prime };\\hbar \\beta \\right]=\\sum _{\\lambda }\\left\\lbrace \\left[\\frac{\\omega _{\\lambda }}{\\sinh \\left(\\omega _{\\lambda }\\hbar \\beta \\right)}\\bigg [\\frac{1}{2}\\left(\\bar{x}_{\\lambda }^{2}+\\bar{x}_{\\lambda }^{\\prime 2}\\right)\\cosh \\left(\\omega _{\\lambda }\\hbar \\beta \\right)-\\bar{x}_{\\lambda }\\bar{x}_{\\lambda }^{\\prime }\\right.\\right.$ $-\\frac{\\bar{x}_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\,\\ \\bar{g}_{\\lambda }\\left(\\tau \\right)\\sinh \\left(\\omega _{\\lambda }\\tau \\right)-\\frac{\\bar{x}_{\\lambda }^{\\prime }}{\\omega _{\\lambda }}\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\,\\ \\bar{g}_{\\lambda }\\left(\\tau \\right)\\sinh \\left(\\omega _{\\lambda }\\left(\\hbar \\beta -\\tau \\right)\\right)$ $\\left.\\left.-\\frac{1}{\\omega _{\\lambda }^{2}}\\int _{0}^{\\hbar \\beta }\\int _{0}^{\\tau }\\textrm {d}\\tau \\textrm {d}\\tau ^{\\prime }\\ \\,\\bar{g}_{\\lambda }\\left(\\tau \\right)\\bar{g}_{\\lambda }\\left(\\tau ^{\\prime }\\right)\\sinh \\left(\\omega _{\\lambda }\\left(\\hbar \\beta -\\tau \\right)\\right)\\sin \\left(\\omega _{\\lambda }\\tau ^{\\prime }\\right)\\right]\\right\\rbrace $ Following Ref.",
"[54], we now also define a new partition function $Z=Z_{\\beta }/Z_{B}$ in terms of the partition functions of the total system $Z_{\\beta }$ and the (isolated) environment $Z_{B}=\\prod _{\\lambda }\\frac{1}{2\\sinh \\left(\\frac{1}{2}\\omega _{\\lambda }\\hbar \\beta \\right)}$ After substituting the path integral and partition function expressions into Eq.",
"(REF ), we obtain an expression for the reduced density matrix after integrating over the environmental trajectories: $\\rho _{t_{f}}\\left(q;q^{\\prime }\\right)=\\frac{1}{Z}\\int \\mathrm {d}\\bar{q}\\mathrm {d}\\bar{q}^{\\prime }\\mathcal {\\mathcal {D}}q\\left(t\\right)\\mathcal {D}\\bar{q}\\left(\\tau \\right)\\mathcal {D}q^{\\prime }\\left(t\\right)\\,\\mathcal {F}\\left[q\\left(t\\right),q^{\\prime }\\left(t\\right),\\bar{q}\\left(\\tau \\right)\\right]$ $\\times \\exp \\left[\\frac{i}{\\hbar }\\int _{0}^{t_{f}}\\mathrm {d}t\\,L_{q}\\left[q\\left(t\\right)\\right]-\\frac{i}{\\hbar }\\int _{0}^{t_{f}}\\mathrm {d}t\\,L_{q}\\left[q^{\\prime }\\left(t\\right)\\right]-\\frac{1}{\\hbar }\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\,L_{q}^{E}\\left(\\bar{q}\\left(\\tau \\right)\\right)\\right]$ The limits of the path integrals here are the same as above.",
"The normalising constant $Z$ in the equilibrium density operator is not generally known, and this issue will be discussed in Section .",
"The influence functional $\\mathcal {F}\\left[q,q^{\\prime },\\bar{q}\\right]$ contains the full path integral over the environment.",
"It is fully factorised over the normal modes $\\lambda $ , and for each mode is composed of a product of three terms: $\\mathcal {F}\\left[q\\left(t\\right),q^{\\prime }\\left(t\\right),\\bar{q}\\left(\\tau \\right)\\right]=\\frac{1}{Z_{B}}\\prod _{\\lambda }\\int \\mathrm {d}x_{\\lambda }\\mathrm {d}\\bar{x}_{\\lambda }\\mathrm {d}\\bar{x}_{\\lambda }^{\\prime }\\ F_{\\lambda }\\left[q_{\\lambda }\\left(t\\right),x_{\\lambda },\\bar{x}_{\\lambda }\\right]F_{\\lambda }^{E}\\left[\\bar{q}_{\\lambda }\\left(\\tau \\right),\\bar{x}_{\\lambda },\\bar{x}_{\\lambda }^{\\prime }\\right]F_{\\lambda }^{*}\\left[q_{\\lambda }^{\\prime }\\left(t\\right),x_{\\lambda },\\bar{x}_{\\lambda }^{\\prime }\\right]$ where $F_{\\lambda }\\left[q_{\\lambda }\\left(t\\right),x_{\\lambda },\\bar{x}_{\\lambda }\\right]=A_{\\lambda }\\exp \\left\\lbrace \\frac{i\\omega _{\\lambda }}{\\hbar \\sin \\left(\\omega _{\\lambda }t_{f}\\right)}\\left[\\frac{1}{2}\\left(x_{\\lambda }^{2}+\\bar{x}_{\\lambda }^{2}\\right)\\cos \\left(\\omega _{\\lambda }t_{f}\\right)-x_{\\lambda }\\bar{x}_{\\lambda }\\right.\\right.$ $+\\frac{x_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{t_{f}}\\textrm {d}t\\,\\ g_{\\lambda }\\left(t\\right)\\sin \\left(\\omega _{\\lambda }t\\right)+\\frac{\\bar{x}_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{t_{f}}\\textrm {d}t\\,\\ g_{\\lambda }\\left(t\\right)\\sin \\left(\\omega _{\\lambda }\\left(t_{f}-t\\right)\\right)$ $\\left.\\left.-\\frac{1}{\\omega _{\\lambda }{}^{2}}\\int _{0}^{t_{f}}\\int _{0}^{t}\\textrm {d}t\\textrm {d}t^{\\prime }\\,\\ g_{\\lambda }\\left(t\\right)g_{\\lambda }\\left(t^{\\prime }\\right)\\sin \\left(\\omega _{\\lambda }\\left(t_{f}-t\\right)\\right)\\sin \\left(\\omega _{\\lambda }t^{\\prime }\\right)\\right]\\right\\rbrace $ $F_{\\lambda }^{*}\\left[q_{\\lambda }^{\\prime }\\left(t\\right),x_{\\lambda },\\bar{x}_{\\lambda }^{\\prime }\\right]=A_{\\lambda }^{*}\\exp \\left\\lbrace -\\frac{i\\omega _{\\lambda }}{\\hbar \\sin \\left(\\omega _{\\lambda }t_{f}\\right)}\\left[\\frac{1}{2}\\left(x_{\\lambda }^{2}+\\bar{x}{}_{\\lambda }^{\\prime 2}\\right)\\cos \\left(\\omega _{\\lambda }t_{f}\\right)-x_{\\lambda }\\bar{x}{}_{\\lambda }^{\\prime }\\right.\\right.$ $+\\frac{x_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{t_{f}}\\textrm {d}t\\,\\ g_{\\lambda }^{\\prime }\\left(t\\right)\\sin \\left(\\omega _{\\lambda }t\\right)+\\frac{\\bar{x}{}_{\\lambda }^{\\prime }}{\\omega _{\\lambda }}\\int _{0}^{t_{f}}\\textrm {d}t\\,\\ g_{\\lambda }^{\\prime }\\left(t\\right)\\sin \\left(\\omega _{\\lambda }\\left(t_{f}-t\\right)\\right)$ $\\left.\\left.-\\frac{1}{\\omega _{\\lambda }{}^{2}}\\int _{0}^{t_{f}}\\int _{0}^{t}\\textrm {d}t\\textrm {d}t^{\\prime }\\,\\ g_{\\lambda }^{\\prime }\\left(t\\right)g_{\\lambda }^{\\prime }\\left(t^{\\prime }\\right)\\sin \\left(\\omega _{\\lambda }\\left(t_{f}-t\\right)\\right)\\sin \\left(\\omega _{\\lambda }t^{\\prime }\\right)\\right]\\right\\rbrace $ $F_{\\lambda }^{E}\\left[\\bar{q}_{\\lambda }\\left(\\tau \\right),\\bar{x}_{\\lambda },\\bar{x}_{\\lambda }^{\\prime }\\right]=A_{\\lambda }^{E}\\exp \\left\\lbrace -\\frac{\\omega _{\\lambda }}{\\hbar \\sinh \\left(\\omega _{\\lambda }\\hbar \\beta \\right)}\\left[\\frac{1}{2}\\left(\\bar{x}_{\\lambda }^{2}+\\bar{x}{}_{\\lambda }^{\\prime 2}\\right)\\cosh \\left(\\omega _{\\lambda }\\hbar \\beta \\right)-\\bar{x}_{\\lambda }\\bar{x}_{\\lambda }^{\\prime }\\right.\\right.$ $-\\frac{\\bar{x}_{\\lambda }}{\\omega _{\\lambda }}\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\,\\ \\bar{g}_{\\lambda }\\left(\\tau \\right)\\sinh \\left(\\omega _{\\lambda }\\tau \\right)-\\frac{\\bar{x}_{\\lambda }^{\\prime }}{\\omega _{\\lambda }}\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\,\\ \\bar{g}_{\\lambda }\\left(\\tau \\right)\\sinh \\left(\\omega _{\\lambda }\\left(\\hbar \\beta -\\tau \\right)\\right)$ $\\left.\\left.-\\frac{1}{\\omega _{\\lambda }^{2}}\\int _{0}^{\\hbar \\beta }\\int _{0}^{\\tau }\\textrm {d}\\tau \\textrm {d}\\tau ^{\\prime }\\,\\ \\bar{g}_{\\lambda }\\left(\\tau \\right)\\bar{g}_{\\lambda }\\left(\\tau ^{\\prime }\\right)\\sinh \\left(\\omega _{\\lambda }\\left(\\hbar \\beta -\\tau \\right)\\right)\\sin \\left(\\omega _{\\lambda }\\tau ^{\\prime }\\right)\\right]\\right\\rbrace $ In order to calculate the influence functional, we notice that the calculation can be performed for each mode $\\lambda $ separately.",
"Then, the integrand in the triple integral over $x_{\\lambda }$ , $\\overline{x}_{\\lambda }$ and $\\overline{x}_{\\lambda }^{\\prime }$ contains an exponential function with a quadratic polynomial over these variables, and is hence a Gaussian.",
"This can therefore be directly integrated.",
"We first note that all pre-exponential factors in the influence functional after the integration multiply exactly to one.",
"Indeed, the introduction of the partition function of the environment $Z_{B}$ in Eq.",
"(REF ) is to ensure that in the case of no interactions between the system and the environment, the influence functional $\\mathcal {F}\\left[q,q^{\\prime },\\bar{q}\\right]$ is unity.",
"If $P_{\\lambda }$ is the pre-exponential factor appearing after the triple integration over $x_{\\lambda }$ , $\\overline{x}_{\\lambda }$ and $\\overline{x}_{\\lambda }^{\\prime }$ in Eq.",
"(REF ) for one mode, then the overall exponential prefactor $J$ for the influence functional after some simple algebra is one: $J=\\frac{AA^{*}A^{E}}{Z_{B}}\\prod _{\\lambda }P_{\\lambda }=1$ After performing the complete integration of Eq.",
"(REF ), we find the following exponential expression for the influence functional (cf.",
"[29], [54]): $\\mathcal {F}\\left[q,q^{\\prime },\\bar{q}\\right]=\\exp \\left(-\\frac{1}{\\hbar }\\Phi \\left[q,q^{\\prime },\\bar{q}\\right]\\right)\\equiv \\exp \\left(-\\frac{1}{\\hbar }\\sum _{\\lambda }\\Phi _{\\lambda }\\left[q,q^{\\prime },\\bar{q}\\right]\\right)$ where $\\Phi =\\sum _{\\lambda }\\Phi _{\\lambda }$ is the influence phase: $\\Phi _{\\lambda }\\left[q,q^{\\prime },\\bar{q}\\right]=-\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{\\tau }\\mathrm {d}\\tau ^{\\prime }\\,K_{\\lambda }\\left(i\\tau ^{\\prime }-i\\tau \\right)\\bar{g}_{\\lambda }\\left(\\tau \\right)\\bar{g}_{\\lambda }\\left(\\tau ^{\\prime }\\right)-i\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{t_{f}}\\mathrm {d}t\\,K_{\\lambda }\\left(t-i\\tau \\right)\\bar{g}_{\\lambda }\\left(\\tau \\right)\\left(g_{\\lambda }\\left(t\\right)-g_{\\lambda }^{\\prime }\\left(t\\right)\\right)$ $+\\int _{0}^{t_{f}}\\mathrm {d}t\\int _{0}^{t}\\mathrm {d}t^{\\prime }\\,\\left(g_{\\lambda }\\left(t\\right)-g_{\\lambda }^{\\prime }\\left(t\\right)\\right)\\left[K_{\\lambda }\\left(t-t^{\\prime }\\right)g_{\\lambda }\\left(t^{\\prime }\\right)-K_{\\lambda }^{*}\\left(t-t^{\\prime }\\right)g_{\\lambda }^{\\prime }\\left(t^{\\prime }\\right)\\right]$ The term multiplying the various $g_{\\lambda }$ within the integrals is the kernel: $K_{\\lambda }\\left(\\theta \\right)=\\frac{\\cosh \\left(\\omega _{\\lambda }\\left(\\frac{\\hbar \\beta }{2}-i\\theta \\right)\\right)}{2\\omega _{\\lambda }\\sinh \\left(\\frac{1}{2}\\beta \\hbar \\omega _{\\lambda }\\right)}$ Note that the kernel appears in three forms, depending on purely imaginary times, $K_{\\lambda }\\left(i\\tau ^{\\prime }-i\\tau \\right)$ , real times, $K_{\\lambda }\\left(t-t^{\\prime }\\right)$ , and complex times, $K_{\\lambda }\\left(t-i\\tau \\right)$ .",
"It will be useful later in the derivation to split the kernel into its real $K_{\\lambda }^{R}$ and imaginary $K_{\\lambda }^{I}$ parts.",
"For real times this produces, $K_{\\lambda }^{R}\\left(t\\right)=\\frac{1}{2\\omega _{\\lambda }}\\coth \\left(\\frac{1}{2}\\hbar \\beta \\omega _{\\lambda }\\right)\\cos \\left(\\omega _{\\lambda }t\\right)$ $K_{\\lambda }^{I}(t)=-\\frac{1}{2\\omega _{\\lambda }}\\sin \\left(\\omega _{\\lambda }t\\right)$ and for complex times, $K_{\\lambda }^{R}\\left(t-i\\tau \\right)=\\frac{1}{2\\omega _{\\lambda }}\\left[\\coth \\left(\\frac{1}{2}\\omega _{\\lambda }\\hbar \\beta \\right)\\cosh \\left(\\omega _{\\lambda }\\tau \\right)-\\sinh \\left(\\omega _{\\lambda }\\tau \\right)\\right]\\cos \\left(\\omega _{\\lambda } t\\right)$ $K^{I}(t-i\\tau )=-\\frac{1}{2\\omega _{\\lambda }}\\left[\\cosh \\left(\\omega \\tau \\right)+\\sinh \\left(\\omega _{\\lambda }\\tau \\right)\\coth \\left(\\frac{1}{2}\\omega _{\\lambda }\\hbar \\beta \\right)\\right]\\sin \\left(\\omega _{\\lambda } t\\right)$ while for purely imaginary times the kernel is real, $K_{\\lambda }(i\\tau )=K_{\\lambda }^{e}(\\tau )+K_{\\lambda }^{o}(\\tau )$ and consisting of even and odd components: $K_{\\lambda }^{o}(\\tau )=\\frac{1}{2\\omega _{\\lambda }}\\sinh \\left(\\omega _{\\lambda }\\tau \\right)$ $K_{\\lambda }^{e}(\\tau )=\\frac{1}{2\\omega _{\\lambda }}\\cosh \\left(\\omega _{\\lambda }\\tau \\right)\\coth \\left(\\frac{1}{2}\\omega _{\\lambda }\\hbar \\beta \\right)$ If for real times we also define new sum and difference interaction functions [40], $\\epsilon _{\\lambda }\\left(t\\right)=g_{\\lambda }\\left(t\\right)-g_{\\lambda }^{\\prime }\\left(t\\right)\\quad \\mbox{and}\\quad y_{\\lambda }(t)=\\frac{1}{2}\\left(g_{\\lambda }\\left(t\\right)+g_{\\lambda }^{\\prime }\\left(t\\right)\\right)$ and substitute these expressions into Eq.",
"(REF ), the single mode influence phase can now be expressed as: $\\Phi _{\\lambda }\\left[q,q^{\\prime },\\bar{q}\\right]=-\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau ^{\\prime }\\,\\frac{1}{2}\\left[K_{\\lambda }^{e}\\left(\\tau ^{\\prime }-\\tau \\right)-K_{\\lambda }^{o}\\left(\\left|\\tau ^{\\prime }-\\tau \\right|\\right)\\right]\\bar{g}_{\\lambda }\\left(\\tau \\right)\\bar{g}_{\\lambda }\\left(\\tau ^{\\prime }\\right)$ $-i\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{t_{f}}\\mathrm {d}t\\,\\left[K_{\\lambda }^{R}\\left(t-i\\tau \\right)+K_{\\lambda }^{I}\\left(t-i\\tau \\right)\\right]\\bar{g}_{\\lambda }\\left(\\tau \\right)\\epsilon _{\\lambda }\\left(t\\right)$ $+\\frac{1}{2}\\int _{0}^{t_{f}}\\mathrm {d}t\\int _{0}^{t_{f}}\\mathrm {d}t^{\\prime }\\,K_{\\lambda }^{R}\\left(t-t^{\\prime }\\right)\\epsilon _{\\lambda }\\left(t\\right)\\epsilon _{\\lambda }\\left(t^{\\prime }\\right)+2i\\int _{0}^{t_{f}}\\mathrm {d}t\\int _{0}^{t_{f}}\\mathrm {d}t^{\\prime }\\,\\left[\\theta \\left(t-t^{\\prime }\\right)K_{\\lambda }^{I}\\left(t-t^{\\prime }\\right)\\right]\\epsilon _{\\lambda }\\left(t\\right)y_{\\lambda }\\left(t^{\\prime }\\right)$ The final two terms in this expression are a generalisation of the well known Feynman-Vernon influence functional [29], with the remaining terms arising from the incorporation of a non-partitioned initial density matrix.",
"Note that, compared to Eq.",
"(REF ), the above expression was modified to ensure identical limits in the double integrals over the times $t,t^{\\prime }$ and $\\tau ,\\tau ^{\\prime }$ .",
"The influence phase still contains the normal mode interaction term $g_{\\lambda }$ .",
"Using Eq.",
"(REF ), we can re-express the phase in terms of the original interaction given in the site representation.",
"The normal mode transformation did not change the $q$ coordinates themselves, so there is no difference between representations in the path integral measure or action $S_{q}$ in Eq.",
"(REF ).",
"The system-bath interaction term contained in the influence functional will have a different form however, and hence the influence phase has a non-trivial alternative representation in terms of functions $f_{i}(t)\\equiv f_{i}\\left(q\\left(t\\right)\\right)$ rather than $g_{\\lambda }\\left(q\\left(t\\right)\\right)$ .",
"In this representation the sum and difference functions $v_{i}\\left(t\\right)=f_{i}\\left(t\\right)-f_{i}^{\\prime }\\left(t\\right)\\quad \\mbox{and}\\quad r_{i}\\left(t\\right)=\\frac{1}{2}\\left(f_{i}\\left(t\\right)+f{}_{i}^{\\prime }\\left(t\\right)\\right)$ can conveniently be introduced, using $f_{i}^{\\prime }(t)\\equiv f_{i}\\left(q^{\\prime }\\left(t\\right)\\right)$ .",
"Substituting Eq.",
"(REF ) into these, we can relate the sum and difference functions (REF ) between the normal mode and site representations: $v_{i}\\left(t\\right)=\\frac{1}{\\sqrt{m_{i}}}\\sum _{\\lambda }e_{\\lambda i}\\epsilon _{\\lambda }\\left(t\\right)\\ \\ \\mbox{and}\\ \\ r_{i}\\left(t\\right)=\\frac{1}{\\sqrt{m_{i}}}\\sum _{\\lambda }e_{\\lambda i}y_{\\lambda }\\left(t\\right)$ The influence phase in the site representation is most easily expresed by defining new kernels from those derived using normal modes $L_{ij}^{R,I}\\left(t\\right)=\\frac{1}{\\sqrt{m_{i}m_{j}}}\\sum _{\\lambda }e_{\\lambda i}e_{\\lambda j}K_{\\lambda }^{R,I}\\left(t\\right)$ $L_{ij}\\left(t-i\\tau \\right)=\\frac{1}{\\sqrt{m_{i}m_{j}}}\\sum _{\\lambda }e_{\\lambda i}e_{\\lambda j}K_{\\lambda }\\left(t-i\\tau \\right)$ $L_{ij}^{e}\\left(\\tau \\right)=\\frac{1}{\\sqrt{m_{i}m_{j}}}\\sum _{\\lambda }\\frac{e_{\\lambda i}e_{\\lambda j}}{2\\omega _{\\lambda }}\\coth \\left(\\frac{1}{2}\\hbar \\beta \\omega _{\\lambda }\\right)\\cosh \\left(\\omega _{\\lambda }\\tau \\right)$ $L_{ij}^{o}\\left(\\tau \\right)=\\frac{1}{\\sqrt{m_{i}m_{j}}}\\sum _{\\lambda }\\frac{e_{\\lambda i}e_{\\lambda j}}{2\\omega _{\\lambda }}\\sinh \\left(\\omega _{\\lambda }\\tau \\right)$ so that the influence phase can be re-expressed in terms of the site interactions: $\\Phi \\left[q,q^{\\prime },\\bar{q}\\right]=\\sum _{ij}\\Phi _{ij}\\left[q,q^{\\prime },\\bar{q}\\right]$ $\\Phi _{ij}\\left[q,q^{\\prime },\\bar{q}\\right]=-\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau ^{\\prime }\\,\\frac{1}{2}\\bar{f}_{i}(\\tau )\\left[L_{ij}^{e}\\left(\\tau ^{\\prime }-\\tau \\right)-L_{ij}^{o}\\left(\\left|\\tau ^{\\prime }-\\tau \\right|\\right)\\right]\\bar{f}_{j}\\left(\\tau ^{\\prime }\\right)$ $-i\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{t_{f}}\\mathrm {d}t\\,v_{i}\\left(t\\right)L_{ij}\\left(t-i\\tau \\right)\\bar{f}_{j}\\left(\\tau \\right)$ $+\\frac{1}{2}\\int _{0}^{t_{f}}\\mathrm {d}t\\int _{0}^{t_{f}}\\mathrm {d}t^{\\prime }\\,v_{i}\\left(t\\right)L_{ij}^{R}\\left(t-t^{\\prime }\\right)v_{j}\\left(t^{\\prime }\\right)+2i\\int _{0}^{t_{f}}\\mathrm {d}t\\int _{0}^{t_{f}}\\mathrm {d}t^{\\prime }\\,v_{i}\\left(t\\right)\\left[\\theta \\left(t-t^{\\prime }\\right)L_{ij}^{I}\\left(t-t^{\\prime }\\right)\\right]r_{j}\\left(t^{\\prime }\\right)$ where an obvious short-hand notation $f(\\bar{q}(\\tau ))\\equiv \\bar{f}_{i}(\\tau )$ has also been introduced.",
"The influence phase expressed here contains additional complexity compared to one derived using a standard CL model (which does not require a normal mode transformation) [54].",
"After allowing the environment to contain internal couplings, we find that the effect of this generalisation on the form of the influence phase is not trivial: instead of a single sum over the bath lattice in the CL model, we have double sums in Eq.",
"(REF ), and this will have a profound effect on the dimensionality of the stochastic field to be introduced below.",
"In principle, having found the influence phase, Eq.",
"(REF ) can be used to describe the exact dynamics of the open system at all times.",
"Path integrals are however awkward to evaluate outside of certain special cases.",
"The goal now is to use Eq.",
"(REF ) to derive an operator expression, and hence a Liouville-von Neumann type equation for the reduced density matrix instead.",
"Unfortunately the influence phase contains double integrals in two time variables ($t$ and $\\tau $ ), meaning there is no simple method to construct a differential equation directly out of Eq.",
"(REF ).",
"Here we will follow previous work [13], [40], [42], [46], and use a transformation to convert this non-local system into a local one exactly, at the cost of introducing stochastic variables."
],
[
"The Two-Time Hubbard-Stratonovich transformation ",
"In order to progress, we will use a statistical technique known as the Hubbard-Stratonovich (HS) transformation [57].",
"We shall consider the most general form of such a transformation based on a complex multivariate Gaussian distribution (cf.",
"[46]).",
"Consider a Gaussian distribution over $N$ complex random variables (“noises”), $z^{1}\\equiv \\left\\lbrace \\eta _{i}\\right\\rbrace $ , and their $N$ complex conjugates, $z^{2}\\equiv \\left\\lbrace \\eta _{i}^{*}\\right\\rbrace $ : $W\\left[\\eta _{1},\\eta _{1}^{*},...,\\eta _{N},\\eta _{N}^{*}\\right]=\\frac{\\left(2\\pi \\right)^{-N}}{\\sqrt{\\det }}\\,\\exp \\left[-\\frac{1}{2}z^{T}\\Sigma z\\right]$ where $z^{\\alpha }=\\left(\\begin{array}{c}z_{1}^{\\alpha }\\\\z_{2}^{\\alpha }\\\\\\vdots \\\\z_{N}^{\\alpha }\\end{array}\\right)$ is the vector of complex variables ($\\alpha =1$ ) or their conjugate ($\\alpha =2$ ).",
"The total vector $z$ is therefore of size $2N$ and is given by: $z & =\\left(\\begin{array}{c}z^{1}\\\\z^{2}\\end{array}\\right)$ The covariance matrix $\\Sigma $ can also be decomposed into a block form $\\equiv \\left(\\Sigma _{ij}^{\\alpha \\beta }\\right)=\\left(\\begin{array}{cc}\\Sigma ^{11} & \\Sigma ^{12}\\\\\\Sigma ^{21} & \\Sigma ^{22}\\end{array}\\right)$ and the correlation functions are given by the usual Gaussian identity: $\\left\\langle z_{i}^{\\alpha }z_{j}^{\\beta }\\right\\rangle _{z}=\\left(\\Sigma ^{-1}\\right)_{ij}^{\\alpha \\beta }$ The Fourier transform of this distribution is the complementary distribution which can be calculated exactly: $\\kappa \\left[k\\right] & =\\int \\mathrm {d}z\\ W\\left(z\\right)\\,\\exp \\left(iz^{T}k\\right)=\\exp \\left(-\\frac{1}{2}k^{T}\\Sigma ^{-1}k\\right)$ where $k$ is a $2N$ -fold vector, consisting of two size $N$ vectors $k^{1}$ and $k^{2}$ .",
"This equation can be interpreted as an average (with respect to the Gaussian distribution $W$ ) of the exponential function, $\\left\\langle \\exp \\left(iz^{T}k\\right)\\right\\rangle _{z}$ .",
"Using the distribution $W$ , one can also calculate the correlation function between any two stochastic variables.",
"Hence, the elements of the inverse matrix $\\Sigma ^{-1}$ appearing in Eq.",
"(REF ) can be written via the correlation functions.",
"The HS transformation is essentially the relation between these two representations of the complementary distribution: $\\left\\langle \\exp \\left(iz^{T}k\\right)\\right\\rangle _{z}\\equiv \\left\\langle \\exp \\left(i\\sum _{i\\alpha }z_{i}^{\\alpha }k_{i}^{\\alpha }\\right)\\right\\rangle _{z}=\\exp \\left(-\\frac{1}{2}\\sum _{ij\\alpha \\beta }k_{i}^{\\alpha }\\left\\langle z_{i}^{\\alpha }z_{j}^{\\beta }\\right\\rangle _{z}k_{j}^{\\beta }\\right)$ So far, we have considered a finite set of discrete stochastic variables $\\left\\lbrace \\eta _{i},\\eta _{i}^{*}\\right\\rbrace $ .",
"The preceding derivation can be extended to (continuous) Gaussian stochastic processes if different stochastic variables are now associated with time instances $t_{k}$ separated by some small time interval $\\Delta $ , i.e.",
"$z_{i}^{\\alpha }\\rightarrow z_{i}^{\\alpha }\\left(t_{k}\\right)$ .",
"Here $t_{k}=k\\Delta $ with $k$ running from 0 to $n$ , so that $n\\Delta =t_{f}$ .",
"Now in the limit of $\\Delta \\rightarrow 0$ , we obtain the HS transformation for a set of continuous Gaussian stochastic processes as follows: $\\left\\langle \\exp \\left[i\\sum _{i\\alpha }\\int _{0}^{t_{f}}\\mathrm {d}t\\ \\,z_{i}^{\\alpha }\\left(t\\right)k_{i}^{\\alpha }\\left(t\\right)\\right]\\right\\rangle _{z(t)}=\\exp \\left[-\\frac{1}{2}\\sum _{ij\\alpha \\beta }\\int _{0}^{t_{f}}\\textrm {d}t\\int _{0}^{t_{f}}\\textrm {d}t^{\\prime }\\ k_{i}^{\\alpha }\\left(t\\right)\\left\\langle z_{i}^{\\alpha }\\left(t\\right)z_{j}^{\\beta }\\left(t^{\\prime }\\right)\\right\\rangle _{z(t)}k_{j}^{\\beta }\\left(t^{\\prime }\\right)\\right]$ Note that integration over the noises $z(t)=\\lbrace z_{i}^{\\alpha }(t)\\rbrace ,$ appearing in both sides of the above equation, becomes the corresponding path integral in the continuum limit.",
"Using the HS transformation defined above, clear progress can be made.",
"Indeed, the exponent in the right hand side of Eq.",
"(REF ) is of the same form as the Feynman-Vernon terms of the influence phase in Eq.",
"(REF ).",
"The correlation functions and $k$ variables in Eq.",
"(REF ) can be mapped to the terms appearing in the integrands of the Feynman-Vernon influence phase.",
"The HS transformation can therefore be used to equate a deterministic non-local integral exponent to a local phase involving auxiliary stochastic terms, that must be averaged over the distribution $W$ .",
"In a more physical sense, we can also consider the HS transformation as converting a system of two body potentials into a set of independent particles in a fluctuating field.",
"The difficulty using this transformation is that Eq.",
"(REF ) contains two time dimensions - one real and one imaginary, with one term involving an integration over both dimensions - requiring a generalisation of the transformation.",
"When we consider how the HS transformation is derived, continuous processes and multiple variables are incorporated through the addition of extra indices, partitioning the arbitrary sum of random complex variables.",
"The same procedure can be applied to introduce different time dimensions.",
"Starting from a discrete representation, we introduce two sets of times, $\\left\\lbrace t_{k},k=0,\\ldots ,M\\right\\rbrace $ and $\\left\\lbrace \\tau _{k},k=0,\\ldots ,M^{\\prime }\\right\\rbrace $ , so that the exponent on the left hand side of the HS transformation (REF ) has the form $z^{T}k\\:\\Rightarrow \\:\\sum _{\\alpha }\\left(\\sum _{ik}z_{i}^{\\alpha }\\left(t_{k}\\right)k_{i}^{\\alpha }\\left(t_{k}\\right)+\\sum _{ik}\\overline{z}_{i}^{\\alpha }\\left(\\tau _{k}\\right)\\overline{k}_{i}^{\\alpha }\\left(\\tau _{k}\\right)\\right)$ where we assign $t_{M}=t_{f}$ and $\\tau _{M^{\\prime }}=\\hbar \\beta $ , and we place a bar above quantities associated with the second set of times (denoted with the real time $\\tau _{k}$ ).",
"Note that the number of stochastic variables in each set (as counted by the index $i$ for the given time index $k$ ) may be different for barred and unbarred fields.",
"In the continuum limit $M,M^{\\prime }\\rightarrow \\infty $ we obtain for the left hand side of the HS transformation: $\\kappa \\left[k\\left(t\\right),\\bar{k}\\left(t\\right)\\right]=\\left\\langle \\exp \\left[i\\sum _{\\alpha i}\\int _{0}^{t_{f}}\\mathrm {d}t\\,z_{i}^{\\alpha }\\left(t\\right)k_{i}^{\\alpha }\\left(t\\right)+i\\sum _{\\alpha i}\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\,\\bar{z}_{i}^{\\alpha }\\left(\\tau \\right)\\bar{k}_{i}^{\\alpha }\\left(\\tau \\right)\\right]\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }$ Correspondingly, the exponent on the right hand side of Eq.",
"(REF ) (after the time labels are introduced), in the continuous limit becomes: $\\kappa \\left[k\\left(t\\right),\\bar{k}\\left(\\tau \\right)\\right]=\\exp \\left\\lbrace -\\frac{1}{2}\\sum _{\\alpha \\beta ij}\\left(\\int _{0}^{t_{f}}\\textrm {d}t\\int _{0}^{t_{f}}\\textrm {d}t^{\\prime }\\,k_{i}^{\\alpha }\\left(t\\right)^{T}A_{ij}^{\\alpha \\beta }\\left(t,t^{\\prime }\\right)k_{j}^{\\beta }\\left(t^{\\prime }\\right)\\right.\\right.$ $\\left.\\left.+\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau ^{\\prime }\\,\\bar{k}_{i}^{\\alpha }\\left(\\tau \\right)^{T}A_{ij}^{\\alpha \\beta }\\left(\\tau ,\\tau ^{\\prime }\\right)\\bar{k}_{j}^{\\beta }\\left(\\tau ^{\\prime }\\right)+2\\int _{0}^{t_{f}}\\textrm {d}t\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\,k_{i}^{\\alpha }\\left(t\\right)^{T}A_{ij}^{\\alpha \\beta }\\left(t,\\tau \\right)\\bar{k}_{j}^{\\beta }\\left(\\tau \\right)\\right)\\right\\rbrace $ where, because of the three possible combinations of times, we introduce three types of correlation functions: $A_{ij}^{\\alpha \\beta }\\left(t,t^{\\prime }\\right)=\\left\\langle z_{i}^{\\alpha }\\left(t\\right)z_{j}^{\\beta }\\left(t^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }$ $A_{ij}^{\\alpha \\beta }\\left(\\tau ,\\tau ^{\\prime }\\right)=\\left\\langle \\bar{z}_{i}^{\\alpha }\\left(\\tau \\right)\\bar{z}_{j}^{\\beta }\\left(\\tau ^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }$ $A_{ij}^{\\alpha \\beta }\\left(t,\\tau \\right)=\\left\\langle z_{i}^{\\alpha }\\left(t\\right)\\bar{z}_{j}^{\\beta }\\left(\\tau \\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }$ In the full multivariate form, the two-time transformation is therefore given by: $\\left\\langle \\exp \\left[i\\sum _{i\\alpha }\\left(\\int _{0}^{t_{f}}\\textrm {d}t\\ z_{i}^{\\alpha }\\left(t\\right)k_{i}^{\\alpha }\\left(t\\right)+\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\ \\bar{z_{i}}^{\\alpha }\\left(\\tau \\right)\\bar{k}_{i}^{\\alpha }\\left(\\tau \\right)\\right)\\right]\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }$ $=\\exp \\left[-\\frac{1}{2}\\sum _{ij\\alpha \\beta }\\left(\\int _{0}^{t_{f}}\\mathrm {\\textrm {d}}t\\int _{0}^{t_{f}}\\textrm {d}t^{\\prime }\\,k_{i}^{\\alpha }\\left(t\\right)A_{ij}^{\\alpha \\beta }\\left(t,t^{\\prime }\\right)k_{j}^{\\beta }\\left(t^{\\prime }\\right)\\right.\\right.$ $+\\left.\\left.\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau ^{\\prime }\\,\\bar{k}_{i}^{\\alpha }\\left(\\tau \\right)A_{ij}^{\\alpha \\beta }\\left(\\tau ,\\tau ^{\\prime }\\right)\\bar{k}_{j}^{\\beta }\\left(\\tau ^{\\prime }\\right)+2\\int _{0}^{t_{f}}\\mbox{d}t\\int _{0}^{\\hbar \\beta }\\textrm {d}\\tau \\,k_{i}^{\\alpha }\\left(t\\right)A_{ij}^{\\alpha \\beta }\\left(t,\\tau \\right)\\bar{k}_{j}^{\\beta }\\left(\\tau \\right)\\right)\\right]$ The connection between the influence phase and the two-time Hubbard-Stratonovich transformation should now be transparent.",
"Notice that here in the exponential all time integrals have either $t_{f}$ or $\\hbar \\beta $ as their upper limits, exactly as in the influence functional expression (REF ) for the phase.",
"The choice for the second time dimension to run up to $\\hbar \\beta $ has been made to highlight the closeness between the influence phase in Eq.",
"(REF ) and the two-time HS transformation presented here.",
"Now we would like to apply the HS transformation to the influence functional expression given by Eqs.",
"(REF ), (REF ) and (REF ).",
"It is clear from the structure of the exponent in the influence functional in Eq.",
"(REF ), that auxiliary stochastic fields should be introduced separately for each lattice site index $i$ .",
"Moreover, there should be two pairs of the stochastic processes for the set associated with the real time $t$ , $z_{i}\\left(t\\right) & \\;\\Rightarrow \\;\\left(\\begin{array}{c}\\eta _{i}\\left(t\\right)\\\\\\eta _{i}^{*}\\left(t\\right)\\\\\\nu _{i}\\left(t\\right)\\\\\\nu _{i}^{*}\\left(t\\right)\\end{array}\\right)$ and one such set for the imaginary time $i\\tau $ : $\\bar{z}_{i}\\left(\\tau \\right)\\;\\Rightarrow \\;\\left(\\begin{array}{c}\\bar{\\mu }_{i}\\left(\\tau \\right)\\\\\\bar{\\mu }_{i}^{*}\\left(\\tau \\right)\\end{array}\\right)$ where we have redefined the size $M$ (number of environmental oscillators) complex vector $z\\equiv \\left\\lbrace z_{i}\\right\\rbrace $ to include two noises and their conjugates.",
"Next, we make the following correspondence between the functions $k_{i}(t)$ in the HS transformation (REF ) and the functions $v_{i}(t)$ , $r_{i}(t)$ and $\\overline{f}_{i}(\\tau )$ appearing in the phase, Eq.",
"(REF ): $k_{i}(t)\\;\\Rightarrow \\;\\left(\\begin{array}{c}v_{i}(t)/\\hbar \\\\0\\\\r_{i}(t)\\\\0\\end{array}\\right)$ and $\\overline{k}_{i}(\\tau )\\;\\Rightarrow \\;\\left(\\begin{array}{c}i\\overline{f}_{i}(\\tau )/\\hbar \\\\0\\end{array}\\right)$ The three pairs of stochastic processes we have introduced must ensure that the influence functional given by Eqs.",
"(REF ), (REF ) and (REF ) coincides exactly with the right hand side of the HS transformation (REF ).",
"Therefore, comparing the exponents in the right hand side of Eq.",
"(REF ) and Eq.",
"(REF ), explicit formulas can be established for the correlation functions $A_{ij}^{\\alpha \\beta }$ between the noises.",
"These are: $\\left\\langle \\eta _{i}\\left(t\\right)\\eta _{j}\\left(t^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }=\\hbar L_{ij}^{R}\\left(t-t^{\\prime }\\right)$ $\\left\\langle \\eta _{i}(t)\\nu _{j}\\left(t^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }=2i\\Theta \\left(t-t^{\\prime }\\right)L_{ij}^{I}\\left(t-t^{\\prime }\\right)$ $\\left\\langle \\eta _{i}\\left(t\\right)\\bar{\\mu }_{j}\\left(\\tau \\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace } & =-\\hbar \\left[L_{ij}^{R}\\left(t-i\\tau \\right)+iL_{ij}^{I}\\left(t-i\\tau \\right)\\right]$ $\\left\\langle \\bar{\\mu }_{i}\\left(\\tau \\right)\\bar{\\mu }_{j}\\left(\\tau ^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }=\\hbar \\left[L_{ij}^{e}\\left(\\tau -\\tau ^{\\prime }\\right)-L_{ij}^{o}\\left(\\left|\\tau -\\tau ^{\\prime }\\right|\\right)\\right]$ $\\left\\langle \\nu _{i}\\left(t\\right)\\nu _{j}\\left(t^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }=\\left\\langle \\nu _{i}\\left(t\\right)\\bar{\\mu }_{j}\\left(\\tau \\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}(\\tau )\\right\\rbrace }=0$ Note that the correlation functions (REF ) and (REF ) are to be symmetric functions with respect to the permutation $i,t\\leftrightarrow j,t^{\\prime }$ and $i,\\tau \\leftrightarrow j,\\tau ^{\\prime }$ , respectively, and the corresponding functions $L_{ij}^{R}$ and $L_{ij}^{o,e}$ provide exactly this.",
"Taking the above results and applying them to Eq.",
"(REF ), we find that the influence functional can be described as an average over multivariate complex Gaussian processes as follows: $\\mathcal {F}\\left[q,q^{\\prime },\\bar{q}\\right]=\\left\\langle \\exp \\left[\\frac{i}{\\hbar }\\sum _{i}\\left(\\int _{0}^{t_{f}}\\mathrm {d}t\\,\\left[\\eta _{i}\\left(t\\right)v_{i}\\left(t\\right)+\\hbar \\nu _{i}\\left(t\\right)r_{i}\\left(t\\right)\\right]+i\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\ \\,\\bar{\\mu }_{i}\\left(\\tau \\right)\\bar{f}_{i}\\left(\\tau \\right)\\right)\\right]\\right\\rangle _{\\left\\lbrace z\\left(t\\right),\\overline{z}\\left(\\tau \\right)\\right\\rbrace }$ where the averaging is made over three pairs of complex noises (or, equivalently, over six real noises) per lattice site of the environment.",
"Importantly, the two-time HS transformation is a purely formal one, and we are free to stipulate that the noises are pure C-numbers; this enables us to avoid the complication of operator-valued noises.",
"Promoting noises to operators has been previously shown to have no effect on the final result, as shown in Ref.",
"[40], [42].",
"Finally it is worth mentioning that the influence phase given above does not uniquely define the Gaussian processes that the influence functional is averaged over after performing the mapping.",
"The influence phase viewed as the right hand side of the HS transformation does not involve every possible correlation defined under the Gaussian distribution.",
"In particular, the conditions we impose on some correlation functions to map the physics to the auxiliary noises do not constrain the correlations between the complex conjugate noises, e.g.",
"$\\left\\langle \\eta _{i}^{*}\\left(t\\right)\\eta _{j}^{*}\\left(t^{\\prime }\\right)\\right\\rangle $ .",
"Therefore any distribution that satisfies Eqs.",
"(REF -REF ) may be used in this transformation."
],
[
"The Extended Stochastic Liouville-von Neumann Equation",
"Now the influence functional $\\mathcal {F}\\left[q,q^{\\prime },\\bar{q}\\right]$ has been evaluated, we are able to write the expression for the reduced density matrix in Eq.",
"(REF ) explicitly.",
"First, having introduced stochastic variables into the equation for the density matrix, we must define a new object $\\tilde{\\rho }_{t}\\left(q;q^{\\prime }\\right)$ to act as an effective, single-trajectory density matrix defined for a particular realisation of the stochastic processes $z(t)$ and $\\bar{z}(\\tau )$ along its path.",
"Inserting Eq.",
"(REF ) into Eqn.",
"(REF ) we obtain: $\\tilde{\\rho }_{t_{f}}\\left(q;q^{\\prime }\\right)=\\frac{1}{Z}\\int \\mathrm {d}\\bar{q}\\mathrm {d}\\bar{q}^{\\prime }\\mathcal {\\mathcal {D}}q(t)\\mathcal {D}\\bar{q}(\\tau )\\mathcal {D}q^{\\prime }(t)\\exp \\left[\\frac{i}{\\hbar }\\tilde{S}^{+}\\left[q\\left(t\\right)\\right]-\\frac{i}{\\hbar }\\tilde{S}^{-}\\left[q^{\\prime }\\left(t\\right)\\right]-\\frac{1}{\\hbar }\\tilde{S}^{E}\\left[\\bar{q}\\left(\\tau \\right)\\right]\\right]$ so that the exact reduced density matrix is recovered as an average over all noises: $\\rho _{t_{f}}\\left(q;q^{\\prime }\\right)=\\left\\langle \\tilde{\\rho }_{t_{f}}\\left(q;q^{\\prime }\\right)\\right\\rangle _{\\left\\lbrace z(t),\\bar{z}\\left(\\tau \\right)\\right\\rbrace }$ Above three effective actions have been introduced: $\\tilde{S}^{+}\\left[q\\left(t\\right)\\right]=\\int _{0}^{t_{f}}\\mathrm {d}t\\,\\left(L_{q}\\left[q\\left(t\\right)\\right]+\\sum _{i}\\left[\\eta _{i}\\left(t\\right)+\\frac{\\hbar }{2}\\nu _{i}\\left(t\\right)\\right]f_{i}\\left(t\\right)\\right)=\\int _{0}^{t_{f}}\\mathrm {d}t\\,L^{+}\\left[q\\left(t\\right)\\right]$ $\\tilde{S}^{-}\\left[q^{\\prime }\\left(t\\right)\\right]=\\int _{0}^{t_{f}}\\mathrm {d}t\\,\\left(L_{q}\\left[q^{\\prime }\\left(t\\right)\\right]+\\sum _{i}\\left[\\eta _{i}\\left(t\\right)-\\frac{\\hbar }{2}\\nu _{i}\\left(t\\right)\\right]f_{i}\\left(t\\right)\\right)=\\int _{0}^{t_{f}}\\mathrm {d}t\\,L^{-}\\left[q^{\\prime }\\left(t\\right)\\right]$ $\\tilde{S}^{E}\\left[\\bar{q}\\left(\\tau \\right)\\right]=\\int _{0}^{\\hbar \\beta }\\mathrm {d}\\tau \\ \\left(L_{q}^{E}\\left[\\bar{q}\\left(\\tau \\right)\\right]+\\ \\bar{\\mu }_{i}\\left(\\tau \\right)\\bar{f}_{i}\\left(\\tau \\right)\\right)$ In the definitions of the effective actions we have reinserted the original forces $f_{i}(t)$ , $f_{i}\\left(t^{\\prime }\\right)$ and $\\overline{f}_{i}(\\tau )$ via Eq.",
"(REF ).",
"It can be seen that the actions $\\tilde{S}^{+}$ and $\\tilde{S}^{-}$ correspond to two different effective Lagrangians, $\\widehat{L}^{\\pm }(t)=\\widehat{L}_{q}\\left(t\\right)+\\sum _{i}\\left[\\eta _{i}\\left(t\\right)\\pm \\frac{\\hbar }{2}\\nu _{i}\\left(t\\right)\\right]\\hat{f}_{i}\\left(t\\right)$ which in turn are associated with two different effective Hamiltonians: $\\widehat{H}^{\\pm }(t)=\\widehat{H}_{q}\\left(t\\right)-\\sum _{i}\\left[\\eta _{i}\\left(t\\right)\\pm \\frac{\\hbar }{2}\\nu _{i}\\left(t\\right)\\right]\\hat{f}_{i}\\left(t\\right)$ As was mentioned in Section , the noises are not promoted to operators but remain as $c$ -numbers.",
"All three path integral coordinates have now been decoupled from each other, and as coordinate functionals may be commuted.",
"The density matrix in Eq.",
"(REF ) can therefore be expressed as: $\\tilde{\\rho }_{t_{f}}(q;q^{\\prime })=\\int \\mathrm {d}\\bar{q}\\mathrm {d}\\bar{q}^{\\prime }\\ U^{+}\\left(q,t_{f};\\bar{q},0\\right)\\tilde{\\rho }_{0}\\left(\\bar{q};\\bar{q}^{\\prime }\\right)U^{-}\\left(\\bar{q}^{\\prime },0;q^{\\prime },t_{f}\\right)\\equiv \\left\\langle q\\right|\\widetilde{\\rho }\\left(t_{f}\\right)\\left|q^{\\prime }\\right\\rangle $ where $U^{+}(q,t_{f};\\bar{q},0)=\\int _{q(0)=\\bar{q}}^{q(t_{f})=q}\\mathcal {\\mathcal {D}}q\\left(t\\right)\\ \\exp \\left[\\frac{i}{\\hbar }\\tilde{S}^{+}\\left[q\\left(t\\right)\\right]\\right]\\equiv \\left\\langle q\\right|\\widehat{U}^{+}\\left(t_{f}\\right)\\left|\\overline{q}\\right\\rangle $ $U^{-}(\\bar{q}^{\\prime },0;q^{\\prime },t_{f})=\\int _{q^{\\prime }(t_{f})=q^{\\prime }}^{q^{\\prime }(0)=\\bar{q}^{\\prime }}\\mathcal {\\mathcal {D}}q^{\\prime }\\left(t\\right)\\ \\exp \\left[-\\frac{i}{\\hbar }\\tilde{S}^{-}\\left[q^{\\prime }\\left(t\\right)\\right]\\right]\\equiv \\left\\langle \\overline{q}^{\\prime }\\right|\\widehat{U}^{-}\\left(t_{f}\\right)\\left|q^{\\prime }\\right\\rangle $ $\\tilde{\\rho }_{0}(\\bar{q};\\bar{q}^{\\prime })=\\frac{1}{Z}\\int _{\\bar{q}(0)=\\bar{q}^{\\prime }}^{\\bar{q}(\\hbar \\beta )=\\bar{q}}\\mathcal {D}\\bar{q}\\left(\\tau \\right)\\exp \\left[-\\frac{1}{\\hbar }\\tilde{S}^{E}\\left[\\bar{q}\\left(\\tau \\right)\\right]\\right]\\equiv \\left\\langle \\bar{q}\\left|\\tilde{\\rho }_{0}\\right|\\bar{q}^{\\prime }\\right\\rangle $ Notice that the forwards propagator is not the Hermitian conjugate of the backwards propagator because of the obvious difference in the their respective Hamiltonians.",
"The consequence of this is that the equation of motion is no longer of the Liouville form, i.e.",
"the time derivative of the density matrix is not solely given by the commutator with some kind of Hamiltonian.",
"Within Eqs.",
"(REF ) and (REF ) we have also introduced the operators $\\widehat{U}^{+}\\left(t_{f}\\right)=\\widehat{T}\\exp \\left(-\\frac{i}{\\hbar }\\int _{0}^{t_{f}}\\widehat{H}^{+}(t)\\mbox{d}t\\right)$ $\\widehat{U}^{-}\\left(t_{f}\\right)=\\widetilde{T}\\exp \\left(\\frac{i}{\\hbar }\\int _{0}^{t_{f}}\\widehat{H}^{-}(t)\\mbox{d}t\\right)$ which correspond to the forward and backward propagation performed with the different Hamiltonians $\\widehat{H}^{+}$ and $\\widehat{H}^{-}$ , respectively, with the corresponding chronological $\\widehat{T}$ and anti-chronological $\\widetilde{T}$ time-ordering operators.",
"It is easy to see that the coordinate representation $\\left\\langle q\\right|\\widehat{U}^{+}\\left(t_{f}\\right)\\left|\\overline{q}\\right\\rangle $ and $\\left\\langle \\overline{q}^{\\prime }\\right|\\widehat{U}^{-}\\left(t_{f}\\right)\\left|q^{\\prime }\\right\\rangle $ of such operators give exactly the paths integrals in these expressions.",
"The propagator operators satisfy the usual equations of motion $i\\hbar \\partial _{t}\\widehat{U}^{+}(t)=\\widehat{H}^{+}(t)\\widehat{U}^{+}(t)$ $i\\hbar \\partial _{t}\\widehat{U}^{-}(t)=-\\widehat{U}^{-}(t)\\widehat{H}^{-}(t)$ Taking Eqs.",
"(REF )-(REF ), the reduced single-trajectory density matrix $\\widetilde{\\rho }\\left(t_{f}\\right)$ of the open system can be written as an operator evolution: $\\widetilde{\\rho }(t)=\\widehat{U}^{+}(t)\\tilde{\\rho }_{0}\\widehat{U}^{-}(t)$ With these definitions it is possible to generate an equation of motion for a single-trajectory reduced density matrix by simply differentiating the above expression with respect to time: $i\\hbar \\partial _{t}\\widetilde{\\rho }\\left(t\\right)=\\widehat{H}^{+}\\left(t\\right)\\tilde{\\rho }\\left(t\\right)-\\tilde{\\rho }\\left(t\\right)\\widehat{H}^{-}\\left(t\\right)$ $=\\left[\\widehat{H}_{q}\\left(t\\right),\\tilde{\\rho }\\left(t\\right)\\right]_{-}-\\sum _{i}\\left(\\eta _{i}\\left(t\\right)\\left[\\hat{f}_{i}\\left(t\\right),\\tilde{\\rho }\\left(t\\right)\\right]_{-}+\\frac{\\hbar }{2}\\nu _{i}\\left(t\\right)\\left[\\hat{f}_{i}\\left(t\\right),\\tilde{\\rho }\\left(t\\right)\\right]_{+}\\right)$ This, together with an equation for $\\widetilde{\\rho }_{0}$ , which provides an initial condition for the reduced density operator $\\widetilde{\\rho }(t)$ , forms the ESLN.",
"It bears a great deal of similarity to the equation derived by Stockburger [46] using the partitioned approach, and while it may be initially surprising to see a similar (albeit generalised) equation of motion, it seems that the partition-free initial density matrix introduced here does not change the dynamics it evolves under.",
"We also note that, as was mentioned above, the obtained equation does not have the usual Liouville form because of an extra anti-commutator term in the right hand side.",
"This originates from the fact that the forward and backward propagations of the reduced density matrix in Eq.",
"(REF ), are governed by different Hamiltonians.",
"We note that the same equation of motion for the reduced density matrix can also be obtained using the method developed by Kleinert and Shabanov in Ref.",
"[40].",
"However, their method requires some care in choosing the correct order of the coordinates and momenta operators.",
"It is a definite advantage of our method that such a problem does not arise.",
"All that remains is to determine the new single-trajectory initial density matrix $\\widetilde{\\rho }_{0}$ .",
"This is the true initial ($t=0$ ) single-trajectory reduced density matrix which is obtained from the canonical density matrix (REF ) by tracing out the degrees of freedom of the bath.",
"There is already a path integral representation for this density, Eq.",
"(REF ), but it is unwieldy and unintuitive.",
"Once again it is best to work backwards to obtain the corresponding effective canonical initial density matrix operator $\\widetilde{\\rho }_{0}$ with the same path integral representation.",
"It is easy to see, however, considering an effective operator Hamiltonian, cf.",
"Eq.",
"(REF ), $\\overline{H}\\left(\\tau \\right)=H_{q}\\left(\\bar{q}\\right)-\\sum _{i}^{M}\\bar{\\mu }_{i}\\left(\\tau \\right)\\bar{f}_{i}\\left(\\tau \\right)$ that the path integral representation of the initial density matrix in Eq.",
"(REF ) is formally identical to the one for the coordinate representation of the evolution operator when time is imaginary and $\\tau $ changes between zero and $\\beta \\hbar $ .",
"Therefore, the initial reduced density operator can be characterised as a propagator through imaginary time: $\\widetilde{\\rho }_{0}\\equiv \\left.\\overline{\\rho }(\\tau )\\right|_{\\tau =\\beta \\hbar }$ using $\\overline{\\rho }(\\tau )=\\frac{1}{Z}\\widehat{\\tau }\\exp \\left[-\\frac{1}{\\hbar }\\int _{0}^{\\tau }\\mathrm {d}\\tau ^{\\prime }\\,\\overline{H}\\left(\\tau ^{\\prime }\\right)\\right]$ This has the form of a time-ordered exponent with $\\widehat{\\tau }$ being the corresponding chronological time-ordering operator.",
"The latter density operator $\\overline{\\rho }(\\tau )$ is responsible for the thermalisation of the open system (when $\\tau \\rightarrow \\beta \\hbar $ ) and will be called the quenched initial density operator.",
"It satisfies the Schrödinger-like equation of motion $-\\hbar \\partial _{\\tau }\\overline{\\rho }(\\tau )=\\overline{H}(\\tau )\\overline{\\rho }(\\tau )$ with the initial condition $\\overline{\\rho }(\\tau =0)=Z^{-1}$ .",
"The initial density $\\overline{\\rho }(\\tau )$ must be normalised when the final value of $\\tau \\equiv \\beta \\hbar $ is reached, i.e.",
"$\\textrm {Tr}_{q}\\left[\\overline{\\rho }(\\beta \\hbar )\\right]=1$ , where the trace is taken with respect to the open system only.",
"Therefore, the correct initial condition for $\\overline{\\rho }(\\tau )$ can be fixed by providing this normalisation at the end of the imaginary time propagation (note that $Z$ , as a ratio of two partition functions, is time independent).",
"We also observe that essentially the same result for the reduced equilibrium density matrix was obtained in Ref.",
"[56].",
"The Hamiltonian $H_{q}$ and the interaction operators in $\\overline{H}(\\tau )$ have no temperature dependence; so the temperature dependence comes entirely from an artificial “propagation” of the quenched density matrix from zero to the “time” $\\tau =\\beta \\hbar $ .",
"This hard limit relating the time to the system temperature is important, as unlike in the real time case, the quenched density matrix may diverge as we take $\\tau \\rightarrow \\infty $ .",
"This is a reflection of the fact that the path integral description of the canonical density matrix is itself only defined for finite temperature.",
"The equations (REF ), (REF ) and (REF ) provide the complete solution for the real time evolution of the reduced density matrix of an open system in our partition-free approach.",
"First of all, the initial density matrix is obtained by propagating in imaginary time $\\tau $ the quenched density $\\overline{\\rho }(\\tau )$ up to the final time $\\tau \\equiv \\beta \\hbar $ (the Euclidean evolution).",
"The initial density is then normalised which fixes the value of the partition function $Z$ .",
"Using the obtained initial density matrix, the actual time dynamics of the reduced density matrix $\\widetilde{\\rho }(t)$ are elucidated by solving Eq.",
"(REF ).",
"Figure REF illustrates the evolution of trajectories through two times, as governed by the two differential equations.",
"First the system evolves through imaginary time according to Eq.",
"(REF ) and some realisation of the imaginary time noise trajectory $\\left\\lbrace \\bar{\\mu }_{i}(\\tau )\\right\\rbrace $ .",
"This state then evolves through real time under Eq.",
"(REF ) using the real time noise trajectories $\\left\\lbrace \\eta _{i}(t)\\right\\rbrace $ and $\\left\\lbrace \\nu _{i}(t)\\right\\rbrace $ , with the requirement that upon averaging over realisations of these trajectories, they satisfy the correlation functions derived in section .",
"The evolution along these two time dimensions is then repeated many times using various realisations of the stochastic noises, and averaging over many trajectories yields the physical reduced density matrix $\\widehat{\\rho }\\left(t\\right)$ appearing in Eq.",
"(REF ).",
"Figure: Representative trajectories for the evolution of the system.",
"Firstthere is an evolution in imaginary time up to τ=βℏ\\tau =\\beta \\hbar ,before evolving in real time from this point up to time t f t_{f}.Different colours correspond to different simulations associated withparticular manifestations of the noises.",
"The average of the finalpoints gives the physical density matrix at that time (indicated attime t f t_{f})."
],
[
"Discussion and conclusions ",
"Having derived the ESLN, we should ask how it differs from previous work.",
"The Hamiltonian we have used is a generalisation of the Caldeira-Leggett model, allowing for a solution in either real or frequency space.",
"The form of the interaction has also been generalised, but is still limited by the essential need for an interaction to be linear in environmental oscillator displacements.",
"In fact, our Hamiltonian emerges naturally from an arbitrary total system Hamiltonian by expanding atomic displacements of the environment up to the second order.",
"Therefore, it can be directly applied to realistic systems.",
"The fundamental result of our paper is the removal of the unphysical partitioned initial condition which implied that the open system and the bath were initially isolated.",
"Following previous procedures to accommodate a more physical partition-free approach, we applied the special variant of the Hubbard-Stratonovich transformation that allowed the initial condition to be determined via an auxiliary differential equation.",
"This allows the ESLN to make exact predictions for the transient behaviour of the primary system when it is perturbed from equilibrium.",
"Additionally, when the total system is in equilibrium, the imaginary time differential equation allows for the exact calculation of the reduced equilibrium density matrix.",
"This is important, as the stationary distribution of dissipative systems with finite couplings has been shown to deviate from that expected under partitioned conditions [58].",
"The true distribution is described by the “Hamiltonian of mean force”, and Eqs.",
"(REF ) and (REF ) provide a route to the exact calculation of the stationary distribution.",
"Indeed, the imaginary time evolution has been independently derived by Moix et al.",
"[56] as an exact description of an open system in interactive equilibrium with its environment.",
"This formulation of the equilibrium density matrix has been used by Tanimura to develop hierarchical equations of motion for fermionic systems [59] under the assumption that the environment spectral density is Ohmic.",
"The ESLN represents a unification and generalisation of the differential equations derived by Stockburger [46] and Moix et al.",
"[56], resulting in additional and highly non-trivial constraints on the correlations between the real and imaginary time noises.",
"The connection between these two pieces of work was not previously apparent, but has emerged naturally from the simultaneous generalisation of the model Hamiltonian and the initial total density matrix.",
"This is the ESLN's principal advantage, and allows for a simpler and more general closed form description of the evolution of the reduced density matrix, as compared to hierarchical equations of motion [59].",
"We also note that our approach can easily be generalised to several environments, e.g., for heat transport problems along similar lines to Ref.",
"[21].",
"Extracting numerical results from the ESLN depends on the feasibility of generating noises that satisfy the correlations outlined in section .",
"Real time noises of the same type can already be efficiently calculated [46], and the outlook for extending this to include the imaginary time noise is promising.",
"Looking forward, a first application of the ESLN is therefore likely to be a calculation of the time evolution of the density matrix for a simple system coupled to a harmonic bath, and the comparison between approximate partitioned and exact partition-free methods.",
"The class of problems that this model may be applied to are rather broad.",
"This includes a two-level spin boson system, coupled to a bath with an arbitrary spectrum [46], or the heat exchange between an arbitrary system and a bath with Ohmic dissipation [45].",
"It is possible that this generalisation may also be applied to numerical schemes for anharmonic bath models [60], and influence functional simulations of complex systems [61].",
"To summarise, the influence functional formalism has been used to generate two stochastic differential equations that together describe the exact evolution of an open system that begins in coupled equilibrium with its harmonic environment.",
"The results presented here are an extension to existing frameworks for thermodynamic analysis in the quantum regime, as well as offering a method for accessing the equilibrium states of arbitrary dissipative systems."
],
[
"Acknowledgements",
"The authors would like to thank Prof Ulrich Weiss for his help in verifying the Feynman-Vernon influence phase at the early stages of this work.",
"GMG is supported by the EPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES, EP/L015854/1).",
"We also would like to thank Ian Ford and Claudia Clarke for thought-provoking and stimulating discussions."
]
] | 1612.05466 |
[
[
"The upper topology and its relation with the projective modules"
],
[
"Abstract In this paper, first we obtain some new and interesting results on projective modules and on the upper topology of an ordinal number.",
"Then it is shown that the rank map of a locally of finite type projective module is continuous with respect to the upper topology (by contract, it is well known this map is not necessarily continuous with respect to the discrete topology).",
"It is also proved that a finitely generated flat module is projective if and only if its rank map is continuous with respect to the upper topology."
],
[
"Introduction",
"This paper grew out of an attempt to understand when the rank map of an infinitely generated projective module is continuous.",
"It is well-known that the rank map of a f.g. projective $R-$ module is continuous whenever $\\operatorname{Spec}(R)$ is equipped with the Zariski topology and the set of natural numbers with the “discrete\" topology.",
"It is worthy to mention that, in Proposition REF , this fact is re-proved by a new method.",
"See also [10] for another proof of it.",
"Note that the “finitely generated” assumption of module is a crucial point in the continuity of the rank map.",
"If we drop this hypothesis then the rank map is no longer necessarily continuous.",
"Namely, there are locally of finite type projective modules whose rank maps are not necessarily continuous w.r.t.",
"the discrete topology, see e.g.",
"[10].",
"Therefore it is natural to ask, under which circumstances, then the rank map of a locally of finite type projective module will be continuous.",
"One of the main aims of the present article is to realize this goal.",
"In order to accomplish this we need a suitable topology to replace instead of the “discrete\" topology of the natural numbers $\\omega =\\lbrace 0, 1, 2,...\\rbrace $ .",
"Finding such a topology requires to have a familiarity with the structure of the natural numbers.",
"The set $\\omega $ is an ordinal number.",
"Indeed, in Theorem REF , a new and natural topology over any ordinal number is introduced.",
"It is called the well-founded topology.",
"Then, in Theorem REF , it is shown that if the “discrete\" topology of $\\omega $ is replaced by the well-founded topology then the rank map remains continuous even if the projective module is infinitely generated.",
"To prove its continuity the main ingredients which are used, in addition to the Kaplansky theorem, are some basic properties of the exterior powers of a module.",
"In this article, we are also interested to investigate the continuity of the rank map of some certain modules.",
"We should mention that the projectivity of modules has very close connection with the continuity of their rank maps.",
"For instance, a f.g. flat module is projective if and only if its rank map is continuous w.r.t.",
"the discrete topology.",
"An analogue of this result is also proved for the well-founded topology, see Theorem REF .",
"This shows that the well-founded topology could be an ideal-theoretic structure.",
"Theorem REF is another interesting result of this article which generalizes some previous results in the literature on the projectivity of f.g. flat modules.",
"In order to get a better understanding of the well-founded topology having a reasonable knowledge from the theory of ordinal numbers is necessary.",
"We refer the reader to [1] for a comprehensive discussion of the ordinal numbers.",
"In this article we also investigate further properties of the well-founded topology.",
"Specially, it is proved that the well-founded topology over an ordinal number is spectral if and only if it is a natural number, see Theorem REF .",
"Throughout the article, all of the rings which are discussed are commutative."
],
[
"Preliminaries",
"Lemma 2.1 Let $(R,\\mathfrak {m})$ be a local ring with the residue field $\\kappa $ and let $M$ be a f.g. $R-$ module.",
"Then a finite subset $\\lbrace x_{1},...,x_{n}\\rbrace $ is a minimal generating set of $M$ if and only if $\\lbrace x_{i}+\\mathfrak {m}M: 1\\le i\\le n\\rbrace $ is a $\\kappa -$ basis of $M/\\mathfrak {m}M$ .",
"Proof.",
"“$\\Rightarrow $ \" Suppose there are elements $a_{i}\\in R$ with $1\\le i\\le n$ such that $\\sum \\limits _{i=1}^{n}a_{i}x_{i}\\in \\mathfrak {m}M$ .",
"Suppose there is some $j$ such that $a_{j}\\notin \\mathfrak {m}$ .",
"Then there is an element $b\\in R$ such that $1=ba_{j}$ .",
"It follows that $x_{j}\\in \\sum \\limits _{\\begin{array}{c}i=1,\\\\i\\ne j\\end{array}}^{n}R x_{i}+\\mathfrak {m}M$ .",
"This means that $M=\\sum \\limits _{\\begin{array}{c}i=1,\\\\i\\ne j\\end{array}}^{n}R x_{i}+\\mathfrak {m}M$ .",
"Thus, by the Nakayama Lemma, $M=\\sum \\limits _{\\begin{array}{c}i=1,\\\\i\\ne j\\end{array}}^{n}R x_{i}$ .",
"But this is a contradiction.",
"Therefore $a_{i}\\in \\mathfrak {m}$ for all $i$ .",
"Conversely, if $\\lbrace x_{i}+\\mathfrak {m}M: 1\\le i\\le n\\rbrace $ is a $\\kappa -$ basis of $M/\\mathfrak {m}M$ then $M=\\sum \\limits _{i=1}^{n}R x_{i}+\\mathfrak {m}M$ .",
"Thus, by the NAK, $M=\\sum \\limits _{i=1}^{n}R x_{i}$ .",
"Moreover, by what we have just proved, we conclude that the generating set $\\lbrace x_{1},...,x_{n}\\rbrace $ is minimal.",
"$\\Box $ The following is an immediate consequence of the above lemma.",
"Corollary 2.2 Any two minimal generating sets of a f.g. module over a local ring have the same number of elements.",
"$\\Box $ The above Corollary does not hold in general.",
"As a specific example, $\\lbrace 1\\rbrace $ and $\\lbrace 2,3\\rbrace $ are two minimal generating sets of $\\mathbb {Z}$ as module over itself with unequal number of elements.",
"Remark 2.3 Let $M$ be a module over a ring $R$ such that $M_{\\mathfrak {p}}$ as $R_{\\mathfrak {p}}-$ module is f.g. for all primes $\\mathfrak {p}$ .",
"In this case we say that $M$ is locally of finite type.",
"If $\\mathfrak {p}$ is a prime ideal of $R$ then we define $\\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})$ as the number of elements of a minimal generating set of the $R_{\\mathfrak {p}}-$ module $M_{\\mathfrak {p}}$ .",
"By Corollary REF , it is well-defined.",
"This leads us to a map from $\\operatorname{Spec}(R)$ into the set of natural numbers $\\omega =\\lbrace 0,1,2,...\\rbrace $ given by $\\mathfrak {p}\\rightsquigarrow \\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})$ .",
"It is called the rank map of $M$ .",
"Note that $\\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})$ , by Lemma REF , is equal to the dimension of the $\\kappa (\\mathfrak {p})-$ space $M\\otimes _{R}\\kappa (\\mathfrak {p})$ where $\\kappa (\\mathfrak {p})$ is the residue field of $R$ at $\\mathfrak {p}$ .",
"The rank map is said to be patch (resp.",
"flat, Zariski) continuous if it is continuous whenever $\\operatorname{Spec}(R)$ is equipped with the patch (resp.",
"flat, Zariski) topology and $\\omega $ with the discrete topology.",
"Finally, we say that the rank map is locally constant if it is patch continuous.",
"Lemma 2.4 Let $S$ and $T$ be two multiplicative subsets of $R$ such that $S\\subseteq T$ .",
"Let $U$ be the image of $T$ under the canonical map $R\\rightarrow S^{-1}R$ .",
"Then the ring $U^{-1}(S^{-1}R)$ is canonically isomorphic to $T^{-1}R$ and for each $R-$ module $M$ , $U^{-1}(S^{-1}M)\\simeq T^{-1}M$ .",
"Proof.",
"It is well-known.",
"$\\Box $"
],
[
"A remark on the Kaplansky theorem",
"Let $M$ be a locally free $R-$ module, i.e., $M_{\\mathfrak {p}}$ is $R_{\\mathfrak {p}}-$ free for all primes $\\mathfrak {p}$ .",
"We define $\\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})$ as the cardinality of a $R_{\\mathfrak {p}}-$ basis of $M_{\\mathfrak {p}}$ .",
"It is well-defined since over a commutative ring, any two bases of a free module have the same cardinality.",
"Note that the projective modules and locally of finite type flat modules are typical examples of locally free modules.",
"Lemma 3.1 Let $M$ be a locally free $R-$ module and let $\\mathfrak {p}\\subseteq \\mathfrak {q}$ be prime ideals of $R$ .",
"Then $\\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})=\\operatorname{rank}_{R_{\\mathfrak {q}}}(M_{\\mathfrak {q}})$ .",
"Proof.",
"By Lemma REF , $M_{\\mathfrak {p}}\\simeq (M_{\\mathfrak {q}})_{\\mathfrak {p}}$ .",
"$\\Box $ Corollary 3.2 Let $M$ be a $R-$ module.",
"Then $\\operatorname{Supp}M$ is stable under the specialization.",
"If moreover $M$ is locally free then $\\operatorname{Supp}M$ is also stable under the generalization.",
"Proof.",
"Let $\\mathfrak {p}\\subseteq \\mathfrak {q}$ be prime ideals of $R$ .",
"If $\\mathfrak {p}\\in \\operatorname{Supp}M$ then there is some $x\\in M$ such that $\\operatorname{Ann}_{R}(x)\\subseteq \\mathfrak {p}$ .",
"Thus $x/1$ is a non-zero element of $M_{\\mathfrak {q}}$ .",
"Hence $\\operatorname{Supp}M$ is stable under the specialization.",
"Now assume that $\\mathfrak {q}\\in \\operatorname{Supp}M$ .",
"Then, by Lemma REF , $M_{\\mathfrak {p}}\\ne 0$ .",
"$\\Box $ Lemma 3.3 Let $M$ be a projective $R-$ module and let $\\lbrace I_{\\alpha }\\rbrace $ be a family of ideals of $R$ .",
"Then $\\bigcap \\limits _{\\alpha }(I_{\\alpha }M)=\\big (\\bigcap \\limits _{\\alpha }I_{\\alpha }\\big )M$ .",
"Proof.",
"There is a free $R-$ module $F$ such that $M$ is a direct summand of it.",
"Thus there exists a $R-$ submodule $N$ of $F$ such that $F=M+N$ and $M\\cap N=0$ .",
"Let $\\lbrace x_{i}\\rbrace $ be a $R-$ basis of $F$ .",
"Consider the bijective map $\\psi :F\\rightarrow \\bigoplus \\limits _{i}R$ given by $x\\rightsquigarrow (r_{i})$ where $x=\\sum \\limits _{i}r_{i}x_{i}$ .",
"If $I$ is an ideal of $R$ then $\\psi (IF)=\\bigoplus \\limits _{i}I$ .",
"Moreover $\\bigcap \\limits _{\\alpha }(\\bigoplus \\limits _{i}I_{\\alpha })=\\bigoplus \\limits _{i}(\\bigcap \\limits _{\\alpha }I_{\\alpha })$ .",
"It follows that $\\psi \\big ((\\bigcap \\limits _{\\alpha }I_{\\alpha })F\\big )=\\psi \\big (\\bigcap \\limits _{\\alpha }(I_{\\alpha }F)\\big )$ .",
"Thus $(\\bigcap \\limits _{\\alpha }I_{\\alpha })F=\\bigcap \\limits _{\\alpha }(I_{\\alpha }F)$ .",
"We also have $IF=IM+IN$ and $\\bigcap \\limits _{\\alpha }(I_{\\alpha }M+I_{\\alpha }N)=\\bigcap \\limits _{\\alpha }(I_{\\alpha }M)+\\bigcap \\limits _{\\alpha }(I_{\\alpha }N)$ .",
"Therefore $(\\bigcap \\limits _{\\alpha }I_{\\alpha })M+(\\bigcap \\limits _{\\alpha }I_{\\alpha })N=\\bigcap \\limits _{\\alpha }(I_{\\alpha }M)+\\bigcap \\limits _{\\alpha }(I_{\\alpha }N)$ .",
"It follows that $(\\bigcap \\limits _{\\alpha }I_{\\alpha })M=\\bigcap \\limits _{\\alpha }(I_{\\alpha }M)$ .",
"$\\Box $ Now we prove the first main result of this article: Theorem 3.4 Let $M$ be either a projective $R-$ module or a locally of finite type flat $R-$ module and let $\\mathfrak {p}$ be a prime ideal of $R$ .",
"Then $M_{\\mathfrak {p}}=0$ if and only if $M=\\mathfrak {p}M$ .",
"Proof.",
"First assume that $M_{\\mathfrak {p}}=0$ .",
"From the exact sequence ${0[r]&R/\\mathfrak {p}[r]&\\kappa (\\mathfrak {p})}$ we obtain the following exact sequence ${0[r]&R/\\mathfrak {p}\\otimes _{R}M[r]&\\kappa (\\mathfrak {p})\\otimes _{R}M}.$ But $\\kappa (\\mathfrak {p})\\otimes _{R}M\\simeq R/\\mathfrak {p}\\otimes _{R}M_{\\mathfrak {p}}=0$ .",
"It follows that $M=\\mathfrak {p}M$ .",
"Conversely, assume that $M=\\mathfrak {p}M$ .",
"We have $M_{\\mathfrak {p}}$ as $R_{\\mathfrak {p}}-$ module is free because apply the Kaplansky theorem [3] whenever $M$ is projective and apply [3] whenever $M$ is locally of finite type.",
"Moreover $M_{\\mathfrak {p}}\\otimes _{R_{\\mathfrak {p}}}\\kappa (\\mathfrak {p})\\simeq M_{\\mathfrak {p}}\\otimes _{R}R/\\mathfrak {p}\\simeq M/\\mathfrak {p}M\\otimes _{R}R_{\\mathfrak {p}}=0$ .",
"It follows that $M_{\\mathfrak {p}}=0$ .",
"$\\Box $ Corollary 3.5 The support of a projective module is both Zariski and flat open.",
"Proof.",
"Let $M$ be a projective $R-$ module, let $X=\\operatorname{Spec}(R)\\setminus \\operatorname{Supp}(M)$ and let $I=\\bigcap \\limits _{\\mathfrak {p}\\in X}\\mathfrak {p}$ .",
"By Theorem REF and Lemma REF , $IM=M$ .",
"Clearly $X\\subseteq V(I)$ .",
"Conversely, assume that $I\\subseteq \\mathfrak {p}$ .",
"Then $M=IM\\subseteq \\mathfrak {p}M\\subseteq M$ .",
"Thus $\\mathfrak {p}M=M$ and so by Theorem REF , $\\mathfrak {p}\\in X$ .",
"Therefore $X=V(I)$ .",
"By using Corollary REF and [8], we conclude that $X$ is flat closed.",
"$\\Box $"
],
[
"On the rank map of some certain modules",
"Lemma 4.1 Let $M$ be a locally free $R-$ module.",
"Then for each natural number $n$ , $\\operatorname{Supp}\\big (\\Lambda ^{n}(M)\\big )=\\lbrace \\mathfrak {p}\\in \\operatorname{Spec}(R): \\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})\\ge n\\rbrace $ where $\\Lambda ^{n}(M)$ is the $n$ -th exterior power of $M$ .",
"Proof.",
"By Theorem REF , $\\Lambda ^{n}(M)\\otimes _{R}R_{\\mathfrak {p}}\\simeq \\Lambda _{R_{\\mathfrak {p}}}^{n}(M_{\\mathfrak {p}})$ .",
"Therefore, by Corollary REF , $\\mathfrak {p}\\in \\operatorname{Supp}\\Lambda ^{n}(M)$ if and only if $\\operatorname{rank}_{R_{\\mathfrak {p}}}(M_{\\mathfrak {p}})\\ge n$ .",
"$\\Box $ The following is a well-known result, e.g.",
"see [10].",
"Here we present a new proof for it: Proposition 4.2 The rank map of a f.g. projective module is both flat and Zariski continuous.",
"Proof.",
"Let $M$ be a f.g. projective module over a ring $R$ .",
"For each natural number $n$ , then by Lemma REF , $\\psi ^{-1}(\\lbrace n\\rbrace )=\\operatorname{Supp}N\\cap \\big (\\operatorname{Spec}(R)\\setminus \\operatorname{Supp}N^{\\prime }\\big )$ where $\\psi $ is the rank map of $M$ , $N=\\Lambda ^{n}(M)$ and $N^{\\prime }=\\Lambda ^{n+1}(M)$ .",
"By Lemmata REF and REF , $N$ and $N^{\\prime }$ are f.g. projective $R-$ modules.",
"Thus, by [9], there are idempotent elements $e,e^{\\prime }\\in R$ such that $\\operatorname{Ann}_{R}(N)=Re$ and $\\operatorname{Ann}_{R}(N^{\\prime })=Re^{\\prime }$ .",
"It follows that $\\psi ^{-1}(\\lbrace n\\rbrace )=V(e)\\cap V(1-e^{\\prime })=D\\big (e^{\\prime }(1-e)\\big )$ .",
"Therefore $\\psi ^{-1}(\\lbrace n\\rbrace )$ is both flat and Zariski open.",
"$\\Box $ Lemma 4.3 Let $M$ be a locally free $R-$ module which is also locally of finite type.",
"If the rank map of $M$ is patch continuous then it is both flat and Zariski continuous.",
"Proof.",
"For each natural number $n$ , by Lemma REF , $\\psi ^{-1}(\\lbrace n\\rbrace )$ is stable under the generalization and specialization where $\\psi $ is the rank map of $M$ .",
"Now if $\\psi $ is patch continuous then, by [8], it is both flat and Zariski continuous.",
"$\\Box $ The following is the second main result of this article.",
"Theorem 4.4 Let $R$ be a ring which has either a finitely many minimal primes or a finitely many maximal ideals.",
"Then the rank map of a locally of finite type flat $R-$ module is patch continuous.",
"In particular, every f.g. flat $R-$ module is $R-$ projective.",
"Proof.",
"Let $M$ be a locally of finite type flat $R-$ module, let $n$ be a natural number and let $E=\\psi ^{-1}(\\lbrace n\\rbrace )$ where $\\psi $ is the rank map of $M$ .",
"First assume that $\\operatorname{Min}(R)=\\lbrace \\mathfrak {p}_{1},...,\\mathfrak {p}_{k}\\rbrace $ .",
"There exists some $s$ with $1\\le s\\le k$ such that $\\mathfrak {p}_{s},\\mathfrak {p}_{s+1},...,\\mathfrak {p}_{k}\\notin E$ but $\\mathfrak {p}_{i}\\in E$ for all $i<s$ .",
"By the proof of Lemma REF , $\\operatorname{Spec}(R)\\setminus E=\\bigcup \\limits _{i=s}^{k}V(\\mathfrak {p}_{i})$ .",
"Thus $E$ is Zariski open in the case of finitely many minimal primes and so it is patch open.",
"Now let $\\operatorname{Max}(R)=\\lbrace \\mathfrak {m}_{1},...,\\mathfrak {m}_{d}\\rbrace $ .",
"Similarly, there exists some $\\ell $ with $1\\le \\ell \\le d$ such that $\\mathfrak {m}_{\\ell },\\mathfrak {m}_{\\ell +1},...,\\mathfrak {m}_{d}\\notin E$ but $\\mathfrak {m}_{i}\\in E$ for all $i<\\ell $ .",
"Again by the proof of Lemma REF , $\\operatorname{Spec}(R)\\setminus E=\\bigcup \\limits _{i=k}^{d}\\Lambda (\\mathfrak {m}_{i})$ where $\\Lambda (\\mathfrak {m}_{i})=\\lbrace \\mathfrak {p}\\in \\operatorname{Spec}(R) : \\mathfrak {p}\\subseteq \\mathfrak {m}_{i}\\rbrace $ .",
"Therefore, by [8], $E$ is a flat open in the case of finitely many maximal ideals and so it is patch open.",
"The latter statement is an immediate consequence of [3] and Lemma REF .",
"$\\Box $ Theorem REF , in particular, generalizes some previous results in the literature on the projectivity of f.g. flat modules specially including [3], [6], [4], [7]."
],
[
"The well-founded topology and its applications",
"Lemma 5.1 Let $\\alpha $ and $\\beta $ be two ordinal numbers.",
"Then $\\beta \\in \\alpha $ if and only if $\\beta $ is a proper subset of $\\alpha $ .",
"Proof.",
"“$\\Rightarrow $ \" every ordinal number is a transitive set, see [1].",
"For the converse, see [1].",
"$\\Box $ We say that $\\beta <\\alpha $ if $\\beta \\in \\alpha $ or equivalently $\\beta \\subset \\alpha $ .",
"Theorem 5.2 Let $\\alpha $ be an ordinal number.",
"Then the collection of all ordinal numbers $\\beta $ with $\\beta \\subseteq \\alpha $ , as closed subsets, constitutes a topology over $\\alpha $ .",
"Proof.",
"Let $\\lbrace \\beta _{i}\\rbrace _{i\\in I}$ be a non-empty family of ordinal numbers.",
"By [1], there exists some $k\\in I$ such that $\\beta _{k}\\subseteq \\beta _{i}$ for all $i$ .",
"It follows that $\\beta _{k}=\\bigcap \\limits _{i\\in I}\\beta _{i}$ .",
"Finally, let $\\lbrace \\beta _{1},...,\\beta _{n}\\rbrace $ be a finite set of ordinal numbers.",
"By [1], there exists some $j$ such that $\\beta _{i}\\subseteq \\beta _{j}$ for all $i$ .",
"Therefore $\\beta _{j}=\\bigcup \\limits _{i=1}^{n}\\beta _{i}$ .",
"$\\Box $ Definition 5.3 The resulting topology of Theorem REF over an ordinal number $\\alpha $ is called the well-founded topology of $\\alpha $ .",
"By Lemma REF , the first interesting property of the well-founded topology is that the proper closed subsets of an ordinal number $\\alpha $ are precisely the points of $\\alpha $ .",
"Theorem 5.4 Let $\\alpha $ be an ordinal number and consider the well-founded topology over it.",
"Then the following conditions hold.",
"$\\textbf {(i)}$ The closure of the point $\\beta \\in \\alpha $ is its successor ordinal, i.e., $\\overline{\\lbrace \\beta \\rbrace }=\\beta ^{+}=\\beta \\cup \\lbrace \\beta \\rbrace $ .",
"$\\textbf {(ii)}$ A closed subset of $\\alpha $ has a generic point if and only if it is not a limit ordinal.",
"Moreover, the generic point, if it exists, is unique.",
"$\\textbf {(iii)}$ If $\\beta \\subset \\alpha $ is an ordinal number then the well-founded and subspace topologies over $\\beta $ are the same.",
"$\\textbf {(iv)}$ If $\\alpha \\ne 0$ then $\\alpha $ is irreducible.",
"$\\textbf {(v)}$ The closed subsets of $\\alpha $ are stable under the arbitrary unions.",
"$\\textbf {(vi)}$ Every open subset of $\\alpha $ is quasi-compact.",
"$\\textbf {(vii)}$ The well-founded topology is noetherian.",
"Recall that an ordinal number is said to be a limit ordinal if it is not the successor of an ordinal number.",
"Proof.",
"$\\textbf {(i)}:$ By [1], the set $\\beta ^{+}=\\beta \\cup \\lbrace \\beta \\rbrace $ is an ordinal number.",
"Moreover, by [1], $\\beta ^{+}\\le \\alpha $ because if $\\alpha \\in \\beta ^{+}=\\beta \\cup \\lbrace \\beta \\rbrace $ then either $\\alpha <\\beta $ or $\\alpha =\\beta $ , a contradiction.",
"Therefore $\\beta ^{+}$ is a closed subset of $\\alpha $ .",
"It follows that $\\overline{\\lbrace \\beta \\rbrace }\\subseteq \\beta ^{+}$ .",
"Conversely, let $\\gamma $ be a closed subset of $\\alpha $ such that $\\beta \\in \\gamma $ .",
"The set $\\gamma $ is an ordinal number and again by [1], $\\beta ^{+}\\le \\gamma $ .",
"Therefore the well-founded closure of the point $\\beta $ is equal to $\\beta ^{+}$ .",
"$\\textbf {(ii)}:$ The first assertion is obvious from (i).",
"Suppose $\\overline{\\lbrace \\beta \\rbrace }=\\overline{\\lbrace \\gamma \\rbrace }$ .",
"Then $\\beta ^{+}=\\gamma ^{+}$ and so, by [1], $\\beta =\\gamma $ .",
"$\\textbf {(iii)}$ and $\\textbf {(iv)}:$ See [1].",
"$\\textbf {(v)}:$ If $\\lbrace \\beta _{i}\\rbrace $ is a family of ordinal numbers then, by [1], $\\bigcup \\limits _{i}\\beta _{i}$ is also an ordinal number.",
"$\\textbf {(vi)}:$ Let $U$ be an open subset of $\\alpha $ and let $\\lbrace U_{i}\\rbrace _{i\\in I}$ be an open covering of it.",
"We may assume that $I\\ne \\emptyset $ .",
"For each $i$ there exists an ordinal number $\\beta _{i}$ with $\\beta _{i}\\le \\alpha $ such that $U_{i}^{c}=\\beta _{i}$ .",
"Let $\\beta _{k}$ be the least element of the set $\\lbrace \\beta _{i} : i\\in I\\rbrace $ .",
"It follows that $U=U_{k}$ .",
"$\\textbf {(vii)}:$ It is obvious from (vi).",
"$\\Box $ Remark 5.5 By [1], every natural number is an ordinal number where $0=\\emptyset $ , $1=0^{+}=\\lbrace 0\\rbrace $ , $2=1^{+}=\\lbrace 0,1\\rbrace $ and so on.",
"Moreover, by [1] and [1], the set of natural numbers $\\omega =\\lbrace 0,1,2,...\\rbrace $ is also an ordinal number.",
"The well-founded topology over an ordinal number $\\alpha $ is discrete if and only if either $\\alpha =0$ or $\\alpha =1$ .",
"The well-founded topology over $2=\\lbrace 0,1\\rbrace $ is just the Sierpiński topology.",
"Theorem 5.6 The well-founded topology over an ordinal number is spectral if and only if it is a natural number.",
"Proof.",
"Suppose that the well-founded topology over the ordinal number $\\alpha $ is spectral.",
"Let $\\alpha \\ge \\omega $ .",
"By Remark REF , $\\omega $ is a closed subset of $\\alpha $ and so by Theorem REF , it is irreducible.",
"But $\\omega $ does not have any generic point.",
"Because, suppose there is some $m\\in \\omega $ such that $\\omega =\\overline{\\lbrace m\\rbrace }=m^{+}$ .",
"By [1], $m^{+}=(m+0)^{+}=m+0^{+}=m+1$ is a natural number, a contradiction.",
"It follows that $\\alpha <\\omega $ , i.e., $\\alpha $ is a natural number.",
"Conversely, let $n$ be a natural number.",
"To prove the assertion, by Theorem REF , it suffices to show that every closed and irreducible subset of $n$ has a generic point.",
"We may assume that $n>0$ .",
"If $m$ is a closed and irreducible subset of $n$ then $m$ is a natural number with $0<m\\le n$ .",
"By [1], there exists a natural number $k$ such that $m=k^{+}$ .",
"Therefore $m=\\overline{\\lbrace k\\rbrace }$ .",
"$\\Box $ As stated in the Introduction, the rank map of a locally of finite type projective module is not necessarily continuous w.r.t.",
"the discrete topology.",
"But its continuity will be recovered if the discrete topology is replaced by the well-founded topology: Theorem 5.7 Let $M$ be a locally of finite type flat $R-$ module and consider the well-founded topology on the set of natural numbers.",
"If $M$ is $R-$ projective then the rank map of $M$ is both flat and Zariski continuous.",
"Proof.",
"Let $n$ be a natural number.",
"By [1], $n$ is either the empty set or there exists a natural number $m$ such that $n=m+1$ .",
"Thus either $n=\\emptyset $ or $n=\\lbrace 0,1,2,...,m\\rbrace $ .",
"By applying this and Lemma REF , we obtain that $\\psi ^{-1}(n)=\\operatorname{Spec}(R)\\setminus \\operatorname{Supp}\\big (\\Lambda ^{n}(M)\\big )$ where $\\psi $ is the rank map of $M$ .",
"Therefore, by Lemma REF and Corollary REF , the assertion concludes.",
"$\\Box $ It will be a great surprise to the author if also the converse of Theorem REF holds.",
"Theorem 5.8 Let $M$ be a f.g. flat module over a ring $R$ .",
"Then $M$ is $R-$ projective if and only if the rank map of $M$ is patch continuous w.r.t.",
"the well-founded topology.",
"Proof.",
"The implication “$\\Rightarrow $ \" implies from Theorem REF or Proposition REF .",
"Conversely, let $n$ be a natural number.",
"By Lemma REF , $\\psi ^{-1}(\\lbrace n\\rbrace )=\\operatorname{Supp}N\\cap \\big (\\operatorname{Spec}(R)\\setminus \\operatorname{Supp}N^{\\prime }\\big )$ and $\\operatorname{Supp}N=\\operatorname{Spec}(R)\\setminus \\psi ^{-1}(n)$ where $\\psi $ is the rank map of $M$ , $N=\\Lambda ^{n}(M)$ and $N^{\\prime }=\\Lambda ^{n+1}(M)$ .",
"Thus, $\\psi $ is patch continuous w.r.t.",
"the discrete topology.",
"Therefore, by [3] and Lemma REF , $M$ is $R-$ projective.",
"$\\Box $ An $R-$ module $M$ is said to be locally of countable rank if for each prime ideal $\\mathfrak {p}$ of $R$ then $M_{\\mathfrak {p}}$ as $R_{\\mathfrak {p}}-$ module is countably generated (possibly infinite).",
"Corollary 5.9 Let $R$ be a ring such that $\\operatorname{Spec}(R)$ is noetherian w.r.t.",
"the flat topology.",
"Let $M$ be a projective $R-$ module which is locally of countable rank.",
"Consider the well-founded topology over $\\omega ^{+}$ and the Zariski topology over $\\operatorname{Spec}(R)$ .",
"Then the rank map of $M$ is continuous.",
"Proof.",
"We have $\\psi ^{-1}(\\omega )=\\bigcup \\limits _{n\\in \\omega }\\psi ^{-1}(n)$ where $\\psi $ is the rank map of $M$ .",
"By the proof of Theorem REF , $\\psi ^{-1}(n)=\\operatorname{Spec}(R)\\setminus \\operatorname{Supp}\\Lambda ^{n}(M)$ .",
"Thus, by Corollary REF , it is Zariski closed.",
"Therefore, by [8], $\\psi ^{-1}(\\omega )$ is also Zariski closed.",
"$\\Box $ Lemma 5.10 Let $R$ be a ring such that $\\operatorname{Spec}(R)$ is noetherian w.r.t.",
"the Zariski topology.",
"Then the flat opens of $\\operatorname{Spec}(R)$ are stable under the arbitrary intersections.",
"Proof.",
"If $I$ is an ideal of $R$ then there is a finite subset $\\lbrace f_{1},...,f_{n}\\rbrace $ of elements of $R$ such that $\\operatorname{Spec}(R)\\setminus V(I)=\\bigcup \\limits _{i=1}^{n}D(f_{i})$ since every Zariski open subset of $\\operatorname{Spec}(R)$ is quasi-compact.",
"It follows that $V(I)=\\bigcap \\limits _{i=1}^{n}V(f_{i})$ .",
"This means that $V(I)$ is a flat open.",
"Now let $\\lbrace I_{\\alpha }\\rbrace $ be a family of ideals of $R$ .",
"If $\\mathfrak {p}\\in \\bigcap \\limits _{\\alpha }V(I_{\\alpha })$ then $V(\\mathfrak {p})\\subseteq \\bigcap \\limits _{\\alpha }V(I_{\\alpha })$ .",
"Therefore, by [8], the intersection of every family of basis flat opens of $\\operatorname{Spec}(R)$ is a flat open.",
"$\\Box $ As a dual of Corollary REF , we have: Corollary 5.11 Let $R$ be a ring such that $\\operatorname{Spec}(R)$ is noetherian w.r.t.",
"the Zariski topology.",
"Let $M$ be a projective $R-$ module which is locally of countable rank.",
"Consider the well-founded topology over $\\omega ^{+}$ and the flat topology over $\\operatorname{Spec}(R)$ .",
"Then the rank map of $M$ is continuous.",
"Proof.",
"We have $\\psi ^{-1}(\\omega )=\\bigcup \\limits _{n\\in \\omega }\\psi ^{-1}(n)$ where $\\psi $ is the rank map of $M$ .",
"By Corollary REF , $\\psi ^{-1}(n)=\\operatorname{Spec}(R)\\setminus \\operatorname{Supp}\\Lambda ^{n}(M)$ is a flat closed.",
"Then apply Lemma REF .",
"$\\Box $"
],
[
"Appendix- exterior powers of a module",
"The literature on the exterior powers is in a somewhat unsatisfactory state.",
"Hence this section is devoted to this topic up to level which is needed in the article and in the subsequent work [9].",
"All of the presentation specially the proofs due to the author.",
"Let $M, N$ be $R-$ modules and let $n\\ge 2$ .",
"A map $f:M^{n}\\rightarrow N$ is called alternative if $f$ vanishes on each $n$ -tuple with at least two distinct equal coordinates.",
"The map $f$ is called skew-symmetric if $f(x_{\\sigma (1)},...,x_{\\sigma (n)})=(\\operatorname{sgn}\\sigma )f(x_{1},...,x_{n})$ for all $\\sigma \\in S_{n}$ .",
"Lemma 6.1 Every alternative multi-linear map is skew-symmetric.",
"Proof.",
"Let $f:M^{n}\\rightarrow N$ be a alternative multi-linear map and let $\\sigma \\in S_{n}$ where $n\\ge 2$ .",
"Each permutation can be written as a product of finitely many transpositions, see [2].",
"Thus we may write $\\sigma =\\tau _{1}...\\tau _{s}$ where $\\tau _{k}$ is a transposition for all $k$ .",
"To prove the assertion we use an induction argument on $s$ .",
"If $s=1$ then $\\sigma =(i,j)$ with $i<j$ .",
"But $f(x_{1},...,x_{i}+x_{j},...,x_{j}+x_{i},...,x_{n})=0$ .",
"It follows that $f(x_{\\sigma (1)},...,x_{\\sigma (n)})=f(x_{1},...,x_{j},...,x_{i},...,x_{n})=-f(x_{1},...,x_{i},...,x_{j},...,x_{n})$ .",
"Let $s>1$ .",
"We may write $\\sigma =(i,j)\\tau $ .",
"Using the induction hypothesis, then $f(x_{\\sigma (1)},...,x_{\\sigma (n)})=f(x_{\\tau (1)},...,x_{\\tau (j)},...,x_{\\tau (i)},...,x_{\\tau (n)})=(\\operatorname{sgn}\\tau ) f(x_{1},...,x_{j},...,x_{i},...,x_{n})=(\\operatorname{sgn}\\sigma )f(x_{1},...,x_{i},...,x_{j},...,x_{n})$ .",
"$\\Box $ Lemma 6.2 A multi-linear map $f:M^{n}\\rightarrow N$ is alternative if and only if the map $f$ vanishes on each $n$ -tuple with a pair of adjacent equal coordinates.",
"Proof.",
"Suppose for a given $n$ -tuple $(x_{1},...,x_{n})\\in M^{n}$ , $x_{i}=x_{j}$ with $i<j$ .",
"If $j=i+1$ then there is nothing to prove.",
"Let $j>i+1$ .",
"By the hypothesis, $f(x_{1},...,x_{i},...,x_{j-1}+x_{j}, x_{j}+x_{j-1},...,x_{n})=0$ .",
"By the induction hypothesis, $f(x_{1},...,x_{i},...,x_{j},x_{j-1},...,x_{n})=0$ .",
"It follows that $f(x_{1},...,x_{i},...,x_{j-1},x_{j},...,x_{n})=0$ .",
"$\\Box $ Let $J_{n}$ be the $R-$ submodule of $M^{\\otimes n}$ generated by the collection of pure tensors of the form $x_{1}\\otimes ...\\otimes x_{n}$ with $x_{i}=x_{j}$ for some $i\\ne j$ .",
"The quotient module $\\Lambda ^{n}(M):=M^{\\otimes n}/J_{n}$ is called the $n$ -th exterior power of $M$ .",
"Write $\\Lambda ^{0}(M)=R$ and $\\Lambda ^{1}(M)=M$ .",
"The canonical multi-linear map $\\eta :M^{n}\\rightarrow \\Lambda ^{n}(M)$ given by $(x_{1},...,x_{n})\\rightsquigarrow x_{1}\\wedge ...\\wedge x_{n}:=x_{1}\\otimes ...\\otimes x_{n}+J_{n}$ is clearly alternative and the module $\\Lambda ^{n}(M)$ together with the map $\\eta $ satisfy in the following universal property: for each alternative multi-linear map $\\psi :M^{n}\\rightarrow N$ then there is a unique $R-$ linear map $\\varphi :\\Lambda ^{n}(M)\\rightarrow N$ such that $\\psi =\\varphi \\circ \\eta $ .",
"Specially, consider the map $M^{n}\\rightarrow M^{\\otimes n}$ given by $(x_{1},...,x_{n})\\rightsquigarrow \\sum \\limits _{\\sigma \\in S_{n}}(\\operatorname{sgn}\\sigma )x_{\\sigma (1)}\\otimes ...\\otimes x_{\\sigma (n)}$ .",
"This map is multi-linear since multi-linear maps are stable under the finite sums.",
"It is also alternative.",
"Because suppose $x_{i}=x_{i+1}$ for some $i$ .",
"Let $\\sigma \\in S_{n}$ .",
"We have $\\sigma (j)=i$ and $\\sigma (k)=i+1$ for some $j,k$ .",
"Consider the permutation $\\sigma ^{\\ast }=(j,k)\\sigma $ .",
"Then clearly $\\operatorname{sgn}\\sigma ^{\\ast }=-\\operatorname{sgn}\\sigma $ , $x_{\\sigma (1)}\\otimes ...\\otimes x_{\\sigma (n)}=x_{\\sigma ^{\\ast }(1)}\\otimes ...\\otimes x_{\\sigma ^{\\ast }(n)}$ and the assignment $\\sigma \\rightsquigarrow \\sigma ^{\\ast }$ is an injective map from $A_{n}$ , the set of even permutations of degree $n$ , onto the set of odd permutations of the same degree.",
"It follows that $\\sum \\limits _{\\sigma \\in S_{n}}(\\operatorname{sgn}\\sigma )x_{\\sigma (1)}\\otimes ...\\otimes x_{\\sigma (n)}=\\sum \\limits _{\\sigma \\in A_{n}}(x_{\\sigma (1)}\\otimes ...\\otimes x_{\\sigma (n)}-x_{\\sigma ^{\\ast }(1)}\\otimes ...\\otimes x_{\\sigma ^{\\ast }(n)})=0$ .",
"Therefore, by the universal property of the exterior powers, there is a unique $R-$ linear map $\\delta :\\Lambda ^{n}(M)\\rightarrow M^{n}$ which maps each pure wedge $x_{1}\\wedge ...\\wedge x_{n}$ into $\\sum \\limits _{\\sigma \\in S_{n}}(\\operatorname{sgn}\\sigma )x_{\\sigma (1)}\\otimes ...\\otimes x_{\\sigma (n)}$ .",
"The map $\\delta $ is injective whenever $M$ is a free $R-$ module.",
"Because if $\\lbrace x_{\\alpha } : \\alpha \\in I\\rbrace $ is a basis of $M$ then the collection of pure tensors $x_{i_{1}}\\otimes ...\\otimes x_{i_{n}}$ with $\\lbrace i_{1},...,i_{n}\\rbrace \\subseteq I$ is a basis of $M^{\\otimes n}$ .",
"In particular, we have shown that: Lemma 6.3 Let $M$ be a (resp.",
"free) $R-$ module and let $\\lbrace x_{\\alpha } : \\alpha \\in I\\rbrace $ be a generating set (resp.",
"basis) of $M$ .",
"Consider a well-ordering relation $<$ on the index set $I$ .",
"Then the collection of pure wedges of the form $x_{i_{1}}\\wedge ...\\wedge x_{i_{n}}$ where $\\lbrace i_{1},...,i_{n}\\rbrace \\subseteq I$ with $i_{1}<...<i_{n}$ is a generating set (resp.",
"basis) of the $R-$ module $\\Lambda ^{n}(M)$ .",
"$\\Box $ Corollary 6.4 If $M$ is a free $R-$ module of rank $n$ then $\\Lambda ^{i}(M)$ is a free $R-$ module of rank $\\binom{n}{i}$ for $0\\le i\\le n$ and $\\Lambda ^{i}(M)=0$ for all $i>n$ .",
"$\\Box $ Let $\\varphi :M\\rightarrow N$ be a $R-$ linear map.",
"By the universal property of the exterior powers, then there is a unique $R-$ linear map $\\Lambda ^{n}(\\varphi ):\\Lambda ^{n}(M)\\rightarrow \\Lambda ^{n}(N)$ which maps each pure wedge $x_{1}\\wedge ...\\wedge x_{n}$ into $\\varphi (x_{1})\\wedge ...\\wedge \\varphi (x_{n})$ .",
"In fact, $\\Lambda ^{n}(-)$ is an additive functor from the category of $R-$ modules into itself.",
"Lemma 6.5 Let $(M_{i}, \\varphi _{i,j})$ be an inductive (direct) system of $R-$ modules and $R-$ homomorphisms on the directed set $(I,<)$ .",
"Then for each fixed $n\\ge 0$ , $\\big (\\Lambda ^{n}(M_{i}), \\Lambda ^{n}(\\varphi _{i,j})\\big )$ is an inductive system of $R-$ modules and $R-$ homomorphisms over the same directed set.",
"Moreover $\\operatorname{colim}\\limits _{i\\in I}\\Lambda ^{n}(M_{i})$ as $R-$ module is canonically isomorphic to $\\Lambda ^{n}(\\operatorname{colim}\\limits _{i\\in I}M_{i})$ .",
"Proof.",
"Easy.",
"$\\Box $ Lemma 6.6 The $R-$ module $\\Lambda ^{n}(M)$ is projective (resp.",
"flat ) whenever $M$ is projective (resp.",
"flat).",
"Proof.",
"First assume that $M$ is projective.",
"Consider the exact sequence ${0[r]&N[r]^{\\varphi }&F[r]^{\\psi }&M[r]&0}$ where $F$ is a free $R-$ module.",
"It is split since $M$ is projective.",
"In an abelian category, every exact and split sequence is left exact and split by an additive functor.",
"Therefore the following sequence is exact and split ${0[r]&\\Lambda ^{n}(N)[r]^{\\Lambda ^{n}(\\varphi )}&\\Lambda ^{n}(F)[r]^{\\Lambda ^{n}(\\psi )}&\\Lambda ^{n}(M)[r]&0.",
"}$ It follows that $\\Lambda ^{n}(F)\\simeq \\Lambda ^{n}(M)\\oplus \\Lambda ^{n}(N)$ .",
"By Lemma REF , $\\Lambda ^{n}(F)$ is a free $R-$ module.",
"Thus $\\Lambda ^{n}(M)$ is $R-$ projective.",
"Now suppose $M$ is a flat $R-$ module.",
"The flatness of $\\Lambda ^{n}(M)$ is an immediate consequence of Corollary REF , Lemma REF and [3].",
"$\\Box $ Theorem 6.7 Let $R\\rightarrow S$ be a ring homomorphism and let $M$ be a $R-$ module.",
"Then $\\Lambda ^{n}(M)\\otimes _{R}S$ as $S-$ module is canonically isomorphic to $\\Lambda _{S}^{n}(M\\otimes _{R}S)$ .",
"Proof.",
"Let $N=M\\otimes _{R}S$ .",
"Consider the canonical map $M^{n}\\rightarrow \\Lambda _{S}^{n}(N)$ which maps each $n$ -tuple $(x_{1},...,x_{n})$ into $(x_{1}\\otimes 1)\\wedge ...\\wedge (x_{n}\\otimes 1)$ .",
"Clearly it is $R-$ multilinear and alternative.",
"Therefore there is a (unique) $R-$ linear map $\\Lambda ^{n}(M)\\rightarrow \\Lambda _{S}^{n}(N)$ which maps each pure wedge $x_{1}\\wedge ...\\wedge x_{n}$ into $(x_{1}\\otimes 1)\\wedge ...\\wedge (x_{n}\\otimes 1)$ .",
"Then we obtain the $S-$ linear map $\\varphi :\\Lambda ^{n}(M)\\otimes _{R}S\\rightarrow \\Lambda _{S}^{n}(N)$ which maps each pure tensor $(x_{1}\\wedge ...\\wedge x_{n})\\otimes s$ into $(x_{1}\\otimes s)\\wedge (x_{2}\\otimes 1)\\wedge ...\\wedge (x_{n}\\otimes 1)$ .",
"We will find the inverse of $\\varphi $ as follows.",
"Consider the canonical map $f:N^{n}\\rightarrow \\Lambda ^{n}(M)\\otimes _{R}S$ which maps each $n$ -tuple of pure tensors $(x_{1}\\otimes s_{1},...,x_{n}\\otimes s_{n})$ into $(x_{1}\\wedge ...\\wedge x_{n})\\otimes s_{1}...s_{n}$ .",
"Clearly it is $S-$ multilinear.",
"It is also alternative.",
"Because by Lemma REF , it suffices to show that $f(z_{1},...,z_{n})=0$ whenever $z_{k}=z_{k+1}=\\sum \\limits _{i=1}^{d}x_{i}\\otimes s_{i}$ for some $k$ .",
"But $f(z_{1},...,z_{n})$ is a finite sum of elements of the form $\\sum \\limits _{1\\le i,j\\le d}(x^{\\prime }_{1}\\wedge ...\\wedge x^{\\prime }_{k-1}\\wedge x_{i}\\wedge x_{j}\\wedge x^{\\prime }_{k+2}\\wedge ...\\wedge x^{\\prime }_{n})\\otimes s^{\\prime }_{1}...s^{\\prime }_{k-1}s_{i}s_{j}s^{\\prime }_{k+2}...s^{\\prime }_{n}$ .",
"We show that each of them is zero.",
"In fact, it suffices to show that $\\sum \\limits _{1\\le i,j\\le d}(x_{i}\\wedge x_{j})\\otimes s_{i}s_{j}=0$ .",
"We act by induction on $d$ .",
"If $d\\ge 2$ then we may write $\\sum \\limits _{1\\le i,j\\le d}(x_{i}\\wedge x_{j})\\otimes s_{i}s_{j}=\\sum \\limits _{j=1}^{d-1}\\Big (\\sum \\limits _{i=1}^{d-1}(x_{i}\\wedge x_{j})\\otimes s_{i}s_{j}+(x_{d}\\wedge x_{j})\\otimes s_{d}s_{j}\\Big )+\\sum \\limits _{i=1}^{d}(x_{i}\\wedge x_{d})\\otimes s_{i}s_{d}=\\sum \\limits _{j=1}^{d-1}(x_{d}\\wedge x_{j})\\otimes s_{d}s_{j}-\\sum \\limits _{i=1}^{d-1}(x_{d}\\wedge x_{i})\\otimes s_{d}s_{i}=0$ .",
"Therefore there is a (unique) $S-$ linear map $\\psi :\\Lambda ^{n}(N)\\rightarrow \\Lambda ^{n}(M)\\otimes _{R}S$ which maps each pure wedge of the form $(x_{1}\\otimes s_{1})\\wedge ...\\wedge (x_{n}\\otimes s_{n})$ into $(x_{1}\\wedge ...\\wedge x_{n})\\otimes s_{1}...s_{n}$ .",
"Clearly $\\varphi \\circ \\psi $ and $\\psi \\circ \\varphi $ are the identity maps.",
"$\\Box $"
]
] | 1612.05745 |
[
[
"A New Class of Anisotropic Charged Compact Star"
],
[
"Abstract A new model of charged compact star is reported by solving the Einstein-Maxwell field equations by choosing a suitable form of radial pressure.",
"The model parameters $\\rho$, $p_r$, $p_{\\perp}$ and $E^{2}$ are in closed form and all are well behaved inside the stellar interior.",
"A comparative study of charged and uncharged model is done with the help of graphical analysis."
],
[
"Introduction",
"To find the exact solution of Einstein's field equations is difficult due to its non-linear nature.",
"A large number of exact solutions of Einstein’s field equations in literature but not all of them are physically relevant.",
"A comprehensive collection of static, spherically symmetric solutions are found in [2] and [1].",
"A large collection of models of stellar objects incorporating charge can be found in literature.",
"[3] proposed that a fluid sphere of uniform density with a net surface charge is more stable than without charge.",
"An interesting observation of [4] is that in the presence of charge, the gravitational collapse of a spherically symmetric distribution of matter to a point singularity may be avoided.",
"Charged anisotropic matter with linear equation of state is discussed by [5].",
"[6] found that the solutions of Einstein-Maxwell system of equations are important to study the cosmic censorship hypothesis and the formation of naked singularities.",
"The presence of charge affects the values for redshifts, luminosities, and maximum mass for stars.",
"Charged perfect fluid sphere satisfying a linear equation of state was discussed by [7].",
"Regular models with quadratic equation of state was discussed by [8].",
"They obtained exact and physically reasonable solution of Einstein-Maxwell system of equations.",
"Their model is well behaved and regular.",
"In particular there is no singularity in the proper charge density.",
"[9] considered a self gravitating, charged and anisotropic fluid sphere.",
"To solve Einstein-Maxwell field equation they have assumed both linear and nonlinear equation of state and discussed the result analytically.",
"[10] extend the work of [11] by considering quadratic equation of state for the matter distribution to study the general situation of a compact relativistic body in presence of electromagnetic field and anisotropy.",
"[12] investigated that for highly compact astrophysical objects like X-ray pulsar, Her-X-1, X-ray buster 4U 1820-30, millisecond pulsar SAX J 1804.4-3658, PSR J1614-2230, LMC X-4 etc.",
"having core density beyond the nuclear density $(\\sim ~10^{15}gm/cm^{3})$ there can be pressure anisotropy, i.e, the pressure inside these compact objects can be decomposed into two parts radial pressure $p_r$ and transverse pressure $p_\\perp $ perpendicular direction to $p_r$ .",
"$\\Delta =p_r-p_\\perp $ is called the anisotropic factor which measures the anisotropy.",
"The reason behind these anisotropic nature are the existence of solid core, in presence of type 3A superfluid [13], phase transition [14], pion condensation [15], rotation, magnetic field, mixture of two fluid, existence of external field etc.",
"Local anisotropy in self gravitating systems were studied by [16].",
"[17] demonstrated that pressure anisotropy affects the physical properties, stability and structure of stellar matter.",
"Relativistic stellar model admitting a quadratic equation of state was proposed by [18] in finch-skea spacetime.",
"[19] has generalized earlier work in modified Finch-Skea spacetime by incorporating a dimensionless parameter n. In a very recent work [20] obtained a new model of an anisotropic superdense star which admits conformal motions in the presence of a quintessence field which is characterized by a parameter $\\omega _q$ with $-1 < \\omega _q < -1/3$ .",
"The model has been developed by choosing [21] ansatz.",
"[22] have studied the behavior of static spherically symmetric relativistic objects with locally anisotropic matter distribution considering the Tolman VII form for the gravitational potential $g_{rr}$ in curvature coordinates together with the linear relation between the energy density and the radial pressure.",
"Charged anisotropic star on paraboloidal spacetime was studied by [29].",
"[30] studied anisotropic star on pseudo-spheroidal spacetime.",
"Charged anisotropic star on pseudo-spheroidal spacetime was studied by [31].",
"The study of compact stars having Matese and Whitman mass function was carried out by [32].",
"Motivated by these earlier works in the present paper we develop a model of compact star by incorporating charge.",
"Our paper is organized as follows: In section 2, interior spacetime and the Einstein-Maxwell system is discussed.",
"Section 3 deals with solution of field equations.",
"Section 4 contains exterior spacetime and matching conditions.",
"Physical analysis of the model is discussed in section 5.",
"Section 6 contains conclusion."
],
[
"Interior Spacetime",
"We consider the static spherically symmetric spacetime metric as, $ds^{2}=e^{\\nu (r)}dt^{2}-e^{\\lambda (r)}dr^{2}-r^{2}\\left(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2} \\right).$ Where $\\nu $ and $\\lambda $ are functions of the radial coordinate `r' only.",
"Einstein-Maxwell Field Equations is given by $R_{i}^{j}-\\frac{1}{2}R\\delta _{i}^{j}=8\\pi \\left(T_{i}^{j}+\\pi _{i}^{j}+E_{i}^{j} \\right),$ where, $T_{i}^{j}=\\left(\\rho +p \\right)u_{i}u^{j}-p\\delta _{i}^{j},$ $\\pi _{i}^{j}=\\sqrt{3}S\\left[c_{i}c^{j}-\\frac{1}{2}\\left(u_{i}u^{j}-\\delta _{i}^{j} \\right) \\right],$ and $E_{i}^{j}=\\frac{1}{4\\pi }\\left(-F_{ik}F^{jk}+\\frac{1}{4}F_{mn}F^{mn}\\delta _{i}^{j} \\right).$ Here $\\rho $ is proper density, $p$ is fluid pressure, $u_{i}$ is unit four velocity, $S$ denotes magnitude of anisotropic tensor and $C^{i}$ is radial vector given by $\\left(0,-e^{-\\lambda /2},0,0 \\right)$ .",
"$F_{ij}$ denotes the anti-symmetric electromagnetic field strength tensor defined by $F_{ij}=\\frac{\\partial A_{j}}{\\partial x_{i}}-\\frac{\\partial A_{i}}{\\partial x_{j}},$ that satisfies the Maxwell equations $F_{ij,k}+F_{jk,i}+F_{ki,j}=0,$ and $\\frac{\\partial }{\\partial x^{k}}\\left(F^{ik}\\sqrt{-g} \\right)=4\\pi \\sqrt{-g}J^{i},$ where $g$ denotes the determinant of $g_{ij}$ , $A_{i}=\\left(\\phi (r), 0, 0, 0 \\right)$ is four-potential and $J^{i}=\\sigma u^{i},$ is the four-current vector where $\\sigma $ denotes the charge density.",
"The only non-vanishing components of $F_{ij}$ is $F_{01}=-F_{10}$ .",
"Here $F_{01}=-\\frac{e^{\\frac{\\nu +\\lambda }{2}}}{r^{2}}\\int _{0}^{r} 4\\pi r^{2}\\sigma e^{\\lambda /2}dr,$ and the total charge inside a radius $r$ is given by $q(r)=4\\pi \\int _{0}^{r} \\sigma r^{2}e^{\\lambda /2}dr.$ The electric field intensity $E$ can be obtained from $E^{2}=-F_{01}F^{01}$ , which subsequently reduces to $E=\\frac{q(r)}{r^{2}}.$ The field equations given by (REF ) are now equivalent to the following set of the non-linear ODE's $\\frac{1-e^{-\\lambda }}{r^{2}}+\\frac{e^{-\\lambda }\\lambda ^{\\prime }}{r}=8\\pi \\rho +E^{2}, $ $\\frac{e^{-\\lambda }-1}{r^{2}}+\\frac{e^{-\\lambda }\\nu ^{\\prime }}{r}=8\\pi p_{r}-E^{2},$ $e^{-\\lambda }\\left(\\frac{\\nu ^{\\prime \\prime }}{2}+\\frac{\\nu ^{\\prime 2}}{4}-\\frac{\\nu ^{\\prime }\\lambda ^{\\prime }}{4}+\\frac{\\nu ^{\\prime }-\\lambda ^{\\prime }}{2r} \\right)=8\\pi p_{\\perp }+E^{2},$ where we have taken $p_{r}=p+\\frac{2S}{\\sqrt{3}},$ $p_{\\perp }=p-\\frac{S}{\\sqrt{3}}.$ $8\\pi \\sqrt{3}S=p_{r}-p_{\\perp }.$"
],
[
"Solution of Field Equations",
"To solve the above set of equations (REF )-(REF ) we take the mass function of the form $m(r)=\\frac{br^{3}}{2(1+ar^{2})},$ where `a' and `b' are two positive constants.",
"The mass function given in (REF ) is known as Matese & Whitman [23] mass function that gives a monotonic decreasing matter density which was used by [24] to model an anisotropic fluid star, [25] to develop a model of dark energy star, [26] to model a class of relativistic stars with a linear equation of state and [11] to model a charged anisotropic matter with linear equation of state.",
"Using the relationship $e^{-\\lambda }=1-\\frac{2m}{r}$ and equation (REF ) we get, $e^{\\lambda }=\\frac{1+ar^{2}}{1+(a-b)r^{2}}.$ From equation (REF ) and (REF ) we obtain $8\\pi \\rho =\\frac{3b+abr^{2}}{(1+ar^{2})^{2}}-E^{2}.$ We choose $E^{2}$ of the form $E^{2}=\\frac{\\alpha ar^{2}}{(1+ar^{2})^2},$ which is regular at the center of the star.",
"Substituting the expression of $E^{2}$ into (REF ) we get, $8\\pi \\rho =\\frac{3b+a(b-\\alpha )r^{2}}{(1+ar^{2})^{2}}.$ To integrate the equation (REF ) we take radial pressure of the form, $8\\pi p_{r}=\\frac{bp_{0}(1-ar^{2})}{(1+ar^{2})^{2}},$ where $p_{0}$ is a positive constant, the choice of $p_r$ is reasonable due to the fact that it is monotonic decreasing function of `r' and the radial pressure vanishes at $r=\\frac{1}{\\sqrt{a}}$ which gives the radius of the star.",
"From (REF ) and (REF ) we get, $\\nu ^{\\prime }=\\frac{(bp_{0}+b)r-a(bp_{0}+\\alpha -b)r^{3}}{(1+ar^{2})\\left[1+(a-b)r^{2} \\right]}.$ Integrating we get, $\\nu =log\\left\\lbrace \\frac{C\\left(1+ar^{2} \\right)^{\\left(\\frac{2bp_{0}+\\alpha }{2b} \\right)} }{\\left[\\left(b-a \\right)r^{2}-1 \\right]^{\\left[\\frac{\\left(b^{2}-2ab \\right)p_{0}+b^{2}-\\alpha a }{2b^{2}-2ab} \\right]} } \\right\\rbrace ,$ where $C$ is contant of integration, and the spacetime metric in the interior is given by $ds^{2}=\\left\\lbrace \\frac{C\\left(1+ar^{2} \\right)^{\\left(\\frac{2bp_{0}+\\alpha }{2b} \\right)} }{\\left[\\left(b-a \\right)r^{2}-1 \\right]^{\\left[\\frac{\\left(b^{2}-2ab \\right)p_{0}+b^{2}-\\alpha a }{2b^{2}-2ab} \\right]} } \\right\\rbrace dt^{2}-\\left[\\frac{1+ar^{2}}{1+\\left(a-b \\right)r^{2}} \\right]dr^{2}-r^{2}\\left(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2} \\right).$ From (REF ), (REF ) and (REF ), we have $8\\pi \\sqrt{3}S=\\frac{r^{2}\\left[A_{1}+A_{2}r^{2}+A_{3}r^{4} \\right]}{\\left[-4+B_{1}r^{2}+B_{2}r^{4}+B_{3}r^{6}+B_{4}r^{8} \\right]},$ where $A_{1}=b^{2}p_{0}^{2}+14b^{2}p_{0}-12abp_{0}+3b^{2}-12\\alpha a$ , $A_{2}=-2ab^{2}p_{0}^{2}+8ab^{2}p_{0}-8a^{2}bp_{0}-2\\alpha abp_{0}+2ab^{2}+8\\alpha ab-16\\alpha a^{2}$ , $A_{3}=a^{2}b^{2}p_{0}^{2}-4a^{2}b^{2}p_{0}+4a^{3}bp_{0}+2\\alpha a^{2}bp_{0}-a^{2}b^{2}+4\\alpha a^{2}b-4\\alpha a^{3}+\\alpha ^{2}a^{2}$ , $B_{1}=4b-16a$ , $B_{2}=12ab-24a^{2}$ , $B_{3}=12a^{2}b-16a^{3}$ and $B_{4}=4a^{3}b-4a^{4}$ .",
"From (REF ) we obtain, $8\\pi p_{\\perp }=\\frac{\\left[4bp_{0}+C_{1}r^{2}+C_{2}r^{4}+C_{3}r^{6}\\right]}{\\left[4-B_{1}r^{2}-B_{2}r^{4}-B_{3}r^{6}-B_{4}r^{8} \\right]},$ where, $C_{1}=b^{2}p_{0}^{2}-8abp_{0}+3b^{2}-12\\alpha a$ , $C_{2}=-2ab^{2}p_{0}^{2}+8ab^{2}p_{0}-12a^{2}bp_{0}-2\\alpha abp_{0}+2ab^{2}+8\\alpha ab-16\\alpha a^{2}$ , $C_{3}=a^{2}b^{2}p_{0}^{2}+2\\alpha a^{2}bp_{0}-a^{2}b^{2}+4\\alpha a^{2}b-4\\alpha a^{3}+\\alpha ^{2}a^{2}$ ."
],
[
"Exterior Spacetime and Matching Condition",
"we match our interior spacetime (REF ) to the exterior Reissner-Nordström spacetime at the boundary $r=r_b$ (where $r_b$ is the radius of the star.).",
"The exterior spacetime is given by the line element $ds^{2}=\\left(1-\\frac{2M}{r}+\\frac{q^{2}}{r^{2}} \\right)dt^{2}-\\left(1-\\frac{2M}{r}+\\frac{q^{2}}{r^{2}} \\right)^{-1}dr^{2}-r^{2}\\left(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2} \\right).$ By using the continuity of the metric potential $g_{rr}$ and $g_{tt}$ at the boundary $r=r_b$ we get, $e^{\\nu (r_b)}=1-\\frac{2M}{r_b}+\\frac{q^{2}}{r^{2}},$ $e^{\\lambda (r_b)}=\\left(1-\\frac{2M}{r_b}+\\frac{q^{2}}{r^{2}}\\right)^{-1}.$ The radial pressure should vanish at the boundary of the star, hence from equation (REF ) we obtain $a=\\frac{1}{r_b^{2}}.$ Using (REF ) & (REF ) we obtain $b=\\frac{4m}{r_b^{3}}.$ We compute the values of `a' and `b' for different compact stars which is given in table REF .",
"Figure: The matter density is plotted against r for the star PSR J1614-2230.Figure: The transverse pressure p t p_t is plotted against r for the star PSR J1614-2230."
],
[
"Physical Analysis",
"To be a physically acceptable model matter density $(\\rho )$ , radial pressure ($p_r$ ), transverse pressure ($p_\\bot $ ) all should be non-negative inside the stellar interior.",
"It is clear from equations (REF ) and (REF ) it is clear that $pr$ is positive throughout the distribution.",
"The profile of $\\rho $ and $p_\\bot $ are shown in fig.",
"1 and fig.",
"2 respectively.",
"From the figure it is clear that all are positive inside the stellar interior.",
"The profile of $\\frac{d \\rho }{dr},~\\frac{d p_r}{dr}$ and $\\frac{dp_{\\perp }}{dr}$ are shown in fig.",
"3, it is clearly indicates that $\\rho $ , $p_{r}$ and $p_{\\perp }$ are descreasing in radially outward direction.",
"According to [27] for an anisotropic fluid sphere the trace of the energy tensor should be positive.",
"To check this condition for our model we plot $\\rho -p_r-2p_{\\perp }$ against r in Fig.",
"4.",
"From the figure it is clear that our proposed model of compact star satisfies Bondi's conditions.",
"Figure: dρ dr\\frac{d\\rho }{dr}, dp r dr\\frac{dp_r}{dr} and dp ⊥ dr\\frac{dp_{\\perp }}{dr} are plotted against r for the star PSR J1614-2230.Figure: ρ-p r -2p t \\rho -p_r-2p_t is plotted against r for the star PSR J1614-2230.For a physically acceptable model of anisotropic fluid sphere the radial and transverse velocity of sound should be less than 1 which is known as causality conditions.",
"Where the radial velocity $(v_{sr}^{2})$ and transverse velocity $(v_{st}^{2})$ of sound can be obtained as $\\frac{dp_{r}}{d\\rho }=\\frac{bp_{0}(3-ar^{2})}{5b+\\alpha +a(b-\\alpha )r^{2}}.$ $\\frac{dp_{\\perp }}{d\\rho }=\\frac{(1+ar^{2})^{3}\\left[D_{1}+D_{2}r^{2}+D_{3}r^{4}+D_{4}r^{6}+D_{5}r^{8} \\right]}{\\left[-10ab-2a\\alpha -2a^{2}(b-\\alpha )r^{2} \\right]\\left[2+E_{1}r^{2}+E_{2}r^{4}+E_{3}r^{6}+E_{4}r^{8}+E_{5}r^{10}+E_{6}r^{12} \\right]}.$ where, $D_{1}=b^{2}p_{0}^{2}+4b^{2}p_{0}-24abp_{0}+3b^{2}-12\\alpha a$ , $D_{2}=-6ab^{2}p_{0}^{2}+32ab^{2}p_{0}-24a^{2}bp_{0}-4\\alpha abp_{0}-2ab^{2}+16\\alpha ab-8\\alpha a^{2}$ , $D_{3}=5ab^{3}p_{0}^{2}-8ab^{3}p_{0}+2\\alpha ab^{2}p_{0}-12a^{2}b^{2}p_{0}+24a^{3}bp_{0}+6\\alpha a^{2}bp_{0}+7ab^{3}-12a^{2}b^{2}-8\\alpha ab^{2}-8\\alpha a^{2}b+24\\alpha a^{3}+3\\alpha ^{2}a^{2}$ , $D_{4}=6a^{3}b^{b}p_{0}^{2}-6a^{2}b^{3}p_{0}^{2}+16a^{2}b^{3}p_{0}-40a^{3}b^{2}p_{0}-8\\alpha a^{2}b^{2}p_{0}+24a^{4}bp_{0}+8\\alpha a^{3}bp_{0}+6a^{2}b^{3}+8\\alpha a^{2}b^{2}-6a^{3}b^{2}-32\\alpha a^{3}b-2\\alpha ^{2}a^{2}b+24\\alpha a^{4}+2\\alpha ^{2}a^{3}$ , $D_{5}=a^{3}b^{3}p_{0}^{2}-a^{4}b^{2}p_{0}^{2}+2\\alpha a^{3}b^{2}p_{0}-2\\alpha a^{4}bp_{0}-a^{3}b^{3}+a^{4}b^{2}+4\\alpha a^{3}b^{2}+\\alpha ^{2}a^{3}b-8\\alpha a^{4}b+4\\alpha a^{5}-\\alpha ^{2}a^{4}$ , $E_{1}=12a-4b$ , $E_{2}=2b^{2}-20ab+30a^{2}$ , $E_{3}=8ab^{2}-40a^{2}b+40a^{3}$ , $E_{4}=12a^{2}b^{2}-40a^{3}b+30a^{4}$ , $E_{5}=8a^{3}b^{2}-20a^{4}b+12a^{5}$ and $E_{6}=2a^{4}b^{2}-4a^{5}b+2a^{6}$ .",
"Due to the complexity of the expression of $v_{st}^{2}$ we prove the causality conditions with the help of graphical representation.",
"The graphs of $(v_{sr}^{2})$ and $(v_{st}^{2})$ have been plotted in fig.",
"5 and fig.",
"6 respectively.",
"From the figure it is clear that $0 <v_{sr}^{2}\\le 1$ and $0<v_{st}^{2} \\le 1$ everywhere within the stellar configuration.",
"Moreover $\\frac{dp_t}{d\\rho } $ and$\\frac{dp_r}{d\\rho } $ are monotonic decreasing function of radius `r' for $0\\le r \\le r_b$ which implies that the velocity of sound is increasing with the increase of density.",
"A relativistic star will be stable if the relativistic adiabatic index $\\Gamma >\\frac{4}{3}$ .",
"where $\\Gamma $ is given by $\\Gamma =\\frac{\\rho +p_r}{p_r}\\frac{dp_r}{d\\rho }$ To see the variation of the relativistic index we plot $\\Gamma $ for our present of compact star which is plotted in fig.",
"7.",
"The figure ensures that our model is stable.",
"Figure: v sr 2 =dp r dρv_{sr}^{2}=\\frac{dp_r}{d\\rho } is plotted against r for the star PSR J1614-2230.Figure: v st 2 =dp ⊥ dρv_{st}^{2}=\\frac{dp_\\perp }{d\\rho } is plotted against r for the star PSR J1614-2230.Figure: The adiabatic index Γ\\Gamma is plotted against r for the star PSR J1614-2230.Table: The values of `a' and `b' obtained from the equation () and ()Table: The values of central density, surface density, central pressure and radial velocity of the sound at the origin for different compact stars are obtained.For an anisotropic fluid sphere all the energy conditions namely Weak Energy Condition (WEC), Null Energy Condition (NEC), Strong Energy Condition (SEC) and Dominant Energy Condition (DEC) are satisfied if and only if the following inequalities hold simultaneously in every point inside the fluid sphere.",
"Figure: The left and middle figures show the dominant energy conditions where as the right figure shows the weak null and strong energy conditions are satisfied by our model for the star PSR J1614-2230.$(i)NEC:\\rho +p_r\\ge 0$ $(ii)WEC:p_r+\\rho \\ge 0,~~~\\rho >0$ $(iii)SEC:\\rho +p_r\\ge 0~~~~\\rho +p_r+2p_{\\perp }\\ge 0$ $(iv)DEC:\\rho >\\left|p_r\\right| ~~~, \\rho >\\left|p_{\\perp }\\right|$ Due to the complexity of the expression of $p_{\\perp }$ we will prove the inequality (REF )-(REF ) with the help of graphical representation.",
"The profiles of the L.H.S of the above inequalities are depicted in fig.",
"8 for the compact star PSR J1614-2230.",
"The figure shows that all the energy conditions are satisfied by our model of compact star.",
"Figure: Variation of anisotropy is shown against r for the star PSR J1614-2230.The ratio of mass to the radius of a compact star can not be arbitrarily large.",
"[28] showed that for a (3+1)-dimensional fluid sphere $\\frac{2M}{r_b}<\\frac{8}{9}$ .",
"To see the maximum ratio of mass to the radius for our model we calculate the compactness of the star given by $u(r)=\\frac{m(r)}{r}=\\frac{br^{2}}{2(1+ar^{2})},$ and the corresponding surface redshift $z_s$ is obtained by, $1+z_s(r_b)=\\left[1-2u(r_b)\\right]^{-1/2}$ .",
"Therefore $z_s$ can be obtained as, $z_s(r_b)=\\left[\\frac{1+(a-b)r_b^{2}}{1+ar_b^{2}}\\right]^{-\\frac{1}{2}}-1.$ The surface redshift of different compact stars are given in table REF .",
"Figure: The variation of electric field is shown against r for the star PSR J1614-2230."
],
[
"Conclusion",
"We have obtained a new class of solution for charged compact stars having [23] mass function.",
"The electric field intensity is increasing in radially outward direction and the adiabatic index $\\Gamma >\\frac{4}{3}$ .",
"The physical requirements are checked for the star PSR J1614-2230 and model satisfies all the physical conditions.",
"Some salient features of the model are (i) In present model if $\\alpha =0$ , the model corresponds to [32] model.",
"(ii) In present model if $\\alpha =0$ , $a=b=\\frac{1}{R^{2}}$ , where $R$ is geometric parameter then the model corresponds to [18] model, which is stable for $\\frac{1}{3}<p_{0}<0.3944$ ."
],
[
"Acknowledgement",
"BSR is thankful to IUCAA, Pune, for providing the facilities and hospitality where the part of this work was done."
]
] | 1612.05417 |
[
[
"On-bird Sound Recordings: Automatic Acoustic Recognition of Activities\n and Contexts"
],
[
"Abstract We introduce a novel approach to studying animal behaviour and the context in which it occurs, through the use of microphone backpacks carried on the backs of individual free-flying birds.",
"These sensors are increasingly used by animal behaviour researchers to study individual vocalisations of freely behaving animals, even in the field.",
"However such devices may record more than an animals vocal behaviour, and have the potential to be used for investigating specific activities (movement) and context (background) within which vocalisations occur.",
"To facilitate this approach, we investigate the automatic annotation of such recordings through two different sound scene analysis paradigms: a scene-classification method using feature learning, and an event-detection method using probabilistic latent component analysis (PLCA).",
"We analyse recordings made with Eurasian jackdaws (Corvus monedula) in both captive and field settings.",
"Results are comparable with the state of the art in sound scene analysis; we find that the current recognition quality level enables scalable automatic annotation of audio logger data, given partial annotation, but also find that individual differences between animals and/or their backpacks limit the generalisation from one individual to another.",
"we consider the interrelation of 'scenes' and 'events' in this particular task, and issues of temporal resolution."
],
[
"Introduction",
"Studying the behaviour of animals in real time and in their natural environments is becoming more and more feasible through the use of animal-borne loggers or other remote sensing technology [1].",
"These technologies have provided insight into different aspects of physiology and behaviour, such as heartbeat [2] or migratory routes [3], [4], which in turn can help us understand basic mechanisms up to evolutionary drivers, as well as support decision-making processes in nature conservation or disease management.",
"To reconstruct daily activity patterns, many remote-sensing studies have used methods that provide information on the location of an animal in space (today most commonly GPS: Global Positioning System).",
"To get more fine-scale information, spatial data have been combined with accelerometry which can shed more light on the actual activities of an animal [5], [1].",
"However, the immediate causes or related contexts of specific animal behaviours were often not identifiable through these technologies, and required additional information sources.",
"Recently, microphone backpacks have become useful tools to investigate different aspects of vocal behaviour in naturalistic contexts, even in small animals [6], [7], [8], [9].",
"By picking up the vocal sounds close to their production origin, researchers are now able to record and identify vocalisations from the signal-emitting individuals, even in physically or acoustically challenging environments.",
"Recording close to the origin also reduces the influence of propagation effects on the audio suchas dispersion or echoes.",
"But in small animals, unlike for example in whales [10], it is often not (yet) possible to apply tags that provide multiple channels of information simultaneously, due to weight limitations—especially in birds.",
"Thus, placing vocal behaviour into relevant context can be limited to specific situations in which a simultaneous collection of further data is possible.",
"Because an on-board microphone moves along with its bearer, most microphone backpacks do not exclusively record vocalisations, but also other sounds.",
"Firstly, depending on their sensitivity, the microphones have the potential to pick up a variety of background sounds.",
"Secondly, specific movement patterns of the animal resulting in characteristic sound patterns might reveal aspects of the animal's behaviour, e.g.",
"“running” or “self-scratching” (noted by [7], [8]).",
"But, to date, this has not been investigated in detail."
],
[
"Automatic Acoustic Recognition",
"Successful identification of animal-related sounds could provide a unique opportunity because it may allow investigating not only the behaviour of the animal itself, but also different aspects of its abiotic and biotic environment—which is currently not possible by recording the spatial position or movement of single individuals, without further data collection.",
"This in turn could be useful for various purposes (as above: from basic research to conservation, e.g.",
"effects of anthropogenic noise), but analysing such signals/soundscapes remains a challenge to date.",
"Manual annotation is possible for small datasets, though hard to scale up; further, for free-flying birds there will usually be no visual/video support for manual annotation.",
"Hence there is strong potential for microphone backpack methodologies to be augmented by automatic acoustic recognition of bird activities and their contexts.",
"The problem of automatic animal context recognition from audio is directly related to the emerging field of sound scene analysis (also termed acoustic scene analysis), and more specifically to the two core problems in the field, namely sound scene analysis and sound event detection [11].",
"Since the context in question can refer either to an animal's current activity or background sounds, the problem can be viewed as either or both of searching for specific acoustic events (e.g.",
"related to flapping wings in the context of flying) or evaluating the overall properties of a continuous sound scene (e.g.",
"background sounds indicating that an individual is based in a nest).",
"The vast majority of approaches in the field of sound scene analysis either fall directly into the problem of sound scene recognition (which typically refers to identifying scenes based on location-specific characteristics, e.g.",
"park, car, kitchen) or the problem of sound event detection (which refers to identifying instances of sound events with a start and end time, e.g.",
"door slam, scream) [11].",
"An approach that is closer to the present work is proposed by Eronen et al.",
"[12], who developed a computationally efficient classification-based system for audio-based context recognition in urban environments, where `context' referred to both locations (e.g.",
"train, street) but also to specific activities (e.g.",
"construction, meeting).",
"In [13], Heittola et al.",
"proposed a system for sound event detection, which is however dependent on the context of each sound scene.",
"A system based on hidden Markov models (HMMs) with multiple Viterbi decoding was proposed, which was able to identify to a relative degree of success 60 types of sound events, being present in 10 different types of location-dependent audio-related contexts.",
"Another related strand of research is speaker diarisation, in which multi-party speech recordings are analysed such as discussions in meetings, and the primary goal is to recover a transcript of which party spoke when [14], [15].",
"In speaker diarisation, the emphasis is primarily on speech and so the range of sound types considered is often highly constrained.",
"Also the targets of transcription are individual speaking sources rather than aggregate contextual categories.",
"Much work in speaker diarisation treats the transcription task as monophonic (only one speaker at a time), although recent directions are beginning to address overlapping speech [15].",
"Generalisation across different domains (e.g.",
"conference meetings versus broadcast news) is also an open topic, indicating the difficulty of these types of problem in general.",
"When placing the present study in context with related work in sound scene analysis, it is important to maintain a focus on the downstream use of the data, which must influence the way we design and evaluate systems.",
"Typical applications in animal behaviour include: (a) aggregating timelines to produce an overall model of a species' diurnal cycle of activity, or creating “time budgets”; (b) data-mining to search for one or many instances of a particular phenomenon.",
"A transcript is rarely the end goal in itself.",
"As an example consequence of this, for the applications just mentioned it may often be helpful to obtain a probabilistic or confidence-weighted output rather than merely a list of events, for optimal combination of information or best guidance of subsequent manual effort.",
"The aims of this study were thus to find out whether the recordings from microphone backpacks could be useful for investigating the immediate context in which individual vocalisations occur, such as an animal's current activity (movement sound) or vocalising conspecifics (background sound), and to investigate the extent to which this could be facilitated by automatic acoustic recognition.",
"To do so, we used video-validated and human-coded on-bird sound recordings from captive and free-flying jackdaws (Corvus monedula), to test the performance of different automatic recognition algorithms.",
"We experimentally compared two different sound recognition paradigms (classification and event detection), as well as combinations and variants, and how they performed in terms of recognising the various categories of activity and context that are of interest for measuring animal behaviour.",
"In the following we describe the data collection process (Section ) before giving details of our two automatic recognition systems (Section ).",
"Our evaluation method and its results are presented in Section , and then in discussion (Section ) we consider the implications of our study for the automatic annotation of animal-attached sound recordings."
],
[
"Birds and microphone backpacks",
"For the current study, we used a subset of on-bird sound recordings obtained during a different study (Gill et al., in preparation).",
"The analysed data were collected in the South of Germany, from 12 individual jackdaws (Corvus monedula, 7 captive-housed and 6 free-living), early in the years of 2014 and 2015.",
"Backpack application was approved by the Government of Upper Bavaria and in compliance with the European directives for the protection of animals used for scientific purposes (2010/63/EU).",
"The backpacks consisted of a commercially available digital voice recorder (Edic Mini Tiny A31, TS-Market Ltd., Russia), a rechargeable battery (ICP581323PA to ICP402035, Renata, Switzerland), a radio transmitter for relocation (BD-2 Holohil, Canada) and a shrinking tube casing.",
"Loggers were charged, programmed and read out via PC connection and the according software (RecManager, version 2.11.19, Telesystems, Russia).",
"They were set to record continuously for a few hours every morning, for a few days, beginning one day post capture (at 22050 Hz sampling rate, uncompressed .wav format).",
"This provided coherent vocalisation data and acoustic background information, as opposed to using amplitude-based triggers (but at a cost of storage and battery).",
"For backpack attachment, birds were either trained to fly inside a smaller compartment of the aviary where they were caught using bird nets (captivity), or trapped inside their nest boxes (wild).",
"The backpacks were fitted using approved attachment methods (glue, or via a harness similar to [16]), and following common recommendations ($<5\\%$ of body weight [17]; close to centre of gravity [18]).",
"Birds were individually identified by colour rings.",
"After capture and backpack attachment (20 mins $\\pm $ 4.1 SD), they were observed using binoculars and/or radio-telemetry, and all of them were immediately able to fly upon release.",
"For further details on procedures and animal welfare, see Gill et al.",
"(in preparation)."
],
[
"Video-validation of sounds",
"For a video-validation of on-bird sound data, video footage was collected from the captive birds during backpack recording hours.",
"For this, an observer sat inside the aviary and video-recorded focal birds using a handheld camcorder (JVC Camcorder Everio GZ-MG77E, Japan).",
"All sound files used for video validation were processed, played back, visualised (waveform or spectrograms: FFT window size 512, Hann, 0–10000 Hz viewing range, gain 20–35 dB, range 45 dB) and annotated in Audacity (Version 2.0.5) by LFG.",
"Corresponding sound and video files were cut to match, and were then played back simultaneously, at normal speed (using Audacity, see above, and using VLC, Version 2.1.5).",
"First, the sounds were annotated step-by-step with the corresponding visual information (see Table REF ).",
"If the focal bird was temporarily out of sight, this was labelled as missing data.",
"Secondly, labels were added for acoustically distinct background sounds, such as vocalising jackdaws.",
"Next, the annotation track (labels, start and end points) of each recording was exported as a text file.",
"To balance between fine detail and sufficient sample size, the original labels were used to create slightly broader behavioural and contextual categories (Table REF ).",
"An example clip of annotated data is visualised in Figure REF (a).",
"In Supplementary Information we provide videos showing the studied birds in some example contexts, along with standard and backpack microphone recordings to illustrate the characteristics of the specific kind of sound recordings dealt with in this work."
],
[
"Annotation of field data",
"Having worked with hours of sound and video recordings from jackdaw backpacks, we had learned a good deal about the acoustic representation of behaviours and were able to annotate the sounds in new files in almost as much detail as in combination with the according visual information (at least at the behavioural category level).",
"Thus, the field recording subset was annotated by LFG based on aural and visual inspection of sounds, as learned from the captive dataset and from observations in the field, but also taking into account differences in the sounds due to different materials in the field (e.g.",
"walking on different substrates), as well as different durations (e.g.",
"prolonged flight).",
"Two labels were added that had not been recorded in captivity: copulations; begging chicks inside the nest (Table REF ).",
"Table: Labelling scheme for the actions/contexts in our recordings.",
"The “Category” column gives the class labels used in the present study, with the other columns indicating the broader or more specific labelling used during manual transcription."
],
[
"Automatic recognition",
"To train and then test recognition algorithms, we used a total of 8.4 hours of video-validated (captive: 43–100 minutes per bird) and 18.5 hours of human-coded (wild: 164–198 minutes per bird) sound recordings and their respective annotations.",
"We next describe the automatic recognition systems that we evaluated, which are summarised in Figure REF .",
"Figure: Overview of the processing workflows used for automatic recognition."
],
[
"Classifier-based System",
"The first system we used for activity and context recognition sits within the classification-based paradigm.",
"We used our feature learning and classification method previously developed for bird species classification from vocalisations [19].",
"Importantly, this approach applies spherical k-means feature learning to Mel-spectrogram patches, in order to transform the input signal into a rich feature space suitable for applying a standard classifier.",
"This particular feature learning algorithm is conceptually related to an unsupervised convolutional neural network, but its simplicity makes it eminently scalable to big data [20], [19].",
"In this work, we segmented input audio into contiguous five-second clips, from which we calculated Mel spectrograms (FFT window size 1024 with 50% overlap), and applied median-clipping noise reduction to each frequency band.",
"Unlike in the cited previous work, for these data we did not apply high-pass filtering, since we expected some classes to be indicated in part by lower-frequency or broadband components.",
"During training we applied a single pass of the feature learning decribed in [19] to these data, learning a high-dimensional projection onto 500 features.",
"We then transformed the training and test data into this new feature space, before summarising each audio clip by the mean and standard deviation of each feature (i.e.",
"1000 summary features).",
"The summary features were used as input to a random forest classifier [21] having 200 trees and trained using an entropy-based criterion for splitting branches.",
"These settings led to good performance in previous work [19].",
"The data in this task is highly unbalanced, with some classes very sparsely represented.",
"A random forest classifier is typically able to handle unbalanced (and high-dimensional) data well.",
"However, an option available to us was to reweight the data to give equal prominence to positive and negative classes.",
"This was particularly pertinent as the subsequent HMM postprocessing (see subsection REF ) also makes use of the relative class balance.",
"We therefore trained the classifier in both modes, equally weighted and balanced-reweighted, to inspect the effect of this choice."
],
[
"Event Detection System",
"The second system used for activity and context recognition is adapted from the system of [22], which was originally proposed for sound event detection in office environments.",
"Thus, this approach attempts to recognize contexts as a collection of acoustic events related to each context, as opposed to the previous approach which was based on modelling the overall characteristics of an acoustic scene.",
"The system extends probabilistic latent component analysis (PLCA) [23], a spectrogram factorisation technique which can be viewed as the probabilistic counterpart of non-negative matrix factorization (NMF) [24].",
"The PLCA-based model assumes that an audio spectrogram can be decomposed as a series of sound activities or contexts, which can potentially overlap over time.",
"Each activity is produced as a combination of sound exemplars, which have been pre-computed from training data.",
"For preprocessing, a time-frequency representation $V_{f,t}$ ($f$ is the frequency index and $t$ is the time index) is computed by processing the input waveform with an equivalent rectangular bandwidth (ERB) filterbank [25], using the approach of [26].",
"The filterbank uses 250 filters which are linearly spaced between 5 Hz and 10.8 kHz on the ERB scale, and has a 23ms time step.",
"Given that in the context of on-bird sound recordings several activities exhibit information in higher frequencies, a linear pre-emphasis filter is applied to $V_{f,t}$ for boosting high frequency content.",
"See Figure REF (b) for an ERB spectrogram of a recording from the captive subset, along with the respective context annotation.",
"Figure: (a) Context annotations for a recording segment from a captive bird.",
"(b) The annotations for focal and non-focal calls and respective ERB spectrogram of the same recording, both corresponding to the temporal region marked with vertical dashed lines in figure (a).The PLCA-based model takes as input $V_{f,t}$ and approximates it as a bivariate probability distribution $P(f,t)$ , which is in turn decomposed into a series of spectral templates per sound activity/context and exemplar index, activations over time for each context class, as well as an auxiliary probability for the activation of each exemplar per context class over time.",
"The model is formulated as: $P(f,t) = P(t)\\sum _{c,e}P(f|c,e)P(c|t)P(e|c,t) $ where $c\\in \\lbrace 1,\\ldots ,C\\rbrace $ denotes the context class and $e\\in \\lbrace 1,\\ldots ,E\\rbrace $ denotes the exemplar index.",
"On model parameters, $P(t)=\\sum _{f}V_{f,t}$ , which is a known quantity.",
"Dictionary $P(f|c,e)$ , which in this system is pre-computed from training data, contains spectral templates per context class $c$ and exemplar $e$ .",
"The main output of the PLCA model is $P(c|t)$ , which is the probability of an active context per time frame $t$ .",
"Finally, the model also contains the auxiliary probability $P(e|c,t)$ , which denotes the contribution of each exemplar $e$ for producing a context $c$ at time $t$ .",
"The unknown model parameters $P(c|t)$ and $P(e|c,t)$ can be iteratively estimated using the expectation-maximization (EM) algorithm [27].",
"For the E-step, the following posterior is computed: $P(c,e|f,t) = \\frac{P(f|c,e)P(c|t)P(e|c,t)}{\\sum _{c,e}P(f|c,e)P(c|t)P(e|c,t)} $ Using the above posterior, $P(c|t)$ and $P(e|c,t)$ can be estimated in the M-step as follows: $P(c|t) = \\frac{\\sum _{e,f}P(c,e|f,t)V_{f,t}}{\\sum _{c,e,f}P(c,e|f,t)V_{f,t}} $ $P(e|c,t) = \\frac{\\sum _{f}P(c,e|f,t)V_{f,t}}{\\sum _{e,f}P(c,e|f,t)V_{f,t}} $ Parameters $P(c|t)$ and $P(e|c,t)$ are initialised in the EM updates with random values between 0 and 1 and are normalised accordingly.",
"Eqs.",
"(REF ) and (REF )-(REF ) are iterated until convergence.",
"In our experiments, we found 30 iterations to be sufficient.",
"In order to extract dictionary $P(f|c,e)$ from training data, first spectra $V^{(c)} \\in \\mathbb {R}^{F\\times T_{c}}$ that correspond to an active context class are collected, where $T_{c}$ corresponds to the number of spectral frames that contain an active context class $c$ .",
"Then, for each context class a list of exemplars is created by performing clustering on $V^{(c)}$ using the k-means algorithm; here, the number of exemplars $E=40$ , following experiments on the training data.",
"The output of the PLCA model is given by $P(c,t) = P(t)P(c|t)$ , i.e.",
"the context activation probability, weighted by the energy of the spectrogram.",
"Since $P(c,t)$ is a non-binary representation, it needs to be converted into a list of estimated contexts per time frame.",
"The first option of post-processing $P(c,t)$ is by performing thresholding, where threshold values were estimated per context class using training data.",
"Finally, active contexts with a small duration (shorter than 120ms) were removed.",
"Additional post-processing options are discussed in the following subsection."
],
[
"Postprocessing",
"Given the output from either the classifier or PLCA detector, we then optionally applied hidden Markov model (HMM) postprocessing to the estimated event sequences.",
"See [28] for an overview of HMMs.",
"HMM-based postprocessing is a common procedure using knowledge about the temporal structure of event sequences (gleaned from the training set) which knowledge may not otherwise be reflected.",
"In particular, in our case the classifier treats each five-second segment as independent, neglecting information from neighbouring segments.",
"Likewise, the PLCA event detection system considers each 23 msec output frame as independent.",
"Since our task was polyphonic, having multiple “channels” in parallel whose activation could be on or off, there was a combinatorially large number of possible states at any time ($2^k$ , with $k$ the number of classes).",
"To deal with this large state space we applied the HMM in two alternative ways: (a) applying a single HMM to the entire system, whose set of possible states is the whole set of state combinations observed in the training data; or (b) independently applying a two-state, on/off HMM to the data of each class.",
"Each approach has advantages and drawbacks.",
"Treating channels as independent may lead to efficient training given a limited amount of data, but it neglects interaction effects which could help to resolve ambiguous situations.",
"Therefore we tested both approaches.",
"We trained the HMMs generatively, using Laplacian smoothing of the transition tables—i.e.",
"initialising each possible transition with a small uniform weight, which yields a prior equivalent to having observed one instance of each possible transition.",
"The emission model for each HMM state was a Gaussian mixture model (GMM).",
"To initialise and to select the number of GMM components, we applied the Dirichlet process GMM approach [29] to the entire training dataset (sometimes called a universal background model or UBM), then for each HMM state we trained its emission model by variational inference initialised from the UBM.",
"We used the GMM implementations provided by scikit-learn 0.17 [30].",
"Having trained a HMM, there are multiple ways to apply it to new data.",
"We explored the use of forward filtering—producing probabilistic “fuzzy” output which may then be thresholded if definite decisions are required—and Viterbi decoding—producing a single definite output, as the maximum likelihood state sequence given the observations.",
"This then resulted in four kinds of HMM postprocessing: filtered or Viterbi-decoded output, from a jointly or independently-trained HMM."
],
[
"Handling Missing Data",
"Occasional time-regions of the data were labelled as missing data (`NA'), when birds were occasionally off-camera.",
"These regions (around 17 minutes total, out of the 8.4 hours of captive audio) were excluded from the training of the classifiers and HMMs.",
"For the PLCA-based system, the NA class was not used to create the pre-extracted dictionary $P(f|c,e)$ , and any spectral frames belonging to the NA class were not used in the training data.",
"In the test phase, any NA regions in the ground truth are set to be non-active, where any time frames $t$ in the PLCA model output that correspond to the NA regions are set so that $P(c,t)=0$ .",
"`NA' regions were excluded from the calculation of our evaluation statistics, due to the lack of ground truth for comparison."
],
[
"Metrics",
"As discussed in Section , the evaluation must be designed with regard to the planned or typical downstream use case—i.e.",
"what tasks or analyses do we expect to follow on from such automatic annotation?",
"For the present task, this bears upon the figures of merit which one calculates, as well as on issues such as the temporal granularity or temporal tolerance.",
"It is desirable for an automatic system to recover exactly-timed transcriptions of every vocalisation, action and context given in the audio, but for some of the downstream tasks we consider the overriding aim does not require the highest resolution, for example when characterising time budgets across large datasets, or locating examples of certain activity.",
"Hence our main evaluation measures were calculated at a five-second granularity (the same granularity as was used for the classifier).",
"The output of the classifier-based system was itself at a five-second granularity; for the PLCA-based system, the output was sampled at 23ms steps, as in the input time-frequency representation $V_{ft}$ .",
"We therefore grouped its outputs into five-second segments, and the output for each 5-sec segment was either the mean or the maximum of the 23 msec-step frames corresponding to that time segment.",
"Evaluation metrics for automatic transcription have been debated in music informatics and in sound scene analysis.",
"Recently Mesaros et al.",
"reviewed such measures for general sound event detection, discussing issues including the use of high-resolution versus segment-based metrics [31].",
"In their terminology our main metrics are segment-based, using five-second segments.",
"However, Mesaros et al.",
"consider only the evaluation of “definite” transcripts, not transcripts with probabilistic/ranked/fuzzy annotations, and as a result their review does not include statistics useful for evaluating the latter type of output.",
"Foster et al., working with probabilistic outputs, use a four-second segment size and use the area under the ROC curve (“AUC”) as their figure of merit [32].",
"The AUC is widely used as an evaluation measure for detection and classification tasks, and has many desirable properties [33]: unlike raw accuracy, it is not impeded by “unbalanced” datasets having an uneven mixture of true-positive and true-negative examples; and it has a standard probabilistic interpretation, in that the AUC statistic tells us the probability that the algorithm will rank a random positive instance higher than a random negative instance.",
"This last feature makes it particularly suitable to evaluating with regard to downstream tasks in which the subsequent postprocessing will for example involve manually confirming/refining the separation of positive and negative instances.",
"Hand criticises the AUC statistic [34], but reluctantly confirms that its use is well-founded when the downstream makes use of the ranking information, for example to allocate a fixed budget of manual postprocessing time.",
"An alternative widely-used evaluation measure is the “F score”: the harmonic mean of precision (robustness against false positives) and recall (robustness against false negatives) of a system [31].",
"The F score is particularly suited to information-retrieval type applications, such as downstream tasks in which the user might for example wish to retrieve a subset of positive examples from a large database.",
"The F score requires definite, binarised output; for fuzzy outputs, this requires postprocessing such as thresholding.",
"In the present work we calculated both the AUCs and the F scores for our systems, yielding slightly different perspectives on their relative performance.",
"Both measures were calculated from the segment-wise output with five-second segment durations.",
"AUCs were calculated separately for each class (our plots will show averages across classes).",
"To use the F score with fuzzy outputs, we chose binarisation thresholds to optimise the score on the training data, before applying the same thresholds to the testing data in each case.",
"This can be done with one threshold per class or with a single threshold; we tested both variants.",
"To summarise the F score we calculated it across all classes, rather than averaging the per-class F scores, since the latter would be numerically unstable especially with sparse data [31]."
],
[
"Evaluation Schemes",
"Our data consisted of annotated long-duration audio from multiple individual birds, one set in captive conditions and one set in field conditions, with multiple recordings from each individual (3–8 per individual for captive; 2 per individual for field, of longer duration).",
"We used this data to evaluate system performance in various crossvalidation scenarios: EachCap: Captive, strictly per-individual.",
"A system was trained with one half of an individual's recordings, and tested with the other.",
"The converse was also done, and then results aggregated over all captive individuals (yielding 14 `folds').",
"X-Y: Captive, pooled.",
"A system was trained with examples from each individual—half of the recordings from each individual—and tested with the remainder.",
"This gave 2 crossvalidation folds.",
"Note that X-Y is constructed so that all the testing files come from birds also seen in the training data.",
"A-B: Captive, pooled and stratified.",
"All recordings from each individual were allocated to one of two partitions.",
"This is similar to X-Y except that no bird used for training is used for testing.",
"Cap-Field: pooled cross-condition.",
"In this case the captive data is used for training, and the field data used for testing.",
"(Here we used only one crossvalidation fold.)",
"It is the most challenging case: as well as the train and test sets having no birds in common, the recording situation is also different.",
"EachField: Field, strictly per-individual.",
"As EachCap, but for the field data (12 folds).",
"Each of these scenarios relates not just to different degrees of generalisation, but to different downstream applications of automatic recognition technology.",
"For example, a researcher may wish to annotate a fraction of a recording and then invoke automatic recognition for the remainder; or to use a fixed system trained on one set of birds, e.g.",
"observed in captivity, and to apply it to new unknown recordings.",
"Finally, since the PLCA-based system produced its output at a higher resolution (i.e.",
"for each 23ms frame), we used this opportunity to explore how the temporal resolution interacts with evaluation procedures and metrics.",
"For this we repeated our evaluation using the segment-based F score, but using a much smaller segment size of 0.1 seconds, as compared with the 5 sec segment size used in the main experiments.",
"In order to ensure a fair comparison, sets of class-specific thresholds were computed from training data for each evaluation segment size (i.e.",
"100 msec and 5 sec) separately.",
"The F-measure was computed directly on the raw high-resolution output of the PLCA-based system."
],
[
"Results",
"As intended, the choice of microphone placement led to high-amplitude recordings for sounds from the focal bird (calls, flying, and other movements) while other background sounds were quiet but still largely audible (see Supplementary Information for examples).",
"The occurrence of the annotated actions and contexts in the collected data was relatively sparse (Figure REF ), with every class being active for less than 16% of the total time in both datasets.",
"Figure: Total ground-truth durations of annotated regions of each category.We evaluated each of our systems in two configurations: the classifier-based system with unbalanced or balanced class-weighting for training; and the PLCA system with mean- or maximum-based temporal downsampling.",
"In each case the differences between configurations were small, and so for clarity of presentation we will plot results from just one of each system (unbalanced classifier, mean-downsampling PLCA).",
"We will refer to differences in outcomes from the system configurations where relevant.",
"Overall, the quality of automatic recognition showed a strong dependency on the choice of crossvalidation setup, i.e.",
"on the relationship between the training data and the test data (Figure REF ).",
"As one clear example: the designs of the X-Y and A-B schemes were very similar except that the latter ensured that birds used for testing were not used for training; this change incurred a substantial penalty both in AUC and F score, implying that individual differences were highly pertinent.",
"The X-Y scheme in turn was similar to the EachCap scheme except that it pooled the training data across individuals.",
"Curiously, this pooling led to very similar F scores as EachCap, but to a marked difference in AUC: judged by AUC, the pooling of training data seems to have led to better generalisation properties, for both of the recognition algorithms tested.",
"Judged by F score, both EachCap and EachField, using systems trained specifically for each individual, attained many of the strongest results.",
"As expected, schemes involving generalising to unseen conditions had lower recognition scores—both A-B (generalising to new birds) and Cap-Field (generalising to new birds and to new recording environments).",
"As this task has not been evaluated before, there are no direct external comparisons for the overall recognition quality.",
"The segment-wise F-measures are broadly comparable to those presented in [31] (for an indoor event-detection task with fewer categories and a different segment duration).",
"In the present comparison of two different approaches, the classifier-based system generally outperformed the PLCA-based system: by an average of 5 percentage points on AUC, and 8 percentage points on F score.",
"Figure REF shows an example of the output from the classifier-based system overlaid with the groundtruth annotation, giving a rough visual indication of the kind of output that corresponds to the results obtained.",
"The effect of HMM postprocessing led to different results when considered via F score or AUC.",
"The F score statistics (Figure REF , upper) often showed a mild improvement when HMM postprocessing is added, particularly for the classifier-based system; while the AUC statistics (Figure REF , lower) unanimously indicated worse results with HMM postprocessing (the leftmost result in each cluster, the unprocessed output, performing best).",
"To binarise continuous-valued output, we found that per-class thresholding was not particularly better than a single threshold in general, except in the case of the raw PLCA output.",
"This exception is because the raw PLCA output is expressed in terms of activation magnitude (i.e.",
"related to the energy of each context class in the spectrogram), which does not have comparable meaning across classes, and so per-class thresholding is highly pertinent in that case.",
"For the HMM-postprocessed outputs, a single threshold often slightly outperformed per-class thresholds, which is probably due to a slight reduction in overfitting the threshold choice.",
"The classes (categories) used in this study are highly diverse in kind, and so to drill further into system performance it is important to inspect performance on a per-class level (Figures REF and REF ).",
"It is immediately clear that detection quality exhibits some correlation with the quantity of positive examples available for training (cf.",
"Figure REF ), although the focal call category is particularly well detected by the classifier system despite being relatively sparse in the training data.",
"Focal calls are behaviourally important; they are also the signal class for which our classifier was originally implemented.",
"The figures also decompose the F score into its components: precision and recall.",
"When the classifier reaches a high F score it is often achieving strong precision, while when the PLCA does well it achieves strong recall.",
"The per-class results for the most difficult evaluation condition, Cap-Field, show that the generalisation to new individuals and new environments has a differential effect on recognition quality (Figure REF ).",
"Importantly, the classifier-based system is able to generalise well on one of the more important categories—focal call—as well as on self-maintenance, yet the performance on some other categories—walking, flying, bg jackdaws—drops off markedly.",
"The performance of the PLCA-based system does generalise on some categories—looking around, self-maintenance—but exhibits lower performance in other categories, including focal calls.",
"Figure: Two examples of automatic annotation from a relatively strongly-performing system (classifier; HMM filtering; per-individual training) for a captive (upper panel) and a field condition (lower panel).",
"The black and white regions are correctly-identified as on and off respectively.",
"Red are false-positive detections, and blue false-negatives.",
"(Best viewed in colour.",
")Figure: F scores (top row) and AUCs (bottom row) for the systems tested.",
"Each panel shows a different crossvalidation setup.",
"In each panel, we show clusters of scores connected by lines; the items in each cluster relate to the different postprocessing options, left-to-right as follows: no postprocessing; unified HMM Viterbi decoding; per-class Viterbi decoding; unified HMM filtering; per-class HMM filtering.",
"Plotted values are the median across crossvalidation folds, with error bars indicating their 5- and 95-percentiles.Figure: F score, Precision and Recall (all in %) for each class separately, for 4 systems tested under the three pooled crossvalidation scenarios (X-Y, A-B, and Cap-Field), using per-class thresholding.Figure: Per-class results as in Figure but for the two per-individual scenarios (EachCap and EachField).Figure: Temporal activity profiles for one of the field recordings, for 8 selected classes.",
"Each panel shows a bar chart plotting, for each subsequent five-minute interval, the proportion of time that the class was active.",
"This was calculated as the proportion of 5-second segments in that interval that were labelled positive; for probabilistic outputs, the `fuzzy' probabilistic decisions were summed.",
"We compare an example of the manually-annotated ground truth (top row), the classifier inference (middle row), and the PLCA inference (bottom row).",
"The two systems were in the EachField condition, with per-class HMM filtering as postprocessing.Figure REF shows a different view of the temporal nature of our data.",
"For selected classes in a chosen recording, it summarises the true or inferred activity levels in broad (five-minute) time-steps.",
"Both systems exhibit some mismatch with the ground-truth, though the output from the classifier-based system can be seen to better match the true contours of activity.",
"In particular the classifier-based system shows a tendency to better match the true sparsity levels of class activations.",
"A final comparative study was made using the higher-resolution 23 msec step raw output of the PLCA-based system, comparing this against the 5 sec mean-pooled segments.",
"Using the X-Y crossvalidation scenario, the performance in terms of segment-based F-measure with 5 sec segment size was 39.07% when using the 23 msec output, and 38.03% when using the 5 sec mean-pooled output.",
"When however the high-resolution output was evaluated using the segment-based F-measure with a 100 msec segment size, performance dropped to 22.19%.",
"These results indicate that the higher-resolution output can lead to a small improvement over the pooled output, and that the numerical value of the chosen evaluation statistic depends strongly on the temporal granularity of evaluation.",
"The reduced performance when evaluated at high resolution may be partly due to issues in the temporal precision of the inferred and/or the ground-truth annotations.",
"Fig.",
"REF shows an example high-resolution output using the PLCA-based system for recording MohawkMOV00F_a from the captive set, which in this case reached a 100 msec segment-based F-measure of 54.1% using the X-Y crossvaliation scheme.",
"A few observations can be made from Fig.",
"REF : the system was able to successfully detect overlapping contexts, in this case background colony sounds and looking around movement.",
"However, the output was often fragmented, as for example can be seen for detected flying events.",
"Another notable issue is the high number of false alarms as compared to missed detections (which translates into high precision and low recall, as shown in Fig.",
"REF ).",
"So for example, flight events present in the recording were correctly detected as flight, but at the same time the output produced false positives for the manipulation and self-maintenance classes.",
"Figure: The 23 msec step output of a recording from the captive set, using the PLCA-based system with the X-Y crossvalidation scheme.",
"The colour scheme is as in Fig.",
"."
],
[
"Discussion",
"Our study has investigated a novel task in animal sound recognition, approaching it via two polyphonic sound recognition methodologies related to those previously studied in environmental and bird sound.",
"Overall evaluation figures are comparable with the state of the art in these neighbouring tasks [11], [31].",
"The details of the timelines recovered (Figures REF , REF , REF ) show that across all conditions, further development is needed before this paradigm can be deployed for fully automatic analysis of animal behaviour patterns from audio data.",
"Of the two recognition systems studied, the classifier-based system consistently led to stronger results, including a better match to the temporal characteristics of the true annotations (Figure REF ); however, the PLCA-based system has an advantage of directly outputting a high-resolution (frame-by-frame) annotation, which may be particularly desirable in some applications, such as investigating the short-time vocal interactions between individuals.",
"Our sequence of crossvalidation tests demonstrated that generalising to new individuals and new environmental conditions remains a critical challenge for automatic sound recognition, certainly when judged by F score (Figure REF ), especially when aiming at extrapolating from captive to field datasets.",
"The present results suggest that to annotate field recordings, the best strategy could be to train a human annotator on the captive data to annotate a small subset of field recordings from individuals which in turn could be used to train the classifier for further field data analyses.",
"Crucially, our study investigated the automatic recognition of a diverse set of classes, each of them pertinent for the study of animal communication and behaviour.",
"The classes vary widely in their acoustic realisations, from single sound events such as calls, to behaviours such as walking heard as compound events or sound textures.",
"Consequently, as expected there were wide variations in recognition performance across classes.",
"The strongest-performing system achieved good F scores for focal calls, flying, self-maintenance and walking.",
"In general, performance levels could be correlated with how well the class of interest was represented in the training data.",
"The sound of flying is quite clear to a human annotator, especially in the field where birds may fly continuously for 15 minutes or longer.",
"Very short flights (less than 1–2 seconds) are more difficult, and require more attention, because they may be confused e.g.",
"with feather ruffling.",
"Especially the captive dataset was characterised by such short flights, which may explain why the relatively good scores for automatic detection of flying were still lower than anticipated.",
"Suitable features and detectors for such noisy, loosely periodic sounds thus remain a topic for further development.",
"In manual inspection, we noted a tendency for systems to output detections for focal call and non-focal call at the same time.",
"This can be attributed partly to acoustic similarities between the classes: the microphone placement was designed to assist with discriminating these categories, though in some instances it remained difficult even for a human annotator.",
"Some acoustic differences included the effects of close-mic recording, giving increased low-frequency energy for the focal call over the non-focal call.",
"We did not adapt our time-frequency representations specifically for this feature, and one future development could include such adaptation.",
"A rival explanation for the confusion of focal and non-focal calls is that the two do tend to co-occur in close temporal proximity ($<1$ seconds), and so the systems may be influenced more by the class co-activation (at the 5-second resolution) rather than acoustics.",
"This highlights the tension inherent in selecting a time resolution for analysis; for studies such as this, in which the different categories operate with rather different temporal characteristics, an option may be for the system—and also the evaluation—to use a class-dependent time resolution.",
"In the present study we found relatively little benefit in HMM postprocessing of system output.",
"Its purpose was to refine per-segment estimates by making use of temporal dependencies between segments.",
"In some configurations it led to a mild improvement in results, though in some other configurations it led to deterioration.",
"We did however find a consistent result that HMM filtering led to better results than Viterbi decoding, and that a per-class HMM was better than a unified HMM.",
"The classifier-based system treated each segment entirely independently, and so should have benefited from some temporal smoothing.",
"One interpretation is that simple Markovian dependency (at the 5-second timescale) does not reflect enough of the temporal structure present in the data, and that more sophisticated temporal models might be investigated.",
"Some of the differences in interpretation implied by the AUC and the F score might be attributed to the fact that F score requires fuzzy/probabilistic outputs to be binarised at a specific threshold, whereas the AUC uses the continuous data and thus generalises over all possible thresholds.",
"In a typical practical application, the user will know the relative cost of false positives and false negatives—i.e.",
"the relative importance of high precision and high recall—and can set a threshold based on this balance.",
"The standard F score weights the two equally.",
"However, downstream applications might imply different priorities, such as high precision in the case of a user retrieving examples of specific behaviour.",
"In those cases it would be desirable to use the generalised F score, sometimes referred to as $F_\\beta $ where $\\beta $ is the desired precision/recall ratio.",
"This would be used not only for evaluation but for threshold-setting.",
"As already discussed, we consider that the current level of performance is not yet at level for blind application to new data.",
"As with tasks in neighbouring disciplines—speaker diarisation and polyphonic music transcription—the task is difficult and the development of full automation will require refinement of methods adapted for the specific characteristics of the signals in question.",
"This is particularly true for categories indirectly represented via clusters of related sound events.",
"The present study with its diverse set of sound categories raises the possibility that a good detection system may benefit from using an entirely different system for each class, perhaps using different timescales.",
"A further possible direction in relation to the timescale is the possibility of using dynamic time resolution.",
"The appropriate time resolution at which to consider animal behaviour is a discussion well-rehearsed in ethology; if time resolutions could be dynamically inferred per-class from data, this might inform debate as well as improving system performance.",
"We investigated the performance of systems using segment-based evaluation measures.",
"Our segment size of 5 seconds was chosen based on manual inspection of pilot data as well as on considerations of the target application.",
"The classifier-based system was also configured to operate at this resolution; such a classifier-based system typically operates over segments of this size (not at `frame-wise' resolution such as 23 ms) in order to make stable classification decisions.",
"Segment-based evaluations aggregate higher-resolution data using a max-pooling approach [31], with the curious side-effect that a single positive item anywhere within the 5 sec segment leads to the whole segment considered active.",
"To mitigate this effect, in future evaluations one might use a smaller (and data-driven) segment size for evaluation, even in the case that the system gives output at a larger segment size; perhaps more fundamentally, the max-pooling could be replaced with a parametric threshold (e.g.",
"percentile-based) to reduce the effect of false-positive `blips' on the evaluation outcome.",
"In the present work we considered interactions between the annotated categories via co-occurrence dependencies (positive or negative) implicitly learnt from the data: the classifier-based system used a single classifier predicting for all classes at once, the PLCA-based system had the opportunity to `explain away' a portion of energy as belonging to one class rather than another, and the HMM postprocessing was able to use a single HMM model across all classes (though this was not found to be better than per-class HMMs).",
"Future work could consider alternative approaches to the relationships between categories.",
"Hierarchical models such as the context-dependent sound event detection of [13] may be suitable, or switching state-space models (switching SSMs), where the discrete “switch” would correspond to a context and the context-dependent SSMs would detect specific sound events or background sounds."
],
[
"Conclusions",
"We have introduced an application of audio recognition specifically for sound recordings from animal-attached microphones, to enable analysis of the activity of a focal animal as well as the context of such activity, i.e.",
"the environment around it as conveyed acoustically.",
"This enables researchers to study the animal's behaviour as well as the context of that behaviour, i.e.",
"the environment around it as conveyed acoustically.",
"We applied automatic recognition to data collected from lightweight backpack loggers carried by free-flying birds (jackdaws) in an aviary and in the field.",
"We directly compared a scene-classification and an event-detection approach approach to this task.",
"The classification method made use of a feature learning method developed for bird vocalisations.",
"For event detection, we introduced a modified PLCA method, improving on previously-published work in related domains.",
"In evaluation, the classifier-based method performed most strongly.",
"We find that the current recognition quality level enables scalable automatic annotation of audio logger data, given partial annotation, but also find that individual differences between animals and/or their backpacks can reduce recognition rates when generalising to previously-unseen individuals.",
"This approach to studying animal behaviour in single individuals requires further development for full automation and application to previously-unseen individuals.",
"However, as on-animal microphones become increasingly common, this seems an effort worth taking to eventually extract meaning from such streams of sounds by facilitating the analyses of vocalisations, as well as some of their associated behaviours and acoustic contexts, without additional data collection and devices.",
"Combining such results with an animal's position in space or relative to its conspecifics, and with detailed acceleration data, would provide us with a more complete picture of what animals do and even provide hints why they do it, to tackle many remaining open questions in mechanistic, evolutionary and conservation-related areas of behavioural research."
],
[
"Author Contributions",
"DS and LFG jointly conceived the study.",
"DS implemented the classifier-based system, led on the evaluation, led on the manuscript writing and wrote parts of the manuscript.",
"LFG provided the data (conducted animal handling and fieldwork, supervised video recordings, performed all initial sound and video analyses and annotations), helped with evaluating the method, and wrote parts of the manuscript.",
"EB implemented the PLCA-based system, collaborated in performing the evaluation, and wrote parts of the manuscript.",
"We would like to thank Tiffany Magdalena Pelayo van Buuren and Magdalena Mair for assistance in the field; Katrin Mayer for collecting video footage and helping with sound files; Auguste von Bayern, Wolfgang Goymann, Andries Ter Maat and Manfred Gahr for their support without which this work had not been possible; and to the von Bayern family for granting access to the premises and facilities.",
"DS is supported by EPSRC Early Career research fellowship EP/L020505/1.",
"EB is supported by a UK Royal Academy of Engineering Research Fellowship (grant no.",
"RF/128).",
"LFG is funded by the Max Planck Society."
]
] | 1612.05489 |
[
[
"Dealing with missing data in the MICROSCOPE space mission: An adaptation\n of inpainting to handle colored-noise data"
],
[
"Abstract The MICROSCOPE space mission, launched on April 25, 2016, aims to test the weak equivalence principle (WEP) with a 10^-15 precision.",
"To reach this performance requires an accurate and robust data analysis method, especially since the possible WEP violation signal will be dominated by a strongly colored noise.",
"An important complication is brought by the fact that some values will be missing -therefore, the measured time series will not be strictly regularly sampled.",
"Those missing values induce a spectral leakage that significantly increases the noise in Fourier space, where the WEP violation signal is looked for, thereby complicating scientific returns.",
"Recently, we developed an inpainting algorithm to correct the MICROSCOPE data for missing values.",
"This code has been integrated in the official MICROSCOPE data processing pipeline because it enables us to significantly measure an equivalence principle violation (EPV) signal in a model-independent way, in the inertial satellite configuration.",
"In this work, we present several improvements to the method that may allow us now to reach the MICROSCOPE requirements for both inertial and spin satellite configurations.",
"The main improvement has been obtained using a prior on the power spectrum of the colored-noise that can be directly derived from the incomplete data.",
"We show that after reconstructing missing values with this new algorithm, a least-squares fit may allow us to significantly measure an EPV signal with a 0.96x10^-15 precision in the inertial mode and 1.2x10^-15 precision in the spin mode.",
"Although, the inpainting method presented in this paper has been optimized to the MICROSCOPE data, it remains sufficiently general to be used in the general context of missing data in time series dominated by an unknown colored-noise.",
"The improved inpainting software, called ICON, is freely available at http://www.cosmostat.org/software/icon."
],
[
"Introduction",
"The MICROSCOPE space mission aims to test the weak equivalence principle (WEP) with a precision of $10^{-15}$ .",
"This is more than two orders of magnitude better than the current ground-based constraints [1], and will allow a precise test of general relativity after LIGO brought a new confirmation of its predictions by the direct detection of gravitational waves [2].",
"Indeed, the equivalence principle is a cornerstone of the general theory of relativity.",
"It states in particular that the inertial mass and the gravitational mass are equivalent; in other words, the acceleration imparted to a body by a gravitational field is independent of its mass and its composition.",
"Any detection of a WEP violation would be paramount since it may question the very basis of general relativity and, more generally, our understanding of the Universe.",
"On the opposite, confirming the WEP to a precision of $10^{-15}$ would place new constraints on unified models for fundamental interactions, since some predict a violation below $10^{-13}$ [3], [4].",
"Estimating a possible equivalence principle violation (EPV), or lack thereof, will require a careful data analysis, and especially a fine characterization of the measurement noise.",
"A possible difficulty, identified in [5], [6], originates from missing data in the time series resulting in irregularly sampled data.",
"Although missing data are a common problem in physics experiments (they occur when data acquisition fails or when a contamination from an external source invalidates some points), they are particularly troublesome when the measurement noise is colored (i.e., frequency-dependent).",
"In particular, [5], [6] have shown that the presence of gaps has a strong impact on the precision of the ordinary least squares fits of harmonic signals.",
"This is due to the spectral leakage of the noise in the frequency domain, which increases the uncertainty of the fit by several orders of magnitude, even for a small fraction of missing data.",
"An important effort has been made to reduce the impact of missing values in the MICROSCOPE data analysis.",
"A Kalman-Auto-Regressive Model Analysis (KARMA) has been proposed in [5], which generalizes least-squares estimation to missing data problems without “filling” missing values.",
"As it has been shown that KARMA allows us to reach MICROSCOPE's requirements, it has been integrated in the official MICROSCOPE data analysis pipeline.",
"However, to strengthen our conclusions after analyzing the MICROSCOPE data, and since missing data are a crucial difficulty in the data analysis, we want to have at least two independent techniques to deal with them.",
"That is why we have proposed an alternative route in [6]: we use the inpainting algorithm to fill in data gaps, which then allows us to use an ordinary least-squares method to look for and characterize a possible EPV.",
"We showed that this method allows us to reach also the MICROSCOPE requirements in the so-called “inertial” mode, where the MICROSCOPE spacecraft is kept fixed with respect to distant stars.",
"However, this method fails in the “spin” mode, where the spacecraft rotates about the axis normal to the orbital plane.",
"Hence, the method needed to be improved so that it can be used in both experimental modes.",
"This is the aim of this paper, which can be seen as the natural sequel to [6] (thereafter Paper I).",
"Paramount to MICROSCOPE, either to characterize the confidence level of an EPV detection, or to characterize a new upper limit on the WEP (if no EPV is detected), is the noise characterization.",
"This characterization will be done in Fourier space, and therefore amounts to estimating the noise power spectral density (PSD).",
"As explained above, the PSD estimation is affected by the spectral leakage created by missing data.",
"Methods like KARMA and inpainting (Paper I), although they allow us to reach MICROSCOPE's requirements in the characterization of a possible EPV detection, fail to fully estimate the PSD [5], [6], and are therefore not fully satisfactory.",
"In [7], the authors extended the KARMA technique into a modified expectation-conditional-maximization (M-ECM) technique; using simulated data, they showed that this technique allows us to fully estimate the MICROSCOPE's noise PSD.",
"This new paper aims to provide a description of the several improvements that have been brought to the previously developed inpainting method (Paper I) in the search of reconstructing the full noise PSD.",
"Those improvements enable us to reach the MICROSCOPE's scientific goals in both inertial and spin modes.",
"This paper is organized as follows.",
"In Sec.",
", we summarize the MICROSCOPE mission and review how missing data affect the data analysis.",
"Sec.",
"describes the improvements brought to the inpainting method and their motivations.",
"The major improvement has been obtained by adding a prior on the noise power spectrum directly derived from the data.",
"The results are presented in Sec.",
"based on MICROSCOPE mock data; in particular, we show the gain in precision that we obtain in the evaluation of a possible EPV signal with a Least Square fit compared to the previous method.",
"We conclude in Sec.",
"."
],
[
"The MICROSCOPE mission",
"MICROSCOPE will test the WEP by measuring the relative acceleration of two test masses of different composition freely falling in the earth's gravitational field.",
"To achieve the highest possible stability and accuracy, the test masses are on-board a drag-compensated and attitude-controlled satellite which screens them from non-gravitational accelerations.",
"The science data will consist of time series of differential accelerations (the half-difference of the test masses accelerations) measured along a sensitive axis by onboard inertial sensors.",
"The source of the gravitational signal used to test the WEP is the earth�s gravitational field modulated by the motion of the satellite as it orbits Earth: hence, the EPV signal is a sine expected at a well known frequency $f_{\\rm EP}$ (which depends on the satellite�s motion) along the most sensitive axial axis of the instrument.\"",
"The measured signal in the MICROSCOPE experiment can thus be written as: $X_{\\rm EP}(t) = \\frac{1}{2} \\mathcal {M}_{\\rm EP} \\delta g_{\\rm EP}(t) + \\mathcal {S}(t) + \\mathcal {N}(t),$ where $\\mathcal {M}_{\\rm EP}$ is an instrumental calibration factor [8], $\\delta $ is the EPV parameter we aim to detect and characterize, $g_{\\rm EP}(t)$ is the Earth gravity field's projection on the measurement-axis (with $g\\approx 8$ m.s$^{-2}$ at MICROSCOPE's altitude –700 km) [9], $\\mathcal {S}$ represents systematics errors, and $\\mathcal {N}$ is the statistical inertial sensor noise.",
"Assuming perfect correction of systematics $\\mathcal {S}$ and instrument's calibration $\\mathcal {M}_{\\rm EP}$ (these corrections are out of the scope of this paper), we will measure a sine-wave signal ($\\delta g_{\\rm EP} (t)/2$ ) at frequency $f_{\\rm EP}$ dominated by a colored-noise ($\\mathcal {N} (t)$ ).",
"Different experimental modes can be used, which will allow us to confirm (or exclude) an EPV detection.",
"In particular, the inertial mode is defined as keeping the satellite's attitude fixed with respect to distant stars; in this case, the WEP test frequency is equal to the orbital frequency $f_{\\rm EP,~iner} = f_{\\rm orb} = 1.8 \\times 10^{-4}$ Hz.",
"In the spin mode, the satellite rotates about the axis normal to the orbital plane in opposite to the rotation due to to the orbital motion, with a frequency $f_{\\rm spin}$ , thereby offset the WEP test frequency to $f_{\\rm EP, spin} = f_{\\rm orb} + f_{\\rm spin}$ ; by choosing $f_{\\rm spin}>0$ , we increase $f_{\\rm EP}$ , moving it to a frequency where the measurement noise is lower.",
"In this paper, we assume $f_{\\rm spin} = 4.5 f_{\\rm orb}$ , thereby $f_{\\rm EP,~spin} = 10^{-3}$ Hz.",
"The orbital frequency can be measured in-flight with precise orbit determination; it is then used in the data analysis to look for the EPV signal through a Least-Square fit.",
"The duration of each session is chosen in such a way to ensure a measurement noise of about $4 \\times 10^{-15}$ ms$^{-2}$ on the differential acceleration at $f_{\\rm EP}$ , as needed to reach a $10^{-15}$ precision on the EPV parameter $\\delta $ .",
"Thus, the inertial and spin sessions last respectively 120 and 20 orbits (inertial and spin sessions will be performed sequentially, one after another).",
"The MICROSCOPE data analysis challenge is then to be able to detect and estimate the amplitude of the periodic signal with a precision of $4 \\times 10^{-15}$ ms$^{-2}$ in both the inertial and spin modes.",
"In case no violation is detected, the challenge is to characterize the noise with the same precision.",
"Touboul [8] computed the expected MICROSCOPE's error budget.",
"We use this model to specify the noise PSD (Fig.",
"REF , black curve).",
"In-flight performance sessions will be dedicated to fully and finely characterize the actual noise.",
"We will then be able to take into account any evolution of the noise characteristics in the data analysis.",
"However, we expect the actual in-flight noise PSD to not differ much from the model [8], so the results of this paper are robust enough for the upcoming MICROSCOPE data analysis.",
"Nominally, the measured accelerations are regularly spaced, with a time sampling of 0.25 second.",
"However, missing data are almost inevitable in long observations.",
"Most expected data alteration in MICROSCOPE come from tank crackles and Multi-Layer Insulation (MLI) coating crackles and are shorter than one second.",
"; micrometeorite impacts are expected to be rare and create very short gaps; tele-transmission losses are wider (up to several seconds) but very rare.",
"If both accelerometers of a differential accelerometer were perfect, such tank crackes and other spacecraft-related disturbance (such as micrometeorit impacts) would not appear in the differential acceleration, since both accelerometer would measure them in the same way.",
"However, the accelerometers are not perfectly identical, and small differences in their electronics's transfer function create small differences in their measurement.",
"Those differences are compensated for in steady-state regime by a posteriori data analysis.",
"However, they currently cannot be easily compensated for in the transient regime created by glitches; an empirical model from the data should allow us to eventually correct for those glitches without the need to mask them.",
"In the case of MICROSCOPE, we expect in the worst-case scenario up to about 3% of missing data (resp.",
"4% of missing data) in the inertial mode (resp.",
"in the spin mode).",
"As mentioned in [10], some crackles come from the temperature varaitions of the satellite's multi-layer insulation cover due to changes in the orientation of the satellite with respect to the Earth (the earth's albedo heats one or another side of the satellite along its orbit); in the spin mode, such changes are more frequent, resulting in more crackles, and therefore more missing data than in the inertial mode.",
"Further discussion about the sources of missing values in MICROSCOPE can be found in [5], [10].",
"Detailing how invalid data points are detected, and how gaps' location and size are set up after the detection such invalid data, goes beyond the scope of this paper.",
"In a few words, we can detect invalid data with a $\\sigma $ -clipping technique; the size of gaps will be set empirically, to remove any transient behavior after crakles (Bergé et al in prep).",
"Rather, we assume that that all such invalid data are correctly detected and masked.",
"We follow [6] to define the size and distribution of gaps.",
"Although the fraction of missing data is very small, the detection and estimation of a possible EPV signal is seriously complicated by this data losses and/or alteration.",
"This is because the spectral leakage induced by missing data makes the noise power from high-power regions spread over the frequency domain; hence, the noise in Fourier space where the EPV signal is looked for is significantly increased, since the measurement noise is strongly colored.",
"Fig.",
"REF shows the effect of missing data in the MICROSCOPE PSD estimate (grey curve).",
"Missing data create an important spectral leakage from $f \\approx 1$ Hz to surrounding frequencies.",
"As a result, the noise in the band $[10^{-4}-10^{-1}]$ Hz, where the EPV signal is looked for, is largely dominated by the spectral leakage from the high-frequency noise.",
"We added to the PSD shown in Fig.",
"REF an EPV signal of $3\\times 10^{-15}$ (red arrow); it is clear that the spectral leakage due to missing data makes its detection extremely difficult in Fourier space.",
"On the opposite, without missing data, the EPV peak emerges clearly from the noise (black curve).",
"Figure: The black curve shows the MICROSCOPE PSD estimate for a 120 orbits simulation.",
"An example of a possible EPV signal of 3×10 -15 3 \\times 10^{-15} in the inertial mode is shown by the peak at 1.8×10 -4 1.8\\times 10^{-4} Hz.",
"The grey curve shows the spectral leakage affecting the PSD estimate when gaps are present in the data."
],
[
"The missing data problem",
"The problem of missing data can be formalized as: $Y(t) = M(t) X(t),$ with $X(t)$ the ideal complete (i.e.",
"regularly sampled) time series, $Y(t)$ the observed time series (with gaps), and $M(t)$ the binary mask (i.e., window function with $M(t) = 1$ , if we have information at data point $X(t)$ ; $M(t) = 0$ otherwise).",
"In Fourier space, the multiplication by the mask becomes a convolution: $\\hat{Y}(t) = \\hat{M}(t) * \\hat{X}(t),$ where $\\hat{}$ denotes the Fourier transform.",
"The convolution by the spectral window $\\hat{M}(t)$ causes the energy at each frequency of the power spectrum to leak into surrounding frequencies, producing a spectral leakage in the Fourier domain."
],
[
"Sparse ",
"In Paper I, we proposed to use a method of sparse inpainting to estimate the missing data.",
"The method proposed was introduced by [11] and consists in recovering $X(t)$ knowing $Y(t)$ and $M(t)$ by imposing a prior of sparsity on the solution $X(t)$ .",
"This inpainting method already had some major successes in astrophysics (e.g.",
"Weak Lensing [12], [13], CMB [14], Asteroseismology, [15], [16]).",
"The sparse inpainting method uses the prior that there is a representation $\\Phi ^T$ of the time series $X(t)$ where most coefficients $\\alpha = \\Phi ^T X$ are close to zero ($^T$ represents the transpose matrix).",
"For example, if the time series $X(t)$ was a single sine wave, the representation $\\Phi ^T$ would be the Fourier transform because all but one coefficient of the Fourier representation of a sine are equal to zero.",
"The solution of this problem is obtained by solving: $\\min \\Vert \\alpha \\Vert _1 \\textrm { subject to } \\parallel Y - MX \\parallel ^2 \\le \\sigma ^2,$ where $||.||_1$ is the convex $l_1$ norm (i.e.",
"$ || z ||_1 = \\sum _k | z_k |$ ), $|| .",
"||$ is the classical $l_2$ norm (i.e.",
"$|| z ||^2 =\\sum _k (z_k)^2$ ) and $\\sigma $ is the standard deviation of the noise in the observed time series.",
"The solution of such an optimization task can be obtained through an iterative algorithm introduced by [11].",
"Let $X_i$ denotes the reconstructed time series at iteration $i$ .",
"If the time series is sparse enough in the representation $\\Phi ^T$ , in this representation the largest coefficients should originate from the time series we want to recover.",
"Thus, the algorithm is based on a threshold that decreases exponentially (at each iteration) from a maximum value to zero.",
"By accumulating more and more high coefficients through each iteration, the gaps in $X_i$ are filling up steadily and the power of the coefficients due to the gaps is decreasing.",
"This algorithm needs as inputs the observed incomplete data $Y$ and the binary mask $M$ .",
"The algorithm can be described as follows: Set the maximum number of iterations $I_{max}=100$ , the solution $X^0$ is initialized to zero, the maximum threshold $\\lambda _{max} = \\max (\\mid \\Phi ^T Y \\mid )$ with $\\Phi ^T$ being a global Discrete Cosine Transform (DCT), and the minimum threshold $\\lambda _{min} = 0$ .",
"Set $i = 0$ , $\\lambda ^0 = \\lambda _{max}$ .",
"Iterate: Set $U^i = X^i + M(Y-X^i)$ to enforce the time series to be equal to the observed data where the mask $M$ is equal to 1.",
"Compute the forward transform of $U^i$ : $\\alpha = \\Phi ^TU^i$ .",
"Compute the threshold level $\\lambda ^i = F(i,\\lambda _{max},\\lambda _{min})$ , where $F$ is a function that describes the decreasing law of the threshold.",
"Compute $\\tilde{\\alpha }$ by keeping only the coefficients $\\alpha $ above the threshold $\\lambda ^i$ and setting the others to zero.",
"Reconstruct $X^{i+1}$ from the remaining coefficients $\\tilde{\\alpha }$ : $X^{i+1} = \\Phi \\tilde{\\alpha }$ .",
"Set $i=i+1$ .",
"If $i<I_{max}$ , return to step 3.",
"In Paper I, $\\Phi ^T$ is chosen to be a global DCT because it provides a sparse representation for the EPV signal.",
"The function $F$ used to describe the threshold decreases (at each iteration $i$ ) from $\\lambda _{max}$ to zero following the empirical law below: $F(i, \\lambda _{\\max }) = \\lambda _{\\max } \\left(1-\\operatorname{erf}\\left(\\frac{i\\beta }{N-1}\\right) \\right),$ with $\\beta = 2.8$ .",
"This law is commonly used because it follows the fast (i.e.",
"exponential) decay of the coefficients that is commonly observed in a sparse representation.",
"Figure: Estimated MICROSCOPE differential acceleration power spectral density (PSD) averaged over 100 120-orbit simulations in the inertial mode (upper panel) and over 100 20-orbit simulations in the spin mode (lower panel).",
"The black line shows the PSD when all the data is available, the blue line shows the effect of missing values and the red line shows the PSD estimated from data filled with the inpainting method developed in Paper I.",
"Note the EPV peaks of 3×10 -15 3 \\times 10^{-15} at 1.8 10 -4 10^{-4} Hz (in the upper panel) and 10 -3 10^{-3} Hz (in the lower panel).In Fig.",
"REF , the black curve shows the estimated MICROSCOPE differential acceleration PSD in the inertial mode averaged over 100 simulations of 120 orbits.",
"The blue curve shows the effect of missing 3% of the data.",
"The red curve shows the PSD estimated after filling missing values using the inpainting algorithm just described above.",
"In the inertial mode (upper panel), a least-squares fit of a sinusoidal function at $f_{\\rm EP} = 1.8 \\times 10^{-4}$ Hz allowed us to detect the EPV peak with a precision of $1.18\\times 10^{-15}$ , which is very close to the MICROSCOPE requirements.",
"However, if we wanted to confirm the detection of the EPV signal in the spin mode (lower panel), there is still an important spectral leakage in the intermediate frequency region where the EPV peak is expected for this second configuration ($f_{\\rm EP} = 10^{-3}$ Hz).",
"Note also that the peak in the spin mode appears smaller in the PSD representation due to the different integration time."
],
[
"Sparse ",
"The inpainting developed in Paper I gives really promising results in the inertial mode.",
"However, it remains unsatisfactory to detect and characterize a possible EPV peak in the spin mode.",
"Fig.",
"REF shows that there is still a residual spectral leakage from the high-frequency noise peak to the low frequency region of the spectrum where we try to detect a possible EPV signal.",
"In this section, we describe the improvements we brought to the algorithm."
],
[
"Noise constraint",
"A major problem in the previous version of the inpainting, is that the minimisation problem of eq.",
"REF is only optimal for white noise data.",
"However, this minimization can be extended to the colored-noise case.",
"An effective way to do so and thus reduce the spectral leakage is to introduce a prior on the noise.",
"This prior could be easily introduced in the algorithm by forcing, at each iteration, the high frequency part of the spectrum to follow an a priori model of the noise.",
"However, this would break the required model-independence of the method.",
"To bypass this problem, so that the method remains model-independent, the noise constraint is based on the data.",
"To do so, at each iteration $i$ of the algorithm, we perform a wavelet transform of the signal $X^i(t)$ at this iteration using the à trous algorithm: $X^i(t) = {c_{J}}(t)+ \\sum _{l=1}^{J} w_{l}(t),$ where $J$ is an input parameter, $c_{J}$ is a smooth version of the original signal $X^i(t)$ and $w_l$ are the wavelet bands that give the details of the signal $X^i(t)$ at different resolutions (see Starck et al.",
"book:starck02, book:starck06 for details).",
"Thus, if the signal $X^i(t)$ is of size $N$ , the algorithm outputs $J+1$ arrays of size $N$ .",
"In this application, J is chosen to be equal to 10 in the spin case and 14 in the inertial case to properly handle the residual spectral leakage at the position of the EPV peak.",
"The wavelet filters $\\psi (x)$ used for this application are defined as $\\frac{1}{2} \\psi (\\frac{x}{2} )=\\varphi (x) - \\frac{1}{2} \\varphi (\\frac{x}{2}),$ the difference between two B$^3$ -Spline functions $\\varphi (x)$ at two different resolutions with: $\\varphi (x)=\\frac{1}{12}(|x-2|^3-4|x-1|^3+6|x|^3-4|x+1|^3+|x+2|^3).$ Thus, at a given scale $l$ , the wavelet filters are defined as $\\frac{1}{2^{l+1}} \\psi (\\frac{x}{2^{l+1}} )=\\frac{1}{2^{l}} \\varphi (\\frac{x}{2^{l}}) - \\frac{1}{2^{l+1}} \\varphi (\\frac{x}{2^{l+1}}).$ Fig.",
"REF shows the shape of the power spectra of the wavelet filters for $l = 0$ to 10 (in colors) and the smoothing filter $\\varphi (x)$ (in black) used for the spin case.",
"Note the position of the EPV peak at $f_{\\rm EP} = 10^{-3}$ Hz, represented in the figure as a vertical dashed red line.",
"The wavelet filters derived in this way have a compact support in real space and are well localized in the Fourier domain.",
"Additionally, the wavelet decomposition is very fast.",
"Figure: Frequency response of the wavelet filters (in different colors) and smoothing filter ϕ(x)\\varphi (x) (in black).",
"The wavelet filters have been used to add a constraint about the power spectrum of the noisy signal.",
"Each wavelet band w l w_l is obtained by the convolution of the signal withthe wavelet filter functions of various characteristic scales as described in the textOnce the signal is decomposed into several wavelet bands $l$ , the estimation of the standard deviation of $w_l(t)$ with $t$ constrained to be outside gaps enables us to estimate the mean power spectrum of the noisy signal in this frequency band.",
"Hence, we have now a way to estimate a broad-band power spectrum of the noisy signal from incomplete data.",
"This being said, for each wavelet band $l$ the code finds the standard deviation of $w_l(t)$ with $t$ constrained to be outside gaps, and does the same for $t$ constrained to be inside gaps.",
"And then, $w_l(t)$ is rescaled inside the gaps by a constant $\\nu _l$ chosen so that the standard deviation inside gaps is the same as outside gaps for that $l$ .",
"Thus, we reduce the spectral leakage by imposing that constraint for each wavelet band.",
"In this process, the f$_{\\rm EP}$ frequency is considered to avoid a rescaling of a possible EPV signal.",
"This is the major improvement of the code."
],
[
"Threshold law",
"The way the threshold is decreased at each iteration has also an impact on the results.",
"Ideally, we would like to have a number of iterations as large as the number of points in the time series and decrease the threshold in such a way so to add one single additional coefficient to $\\alpha $ at each iteration.",
"However, this number is too large and instead, we have to find a trade-off between the speed of the algorithm and its quality.",
"We optimize the decreasing law $F$ for the MICROSCOPE data by modifying slightly the slope ($\\beta = 4.8$ ), and we add a constant $\\rho $ corresponding to the ratio of missing values: $F(i, \\lambda _{max}) = \\rho \\lambda _{max} \\left(1-\\operatorname{erf}\\left(\\frac{i\\beta }{N-1}\\right) \\right).$ The number of iterations $I_{max}$ has been raised to 1000.",
"Higher values for $I_{max}$ have a small impact on the result.",
"The value $\\rho $ has been chosen equal to $0.03$ in the inertial mode (resp.",
"0.04 in the spin mode) which is the ratio of missing values.",
"This value ensures that the maximum threshold $\\lambda _{max}$ will be larger than the “plateau\" due to the spectral leakage before inpainting (see the blue curve in Fig.",
"REF ).",
"Indeed, the algorithm does not need to spend time on high-power density values of the threshold where the spectral leakage is negligible.",
"Thus, the constant $\\rho $ makes the threshold start at a lower value and the new value for the slope of the decreasing law $\\beta $ makes the threshold decrease more quickly to the low amplitudes where the EPV signal is expected.",
"With this new decreasing law, most of the coefficients due to the high-frequency noise are caught at the first iterations, thus saving more iterations for the small coefficients where the spectral leakage is high."
],
[
"Algorithm",
"The new algorithm can be described as follows: Set the maximum number of iterations $I_{max}=1000$ , the solution $X^0$ is initialized to zero, the maximum threshold $\\lambda _{max} = \\max (\\mid \\Phi ^T Y \\mid )$ with $\\Phi ^T$ a global Discrete Cosine Transform (DCT) and the minimum threshold $\\lambda _{min} = 0$ .",
"Set $i = 0$ , $\\lambda ^0 = \\lambda _{max}$ .",
"Iterate: Set $U^i = X^i + M(Y-X^i)$ to enforce the time series to be equal to the observed data where the mask $M$ is equal to 1.",
"Compute the forward transform of $U^i$ : $\\alpha = \\Phi ^TU^i$ .",
"Compute the new threshold level $\\lambda ^i $ : $\\lambda ^i = \\rho \\lambda _{max} \\left(1-\\operatorname{erf}\\left(\\frac{i\\beta }{N-1}\\right) \\right)$ with $\\beta = 4.8$ .",
"Compute $\\tilde{\\alpha }$ by keeping only the coefficients $\\alpha $ above the threshold $\\lambda ^i$ and setting the others to zero.",
"Reconstruct $X^{i+1}$ from the remaining coefficients $\\tilde{\\alpha }$ : $X^{i+1} = \\Phi \\tilde{\\alpha }$ .",
"Set $X^{i+1}(t^{\\prime })= {c_{J}}(t^{\\prime })+ \\sum _{l=1}^{J} \\nu _l w_{l}(t^{\\prime })$ for $t^{\\prime }$ inside the gaps to apply the noise constraint described in section REF .",
"Locate the position of the EPV signal in the DCT and remove the effect of the noise constraint at the position of the peak.",
"Set $i=i+1$ .",
"If $i<I_{max}$ , return to step 3."
],
[
"Simulations",
"To assess inpainting's performance on MICROSCOPE-like data, we design a suite of simulations with the assumption that all nuisance parameters are perfectly corrected for.",
"Hence, the signal consists of just a pure sine at a well known frequency and a noise: $y_{\\rm EP}(t) = \\delta g_{\\rm EP}(t)/2 + \\mathcal {N}(t).$ The simulated time series are sampled at $f_s = 4 \\text{ Hz}$ .",
"For the sake of clarity in this section, and to have an acceptable signal-to-noise ratio, we set $\\delta =3\\times 10^{-15}$ following [5] and Paper I.",
"We then consider the two satellite configurations described in Sect : Inertial mode: 120 orbits, $f_{\\rm EP, iner} =1.8\\times 10^{-4}$ Hz Spin mode: 20 orbits, $f_{\\rm EP, spin} = 10^{-3}$ Hz Following the experimental setup used in Paper I, we define missing values in a worst case scenario, with 3% of missing values (resp.",
"4% of missing values) in the inertial mode (resp.",
"in the spin mode), due to 260 tank crackles per orbit, 24 (resp.",
"111) MLI coating crackles per orbit in the inertial mode (resp.",
"spin mode), 0.2 micrometeorite impacts per orbit and 0.05 telemetry loss per orbit.",
"The mean duration of saturated data due to crackles and micrometeorite impacts is set to 0.75 seconds (corresponding to 3 data points), and the telemetry losses can vary from 1 second to 250 seconds.",
"Gaps are distributed randomly within the time series, following a uniform distribution.",
"Gaps are not pre-defined, but their distribution is drawn randomly for each simulation, therefore we have access to their statistics only.",
"The exact probability of occurrence of these events is unknown at the time of writing.",
"However, the worst case scenario used in the simulations have been estimated by on-ground tests.",
"Finally, we generate 100 similar simulations for each configuration to perform a statistical analysis of our estimates."
],
[
"Missing data interpolation",
"The sets of simulations presented above are analyzed using the inpainting method developed in Paper I (described in Sect.",
"REF ) and the new version of the code, ICON presented in this paper (described in Sect.",
"REF ).",
"The PSD estimates averaged over 100 simulations obtained with these two methods are shown in Fig.",
"REF for the inertial mode (upper panel) and for the spin mode (lower panel).",
"The black curves show the averaged estimated MICROSCOPE differential acceleration.",
"The red curves show the averaged PSD after filling missing values using the inpainting method developed in Paper I and the green curves show the averaged PSD after filling values with the new algorithm proposed in this paper.",
"It is obvious that the original PSD is better recovered with the inpainting method presented in this paper.",
"Indeed, the spectral leakage in the PSD estimate is more than one order of magnitude smaller than the residual spectral leakage obtained with the inpainting method developed in Paper I which already reduces the spectral leakage by more than two order of magnitudes if compared to the PSD estimated from incomplete data (see Fig.",
"REF ).",
"Although the residual spectral leakage has not totally disappeared, the new version of the inpainting algorithm allows the EPV signal to clearly emerge from the noise both in the inertial and spin modes.",
"It is therefore possible to detect and estimate the amplitude of the signal in both configurations after running the improved algorithm.",
"Figure: MICROSCOPE differential acceleration PSD estimates averaged over 100 simulations in the inertial mode (upper panel) and in the spin mode (lower panel).",
"The black lines show the PSD estimated when all the data is available, the red lines show the PSD estimated from data filled with the inpainting method developed in Paper I and the green lines show the PSD estimated from data filled with the new inpainting method (ICON) presented in this paper."
],
[
"Detection and characterization of the EPV ",
"To further quantify the results obtained with inpainting, we can use common techniques to detect and characterize the EPV signal since the inpainting interpolation enables us to recover a regularly sampled time series.",
"In this work, because we are assuming perfect correction of systematics and instrumental's calibration, the signal we are looking for is just a pure sine wave, of known frequency and phase.",
"Thus, a simple least-squares fit to the corrected data is sufficient to estimate the amplitude of the EP parameter $\\delta $ .",
"For more realistic signals, a more general regression technique will be required, like a least-squares fit with a more complex model or a MCMC technique, which will allow us to constrain more parameters.",
"As aforementioned, we have run 100 realizations for the two configurations described in Sect.",
"to perform a statistical analysis of our estimate.",
"The errors we quote are the rms of the least-squares estimators estimated on these 100 simulations; in this way, we are able to quantify the combination of errors coming both from the inpainting interpolation and from the least-squares estimation.",
"In this work, because we have noted a bias in the amplitude of the EPV peak estimated after inpainting, we adopt an ordinary least-squares estimator different from the Paper I.",
"We perform a simple least-squares fit in the temporal domain to remove all possible bias introduced by the method.",
"The results of the least-squares estimation are summarized in Table REF .",
"Table: EPV signal estimation and statistical errors for a simulated EPV peak of 3×10 -15 3 \\times 10^{-15}.",
"These numbers correspond to the mean and the standard deviation of a time-domain least-squares estimators obtained on a set of 100 simulations.Table REF shows that while the inpainting algorithm of paper I strongly decreases the EPV measurement uncertainty $\\sigma _{\\delta }$ , further improvement can be expected from the new (ICON) method presented in this paper, especially for the spin mode.",
"Therefore, the new version of the inpainting code allows us to have a significant measurement of a $3\\times 10^{-15}$ EPV signal in both configurations, which would be impossible by simply performing an Ordinary Least Square fit on the available data.",
"Given our estimated $1\\sigma $ statistical error, we can conclude that with only one measurement run of 120 orbits (inertial mode) or 20 orbits (spin mode), we may be able to characterize a possible EPV signal with a $0.96\\times 10^{-15}$ precision in the inertial mode and $1.20\\times 10^{-15}$ in the spin mode assuming an instrumental noise at the level of the simulated one.",
"However, in view of the number of simulations ($N = 100$ ) used to perform the statistical analysis, the results show a bias on the estimated mean value of the EPV peak in the spin mode.",
"Indeed, the expected standard deviation of the mean should be $\\sigma _{\\rm mean}=\\sigma /\\sqrt{N}$ = $0.12 \\times 10^{-15}$ in the spin mode (resp.",
"$0.096 \\times 10^{-15}$ in the inertial mode) and the estimated mean value of the EPV peak is outside the $3\\sigma _{\\rm mean}$ error bars in the spin mode.",
"We further investigated the bias introduced by inpainting by considering different amplitudes for the EPV signal ($10^{-15}$ , $3 \\times 10^{-15}$ , $8 \\times 10^{-15}$ , $3 \\times 10^{-14}$ , $10^{-13}$ ) in the two satellite configurations (see Fig.",
"REF ).",
"The bias is more important in the spin mode.",
"Indeed, in the inertial mode (upper panel), the bias appears for EPV peaks larger than $3 \\times 10^{-15}$ and in the spin mode (lower panel), a bias is significant even for the small amplitudes of the peak.",
"The likely reason for the bias in the EPV signal estimation is that the sparsity condition in the inpainting method is not fully verified.",
"Although the decomposition into a set of oscillating functions of the DCT is ideal to represent the EP sine-wave signal ($\\delta g_{\\rm EP}(t)/2$ ), the global DCT is much less efficient to represent the continuous spectrum of the colored-noise.",
"Indeed, we checked that the bias disappears in white-noise simulations.",
"Although the bias seems to increase with the amplitude of the EPV peak, the relative bias decreases with the amplitude.",
"The bias is related to the noise spectral leakage: the more the peak is embedded in the noise after spectral leakage, the larger the bias.",
"This explains why the bias is more important in the spin mode.",
"The inpainting code (ICON) presented in this paper is dedicated to reliably asserting the detection of a possible EPV signal in parallel with the other code present in the pipeline, KARMA [5].",
"Having these two independent techniques allow us to cross-check our results.",
"If an EPV peak is detected and confirmed, further work will be needed to fully characterize the peak.",
"Figure: EPV signal estimation for different amplitudes (10 -15 10^{-15}, 3×10 -15 3 \\times 10^{-15}, 8×10 -15 8 \\times 10^{-15}, 3×10 -14 3 \\times 10^{-14}, 10 -13 10^{-13}) in the inertial case (upper panel) and in the spin case (lower panel).",
"The points correspond to the mean and the error bars to the standard deviation of the mean obtained on a set of 100 simulations."
],
[
"Conclusion",
"We have presented an updated version of the inpainting algorithm used to judiciously fill-in the missing values in MICROSCOPE data.",
"Several improvements have been made to lower the noise spectral leakage residuals of the inpainting method.",
"The major improvement was obtained by the introduction of a prior on the noise power spectrum.",
"We showed how this prior can be directly derived from the incomplete data using a multi-scale representation.",
"The second improvement consists in changing the threshold law of the iterative algorithm used to solve the minimization problem in Eq.",
"REF .",
"The idea behind this modification is to spend more time on small coefficients (depending on the missing data ratio) where the spectral leakage is more important.",
"The performance of the new inpainting code was assessed based on MICROSCOPE simulations in a worst-case scenario for missing data assuming perfect correction of systematics and perfect instrumental calibration.",
"Further work is under way to test the code in more realistic simulated data including calibration imperfections and other additional perturbations (Bergé et al in prep).",
"We showed that the performance of the new inpainting algorithm presented in this paper reach the MICROSCOPE requirements for both the inertial and spin modes.",
"With the simulated noise, our estimated statistical $1\\sigma $ error for the detection of a $3\\times 10^{-15}$ EPV signal is $0.96\\times 10^{-15}$ in the inertial mode and $1.20\\times 10^{-15}$ in the spin mode.",
"Thus, the new version of the inpainting code will replace the previous version in the official MICROSCOPE's data processing and analysis pipeline.",
"In the performance study, we noticed a bias in the estimation of the EPV signal that is explained by the fact that the sparsity constraints is not fully verified because the colored-noise is not sparse enough in the DCT representation.",
"This bias in the EPV signal estimation makes the inpainting code suboptimal to characterize a possible EPV signal with an ordinary least-squares method.",
"The major asset of the inpainting technique is that it is model-independent; this allows us to cross-check any EPV signal detection with the KARMA independent method.",
"The characterization of the EPV signal, if any detection is confirmed will require further work.",
"While the inpainting method presented in this paper has been optimized to process MICROSCOPE simulated data, it also provides a robust method to deal with missing data in the general context of time series dominated by an unknown colored-noise.",
"This is because the code is model-independent, fully adaptive to the data and should behave well in more complex data.",
"Following the reproducible research guidelines, the inpainting software presented in this study, named ICON (Inpainting for COlored-Noise dominated signals), is now freely available at the following address: http://www.cosmostat.org/software/icon/.",
"We thank Patrice Carle for his help in incorporating the inpainting code to the official MICROSCOPE data processing software.",
"We wish also to thank Florent Sureau for useful discussions.",
"This work makes use of technical data from the CNES-ESA-ONERA-CNRS-OCA Microscope mission, and has received financial support from ONERA and CNES.",
"We acknowledge the financial support of the UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02)."
]
] | 1612.05452 |
[
[
"Extreme prices in electricity balancing markets from an approach of\n statistical physics"
],
[
"Abstract An increase in energy production from renewable energy sources is viewed as a crucial achievement in most industrialized countries.",
"The higher variability of power production via renewables leads to a rise in ancillary service costs over the power system, in particular costs within the electricity balancing markets, mainly due to an increased number of extreme price spikes.",
"This study focuses on forecasting the behavior of price and volumes of the Italian balancing market in the presence of an increased share of renewable energy sources.",
"Starting from configurations of load and power production, which guarantee a stable performance, we implement fluctuations in the load and in renewables; in particular we artificially increase the contribution of renewables as compared to conventional power sources to cover the total load.",
"We then forecast the amount of provided energy in the balancing market and its fluctuations, which are induced by production and consumption.",
"Within an approach of agent based modeling we estimate the resulting energy prices and costs.",
"While their average values turn out to be only slightly affected by an increased contribution from renewables, the probability for extreme price events is shown to increase along with undesired peaks in the costs."
],
[
"Introduction",
"The increasing environmental awareness, together with the progressive reduction of production and installation costs[1], leads to a considerable growth in the amount of Renewable Energy Sources (RES) that is installed worldwide.",
"Moreover, the increasing propensity to reduce greenhouse gas emissions requires an increment of the energy produced by clean, accessible energy sources such as wind and photovoltaic (PV) generation.",
"Despite the great advantages of these energy sources, their intrinsic variability in power production badly fits to the very hierarchical structure and the strictly dispatch rules of actual power systems.",
"The limited accuracy of the prediction of their energy production profiles makes the management of these intermittent power sources difficult and limits the amount of RES generation that the power system can tolerate.",
"After the network liberalization over the last 15 years, the system balancing in real time is performed via the Electricity Balancing Market (EBM), which is a subphase of the Ancillary Services Market (ASM).",
"This market phase shall ensure the correct balanced state over the system at the transmission level, providing the security of the supply at the lowest possible costs.",
"However, the short time-scale and the volatility of this market phase produce higher energy costs when compared with the day-ahead market phase.",
"Therefore an increase in the EBM volume can lead to very high system maintenance costs.",
"The growing amount of the installed RES generation introduces a high number of partially correlated fluctuations in the power production.",
"Along with that, it becomes more difficult to predict the amount of energy that is needed for balancing the system.",
"In general, an increase in production fluctuations could lead to both an increase in market average volumes and a more frequent occurrence of extremely high values of the volume.",
"Whereas an increase in average volumes could be cured by strengthening the reserve capacity, the occurrence of extreme volumes is more difficult to control.",
"Moreover, given the fact that the relation between price and demand, also known as power stack function [2], [3], is highly nonlinear, large volume events can lead to very high energy prices.",
"Such extreme and unwanted price events have been observed already by Nicolosi in [4] for the German system: if they happen too often, the total costs of the market session increase and undermine the principles on which the electricity market was designed.",
"Therefore, the forecast of the fluctuations' impact on the balancing market can be vital for an optimal planning of the network growth, and for uncovering possible critical situations of the network.",
"So it is not surprising that the evaluation of volumes and prices of EBMs, and in general, of electricity markets, has attracted much interest in the last years.",
"However, the proposed solutions are mostly based on historical data for the market volume together with learning procedures of agents and game theory [5], [6], [7], [8], [9].",
"They need an update to more recent data and an extrapolation towards future increased contributions from RES.",
"In the next section we shall show how a combination of an approach from statistical physics with agent based modeling overcomes the need for using historical data and allows for the prediction of energy prices, the emergence of price peaks, volumes of balancing markets and overall daily costs for different contributions from RES, to cover up to 60% of the load.",
"This combination of methods was proposed and validated for the Italian EBM in [10].",
"Our results together with the methodology should enter planning procedures of how to further increase and control the amount of RES in the future."
],
[
"Methods",
"Before going into detail, let us summarize the procedure, which consists of three steps: (i) Based on real data for production and consumption at a certain representative day in the winter period of 2011-2012 in the Italian grid, we generate a certain set of starting configurations, each one describing a combination of production and load at nodes of the Italian transmission grid, which lead to a stable performance by construction.",
"The real data were taken every 15 minutes over a whole day for all 6 price zones of the power grid in Italy.",
"We then extrapolate these data towards a higher contribution of RES, ranging from the real value of 24% to 60% in the extrapolates sets.",
"In all extrapolated cases we guarantee a stable performance by running optimal DC-power flow equations to adjust the production by conventional generators so as to guarantee an overall balanced power in the grid.",
"(ii) Each of the configurations of the resulting set (6 zones x 96 time instants per day for 24, 30, 40, 50, and 60% of RES) serves as starting point for generating an ensemble of configurations in the spirit of statistical physics.",
"In statistical physics one usually describes a macrostate say of a gas of molecules by an ensemble of microstates; microstates differ by small deviations of the generalized coordinates and momenta, so that an average over many microstates leads to representative macroscopic observables.",
"In analogy here, members of each ensemble differ from the starting configuration and therefore also mutually by small deviations in the load and the renewable energy production, chosen from a Gaussian or Weibull distribution, as explained below.",
"Mean and width of the Gaussian distribution as well as the parameters of the Weibull distribution are chosen from real data and depend on the load and the type of renewable energy.",
"So each configuration j of the ensemble represents a certain realization of fluctuations in load and production, for which quantities like the resulting mismatch $S_i^j$ of power at node i, induced by the various fluctuations, can be measured.",
"$S_i^j$ can be summed over all nodes of the grid to obtain the total mismatch $S^j$ in power production for a given configuration j, which the balancing market is supposed to compensate for.",
"To obtain representative values of the mismatch (later called the market volume), also here (as for the microstates of a gas) a sufficient number of configurations should be included in the ensemble.",
"(iii) The energy balancing market The energy balancing market is modeled by a so-called market authority and a set of agents.",
"The market authority knows the required amount of power $S^j$ , which is needed for balancing consumption and production; it accepts or rejects bids from the agents until an amount of $S^j$ is obtained at the lowest possible price.",
"It then informs the agents about the outcome of their placed bids as to whether they were accepted or not.",
"The agents are assigned to conventional generators in a one-to-one relation (for simplicity).",
"They choose their bids from a distribution of propensities to offer a certain amount of energy at a certain price.",
"The propensity distribution changes with time during the learning phase, using a modified Roth-Erev algorithm[11].",
"For the learning phase we choose 3000 updates of the propensity distribution, in which the agents are trained on the same number of different configurations, chosen from the ensemble around a fixed starting configuration, so differing just by fluctuations among each other.",
"This number of updates turned out to be sufficient for the propensity distribution to converge towards an optimized distribution, resulting from the learning experience of the agents after feedback from the market authority.",
"For the next thousand configurations of a given ensemble the propensity distribution of agents is then kept fixed, and the energy price in this market session can be calculated, for each configuration, leading to a distribution of energy prices over all configurations of the ensemble.",
"The distribution of energy prices refers to a certain time during a day for a given zone in Italy.",
"Repeating the whole procedure for different starting configurations, corresponding to different instants of time at the reference day and different price zones, we can measure histograms of how often a price from a certain price interval was achieved over the day or for a restricted time interval of an hour etc.. We are particularly interested in the shape of these histograms as a function of the percentage of renewables, which contributed to the power production.",
"Details are presented in the following sections."
],
[
"Evaluation of imbalances of real-time systems",
"Let us first estimate the effect of RES and load power fluctuations on the system's power balance.",
"According to the literature[12], [13], [14], wind, PV and load forecasting errors in the power production are often treated as normal-distributed.",
"So their power production or consumption can be modeled in a statistical way, assuming truncated Gaussian-like forecast errors with standard deviations $\\sigma _i$ , where the errors represent the expected power variations at each single fluctuating element $i$ of the power grid at a given time.",
"The associated variables are the following: load power demand $D_l$ , the corresponding standard deviation $\\sigma _l$ of the forecasting error and minimum and maximum values of the distribution, $m_l$ and $M_l$ , respectively, corresponding to the load-power constraints; wind power production $G_w$ and the corresponding $\\sigma _w$ , together with the power constraints $m_w$ and $M_w$ of the generators; photovoltaic power production $G_{PV}$ , the corresponding $\\sigma _{PV}$ , and the production limits $m_{PV}$ and $M_{PV}$ , corresponding to the power constraints of the PV-generators.",
"According to these constraints, a possible state of the system can be sampled numerically by adding a random value to the expected power production and consumption at every node and RES generator $i$ of the grid.",
"The random variable is extracted from the truncated normal distribution, whose probability density function (PDF) is defined in equations REF and REF .",
"$ N_{PDF}^T =\\left\\lbrace \\begin{array}{rl}0 &\\mbox{ if $x<m_i$} \\\\N_{PDF} &\\mbox{ if $m_i<x<M_i$} \\\\0 &\\mbox{ if $x>M_i$}\\end{array} \\right.$ $ N_{PDF}(x) = \\frac{1}{\\sqrt{2\\pi \\sigma _i^2}} e^{\\frac{x^2}{2\\sigma _i^2}}.$ The outcome of this procedure is one of the possible configurations or states $j$ , in which the system in zone k ($k=1,...,6$ ) can be found in real time, due to assumed unavoidable fluctuations of RES and load.",
"In order to check the impact of other than Gaussian-type fluctuations, we complemented the normal distribution of (REF ) in case of wind production by a Weibull distribution, whose PDF is defined in equation REF , while the fluctuations for photovoltaics production and load were kept being chosen from Gaussian distributions.",
"Since the literature gives values between 1.5 and 3 for the value of the parameter $a$ [15], [16], we have chosen $a = 2$ and $\\lambda = \\frac{P_w}{\\Gamma (\\frac{3}{2})}$ , where $P_w$ here is chosen as the wind production from the reference configuration and $\\Gamma $ denotes the Gamma-function.",
"The PDF is then given as $ f(x; \\lambda , a ) ={\\left\\lbrace \\begin{array}{ll}\\frac{a}{\\lambda } (\\frac{x}{\\lambda }) ^{a - 1} e^{-( \\frac{x}{\\lambda } )^{a}},& \\text{if } x\\ge 0\\\\0, & \\text{otherwise}.\\end{array}\\right.",
"}$ Starting from the so generated configuration or network state, we apply the optimal DC-power flow algorithm [17] to calculate the power $S^j$ that is needed for balancing the mismatch in power, induced by the deviations from the starting configuration, for each state $j$ of the ensemble, related to zone k and to each of the 96 time instants a day.",
"Due to the stochastic nature of RES and load fluctuations, also $S^j$ is a random variable.",
"Therefore, to sample sufficient statistics of the market behavior, a significant number of possible balancing requirements is needed.",
"It is obtained by numerical sampling a large number of possible perturbed configurations $j$ , each one with an associated balancing requirement $S^j$ .",
"Its distribution over the ensemble and over the day is then used for describing the daily expected volume of the balancing market in the system."
],
[
"Agent Based Modeling",
"Energy prices and total costs in the EBM are determined by an agent based modeling approach, for which we use a modified Roth-Erev algorithm, introduced by Nicolaisen et al.",
"[11] in 2001 and used by Rastegar et al.",
"[18] already for the simulation of the Italian ODA electricity market.",
"The electricity-market operators are represented by agents, who learn how to place optimal bids in competitive auctions with the aim of buying (or selling) in the most profitable way.",
"In order to simulate how real market operators acquire knowledge about the market in the course of time and adapt their decisions, Roth-Erev algorithms simulate this learning process by adjusting the offer propensities of agents in a self-consistent way with the goal to maximize profits.",
"Market operators pursue economic guidelines, when they represent power plants (or groups of them) in the EBM auction phase.",
"They are allowed to place bids into the EBM auction, in which they must specify how much the corresponding power plants can vary their amount of power supplied to the system, and at what price they will offer this service.",
"For simplicity we represent each conventional power-plant generator by a single agent, although the exact relationship among market operators and brokers may vary over time and can be more involved.",
"Each agent $k$ is allowed to offer an amount of power $g^k_{off}$ that must meet the physical constraints of the power generator k: $G^k_{min} \\le G^k_{given} + g^k_{off} \\le G^k_{max}$ , where $G^k_{min}$ and $G^k_{max}$ are the minimum and maximum power production constraints of the generator, respectively, and $G^k_{given}$ is its actual power production.",
"$-G^k_{ramp} \\le g^k_{off} \\le G^k_{ramp}$ , where $G_{ramp}$ is the generator ramping constraint.",
"(Depending on the technology, each generator has ramping constraints, which limit its maximum change in power production $G_{ramp}$ in time.)",
"In order to define the bids' price, we use the concept of agent propensities, representing the willingness of each agent to place a bid at a certain price on the market.",
"The offer propensities of each operator $k$ are described in terms of a discrete set of probabilities, $q^i_k$ , to be defined below, corresponding to possible bidding strategies $\\lbrace (m^i_k, s^i_k)\\rbrace $ , which roughly speaking differ by how to deal with risks in offering higher prices.",
"The index i, $0<i<N$ , labels the strategy, $N$ is the number of possible strategies, and $s^i_k$ is the k-th operator's propensity to make an offer at a given (so-called markup) value $m^i_k$ ($1\\le m^i_k \\le 10$ for upward bids, $0\\le m^i_k \\le 1$ for downward bids).",
"The number of strategies equals the number of intervals into which the range of $m^i_k$ is divided.",
"Here we have chosen $N=50$ , so that one has to assign 50 propensities to values of $m^i_k$ .",
"The markup value determines the bidding price according to $p_{off}^k = C_{prod}^k\\cdot m^\\star _k$ , where $C_{prod}^k$ is the production cost (per MWh) of each generator k, given by its technology type, labeled by the subscript $prod$ , and $m^\\star _k$ is the actual chosen value from the discrete distribution for the bid of agent k. So the operators' behavior is modeled stochastically, where the probability of placing a bid at a given price $p_{off}^k = C_{prod}^k\\cdot m^\\star _k$ is given by the normalized propensity $q^i_k = s^i_k/\\sum _i s^i_k$ with $i=1,...,N$ and $k=1,...,G$ with G the number of conventional producers equal to the number of agents.",
"It is then the set of propensities, which get optimized when updated in an iterative reinforced learning algorithm.",
"Initially, all propensities $s^i_k$ are set to the same value $s^i_k=1$ .",
"Each learning iteration step is divided into three phases: Bid presentation: Every agent k presents a bid $\\left(g_{off}^k,p_{off}^k\\right)$ , both for the upward and downward market.",
"This bid is given by a feasible quantity of offered energy $g_{off}^k$ (i.e.",
"satisfying the physical constraints) and by a price $p_{off}^k$ , which will be drawn from the agents' propensities.",
"Market session: Given the knowledge of the total balancing needs of the system $S^j$ , all the bids, which are needed to ensure sufficient energy supply, are checked with respect to their economic profit and the physical constraints of the system.",
"Agent update: Market outcomes are communicated by the market authority to each agent, who updates his propensities in relation to the profit, which he made in the previous session.",
"The agents' propensities at iteration step $t$ are updated as follows: $s^i_k(t)=(1-r)\\cdot s^i_k(t-1) + E_i(t),$ where $r \\in [0,1]$ is a memory parameter and $E_i(t)$ is obtained from the relation: $E_i(t)= {\\left\\lbrace \\begin{array}{ll}(p_{off}^k-C_{prod^k}) \\cdot g_{off}^k & \\mbox{if the bid is accepted} \\\\e \\cdot m^i_k(t-1)/\\left(N-1\\right) & \\mbox{otherwise},\\end{array}\\right.",
"}$ for all k, where $e \\in [0,1]$ is an experimental parameter that assigns a different weight to played and non-played actions.",
"To the best of our knowledge, Roth-Erev algorithms were previously applied to training agents based on the exclusive use of historical data with their limited relevance for current and future power distributions in electricity grids.",
"In this paper, following the guidelines presented in [10], we overcome the usage of outdated data by performing the training on realistic system configurations, which are synthetically generated as we explain in the next section."
],
[
"Dataset",
"For a correct representation of the market phase, we need a detailed description of the power transmission system in space and time, in terms of network nodes, branches and generators.",
"The reference configuration of the power system is obtained by combining three datasets.",
"The first dataset is related to the characteristics of the power system (from the TERNA website [19]) and includes the geo-referenced position of every 220 and 380 KV substations together with their electrical characteristics, the geo-referenced position of the conventional generators, together with their power rates and power ramp limits, and the electrical characteristics of the power network.",
"A geographical representation of these data is depicted in figure REF .",
"Figure: Geographical representation of the topology of the reference network, chosen as the 2011 Italian transmission grid with 220 and 380 KV nodes, together with connections pointing to neighboring countries.The second dataset (from the GME website [20]) reports the detailed time evolution of production/consumption every 15 minutes of a reference day in the winter period 2011-2012, so that 96 data sets per day are available.",
"The third dataset is obtained from Atlasole and Atlavento (see the website [21], [22]).",
"These sites were made available by the Italian energy services authority [20].",
"They contain the full georeferenced information on each Italian PV and wind generator, such as the installed power and technology.",
"Combining these datasets, we reconstructed the time evolution of power production and consumption in steps of 15 minutes over a full day in the winter period 2011-2012 for all six market zones (virtual ones are excluded).",
"These data ensured already a balanced grid performance, based on optimal DC-power balancing, when the conventional power production was adjusted accordingly.",
"Moreover, starting from the real distribution of installed RES capacity, we extrapolated the data to starting configurations with a different percentage of RES production.",
"We separately assumed aggregated wind and photovoltaic productions to take desired values, given by $P_{RES}(t) = P_{\\%}\\cdot L(t)$ , with $L(t)$ being the total load that is kept fixed and $P_{\\%}$ being the percentage of load covered by RES.",
"Biomass production as a third type of RES was not considered to contribute to fluctuations, as energy production from biomass is easily controlled.",
"We then estimated the required adapted production by conventional generation to cover the load by means of an optimal DC-power flow.",
"As result we obtained initial configurations that lead to a stable grid performance for daily peak shares $P_{\\%}$ of 24% (which was the actual one used in [23]) up to and including 60%, which is nowadays already temporarily achieved.",
"For values larger than 60% it became increasingly difficult to adjust the conventional power production to compensate for the increased amount of renewables."
],
[
"Results",
"Based on the approach as outlined in the previous section , we estimate the impact of an increasing share of wind and PV generation on the Italian balancing market.",
"In particular, we identify a change in the daily market volumes and costs, due to an increased percentage of fluctuating production, introduced by these sources.",
"We tested the balancing market phase in five scenarios, characterized by a RES power production with a share of $P_{\\%}$ , chosen as 24% (as of the reference day [23]), 30%, 40%, 50% and 60% of the total load $L$ .",
"It should be noticed that the size and variation of fluctuations in the power grid are heterogeneous, as they depend on the nature of the fluctuating quantity being load, wind or photovoltaics.",
"Their respective strengths are described by Gaussian distributions with fluctuation parameters $\\sigma _l = 0.1$ , $\\sigma _{PV} = 0.08$ and $\\sigma _{w} = 0.1$ [14], that is, chosen from real data, and in case of the assumed Weibull-distributed fluctuations of wind, chosen as $a = 2$ and $\\lambda = \\frac{P_w}{\\Gamma (\\frac{3}{2})}$ with $P_w$ the wind production from the reference configuration.",
"Next it is of much interest how these fluctuations sum up over all nodes of the grid and over a whole day.",
"The resulting overall deviation from the total balanced power of the starting reference configuration can be positive or negative.",
"Figure REF shows a distribution of cases when it turns out to be negative, so that the up-energy balancing market has to compensate for this missing power.",
"Its total volume in GWh (sum over all nodes of mismatched energy) measured over all configurations of an ensemble, over the six zones of the Italian grid, and over the reference day yields the histogram of figure REF , Figure: Distribution of daily volumes for the up-market.",
"The results are presented as barplot showing five different values of P % P_\\%.",
"The peak value of each distribution is stable, but the distributions for larger P % P_\\%-values are more skewed to high values than they are for low percentages of RES.normalized over the total number of events, to predict the frequency of the various volume events.",
"Different colors code the different scenarios, characterized by the varying contributions of RES.",
"We present the distributions in two ways, as barplots, and as boxplots [24].",
"While the median (expected volume) only slightly increases for a higher amount of RES, the number of extreme market sizes of more than 10 GWh considerably increases by roughly 50% between 24% and 60% of RES.",
"The fact that the median is relatively insensitive to the amount of RES may be due to the fact that the strongest fluctuations in size are due to load fluctuations.",
"However, the whole distribution gets more skewed for a larger amount of RES.",
"Therefore rare but large market sizes should lead to high energy prices in the corresponding market sessions.",
"This must be expected even more in view of the nonlinear relation between price and demand, which is ruled by the power stack function [2], [3], emerging from the market session (agent based modeling part), so that moderate increases in the demand can lead to high changes in the price.",
"Here a remark is in order about the choice of the ensemble size, over which the observables are measured: We have chosen thousand configurations, after 3000 configurations were reserved for the learning phase of the agents.",
"This size seems to be sufficient, see figure REF , in which we compare the market volume for two samples of an ensemble with the same reference configuration and with 60% of renewables (for which the contribution of fluctuations is most pronounced).",
"Therefore the differences here are of purely statistical nature and due to the ensemble size.",
"As it is seen from figure REF , the median and quartiles of the distributions vary then less than 1%, differences are seen in the outliers.",
"Figure: A comparison between the distributions of market volumes of two different samples (green and blue) for P % =60%P_\\% = 60 \\%.",
"The plot is organized as barplot showing the two distributions in the form of histograms, and as a boxplot in the inset.",
"The two boxes of the boxplot show the distribution's median (red line), the first and third quartiles (blue lines next to the red one), and its statistical minimum and maximum, together with the outliers.Moreover, the same analysis of the market volume was performed by assuming the wind fluctuations to be Weibull-distributed.",
"The results are shown in figure REF .",
"The results for the market do not sensibly change as long as $P_\\% \\le 50\\%$ , while for $P_\\% = 60\\%$ the fat tail of wind fluctuations has an impact on the volume distribution.",
"In particular, the mode of the distribution is shifted from about 9.75 GWh to 10.5 GWh.",
"So for larger contributions of renewables one should be aware of the possibility that the non-Gaussian fluctuations in power generation finally may shift the prices and costs towards higher values, which we here have not pursued.",
"Figure: Distribution of daily volumes for the up-market, with wind fluctuations being Weibull-distributed, for five values of RES.",
"Again the plot is organized as barplot showing the five distributions in the form of histograms, and as a boxplot in the inset.",
"The largest impact is seen in the mode of the P % =60%P_\\% = 60\\% distribution.The actual results for the market prices and costs during one day under the assumption of Gaussian fluctuations are displayed in the following figures.",
"In figure REF we show the probability distribution of the market costs per day, for different values of $P_{\\%}$ .",
"Again different colors represent the different scenarios, the height of the columns is a measure for the probability to have a daily cost in the covered interval.",
"As shown in the boxplot in the inset, the median and quartiles of these distributions do not sensibly differ, similarly to figure REF .",
"This does not come as a surprise, as the balancing market is only sensitive to the power mismatch $S^j$ , where the amount of RES mainly influenced the tails of the distribution rather than the median.",
"So, also here the effect of a larger contribution of renewables is visible in the tail of the histogram of the main figure: high values of RES generation cause more likely very high costs.",
"For example, the number of events, for which the costs exceed 3 million Euro is roughly 4% for $P_\\% = 24\\%$ as compared to 5% for $P_\\% = 60\\%$ renewables.",
"Similar features are observed if the distribution of daily costs are resolved in energy prices averaged over four data per hour over all hours a day, see figure REF , where the third quartile and upper whiskers reflect a fat tail in the price distribution.",
"As a further result of the simulations, figure REF shows the average profit made by generators of different technologies during the day, for different values of $P_{\\%}$ .",
"It was calculated from the outcomes of the market sessions for each configuration, sorted with respect to the used technology of the generators, represented by the agents, and then averaged over the ensembles, the zones and the day.",
"We list generators based on coal, combined cycle, turbogas, and oil technologies.",
"It is worthwhile to stress how the average profit changes by changing $P_{\\%}$ .",
"An increase in $P_{\\%}$ causes an increase in the profit made by fast generators like turbogas or oil.",
"When high market volumes become likely due to strong fluctuations, the market authority will be forced to accept very high bids, this way encouraging a higher risk propensity for this type of generators.",
"Figure: Predicted distribution of the daily balancing market costs, for different values of P % P_{\\%}.",
"Although the prices' medians and quartiles do not sensibly increase with the increase in the share of RES, more events with extreme costs are observed.Figure: Probability distribution of the amount of sold energy, averaged over 4 sessions per hour, recorded over 1000 configurations of the ensembles for each of the 24 hours per day and the six price zones, for different values of P % P_\\%.",
"An increase in P % P_{\\%} causes an increase in the amount of energy that is sold at high prices.Figure: Average profit made by generators of different technologies, for different values of P % P_{\\%}.",
"Generators with high production costs are more likely called in the market when a high amount of renewables is present."
],
[
"Conclusions",
"Following the methodology as proposed in [10], we combined agent based modeling with a description from statistical physics by calculating the power mismatch over an ensemble of configurations that differ by small deviations from a stable power grid configuration.",
"The power mismatch entered as market volume in the energy balancing market, whose agents offer energy from a certain learned distribution of prices, until the missing amount of energy is covered by their offers.",
"The original fluctuations in load and renewables lead to fluctuations in the daily costs, handled in the balancing market, and the fluctuating energy prices.",
"While the average values of prices and actual costs only slightly increased for a higher percentage of shared renewables, the shape of the distributions became more skewed, and the number of accepted offers at extreme prices considerably increased with an increasing amount of renewables.",
"So far we have reduced the production of conventional generators as required by the artificially increased production of RES, while keeping the load and geographical location of production places fixed, that is, without closing any conventional production places.",
"Moreover, power reserve in the background was kept for sale only from online conventional generators, while different technologies may share the reserve in future grid extensions.",
"Also the underlying datasets should be further updated in view of subsequent real extensions of RES in the grid.",
"The methodology can be extended towards including other market phases than the short-time balancing market, different strategies of the agents in placing their bids, different price policies beyond the mere minimization of the energy-balancing market costs; also additional players may be included in the market games.",
"So far they were restricted to agents from conventional production places.",
"At the same time high fluctuations of RES may be damped by installing storage devices: before the fluctuations are fed into the grid, high peaks may be cut-off via storing some part of the production peaks.",
"In any case, our methodology together with these various possible extensions allow for a quantification of the risk of high price peaks.",
"So they aid decisions on whether countermeasures against such peaks are worth the effort, or the rare peaks are so rare to be safely ignored."
],
[
"Acknowledgment",
"The authors gratefully acknowledge the support from the German Federal Ministry of Education and Research (BMBF grant no.",
"03SF0472D (NET-538-167))."
]
] | 1612.05525 |
[
[
"Intra-night optical variability characteristics of different classes of\n narrow line Seyfert 1 galaxies"
],
[
"Abstract In a first systematic effort to characterize the intra-night optical variability (INOV) of different classes of narrow line Seyfert 1 (NLSy1) galaxies, we have carried out observations on a sample of radio-loud (RL) and radio-quiet (RQ) NLSy1 galaxies.",
"The RL-NLSy1 galaxies are further divided into {\\gamma}-ray loud (GL) and {\\gamma}-ray quiet (GQ) NLSy1 galaxies.",
"Our sample consists of four sets, each set consisting of a RQ-NLSy1, a GQ-NLSy1 and a GL-NLSy1 galaxy, closely matched in redshift and optical luminosity.",
"Our observations on both RQ and GQ-NLSy1 galaxies consist of a total of 19 nights, whereas the data for GL-NLSy1 galaxies (18 nights) were taken from literature published earlier by us.",
"This enabled us to do a comparison of the duty cycle (DC) of different classes of NLSy1 galaxies.",
"Using power-enhanced F-test, with a variability threshold of 1%, we find DCs of about 55%, 39% and 0% for GL, GQ and RQ-NLSy1 galaxies respectively.",
"The high DC and large amplitude of INOV (24.0 +/- 13.7%) shown by GL-NLSy1 galaxies relative to the other two classes might be due to their inner aligned relativistic jets having large bulk Lorentz factors.",
"The null DC of RQ-NLSy1 galaxies could mean the presence of low power and/or largely misaligned jets in them.",
"However, dividing RL-NLSy1 galaxies into low and high optical polarization sources, we find that sources with large polarization show somewhat higher DCs (69%) and amplitudes (29%) compared to those with low polarization.",
"This points to a possible link between INOV and optical polarization."
],
[
"Introduction",
"One of the defining characteristics of active galactic nuclei (AGN) is that they show intensity variations over the complete accessible electromagnetic spectrum from low energy radio waves to high energy $\\gamma $ -rays .",
"Such variations are known to occur on time scales ranging from minutes to hours to days , , [3].",
"Variability, particularly those observed at short time scales of the order of minutes, can be an efficient probe of AGN structure as well as the physical processes happening close to their central engine not accessible to conventional imaging techniques.",
"Among AGN, blazars (that comprise both flat spectrum radio quasars (FSRQs) and BL Lacertae (BL Lac) objects) whose jets are pointed close to our line of sight are known to show violent (large amplitude and short time scale) variations.",
"Such short time scale variations in blazars are known across the electromagnetic spectrum including $\\gamma $ -rays [4] and X-rays as well as optical .",
"Apart from blazars, other classes of AGN too are known to exhibit rapid intensity fluctuations in the optical, hereafter termed as intra-night optical variability (INOV).",
"Example includes the reports of INOV in radio-quiet quasars , lobe dominated quasars and the weak line quasars .",
"Thus, it is now convincingly established that different classes of quasars show INOV and among them blazars show large amplitude and high duty cycle (DC) of INOV compared to radio-quiet quasars that show low amplitude and low DC of INOV .",
"The detection of INOV in blazars is usually explained by invoking shocks propagating down their relativistic jets .",
"Alternative models of intrinsic flux variations in blazars include instabilities or fluctuations in relativistic jet flows , .",
"The clear existence of INOV (albeit with low amplitude and low DC) in radio-quiet quasars, had led to the idea of the presence of micro-arcsec scale optical synchrotron jets in them .",
"Hints for the presence of weak radio jets in radio-quiet quasars are known from deep VLA observations [7], as well as from the presence of hard X-ray tail in some radio-quiet quasars .",
"Though the presence of INOV in luminous quasars is well established , , their presence in the less luminous counterparts, the Seyfert galaxies is not very clear.",
"Though Seyfert galaxies are known since the discovery of AGN , a new class of Seyfert galaxies called Narrow-line Seyfert 1 galaxies (NLSy1) were discovered about three decades ago by .",
"In the optical spectrum these source have emission lines from the broad line region narrower than Seyfert 1 galaxies but broader than Seyfert 2 galaxies with FWHM ($H_\\beta $ ) less than 2000 km sec$^{-1}$ .",
"They have weak OIII $(O{\\sc III}/H_{\\beta } <3)$ and strong Fe II lines.",
"In addition to the narrow permitted lines from the broad line region (attributed to low black hole mass), NLSy1 galaxies have excess soft X-ray emission and steeper spectra in X-rays when compared to broad line Seyfert 1 galaxies.",
"They also show high accretion rates, close to the Eddington limit [8].",
"In the radio regime, NLSy1 galaxies are found to be mostly radio-quiet.",
"However, found that about $7\\%$ of the NLSy1 galaxies are radio-loud having the radio loudness parameter $R > 10$ (R is defined as the ratio of the radio flux density at 5 GHz to the optical flux density at 440 nm and among them about $ 2\\%-3\\% $ are very radio-loud (R $>$ 100).",
"This is much smaller than the radio-loud fraction of 15% known in the quasar category of AGN .",
"The small fraction can be due to 'selection effects' as the sample was selected based on $FWHM(H\\beta ) < 2000$ km s$^{-1}$ irrespective of the luminosity of the source .",
"While a majority of radio-loud NLSy1 (RL-NLSy1) galaxies are known to have a flat radio spectrum, some of them are found to be compact steep spectrum sources.",
"There are also a few reports of rapid optical flux variations in NLSy1 galaxies , .",
"After the launch of the Fermi Gamma-ray Space Telescope (hereafter Fermi), in the year 2008, the Large Area Telescope (LAT) aboard Fermi detected for the first time $\\gamma $ -ray emission from the RL-NLSy1 galaxy, PMN J0948+0022 [1].",
"This gave proof of the presence of relativistic jet in PMN J0948+0022.",
"Soon after the detection in $\\gamma $ -rays, INOV with an amplitude of variability of 0.5 mag in few hours was reported by in PMN J0948+0022.",
"It was also found to be variable by .",
"Rapid variations have also been reported in the infra-red bands by .",
"Subsequent to the report on the detection of $\\gamma $ -ray emission in PMN J0948+0022, few more RL-NLSy1 galaxies were also reported to be emitters of $\\gamma $ -rays by Fermi.",
"During the start of this observational program five NLSy1 galaxies were known to be $\\gamma $ -ray emitters [19], [20] with high significance having test statistics (TS) $>$ 100 and the $\\gamma $ -ray flux in the 0.1 $-$ 100 GeV band larger than 5 $\\times $ 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$ .",
"A 10$\\sigma $ detection corresponds to a TS value of 100 .",
"Since then more $\\gamma $ -ray emitting RL-NLSy1 galaxies have been discovered, such as 1FHL J1410.4+7408 , FBQS J1644+2619 [18] and SDSS J122222.55+041315.7 .",
"The RL-NLSy1 galaxies which emit $\\gamma $ -rays are hereafter referred to as $\\gamma $ -ray loud NLSy1 (GL-NLSy1) galaxies.",
"Majority of the GL-NLSy1 galaxies have been studied for INOV and from the observations as of now available in the literature, it is clear that their INOV properties are similar to blazars , , , .",
"Strong and variable optical polarization are also known in three GL-NLSy1 galaxies , , .",
"The broad band spectral energy distribution of GL-NLSy1 galaxies resemble FSRQs [2], , .",
"High quality observations exist on the INOV properties of radio-loud and radio-quiet quasars.",
"These studies present ample indirect evidence of the presence of relativistic miniature (parsec scale) jets in radio-quiet quasars.",
"If this observational evidence is considered to be present in the less luminous NLSy1 galaxies, it is natural to expect both RL and radio-quiet NLSy1 (RQ-NLSy1) galaxies to show INOV, though the amplitude and the prevalence of INOV will be different among them.",
"Incidentally, the high DC and amplitude of INOV detected in GL-NLSy1 galaxies reinforces their similarity to blazars and also argues for the detectability of INOV in RQ-NLSy1 galaxies.",
"However, as of now, INOV studies of other classes of NLSy1 galaxies are limited to few nights of observations , .",
"There is thus a need for systematic INOV observations of different classes of NLSy1 galaxies (a) to establish the presence of INOV in RQ-NLSy1 galaxies and (b) to compare the INOV of RQ-NLSy1 galaxies with their radio-loud counterparts.",
"To achieve the above stated objectives we have carried out a programme of systematic INOV observations of a sample of NLSy1 galaxies.",
"We present here the results of that programme.",
"Our sample consists of radio-loud and radio-quiet NLSy1 galaxies.",
"The RL-NLSy1 galaxies are further divided into $\\gamma $ -ray loud and $\\gamma $ -ray quiet (GQ) NLSy1 galaxies based on the detection of $\\gamma $ -rays by Fermi.",
"There are four sets of NLSy1 galaxies.",
"This was driven by the number of GL-NLSy1 galaxies known as well as their observability using the 2m Himalayan Chandra Telescope (HCT) at Hanle, during the time this monitoring program was initiated.",
"Each set consists of a GL-NLSy1, a GQ-NLSy1 and a RQ-NLSy1 galaxy.",
"To avoid selection biases due to differences in luminosity and redshift, objects were selected such that all sources in a particular set have similar optical brightness and redshift.",
"The four GL-NLSy1 galaxies are taken from the papers already published by us in , .",
"The remaining four RQ NLSy1 galaxies and GQ-NLSy1 galaxies each were taken from the catalogue of NLSy1 galaxies by .",
"Our entire sample thus consists of a total of 12 sources spanning the redshift range 0.06 $< z < $ 0.66.",
"The RQ-NLSy1 galaxies in each set are selected such that they are truly radio-quiet and not-detected at 1.4 GHz at 1 mJy level in the FIRSTFaint Images of the Radio Sky at Twenty cm: http://sundog.stsci.edu survey.",
"However, GQ-NLSy1 galaxies are those that have a significant detection in FIRST.",
"For the GQ and GL-NLSy1 galaxies we estimated the radio-loudness R parameter using the formula given by $log R=log\\left(\\frac{F_r}{F_{opt}}\\right)=0.4(m-t)$ where $m$ is the SDSS r-band magnitude and $t$ is the radio AB magnitude defined as $t=-2.5log\\left(\\frac{F_{int}}{3631 Jy} \\right)$ here, $F_{int}$ is the integrated radio flux and 3631 Jy is the zero point.",
"It is derived using the following relation by $m_{AB} = -2.5 log(f_{\\nu })-48.60 \\\\$ here, $f_{\\nu }$ is flux density in erg cm$^{-2}$ s$^{-1}$ Hz$^{-1}$ .",
"The properties of the objects observed for INOV are given in Table REF .",
"Table: Details of the NLSy1 galaxies that are monitored in this program.",
"Column information are as follows:(1)IAU Name (2) Other name (3)right ascension (4) declination (5) R-band magnitude (6) redshift (7) type (8) integrated flux from FIRST (here ND refers to 'Not detected' in FIRST survey) (9) radio loudness parameter and (10) optical polarization"
],
[
"Observations",
"The observations of the RQ and GQ-NLSy1 galaxies in our sample were carried out over a period of 2 years from 2012 August to 2014 September, using the 2 m HCT operated by the Indian Institute of Astrophysics, Bangalore.",
"It is a Ritchey-Chretien system with f/9 Cassegrain focus.",
"The CCD detector has a gain of 1.22e$^{-}$ and readout noise of 4.8e$^{-}/ADU$ .",
"The field of view is $10^{\\prime }\\times 10^{\\prime }$ .",
"Each pixel of the CCD corresponds to $0.3^{\\prime \\prime }\\times 0.3^{\\prime \\prime }$ of the sky http://www.iiap.res.in/centers/iao.",
"The objects were observed in Bessel R band as it gives the best S/N ratio.",
"The typical S/N ratio in our observations is around 150 and the time resolution is of the order of 300 seconds for most of the objects.",
"According to [10], the probability of detecting INOV in a source is enhanced if it is observed continuously for a duration of about 3-4 hours on any given night.",
"Hence, we tried to monitor each source for greater than 3 hours continuously whenever possible.",
"Also, each source was suitably pointed to have two or more comparison stars within a magnitude of the target NLSy1 galaxies.",
"The log of observations is given in Table REF .",
"Table: Log of observations.",
"Columns information are as follows: (1) IAU name(2) date of observation (3) duration of observation and (4) number of data points"
],
[
"Data Reduction",
"Preliminary processing (bias subtraction and flat fielding) of the observations as well as photometry were done using the IRAF software IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy Inc. under cooperative agreement with the National Science Foundation.",
"Instrumental magnitudes of the target NLSy1 galaxies and the comparison stars were determined by aperture photometry using the phot task in APPHOT package in IRAF.",
"At least three comparison stars (usually more) with similar brightness to the NLSy1 galaxies were considered in each frame.",
"A crucial parameter to the phot task is the radius of the aperture for photometry and this determines the S/N ratio of any object in the observed frames.",
"According to the S/N ratio of a source in a CCD frame is maximum when the radius of the photometric aperture is approximately equal to the full width at half maxima (FWHM) of the point spread function (PSF) of the objects and decreases for both larger and smaller apertures.",
"Hence, to select the optimum aperture that maximizes the S/N, for each night, a range of aperture radii was considered and the aperture that minimizes the standard deviation of the differential light curve (DLC) of the steadiest pair of comparison stars was taken as the optimum aperture for photometry of the target NLSy1 galaxies for that night .",
"The standard deviation of the DLC of the steadiest pair of comparison stars thus constructed will provide the actual error in the photometry, as the photometric errors reported by IRAF are an underestimate .",
"The positions and the apparent B and R magnitudes (the uncertainties of which can be up to 0.25 mag) taken from the USNO-B2 http://www.nofs.navy.mil/data/fchpix/ catalogue, of the comparison stars for each of the sources, are mentioned in Table REF .",
"Table: Properties of the comparison stars used in differential photometry.",
"Column information are as follows: (1) IAU name(2) Star identification(S) (3) right ascension (4) declination (5) B and (6) R magnitudesUsing the instrumental magnitudes obtained from photometry, DLCs of the NLSy1 galaxies were generated relative to a minimum of three comparison stars, using an optimum aperture that is close to the median FWHM on that night.",
"In most of the NLSy1 galaxies in our sample, the host galaxy has negligible effects on the photometry of the objects [12], as the central AGN dominates.",
"However, for the NLSy1 galaxies in Set 1 (lowest redshift bin), it is likely that the underlying host galaxy may affect the flux variations of the NLSy1 galaxy owing to varying seeing conditions.",
"As noted by [11], for sources with underlying host galaxy components, any spurious variations introduced by fluctuations in atmospheric seeing are typically smaller than the observational uncertainties.",
"Even so, for unambiguous detection of INOV and to rule out any spurious variations contributed by the hosts of the NLSy1 galaxies, we have checked for any close correspondence between the seeing variations in any particular night and the DLCs of the NLSy1 galaxies relative to the comparison stars.",
"We find that the host galaxies have negligible effect on the photometry reported here.",
"We define INOV as flux variations greater than or of the order of 1% which corresponds to 95% confidence (2$\\sigma $ level).",
"This is driven by the error in our photometry which is typically around 0.005 magnitudes.",
"To check for the presence or absence of INOV in a source, we have used the power-enhanced F test ."
],
[
"Power-enhanced F-test",
"This test is proposed to be free from the difficulties associated with the widely used criteria to test INOV, such as the C and F statistics and has found increased usage in recent studies of INOV in AGN , .",
"It consists of transforming the comparison star's DLCs to have the same photometric noise, as if their magnitudes exactly matched the mean magnitude of the AGN under study.",
"The enhanced F, statistical criterion is defined as $F_{enh} = \\frac{s_{qso}^2}{s_c^2}$ Here, $s_{qso}^2$ is the variance of the NLSy1 galaxy-reference star DLC and $s_c^2$ is the stacked variance of the comparison star-reference star DLCs given as $s_c^2=\\frac{1}{(\\sum _{j=1}^k N_j - k)}\\sum _{j=1}^{k}\\sum _{i=1}^{N_j}s_{j,i}^2$ where $N_j$ is the number of observations of the $j^{th}$ star, $k$ is the total number of comparison stars.",
"$s_{j,i}^2$ is the scaled square deviation, defined as $s_{j,i}^2=\\omega _j(m_{j,i}-\\bar{m_j})^2$ where, $\\omega _j$ is the scaling factor, $m_{j,i}$ 's are the differential instrumental magnitudes and $\\bar{m_j}$ is the mean differential magnitude of the reference star and the $j^{th}$ comparison star.",
"Following , we have taken $\\omega _j$ as the ratio of the averaged square error of the differential instrumental magnitudes in the NLSy1 galaxy - reference star DLC (<$\\sigma _q^2$ >) to the averaged square error of the differential instrumental magnitudes in the comparison star - reference star DLC (<$\\sigma _{s_j}^2$ >).",
"For the $j^{th}$ star DLC $\\omega _j=\\frac{<\\sigma _q^2>}{<\\sigma _{s_j}^2>}$ It is now known that the photometric errors returned by IRAF are underestimated , by factors of about 1.5.",
"But as $\\omega _j$ is the ratio of the errors, this factor gets cancelled out.",
"Thus by stacking the variances of the comparison stars, the degrees of freedom of the denominator of the F-distribution given in Equation 4 increases, which in turn increases the power of the test .",
"In this work, for most of the nights we have three stars from which the star with magnitude as close as possible or brighter than the NLSy1 galaxy in the field is taken as the reference star.",
"Hence we are left with two stars as the comparison stars.",
"The quasar and comparison stars have the same number of observations and hence the number of degrees of freedom of the numerator and denominator in Equation 4 are $\\nu _1=N-1$ and $\\nu _2=k(N-1)$ respectively.",
"The $F_{enh}$ value is then estimated and compared with the critical value of F ($F_{c}$ ) for $\\alpha =$ 0.05 which corresponds to 95% confidence level.",
"A source is considered variable if $F_{enh}$ is greater than or equal to the calculated $F_{c}$ .",
"The estimated values of $F_{enh}$ are given in Table REF ."
],
[
"Amplitude of Variability",
"For objects that are found to satisfy the adopted statistical criterion, we calculated the variability amplitude (Amp) following .",
"Amp is defined as $Amp = \\sqrt{(A_{max}-A_{min})^2-2\\sigma ^2}$ here, $A_{max}$ is the maximum in the NLSy1 galaxy - reference star DLC, $A_{min}$ is the minimum in the NLSy1 galaxy - reference star DLC and $\\sigma $ is the standard deviation of the steadiest comparison star-reference star DLC.",
"The calculated amplitude of variability for the variable NLSy1 galaxies are shown in Table REF .",
"Table REF shows the mean variability amplitude and the associated error in amplitude for the different types of NLSy1 galaxies.",
"Figure: FIRST image of the GQ-NLSy1 galaxy, J2219+1207.Figure: DLCs of the GQ-NLSy1 galaxy, J2219+1207.Figure: DLCs of the RQ-NLSy1 galaxy, J0351-0526.Figure: DLCs of the GQ-NLSy1 galaxy, J1613+5247.Figure: DLCs of the RQ-NLSy1 galaxy, J2123+0102.Figure: DLC of the GQ-NLSy1 galaxy, J2339-0912.Figure: DLC of the RQ-NLSy1 galaxy, J1326+0334.Figure: DLCs of the GQ-NLSy1 galaxy, J1256+3852.Figure: DLCs of the RQ-NLSy1 galaxy, J0037-0933.Table: Results of variability analysis.",
"NV: non-variable and V: variable.",
"Column information are as follows(1) IAU name (2) type (3) date of observation (4) reference star (5) comparison stars(6) degrees of freedom of numerator (ν 1 \\nu _1) and denominator (ν 2 \\nu _2) (7) enhanced F test value (8) critical F value for α=\\alpha =0.05 (9) enhanced F statistics variability status (10) amplitude of variability"
],
[
"Duty Cycle",
"AGN do not show variability at all times they are monitored.",
"Therefore, duty cycle (DC) of variability of each class of NLSy1 galaxy is estimated by the ratio of the time over which each object in a given class is found to be variable to the total observing time spent on monitoring the objects in a class.",
"As per the definition given by , the DC is calculated as $DC=100\\frac{\\sum _{n}^{i=1}N_i(1/\\Delta t_i)}{\\sum _{n}^{i=1}(1/\\Delta t_i)} \\%$ where $\\Delta t_i=\\Delta t_{i,obs}(1+z)^{-1}$ is the duration of monitoring session of a source on the ith night, corrected for its cosmological redshift $z$ .",
"$N_i$ is set to 1 if INOV is detected, else it is set to 0.",
"The results of DC for different classes of NLSy1 galaxies are given in Table REF ."
],
[
"Notes on individual objects monitored in this work",
"Set 1 - J2219+1207: This is a compact radio-loud source having an integrated flux density of 1.51 mJy at 1.4GHz.",
"The radio contour of the source from FIRST survey generated using AIPS AIPS is produced and maintained by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. is shown in Fig.",
"REF .",
"It has a radio loudness parameter (log R) smaller than 1 which is the dividing value between radio-loud and radio-quiet AGN as proposed by .",
"This low value might be attributed to an overestimate of the optical flux due the underlying host galaxy which is clearly visible in the optical images.",
"This source has been observed for three nights over a period of 1 year for durations ranging between about 3 to 7.6 hours.",
"For statistical calculations, S1 is taken as the reference star and S2, S3 are taken as the comparison stars on all the nights except 26 August 2014 on which only S2 is used as the comparison star as S3 varies (Fig.",
"REF ).",
"Seeing is found to be steady on all the three nights of observations.",
"On the first night, the object is found to be non-variable (NV).",
"However it shows unambiguous INOV of the order of 0.03 magnitudes on the second night.",
"On the third night, when it was monitored for the longest duration, the densely sampled light curves with 73 epochs of observation over 7.6 hours shows no INOV.",
"In the long term, it has shown both brightening and dimming behaviour during our observations.",
"Between 29 August 2013 and 26 August 2014 the source has brightened by 0.2 mag and faded by the same amount when observed a month later on 29 September 2016.",
"Set 1 - J0351-0526: This RQ-NLSy1 galaxy was monitored for two consecutive days in February 2014 for about 3 hours on both the nights.",
"For statistical analysis of variability, the reference star used is S2.",
"S1 and S3 are used as the comparison stars (Fig.",
"REF ).",
"It is classified as non-variable.",
"Set 2 - J1613+5247: This radio-loud NLSy1 galaxy with log R $=$ 1.04 was observed for three nights over a period of eight months between July 2013 and March 2014.",
"The DLC of stars S1 and S2 has the lowest standard deviation of 0.0076 on two nights and thus has been considered as the steadiest pair of stars on those two nights for statistical analysis of variability of which S1 is taken as the reference star (Fig.",
"REF ).",
"However, on one night (31 March 2014) stars S2 and S3 are taken as the reference star and comparison star respectively, as the star S1 is found to be unsteady.",
"This source has shown definite INOV on 03 July 2013 with INOV amplitude of 6.6% and on 31 March 2014 with amplitude 4.4%.",
"On 12 July 2013 it is non-variable.",
"Set 2 - J2123+0102: This radio-quiet source was monitored for two nights.",
"The reference star used for the statistical analysis is S2.",
"Stars S1, S3 are used for comparison except on 11 August 2012 where only S1 is used as S3 varies (Fig.",
"REF ).",
"The source is non-variable on both the nights.",
"Long term optical variability is seen in this source, wherein, it faded by about 0.2 magnitudes over a 10 month period between 11 August 2012 and 3 June 2013.",
"Set 3 - J2339-0912: This radio-loud NLSy1 galaxy has the highest radio flux density of 4.39 mJy at 1.4 GHz with a radio-loudness parameter (log R) of 1.79, close to the dividing line between radio-loud and very radio-loud NLSy1 galaxies .",
"It has a steep radio spectral index $\\alpha $ = $-$ 1.4 between 6 and 20cm and the NVSS contour map points to an extended structure.",
"It did not show any unambiguous evidence of INOV on the single night it was observed (Fig.",
"REF ).",
"For statistical tests, the comparison stars used are S1 and S2, whereas S3 is taken as the reference star.",
"Set 3 - J1326+0334: This RQ-NLSy1 was observed for only one epoch.",
"Based on the statistical test, we classify this source as NV on that night (Fig.",
"REF ).",
"The reference star used is S3 and S1, S2 are taken as the comparison stars.",
"Set 4 - J1256+3852: This radio-loud source was monitored for five epochs between April 2013 and May 2014.",
"The steadiest pair of stars are S2 and S3 (standard deviation $\\sim $ 0.0053) of which S2 is taken as the reference star.",
"S1 is found to vary on the first three nights and hence is not used in the analysis for those nights.",
"However, on the remaining two nights (2 February 2014 and 1 May 2014) it is included as the second comparison star.",
"This NLSy1 galaxy shows clear evidence of INOV on two nights, 5 January 2014 and 1 February 2014, with amplitude of variability of 7.1% and 4.8% respectively (Fig.",
"REF ).",
"Over a period of five months between 5 January and 01 May, 2014, the object has steadily brightened by about 0.075 mag.",
"Set 4 - J0037-0933: Classified as radio-quiet, this source was observed for two epochs, separated by a year.",
"On both the nights it is NV.",
"For characterizing the variability, the reference star used is S2 and stars S1, S3 are taken as the comparison stars (Fig.",
"REF ).",
"Table: Duty cycles, mean amplitude of variability and the corresponding error in amplitude (in%), for different classes of NLSy1 galaxies"
],
[
"Results and Discussions",
"Our results of INOV show clear contrast between RQ and RL-NLSy1 galaxies that includes GL-NLSy1 and GQ-NLSy1 galaxies.",
"None of the RQ-NLSy1 galaxies show INOV whereas three out of the four GQ-NLSy1 galaxies show INOV.",
"When compared with the INOV characteristics of GL-NLSy1 galaxies, all the four GL-NLSy1 galaxies show INOV, the results of which are already available in literature.",
"The DC of INOV is the highest in the case of GL-NLSy1 galaxies with a value of about 55%.",
"This is followed by GQ-NLSy1 galaxies with a DC of about 39%.",
"Thus, the observations reported here find clear differences between the duty cycles of INOV in different classes of NLSy1 galaxies.",
"Also, GL-NLSy1 galaxies show large amplitude variations compared to the other two classes.",
"The amplitude of INOV shown by GL-NLSy1 galaxies range from 7% to 52% with a mean of 24.0 $\\pm $ 13.7%.",
"In contrast, GQ-NLSy1 galaxies show milder INOV ranging from 3% to 7% with an average of 5.2 $\\pm $ 1.6%.",
"The non-detection of INOV in the limited sample RQ-NLSy1 galaxies studied here, need not necessarily imply the absence of INOV in this class of NLSy1 galaxies.",
"A larger sample with more monitoring observations could result in INOV detection.",
"The results obtained for NLSy1 galaxies in this work are in close agreement with that known for quasars, with blazars showing large amplitude and high DC of variability than radio-quiet quasars that show low DC.",
"The contrasting INOV behaviour between RQ and RL-NLSy1 galaxies as well as between RQ and RL-quasars can naturally be explained by relativistic beaming arguments .",
"However, optical polarization observations in the recent past on FSRQs, separated in two groups namely low polarization FSRQs and high polarization FSRQs, clearly show that high polarization FSRQs are more variable (high DC and amplitude) than their less polarized counterparts.",
"This suggest that mere presence of relativistically beamed jet is not sufficient to explain high INOV.",
"Alternatively, the close linkage between optical polarization and INOV indicates that optical polarization is the crucial factor for causing INOV.",
"This is quite on expected lines as optical polarization is closely connected to shocks in relativistically beamed jets in blazars.",
"Three of the four GL-NLSy1 galaxies considered in this work, have optical polarization observations.",
"1H 0323+343 has polarization value around 1%.",
"However it does show a flaring state with polarization value close to 3% .",
"For SBS 0846+513 and PMN J0948+0022, polarization as high as 10% and 36% have been known.",
"Optical polarization measurements are not known for other sources in our sample.",
"Dividing the GL-NLSy1 galaxies into high polarization (HP) (with P $>$ 3%) and low polarization (LP) NLSy1 galaxies, we find a DC cycle of about 69% and 59% for HP and LP-NLSy1 galaxies respectively.",
"This is in close analogy with what is seen in blazars.",
".",
"Also, GL-blazars are found to show more polarization in the optical band than GQ-blazars [5].",
"In the population of GL-NLSy1 galaxies, sources with strong optical polarization show large amplitude INOV (29.2 $\\pm $ 12.4%) compared to their less polarized counterparts (10.0 $\\pm $ 3.8%).",
"However, the caveat here is that the polarization data used here (that are compiled from literature) has not been acquired simultaneous to our INOV observation.",
"In spite of this short coming, results found here clearly suggest that high polarization GL-NLSy1 galaxies are more variable within a night than their counterparts with low optical polarization.",
"To firmly establish this point, simultaneous optical flux and polarization observations on a large number of sources are needed.",
"This is not feasible at present as there are only about half a dozen of $\\gamma $ -ray emitting NLSy1 galaxies known.",
"However, it is expected to be achievable in the future when more number of $\\gamma $ -ray emitting NLSy1 galaxies would be detected.",
"In this work we want to quantify the INOV characteristics of the different categories of NLSy1 galaxies to understand the similarities and differences among them as well as to compare with that of their luminous counterparts, the quasars.",
"This is motivated by the detection of $\\gamma $ -ray emission from few RL-NLSy1 galaxies by Fermi.",
"Available studies do indicate that RL-AGN are hosted by black holes having high masses in excess of 10$^8$ $M_{\\odot }$ , [13].",
"The detection of $\\gamma $ -ray emission in RL-NLSy1 galaxies believed to have BH masses of the order of $10^6$ - $10^7$ $M_{\\odot }$ is a direct challenge to this idea.",
"Irrespective of this, GL-NLSy1 galaxies have now emerged as a new class of $\\gamma $ -ray emitting AGN, though the number of objects known as of now are limited.",
"This is likely due the small number of NLSy1 galaxies known till date .",
"A multitude of observations on most of the GL-NLSy1 galaxies known as of now, do indicate that most of the properties of these sources are analogous to blazars, such as large INOV , , broad band SED , compact core-jet structure in the radio [14], , [15], [17], super-luminal motion [16], optical polarization , , etc.",
"All these properties do indicate the presence of relativistic jets in these sources, similar to blazars.",
"As the broad band SED and gamma-ray spectral properties of these sources have close resemblance to FSRQs, it has been postulated that they are the low mass black hole counterparts to FSRQs .",
"However, this assertion is unlikely to hold long if it could be conclusively established that these sources do have BH masses like blazars.",
"Recent reports do indicate little differences between the black hole masses of GL-NLSy1 galaxies and blazars.",
"Examples are the determination of BH mass of $6 \\times 10^8 M_{\\odot }$ in the GL-NLSy1 galaxy PKS 2004-0447 from spectro-polarimetry [6] using the broad $H_{\\alpha }$ emission line (FWHM = 9000 km/sec) in polarized light, and masses of $1.6 \\times 10^9 M_{\\odot }$ and $3.2 \\times 10^8 M_{\\odot }$ respectively in PMN J0948+0022 and PKS 1502+036 via accretion disk modeling [9].",
"Thus, recent observations including the one presented here point to close similarity between GL-NLSy1 galaxies and blazars.",
"However, ambiguity still remains on the host galaxy of GL-NLSy1 galaxies (spirals v/s ellipticals) compared to blazars that are hosted by ellipticals.",
","
],
[
"Summary",
"We have reported results of a first comparative study on the INOV behaviour of three different classes of NLSy1 galaxies, namely, RQ, GQ and GL-NLSy1 galaxies.",
"Though the observations of RQ and GQ-NLSy1 galaxies are reported for the first time, INOV studies of GL-NLSy1 galaxies were available in the literature , , .",
"Taking the observations of GL-NLSy1 galaxies reported by us earlier, the mean observing time for the sample of GL-NLSy1 galaxies is 3.8 hours over 18 nights.",
"For RQ-NLSy1 and GQ-NLSy1 galaxies sample the mean monitoring time are 4.6 hours over 7 nights and 4.4 hours over 12 nights respectively.",
"Considering the error in the photometry ($\\sigma $ = 0.0054), our observations are sensitive to detect INOV above the 1% level corresponding to 95% confidence (2$\\sigma $ ), which forms the detection threshold of our observations reported here.",
"The power-enhanced F-test was adopted to detect INOV.",
"The findings of this study are: INOV has been detected in RL-NLSy1 galaxies but not in RQ-NLSy1 galaxies.",
"RL-NLSy1 galaxies that include both GL and GQ-NLSy1 galaxies show large DC of INOV of 50% in sharp contrast to the null DC shown by RQ-NLSy1 galaxies.",
"Thus there is a marked difference in the INOV properties between RL and RQ-NLSy1 galaxies.",
"This is also in close agreement with what is known of the INOV properties of the luminous counterparts to NLSy1 galaxies, i.e.",
"RL-quasars show INOV more frequently than RQ-quasars.",
"Among RL-NLSy1 galaxies, GL-NLSy1 show the highest DC of variability of about 55.2% using the adopted variability statistics.",
"Their amplitude of variability is also larger than the other classes of NLSy1 galaxies.",
"They show a large mean variability amplitude of 24.0 $\\pm $ 13.7% compared to GQ-NLSy1 galaxies (5.2 $\\pm $ 1.6%).",
"Such extreme INOV behaviour shown by GL-NLSy1 galaxies might be due to the high bulk Lorentz factor of their inner jets.",
"The null DC of INOV shown by RQ-NLSy1 galaxies could be due to a less powerful or a misaligned (and consequently a lower $\\delta $ ) jet compared to the other classes of NLSy1 galaxies.",
"Dividing the GL-NLSy1 galaxies into HP-NLSy1 and LP-NLSy1 galaxies based on their optical polarization properties, we find that HP-NLSy1 galaxies show DC (amplitude) of variations of about 69.0(29.2 $\\pm $ 12.4)%, which appear to be somewhat higher than the values of 58.9(10.0 $\\pm $ 3.8)% shown by the LP-NLSy1 galaxies."
],
[
"Acknowledgements",
"We thank the referee Prof. Paul Wiita for his critical comments on the manuscript.",
"We also thank the staff of IAO, Hanle and CREST, Hoskote, who made these observations possible.",
"The facilities at IAO and CREST are operated by the Indian Institute of Astrophysics, Bangalore."
]
] | 1612.05589 |
[
[
"On the best constant matrix approximating an oscillatory matrix-valued\n coefficient in divergence-form operators"
],
[
"Abstract We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients.",
"We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the equation can be incomplete.",
"A theoretical foundation of the approach in the limit of infinitely small oscillations of the coefficients is provided, using the classical theory of homogenization.",
"We present a comprehensive study of the implementation aspects of our method, and a set of numerical tests and comparisons that show the potential practical interest of the approach.",
"The approach detailed in this article improves on an earlier version briefly presented in [C. Le Bris, F. Legoll and K. Li, C. R. Acad.",
"Sci.",
"Paris 2013]."
],
[
"Context",
"Consider the simple, linear, elliptic equation $ -{\\rm div}(A_\\varepsilon \\nabla u_\\varepsilon ) = f \\ \\ \\text{in ${\\cal D}$}, \\qquad u_\\varepsilon = 0 \\ \\ \\text{on $\\partial {\\cal D}$},$ in divergence-form, where ${\\cal D}\\subset \\mathbb {R}^d$ , $d\\ge 1$ , is an open, bounded domain which delimits what we hereafter call 'the physical medium', and where $A_\\varepsilon $ is a possibly random oscillatory matrix-valued coefficient.",
"We suppose that all the requirements are satisfied so that problem (REF ) is well-posed.",
"In particular, we assume that $A_\\varepsilon $ is bounded and bounded away from zero uniformly in $\\varepsilon $ .",
"Our assumptions will be detailed in Section REF below.",
"The subscript $\\varepsilon $ encodes the characteristic scale of variation of the matrix field $A_\\varepsilon $ .",
"For instance, one may think of the case $A_\\varepsilon (x)=A^{\\rm per}(x/\\varepsilon )$ for a fixed $\\mathbb {Z}^d$ -periodic matrix field $A^{\\rm per}$ , although all what follows is not restricted to that particular case.",
"It is well-known that, for $\\varepsilon $ small (comparatively to the size of ${\\cal D}$ ), and not necessarily infinitesimally small, the direct computation of the solution to (REF ) is expensive since, in order to capture the oscillatory behavior of $A_\\varepsilon $ and $u_\\varepsilon $ , one has to discretize the domain ${\\cal D}$ with a meshsize $h\\ll \\varepsilon $ .",
"The computation becomes prohibitively expensive in a multi-query context where the solution $u_\\varepsilon (f)$ is needed for a large number of right-hand sides $f$ (think, e.g., of a time-dependent model where (REF ), or a similar equation, should be solved at each time step $t^n$ with a right-hand side $f(t^n)$ , or of an optimization loop with $f$ as an unknown variable, where (REF ) would encode a distributed constraint).",
"Alternatives to the direct computation of $u_\\varepsilon $ exist.",
"Depending on the value of $\\varepsilon $ , the situation is schematically as follows.",
"$\\bullet $ For $\\varepsilon <\\overline{\\varepsilon }$ , where $\\overline{\\varepsilon }$ is a given, medium-dependent threshold (typically $\\overline{\\varepsilon }\\approx {\\rm size}({\\cal D})/10$ ), one can consider that homogenization theory [3], [13], [19] provides a suitable framework to address problem (REF ).",
"That theory ensures the existence of a limit problem for infinitely small oscillations of the coefficient $A_\\varepsilon $ .",
"The limit problem reads $ -{\\rm div}(A_\\star \\nabla u_\\star ) = f \\ \\ \\text{in ${\\cal D}$}, \\qquad u_\\star = 0 \\ \\ \\text{on $\\partial {\\cal D}$}.$ The matrix-valued coefficient $A_\\star $ is (i) non-oscillatory, (ii) independent of $f$ , and (iii) given by an abstract definition that can become more or less explicit, depending on the assumptions concerning the structure of $A_\\varepsilon $ (and the probabilistic setting in the random case).",
"The solution to the homogenized problem (REF ) can be considered an accurate $L^2$ -approximation of the oscillatory solution to (REF ) as soon as the size $\\varepsilon $ of the oscillations of $A_\\varepsilon $ is sufficiently small.",
"There are several cases for which the abstract definition giving $A_\\star $ can be made explicit.",
"The simplest examples are (i) periodic coefficients of the form $A_\\varepsilon (x)=A^{\\rm per}(x/\\varepsilon )$ , with $A^{\\rm per}$ a $\\mathbb {Z}^d$ -periodic matrix field, and (ii) stationary ergodic coefficients of the form $A_\\varepsilon (x,\\omega )=A^{\\rm sto}(x/\\varepsilon ,\\omega )$ , with $A^{\\rm sto}$ a (continuous or discrete) stationary matrix field.",
"In both cases, one can prove that $A_\\star $ is a deterministic constant (i.e.",
"independent of $x$ ) matrix, for which a simple explicit expression is available.",
"Whenever a corrector (in the terminology of homogenization theory, see [3], [13], [19] and (REF )–(REF ) below) exists, it is in addition possible to reconstruct an $H^1$ -approximation of the solution to (REF ), using the solutions to the corrector problem and to the homogenized problem (REF ).",
"Practically, whenever an explicit definition is available for $A_\\star $ , one can compute an approximation of the oscillatory solution to (REF ) by solving the non-oscillatory problem (REF ).",
"The advantage is obviously that the latter can be solved on a coarse mesh.",
"The cost of the method then lies in the offline computation of $A_\\star $ .",
"$\\bullet $ For $\\varepsilon \\ge \\overline{\\varepsilon }$ , the size of the oscillations is too large to consider that homogenization theory provides a suitable framework to approximate problem (REF ), and one may use, in order to efficiently compute an approximation of $u_\\varepsilon $ , dedicated numerical approaches.",
"Classical examples include the Variational Multiscale Method (VMM) introduced by Hughes et al.",
"[12], and the Multiscale Finite Element Method (MsFEM) introduced by Hou and Wu [11] (see also the textbook [9]).",
"We also refer to the more recent works by Målqvist and Peterseim [17] (on the Local Orthogonal Decomposition (LOD) method), or Kornhuber and Yserentant [14], on localization and subspace decomposition.",
"Many more examples of approaches are available in the literature.",
"The MsFEM approach (as well as the LOD approach) is essentially based on an offline/online decomposition of the computations.",
"In the first step, local problems are solved at the microscale, in order to compute oscillatory basis functions.",
"Each basis function is obtained by solving an oscillatory problem posed on a macro-element or on a patch of macro-elements.",
"These oscillatory problems do not depend on the right-hand side $f$ , and are independent one from another.",
"In the second step, the global problem, which depends on the right-hand side $f$ , is solved.",
"The second step is performed, e.g., by considering a Galerkin approximation on the multiscale discrete space built in the offline step.",
"The original online cost of solving an oscillatory problem on a fine mesh (using a discrete space at one single fine scale) is reduced to solving an oscillatory problem on a coarse mesh consisting of macro-elements (using a multiscale discrete space).",
"These methods provide an $H^1$ -approximation of the oscillatory solution $u_\\varepsilon $ .",
"Note that they are (a priori) applicable without any restriction on the structure of $A_\\varepsilon $ , and are also applicable, and indeed applied, in the regime $\\varepsilon <\\overline{\\varepsilon }$ .",
"Note also that, in the stochastic setting, the computations must be performed $\\omega $ by $\\omega $ , for “each” realization $\\omega $ of the random environment.",
"The finite element Heterogeneous Multiscale Method (HMM) introduced by E and Engquist [8] is another popular multiscale technique.",
"It is however based on a different perspective.",
"Its aim is to compute an approximation of the coarse solution $u_\\star $ by means of local averages of the oscillatory coefficient $A_\\varepsilon $ .",
"One way or another, all these approaches rely on the knowledge of the coefficient $A_\\varepsilon $ .",
"It turns out that there are several contexts where such a knowledge is incomplete, or sometimes merely unavailable.",
"From an engineering perspective (think, e.g., of experiments in Mechanics), there are numerous prototypical situations where the response $u_\\varepsilon (f)$ can be measured for some loadings $f$ , but where $A_\\varepsilon $ is not completely known.",
"In these situations, it is thus not possible to use homogenization theory, nor to proceed with any MsFEM-type approach or with the similar approaches mentioned above.",
"We have discussed above two possibilities to address multiscale problems such as (REF ), using either the homogenization theory or dedicated numerical approaches.",
"Restricting our discussion to homogenization theory, we can identify three limitations, quite different in nature, to the practical application of the theory: $\\bullet $ First, homogenization theory has been developed in order to address the case of infinitely small oscillations of the coefficients, and is hence not appropriate for media such that $\\varepsilon \\ge \\overline{\\varepsilon }$ .",
"In practice, one may for instance want to evaluate the effective coefficients (such as the Poisson ratio and the Young modulus for problems in Mechanics) of a medium for which $\\varepsilon \\ge \\overline{\\varepsilon }$ .",
"It is always possible (if an explicit definition is available) to compute $A_\\star $ , considering on purpose the (fictitious) limit of infinitely small oscillations, but there is no reason for that $A_\\star $ to be an accurate approximation of the medium it is supposed to describe.",
"$\\bullet $ Assume that an explicit expression is available for $A_\\star $ .",
"A practical limitation is that, in most cases except for the somewhat ideal case of periodic coefficients (with a known period), the computation of $A_\\star $ by classical methods is expensive.",
"For instance, in the stochastic setting, the computation of $A_\\star $ requires to solve, many times, a corrector problem set on a truncated approximation of an asymptotically infinitely large domain.",
"This is especially challenging in the stationary ergodic case with long-range correlations.",
"Note that equivalent limitations appear for MsFEM-type or similar approaches in the stochastic setting.",
"$\\bullet $ Another evident limitation shows up when one examines the homogenized limit of (REF ) for a coefficient $A_\\varepsilon $ such that no explicit expression is available for $A_\\star $ (although $A_\\varepsilon $ is well-known, and although the homogenized limit of (REF ) is known to read as (REF )).",
"This case might occur as soon as $A_\\varepsilon $ is not the rescaling $A(\\cdot /\\varepsilon )$ of a simple (periodic, quasi-periodic, random stationary, ...) function $A$ .",
"Finding a pathway alternate to standard approaches is thus a practically relevant question.",
"Given our discussion above, we are interested in approaches valid for the different regimes of $\\varepsilon $ , which make no use of the knowledge on the coefficient $A_\\varepsilon $ , but only use some (measurable) responses of the medium (obtained for certain given solicitations).",
"Questions similar in spirit, but different in practice, have been addressed two decades ago by Durlofsky in [7].",
"They are similar in spirit because the point is to define an effective coefficient only using outputs of the system.",
"They are however different in practice because the effective matrix is defined by upscaling, and hence the approach of [7] is local.",
"This approach is indeed based on considering, in a representative elementary volume, some particular problems (with zero loading and suitable boundary conditions), for which the solutions in the case of homogeneous coefficients are affine and write as independent of these homogeneous coefficients.",
"Considering $d$ choices of such problems (that is, $d$ choices of boundary conditions), and postulating the equality of the fluxes respectively resulting from the original oscillatory and homogeneous equivalent problems, one determines the coefficients of an “effective” matrix.",
"Several variants exist in the literature, as well as many other approaches.",
"The original approach we introduce in this article improves on an earlier version briefly presented in [16].",
"Our approach is global, in the sense that it uses the responses of the system in the whole domain ${\\cal D}$ .",
"Note of course that it can be used locally as an upscaling technique, for instance in problems featuring a prohibitively large number of degrees of freedom.",
"In passing, we note that our approach provides, at least in some settings, a characterization of the homogenized matrix which is an alternative to the standard characterization of homogenization theory (see Proposition REF below).",
"To the best of our knowledge, this characterization has never been made explicit in the literature.",
"Throughout this article, we restrict ourselves to cases when problem (REF ) admits (possibly up to some extraction) a homogenized limit that reads as problem (REF ), where the homogenized matrix coefficient $ A_\\star \\text{ is {\\em deterministic} and {\\em constant}.",
"}$ This restrictive assumption on the class of $A_\\star $ (and thus on the structure of the coefficient $A_\\varepsilon $ in (REF ), and on the probabilistic setting in the random case) is useful for our theoretical justifications, but not mandatory for the approach to be applicable (see Section REF below)."
],
[
"Presentation of our approach",
"We now sketch, for a coefficient $A_\\varepsilon $ that we take for simplicity deterministic, the idea underlying our approach.",
"Let ${\\cal S}$ denote the set of real-valued $d\\times d$ positive-definite symmetric matrices.",
"For any constant matrix $\\overline{A} \\in {\\cal S}$ , consider generically the problem with constant coefficients $ -{\\rm div}(\\overline{A}\\nabla \\overline{u})=f \\ \\ \\text{in ${\\cal D}$}, \\qquad \\overline{u} = 0 \\ \\ \\text{on $\\partial {\\cal D}$}.$ We investigate, for any value of the parameter $\\varepsilon $ , how we may define a constant matrix $\\overline{A}_\\varepsilon \\in {\\cal S}$ such that the solution $\\overline{u}_\\varepsilon $ to problem (REF ) with matrix $\\overline{A} = \\overline{A}_\\varepsilon $ best approximates the solution $u_\\varepsilon $ to (REF ).",
"Note that, since $\\overline{A}_\\varepsilon $ is constant, its skew-symmetric part plays no role in (REF ).",
"We hence cannot hope for characterizing the skew-symmetric part of $\\overline{A}_\\varepsilon $ .",
"Without loss of generality, we henceforth make the additional assumption that the homogenized matrix $A_\\star $ is symmetric and that we seek a best (constant) symmetric matrix.",
"Should $A_\\star $ not be symmetric, it is replaced in the sequel by its symmetric part.",
"In [16], the constant matrix $\\overline{A}_\\varepsilon $ is defined as a minimizer of $ \\inf _{\\overline{A} \\in {\\cal S}} \\; \\sup _{f\\in L^2({\\cal D}), \\, {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}=1} \\ {\\left\\Vert u_\\varepsilon (f)-\\overline{u}(f)\\right\\Vert }_{L^2({\\cal D})}^2,$ where we have emphasized the dependency upon the right-hand side $f$ of the solutions to (REF ) and (REF ).",
"The use of a $L^2$ norm in (REF ) (and not of e.g.",
"a $H^1$ norm) is reminiscent of the fact that, for sufficiently small $\\varepsilon $ , we wish the best constant matrix to be close to $A_\\star $ , and that $u_\\varepsilon $ converges to $u_\\star $ in the $L^2$ norm but not in the $H^1$ norm.",
"Note that problem (REF ) is only based on the knowledge of the outputs $u_\\varepsilon (f)$ (that could be, e.g., experimentally measured), and not on that of $A_\\varepsilon $ itself.",
"Note also that, in practice, we cannot maximize upon all right-hand sides $f$ in $L^2({\\cal D})$ (with unit norm).",
"We therefore have to replace the supremum in (REF ) by a maximization upon a finite-dimensional set of right-hand sides, which we will have to select thoughtfully (see Section REF ).",
"In this article, we keep the same type of characterization for $\\overline{A}_\\varepsilon $ as in [16] (that is, through an inf-sup problem), but we use a slightly different cost function than in (REF ).",
"The constant matrix $\\overline{A}_\\varepsilon $ is here defined as a minimizer of $ \\inf _{\\overline{A} \\in {\\cal S}} \\ \\sup _{f\\in L^2({\\cal D}), \\, {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}=1} \\ {\\left\\Vert (-\\Delta )^{-1}\\left({\\rm div}(\\overline{A}\\nabla u_\\varepsilon (f))+f\\right)\\right\\Vert }_{L^2({\\cal D})}^2,$ where $(-\\Delta )^{-1}$ is the inverse laplacian operator supplied with homogeneous Dirichlet boundary conditions: for any $g\\in H^{-1}({\\cal D})$ , $z = (-\\Delta )^{-1}g$ is the unique solution in $H^1_0({\\cal D})$ to $ -\\Delta z=g \\ \\ \\text{in ${\\cal D}$}, \\qquad z=0 \\ \\ \\text{on $\\partial {\\cal D}$}.$ The cost function of (REF ) is related to the one of (REF ) through the application, inside the $L^2$ norm of the latter, of the zero-order differential operator $(-\\Delta )^{-1}\\left({\\rm div}(\\overline{A}\\nabla \\cdot )\\right)$ .",
"Note that, in sharp contrast with (REF ), the function $\\displaystyle {\\left\\Vert (-\\Delta )^{-1}\\left({\\rm div}(\\overline{A}\\nabla u_\\varepsilon (f))+f\\right)\\right\\Vert }_{L^2({\\cal D})}^2$ used in (REF ) is a polynomial function of degree 2 in terms of $\\overline{A}$ , a property which brings stability and significantly speeds up the computations.",
"The specific choice (REF ) has been suggested to us by Albert Cohen (Université Pierre et Marie Curie).",
"Remark 1 The reason to choose $f \\in L^2({\\cal D})$ in (REF ), rather than $f \\in H^{-1}({\\cal D})$ , is discussed in Remark REF below.",
"Several criteria can be considered to assess the quality and the usefulness of our approach: asymptotic consistency: does the sequence $\\left\\lbrace \\overline{A}_\\varepsilon \\right\\rbrace _{\\varepsilon >0}$ of best matrices, defined as minimizers of (REF ), converge, when $\\varepsilon $ goes to 0, to the homogenized matrix $A_\\star $ ?",
"If this is indeed the case, the approach provides an approximation for the homogenized matrix alternate to standard homogenization (note, in particular, that our approach does not require solving a corrector problem).",
"efficiency: practically, is this best matrix $\\overline{A}_\\varepsilon $ efficiently computable?",
"In particular, how many right-hand sides does its computation really require?",
"$L^2$ -approximation: for any fixed $\\varepsilon $ , not necessarily small, how well does the solution $\\overline{u}_\\varepsilon $ to (REF ) with matrix $\\overline{A}_\\varepsilon $ approximate the reference solution $u_\\varepsilon $ to (REF ) in the $L^2$ norm?",
"$H^1$ -approximation: using $\\overline{A}_\\varepsilon $ , is it possible to reconstruct (if possible for a marginal additional cost) an accurate approximation of $u_\\varepsilon $ in the $H^1$ norm?",
"Recall that in homogenization theory, a corrector problem must be solved to compute the homogenized matrix, but once this is performed, one can reconstruct an $H^1$ -approximation of $u_\\varepsilon $ using the solution of the latter problem at no additional cost."
],
[
"Outline and perspectives",
"The article is organized as follows.",
"To begin with, we introduce in Section the assumptions we will make throughout the article, and we recall the basics of homogenization.",
"We formalize our approach in Section .",
"We establish an asymptotic consistency result (thereby positively answering to Question (REF ) above, see Proposition REF ), and we explain how the best matrix we compute can be used to construct an approximation in the $H^1$ norm of the oscillatory solution (hence addressing Question (REF ) above).",
"We also detail how to approximate the infinite-dimensional space $\\displaystyle \\left\\lbrace f\\in L^2({\\cal D}), \\ \\ {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}=1 \\right\\rbrace $ present in (REF ) by a finite-dimensional space of the form $\\displaystyle \\text{Span} \\, \\left\\lbrace f_p, \\ 1 \\le p \\le P \\right\\rbrace $ for some appropriate functions $f_p$ (see (REF ) below).",
"In Section , we explain how the problem of finding the best constant matrix can be efficiently solved in practice (thereby answering to Question (REF )).",
"Finally, in Section , we present, as a practical answer to Questions (REF ), (REF ), (REF ) and (REF ), a number of representative numerical experiments, both in the periodic and stationary ergodic settings, and we provide some comparison with the classical homogenization approach.",
"We show in particular that choosing a small number $P$ of right-hand sides (in practice, we often set $P=d(d+1)/2$ ) is sufficient for our approach to provide accurate results.",
"We emphasize that the aim of the numerical experiments described in Section is different in the periodic setting and in the stochastic setting.",
"In the former case, computing the homogenized matrix is inexpensive, and thus we cannot hope for our approach (which requires solving highly oscillatory equations) to outperform the classical homogenization approach in terms of efficiency.",
"The periodic setting is hence to be considered as a validation setting.",
"The situation is entirely different in the stochastic setting, which is much more challenging.",
"In that setting, our approach can compete as far as Questions (REF ), (REF ) and (REF ) are concerned.",
"We show that, for an essentially identical computational cost compared to the standard homogenization approach, our approach allows us to compute a more accurate approximation of the solution $u_\\varepsilon $ to the highly oscillatory equation, both in $L^2$ and in $H^1$ norms.",
"More importantly, the reader should bear in mind that our approach targets practical situations where the information on the oscillatory coefficients in the equation may be incomplete.",
"The comparison with standard homogenization approaches which is performed in Section is hence somewhat unfair for our approach, as the former approaches need a complete knowledge of the coefficient $A_\\varepsilon $ , whereas ours does not.",
"There are several possible follow-ups for this work: First, one can perform a detailed study of the robustness of the approach with respect to imprecise data, assuming for instance that we only have access locally to coarse averages of the outputs $u_\\varepsilon (f)$ or $\\nabla u_\\varepsilon (f)$ .",
"Second, the extension to nonlinear equations may be studied, where the oscillatory problem is formulated as the optimization problem $\\inf \\left\\lbrace \\int _{\\cal D}K\\left(\\frac{x}{\\varepsilon },\\nabla u(x)\\right){\\rm d}x-\\int _{\\cal D}f(x)u(x)\\,{\\rm d}x, \\quad u \\in W^{1,p}_0({\\cal D})\\right\\rbrace ,$ where the function $\\xi \\in \\mathbb {R}^d \\mapsto K(\\cdot ,\\xi )$ is strictly convex.",
"In a multi-query context, our approach (and this is also true for other approaches) is even more interesting for nonlinear equations than for linear ones.",
"Indeed, however large the parameter $\\varepsilon $ is, solving a nonlinear oscillatory equation for a large number of right-hand sides is prohibitively expensive.",
"In contrast, in the linear case, as soon as the LU decomposition of the stiffness matrix can be computed and stored, i.e.",
"as soon as $\\varepsilon $ is not too small, the cost for computing several solutions becomes almost equal to the cost for computing one.",
"The computational workload thus remains affordable.",
"This is not the case in a nonlinear context.",
"Third, the approach may be extended to homogenized matrices that are not constant.",
"Indeed, as soon as some additional information is available on $A_\\star $ , one could adequately modify the search space for $\\overline{A}$ in (REF ) or (REF ).",
"For instance, the case of a slowly varying matrix $A_\\star (x)$ , depending upon $x \\in {\\cal D}$ in a sense to be made precise, can be considered.",
"Following a suggestion by Albert Cohen, it may also be possible to balance the dimension of the space in which $\\overline{A}$ is searched with the amount of noise present in the problem (which is related to the value of $\\varepsilon $ ) and the number of fine-scale solutions that are available (here the dimension $P$ of the space (REF ) introduced below)."
],
[
"Preliminaries",
"We describe the stationary ergodic setting we adopt.",
"This setting includes, as a particular case, the periodic case.",
"For a more detailed presentation of the particular stochastic setting we here consider, we refer to the theoretically-oriented articles [4], [5], to the numerically-oriented articles [6], [15], and to the review article [2] (as well as to the extensive bibliography contained therein).",
"For more insight on stochastic homogenization in general, we refer the reader to the seminal contribution [18], to [10] for a numerically-oriented presentation, as well as to the classical textbooks [3], [13].",
"The reader familiar with that theory may easily skip this section and directly proceed to Section ."
],
[
"Assumptions",
"Recall that ${\\cal D}$ denotes an open, bounded subset of $\\mathbb {R}^d$ , $d\\ge 1$ .",
"Let $(\\Omega ,{\\cal Z},\\mathbb {P})$ be a probability space, on which we assume an ergodic structure, and let $\\displaystyle \\mathbb {E}(X) = \\int _\\Omega X(\\omega )\\,{\\rm d}\\mathbb {P}(\\omega )$ be the expectation of any random variable $X\\in L^1(\\Omega ,{\\rm d}\\mathbb {P})$ .",
"We consider problem (REF ), which reads, in the stochastic setting, as $ -{\\rm div}(A_\\varepsilon (\\cdot ,\\omega )\\nabla u_\\varepsilon (\\cdot ,\\omega ))=f \\ \\ \\text{a.s.~in ${\\cal D}$}, \\qquad u_\\varepsilon (\\cdot ,\\omega )=0 \\ \\ \\text{a.s.~on $\\partial {\\cal D}$},$ where the function $f \\in {L^2(\\cal D)}$ is independent of $\\varepsilon $ and deterministic (see Remark REF below for a discussion on the choice of taking $f$ in ${L^2(\\cal D)}$ ).",
"We assume that $ A_\\varepsilon (x,\\omega )=A^{\\rm sto}(x/\\varepsilon ,\\omega ),$ where $A^{\\rm sto}$ is such that there exist deterministic real numbers $\\alpha ,\\beta >0$ such that $ A^{\\rm sto}(\\cdot ,\\omega )\\in L^\\infty (\\mathbb {R}^d;{\\cal S}_{\\alpha ,\\beta })\\quad \\text{almost surely},$ with ${\\cal S}_{\\alpha ,\\beta } = \\left\\lbrace M\\in \\mathbb {R}^{d \\times d}, \\ \\ \\text{$M$ is symmetric}, \\ \\ \\alpha \\, |\\xi |^2 \\le \\xi ^T M \\xi \\le \\beta \\, |\\xi |^2 \\ \\text{for any $\\xi \\in \\mathbb {R}^d$} \\right\\rbrace .$ In addition, we assume that $A^{\\rm sto}$ is a discrete stationary matrix field.",
"A complete description of the discrete stationary ergodic setting we here consider can be found, e.g., in the review article [2].",
"For brevity, we only mention here that the purpose of this setting is to formalize the fact that, even though realizations may vary, the matrix $A^{\\rm sto}$ at point $y \\in \\mathbb {R}^d$ and the matrix $A^{\\rm sto}$ at point $y+k$ , $k \\in \\mathbb {Z}^d$ , share the same probability law.",
"The local, microscopic environment (encoded in the oscillatory matrix field $A_\\varepsilon (x,\\omega )=A^{\\rm sto}(x/\\varepsilon ,\\omega )$ ) has a $\\varepsilon \\mathbb {Z}^d$ -periodic structure on average.",
"Assumption (REF ) ensures the existence and uniqueness of the solution to (REF ) in $H^1_0({\\cal D})$ , almost surely.",
"Furthermore, almost surely, the solution $u_\\varepsilon (\\cdot ,\\omega )$ to (REF ) converges (strongly in $L^2({\\cal D})$ and weakly in $H^1({\\cal D})$ ) to some $u_\\star \\in H^1_0({\\cal D})$ solution to (REF ), where the homogenized matrix $A_\\star $ is deterministic, constant and belongs to ${\\cal S}_{\\alpha ,\\beta }$ .",
"As is well-known, $A_\\star $ is independent of the right-hand side $f$ in (REF ).",
"Remark 2 The above discussion is not restricted to the discrete stationary setting.",
"We could as well have considered the continuous stationary setting, where the probability law of $A(y,\\omega )$ does not depend on $y$ .",
"Remark 3 The form of the homogenized equation (REF ) is in this context identical to that of the original equation (REF ).",
"This is not a general fact.",
"Although definite conclusions are yet to be obtained, there are all reasons to believe that the practical approach we introduce in this article carries over to cases where the homogenized equation is of a different form.",
"The periodic setting is a particular case of the above discrete stationary setting, when $A$ is independent of $\\omega $ .",
"This amounts to assuming that $ A_\\varepsilon (x)=A^{\\rm per}(x/\\varepsilon ),$ with $A^{\\rm per}$ a $\\mathbb {Z}^d$ -periodic matrix field such that $ A^{\\rm per}\\in L^\\infty (\\mathbb {R}^d;{\\cal S}_{\\alpha ,\\beta }).$"
],
[
"Classical homogenization approach",
"We briefly recall here the basics of homogenization.",
"We focus the presentation on the stationary ergodic setting.",
"The easy adaptation to the periodic setting is briefly commented upon.",
"Let $Q = (0,1)^d$ .",
"In the discrete stationary ergodic setting, the (deterministic, constant and symmetric) homogenized matrix $A_\\star $ reads, for all $1\\le i,j\\le d$ , as $ \\left[ A_\\star \\right]_{i,j} = \\mathbb {E} \\left( \\int _Q \\left( e_i + \\nabla w_{e_i}(y,\\cdot ) \\right)^T \\, A^{\\rm sto}(y,\\cdot ) \\, \\left( e_j + \\nabla w_{e_j}(y,\\cdot ) \\right) \\, {\\rm d}y \\right),$ where $(e_1,\\ldots ,e_d)$ denotes the canonical basis of $\\mathbb {R}^d$ , and where, for any $p\\in \\mathbb {R}^d$ , $w_{p}$ is the solution (unique up to the addition of a random constant) to the so-called corrector equation $ \\left\\lbrace \\begin{array}{l}-{\\rm div}\\left(A^{\\rm sto}(\\cdot ,\\omega )(p + \\nabla w_{p}(\\cdot ,\\omega ))\\right)=0\\quad \\text{a.s.~in~$\\mathbb {R}^d$},\\\\\\nabla w_{p}\\text{ is stationary},\\qquad \\displaystyle \\mathbb {E}\\left(\\int _Q\\nabla w_{p}(y,\\cdot )\\,{\\rm d}y\\right)=0.\\end{array}\\right.$ In the periodic case $A_\\varepsilon (x)=A^{\\rm per}(x/\\varepsilon )$ , the corrector equation reads as $ \\left\\lbrace \\begin{array}{l}-{\\rm div}\\left(A^{\\rm per}(p + \\nabla w_{p})\\right)=0\\quad \\text{in~$\\mathbb {R}^d$},\\\\w_{p}\\text{ is $\\mathbb {Z}^d$-periodic},\\end{array}\\right.$ and the homogenized matrix $A_\\star $ is given by $ \\left[A_\\star \\right]_{i,j} = \\int _Q \\left( e_i + \\nabla w_{e_i}(y) \\right)^T \\, A^{\\rm per}(y) \\, \\left( e_j + \\nabla w_{e_j}(y) \\right) \\, {\\rm d}y.$ In sharp contrast with the periodic case where, precisely by periodicity, it is sufficient to solve the corrector equation (REF ) on the unit cell $Q$ , the corrector equation (REF ) must be solved in the discrete stationary ergodic setting on the entire space $\\mathbb {R}^d$ .",
"As pointed out in the introduction, this is computationally challenging.",
"In practice, one often considers a truncated corrector equation posed, for an integer $N\\ne 0$ , on a large domain $Q^N = (-N,N)^d$ : $ -{\\rm div}\\left(A^{\\rm sto}(\\cdot ,\\omega )(p + \\nabla w^N_{p}(\\cdot ,\\omega ))\\right)=0 \\ \\ \\text{a.s.~in $Q^N$}, \\qquad w^N_{p}(\\cdot ,\\omega ) \\text{ is a.s.~$Q^N$-periodic.",
"}$ The random matrix $A_\\star ^N(\\omega )$ , approximation of the deterministic homogenized matrix $A_\\star $ given by (REF ), is defined, for all $1\\le i,j\\le d$ , by $ \\left[A_\\star ^N(\\omega )\\right]_{i,j} = \\frac{1}{|Q^N|} \\int _{Q^N} \\left( e_i + \\nabla w^N_{e_i}(y,\\omega ) \\right)^T \\, A^{\\rm sto}(y,\\omega ) \\, \\left( e_j + \\nabla w^N_{e_j}(y,\\omega ) \\right) \\, {\\rm d}y.$ Almost surely, it converges, in the limit of infinitely large domains $Q^N$ , i.e.",
"when $N\\rightarrow +\\infty $ , to the (deterministic) matrix $A_\\star $ (see [6]).",
"Since $A_\\star ^N(\\omega )$ is random, it is natural to consider $M$ independent and identically distributed (i.i.d.)",
"realizations of the field $A^{\\rm sto}$ , say $\\left\\lbrace A^{\\rm sto}(\\cdot ,\\omega _m) \\right\\rbrace _{1\\le m\\le M}$ , solve (REF ) and compute (REF ) for each of them, and define $ A_\\star ^{N,M} = \\frac{1}{M}\\sum _{m=1}^M A_\\star ^N(\\omega _m)$ as a practical approximation to $A_\\star $ .",
"Owing to the strong law of large numbers, we have that $\\displaystyle \\lim _{N \\rightarrow \\infty } \\lim _{M \\rightarrow \\infty } A_\\star ^{N,M} = A_\\star $ almost surely."
],
[
"Formalization of our approach",
"The approach we introduce below applies, up to minor changes, to both the periodic and the stationary ergodic settings.",
"We however recall from Section REF that only the stochastic setting (and more difficult cases) is practically relevant for our approach.",
"For simplicity and clarity, we first present the full study of the approach in the periodic setting (see Sections REF , REF and REF ).",
"We next discuss its extension to the stationary ergodic setting in Section REF ."
],
[
"Inf $\\sup $ formulation",
"As exposed in the introduction and expressed in formula (REF ), we are going to seek a constant, symmetric, positive-definite matrix $\\overline{A}_\\varepsilon $ , so that problem (REF ) with matrix $\\overline{A}_\\varepsilon $ best approximates problem (REF ).",
"To do so, we consider the problem introduced in (REF ), that is $ I_\\varepsilon = \\inf _{\\overline{A}\\in {\\cal S}} \\ \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} \\ \\Phi _\\varepsilon (\\overline{A},f),$ where ${L^2_{\\rm n}(\\cal D)}= \\left\\lbrace f\\in {L^2(\\cal D)}, \\ \\ {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}=1\\right\\rbrace $ and where, for any $\\overline{A} \\in \\mathbb {R}^{d \\times d}_{\\rm sym}$ (the space of $d \\times d$ real symmetric matrices) and any $f\\in {L^2(\\cal D)}$ , $ \\Phi _\\varepsilon (\\overline{A},f) = {\\left\\Vert (-\\Delta )^{-1} \\left( {\\rm div}( \\overline{A} \\nabla u_\\varepsilon (f) ) + f \\right) \\right\\Vert }_{{L^2(\\cal D)}}^2.$ Note that formula (REF ) is well-defined since ${\\rm div}(\\overline{A}\\nabla u_\\varepsilon (f))$ clearly belongs to $H^{-1}({\\cal D})$ for all $\\overline{A} \\in \\mathbb {R}^{d \\times d}_{\\rm sym}$ and $f\\in {L^2(\\cal D)}$ .",
"We observe, as briefly mentioned in Section REF , that the cost function $\\Phi _\\varepsilon (\\cdot ,f)$ depends quadratically upon $\\overline{A}$ .",
"From a computational viewpoint, in an iterative algorithm solving (REF ) or (REF ) that successively optimizes on $f$ and $\\overline{A}$ , minimizing $\\Phi _\\varepsilon $ with respect to $\\overline{A}$ for a fixed $f\\in {L^2_{\\rm n}(\\cal D)}$ thus reduces to the simple inversion of a small linear system with $d(d+1)/2$ unknowns (see Section REF ).",
"This is in sharp contrast with our former formulation (REF ).",
"Of course, in both formulations (REF ) or (REF ), for $\\varepsilon $ fixed, it is not guaranteed that our numerical algorithm captures the value $I_\\varepsilon $ defined by (REF ).",
"It only captures an approximation of it.",
"For both approaches (REF ) and (REF ), one can prove an asymptotic consistency result for the sequence $\\left\\lbrace \\overline{A}_\\varepsilon \\right\\rbrace _{\\varepsilon >0}$ : see Proposition REF below in the case of (REF ) and [16] in the case of (REF ).",
"As the proof is essentially identical for both approaches, we only detail it for the present choice (REF ) (see Appendix below) and briefly point out to the case (REF ) considered in [16] in Remark REF below.",
"In order to gain further insight, and before stating the asymptotic consistency result, we first study, separately and for a fixed value of $\\varepsilon $ , the maximization and minimization problems involved in (REF )."
],
[
"The $\\sup $ problem",
"We show here that, for any fixed $\\overline{A}\\in {\\cal S}$ , the maximization problem over $f$ that is involved in (REF ), namely $\\displaystyle \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} \\ \\Phi _\\varepsilon (\\overline{A},f)$ , is attained, and discuss how it can be solved in practice.",
"Let $\\overline{A}\\in {\\cal S}$ be given.",
"We introduce the notation $\\Delta _{\\overline{A}} = {\\rm div}(\\overline{A}\\nabla \\cdot ),$ and let $(-\\Delta _{\\overline{A}})^{-1}$ be the operator defined by: for any $g\\in H^{-1}({\\cal D})$ , $z = (-\\Delta _{\\overline{A}})^{-1}g$ is the unique solution in $H^1_0({\\cal D})$ to $ -{\\rm div}(\\overline{A}\\nabla z) = g \\ \\ \\text{in ${\\cal D}$}, \\qquad z=0 \\ \\ \\text{on $\\partial {\\cal D}$}.$ We denote by ${\\rm L}^{-1}_{\\varepsilon }$ the linear, compact and positive-definite operator from ${L^2(\\cal D)}$ to ${L^2(\\cal D)}$ such that, for any $f\\in {L^2(\\cal D)}$ , ${\\rm L}^{-1}_{\\varepsilon }f = u_\\varepsilon (f)$ , where $u_\\varepsilon (f)$ is the unique solution in $H^1_0({\\cal D})$ to (REF ).",
"Starting from (REF ), it can be easily shown that $ \\Phi _\\varepsilon (\\overline{A},f)=\\int _{\\cal D}{\\cal H}_\\varepsilon ^{\\overline{A}}(f)\\;f,$ where $ {\\cal H}_\\varepsilon ^{\\overline{A}}(f) = \\Big ( \\left( {\\rm L}^{-1}_{\\varepsilon } \\right)^\\star \\, \\Delta _{\\overline{A}} \\, (-\\Delta )^{-1} + (-\\Delta )^{-1} \\Big ) \\Big ( (-\\Delta )^{-1} \\, \\Delta _{\\overline{A}} \\, {\\rm L}^{-1}_{\\varepsilon } + (-\\Delta )^{-1} \\Big ) \\, f$ is a compact, self-adjoint and positive semi-definite linear operator from ${L^2(\\cal D)}$ to ${L^2(\\cal D)}$ .",
"The eigenvalues of ${\\cal H}_\\varepsilon ^{\\overline{A}}$ are thus nonnegative real numbers forming a sequence that converges to zero.",
"We denote by $\\lambda _{\\varepsilon ,{\\rm m}}^{\\overline{A}}$ and $f_{\\varepsilon ,{\\rm m}}^{\\overline{A}}$ the largest eigenvalue of ${\\cal H}_\\varepsilon ^{\\overline{A}}$ and an associated normalized eigenvector, respectively.",
"In view of (REF ), we have $\\sup _{f\\in {L^2_{\\rm n}(\\cal D)}}\\Phi _\\varepsilon (\\overline{A},f)=\\lambda _{\\varepsilon ,{\\rm m}}^{\\overline{A}}$ and the supremum is attained at $f_{\\varepsilon ,{\\rm m}}^{\\overline{A}}$ , which is hence a solution to the $\\sup $ problem involved in (REF ).",
"In practice, instead of looking for the largest eigenvalue (and the associated eigenvector) of ${\\cal H}_\\varepsilon ^{\\overline{A}}$ in the infinite-dimensional space ${L^2_{\\rm n}(\\cal D)}$ , our approach consists in approximating this space ${L^2_{\\rm n}(\\cal D)}$ by a finite-dimensional subspace of the form $ V^P_{\\rm n}({\\cal D}) = \\left\\lbrace f\\in {L^2_{\\rm n}(\\cal D)}\\ \\text{s.t.",
"there exists} \\ c = \\lbrace c_p \\rbrace _{1\\le p\\le P} \\in \\mathbb {R}^P, \\ \\ |c|^2=1, \\ \\ f=\\sum _{p=1}^Pc_pf_p\\right\\rbrace ,$ where $(f_1,\\ldots ,f_P)$ is an orthonormal family of functions in ${L^2(\\cal D)}$ .",
"We discuss the choice of the dimension $P$ and of the family of functions $\\lbrace f_p \\rbrace _{1\\le p\\le P}$ .",
"First of all, in the light of Lemma REF below (see also Section REF ), it seems in order to choose the dimension of $V^P_{\\rm n}({\\cal D})$ such that $\\displaystyle P \\ge d(d+1)/2$ .",
"We now proceed, considering the regime $\\varepsilon $ small.",
"Let $\\overline{A} \\ne A_\\star $ be fixed.",
"Homogenization theory states that, for $\\varepsilon $ sufficiently small, the operator ${\\rm L}_\\varepsilon ^{-1}$ (considered as an operator from ${L^2(\\cal D)}$ to ${L^2(\\cal D)}$ ) is close to the operator $(-\\Delta _{A_\\star })^{-1}$ .",
"Thus the operator ${\\cal H}_\\varepsilon ^{\\overline{A}}$ defined by (REF ) is expected to be well-approximated by $ {\\cal H}^{\\overline{A}}_\\star =\\Big ( (-\\Delta _{A_\\star })^{-1} \\, \\Delta _{\\overline{A}} \\, (-\\Delta )^{-1} + (-\\Delta )^{-1} \\Big ) \\Big ( (-\\Delta )^{-1} \\, \\Delta _{\\overline{A}} \\, (-\\Delta _{A_\\star })^{-1} + (-\\Delta )^{-1} \\Big ).$ Up to the extraction of a subsequence, the eigenvector $f^{\\overline{A}}_{\\varepsilon ,{\\rm m}}$ we are seeking thus satisfies, by homogenization theory on eigenvalue problems, $\\lim _{\\varepsilon \\rightarrow 0}{\\left\\Vert f^{\\overline{A}}_{\\varepsilon ,{\\rm m}}-f^{\\overline{A}}_{\\star ,{\\rm m}}\\right\\Vert }_{{L^2(\\cal D)}}=0,$ where $f^{\\overline{A}}_{\\star ,{\\rm m}}$ is a normalized eigenvector associated with the largest eigenvalue of ${\\cal H}^{\\overline{A}}_\\star $ .",
"In view of the expression (REF ) of the limit operator, it seems natural to choose for the family of functions $\\lbrace f_p \\rbrace _{1\\le p\\le P}$ the first $P$ (normalized) eigenvectors of the laplacian operator in the domain ${\\cal D}$ .",
"For small values of $\\varepsilon $ , say $\\varepsilon <\\overline{\\varepsilon }$ , we show that considering $\\displaystyle P = d(d+1)/2$ functions $f_p$ is sufficient.",
"This threshold $d(d+1)/2$ is at least intuitive thinking at the case of a constant symmetric matrix $\\overline{A}$ and the set of equations $\\displaystyle \\sum _{1 \\le i,j \\le d} -\\overline{A}_{i,j} \\ \\partial _{ij} u_p = f_p$ .",
"In order to determine the $d(d+1)/2$ coefficients $\\overline{A}_{i,j}$ , the correct number of right-hand sides $f_p$ to consider is $d(d+1)/2$ .",
"The fact that it is indeed sufficient is made precise in the proof of Proposition REF below (see in particular Lemma REF ) and in Remark REF below.",
"When the parameter $\\varepsilon $ takes larger values, say $\\varepsilon \\ge \\overline{\\varepsilon }$ , the operator ${\\cal H}^{\\overline{A}}_\\varepsilon $ cannot be anymore approximated by the operator (REF ) (with constant coefficients), and it may thus be necessary in that case to consider a larger number $\\displaystyle P > d(d+1)/2$ of functions.",
"We refer to Section for concrete examples.",
"Remark 4 We discuss here why we have chosen to work with right-hand sides $f$ of the equation (e.g.",
"(REF )) in ${L^2(\\cal D)}$ rather than in $H^{-1}({\\cal D})$ .",
"We have here considered $\\displaystyle \\sup _{f \\in L^2({\\cal D})} \\ \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{L^2({\\cal D})}}$ , and we could have considered $\\displaystyle \\sup _{f \\in H^{-1}({\\cal D})} \\ \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{H^{-1}({\\cal D})}}$ .",
"Since ${L^2(\\cal D)}\\subset H^{-1}({\\cal D})$ , we of course have $\\displaystyle \\sup _{f \\in H^{-1}({\\cal D})} \\ \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{H^{-1}({\\cal D})}} \\ge \\sup _{f \\in L^2({\\cal D})} \\ \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{H^{-1}({\\cal D})}}$ .",
"Using the density of ${L^2(\\cal D)}$ in $H^{-1}({\\cal D})$ and the continuity of $\\Phi _\\varepsilon (\\overline{A},\\cdot )$ in $H^{-1}({\\cal D})$ , we actually get $\\sup _{f \\in H^{-1}({\\cal D})} \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{H^{-1}({\\cal D})}} = \\sup _{f \\in L^2({\\cal D})} \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{H^{-1}({\\cal D})}}.$ The right-hand side of (REF ) is of course different from the quantity $\\displaystyle \\sup _{f \\in L^2({\\cal D})} \\frac{\\Phi _\\varepsilon (\\overline{A},f)}{\\Vert f \\Vert ^2_{L^2({\\cal D})}}$ , which we have considered in this article.",
"Our choice is motivated by the fact that it is easier in practice to manipulate functions of unit $L^2$ -norm.",
"From the theoretical viewpoint, similar results would have been obtained with the left-hand side of (REF )."
],
[
"The $\\inf $ problem",
"We discuss here how to efficiently solve the minimization problem over $\\overline{A}$ that is involved in (REF ), namely $\\inf _{\\overline{A}\\in {\\cal S}} \\ \\Phi _\\varepsilon (\\overline{A},f).$ Let $f\\in {L^2_{\\rm n}(\\cal D)}$ be fixed.",
"It can be easily shown, starting from (REF ) and using the linearity of both the divergence and inverse laplacian operators, that $ \\Phi _\\varepsilon (\\overline{A},f) = \\frac{1}{2} \\sum _{1\\le i,j,k,l\\le d} \\left[ \\mathbb {B}_\\varepsilon (f)\\right]_{i,j,k,l} \\, \\overline{A}_{i,j} \\, \\overline{A}_{k,l}-\\sum _{1\\le i,j\\le d} \\left[B_\\varepsilon (f)\\right]_{i,j} \\, \\overline{A}_{i,j}+b(f),$ where the fourth-order tensor $\\mathbb {B}_\\varepsilon (f)$ , the matrix $B_\\varepsilon (f)$ and the scalar $b(f)$ , which all depend on $f$ , are given, for integers $1\\le i,j,k,l\\le d$ , by $\\left[\\mathbb {B}_\\varepsilon (f)\\right]_{i,j,k,l}& = &\\displaystyle 2 \\int _{\\cal D} \\left[ (-\\Delta )^{-1}(\\partial _{ij} u_\\varepsilon (f)) \\right] \\ \\left[ (-\\Delta )^{-1}(\\partial _{kl} u_\\varepsilon (f))\\right],\\\\\\left[B_\\varepsilon (f)\\right]_{i,j}& = &\\displaystyle -2 \\int _{\\cal D} \\left[ (-\\Delta )^{-1}(\\partial _{ij} u_\\varepsilon (f))\\right] \\ \\left[ (-\\Delta )^{-1}f\\right],\\\\b(f)& = &{\\left\\Vert (-\\Delta )^{-1}f\\right\\Vert }_{{L^2(\\cal D)}}^2.$ Practically, the $\\inf $ problem (REF ) (with fixed $f$ ) is solved on the whole set $\\mathbb {R}^{d \\times d}_{\\rm sym}$ of symmetric matrices, instead of considering the subset ${\\cal S}$ of positive-definite symmetric matrices.",
"Under this simplification, solving the $\\inf $ problem (REF ) amounts to considering the linear system $\\forall \\,1\\le i, j\\le d, \\quad \\sum _{1\\le k,l\\le d} \\left[\\mathbb {B}_\\varepsilon (f)\\right]_{i,j,k,l} \\ \\overline{A}_{k,l} = \\left[B_\\varepsilon (f)\\right]_{i,j}.$ This system is low-dimensional and inexpensive to solve.",
"In our numerical experiments, we have observed that the problem (REF ) always has a unique solution in $\\mathbb {R}^{d \\times d}_{\\rm sym}$ , for all the functions $f$ that our algorithm explores.",
"In addition, this solution is in ${\\cal S}$ ."
],
[
"Asymptotic consistency",
"We study here problem (REF ) in the limit of a vanishing parameter $\\varepsilon $ .",
"We introduce the notation $ \\Phi _\\varepsilon (\\overline{A}) = \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}}\\Phi _\\varepsilon (\\overline{A},f).$ Note that $\\Phi _\\varepsilon $ is nonnegative.",
"Consequently, for any $\\varepsilon $ , problem (REF ) admits a quasi-minimizer, namely a matrix $\\overline{A}^\\flat _\\varepsilon \\in {\\cal S}$ such that $ I_\\varepsilon \\le \\Phi _\\varepsilon (\\overline{A}^\\flat _\\varepsilon )\\le I_\\varepsilon +\\varepsilon \\le \\Phi _\\varepsilon (\\overline{A})+\\varepsilon \\qquad \\text{for any }\\overline{A}\\in {\\cal S}.$ The following proposition holds.",
"Proposition 5 (Asymptotic consistency, periodic case) Consider problem (REF ), that is $I_\\varepsilon = \\inf _{\\overline{A}\\in {\\cal S}}\\sup _{f\\in {L^2_{\\rm n}(\\cal D)}}\\Phi _\\varepsilon (\\overline{A},f).$ In the periodic setting, namely under the assumptions (REF ) and (REF ), the following convergence holds: $ \\lim _{\\varepsilon \\rightarrow 0} I_\\varepsilon =0.$ Furthermore, for any sequence $\\left\\lbrace \\overline{A}^\\flat _\\varepsilon \\in {\\cal S} \\right\\rbrace _{\\varepsilon >0}$ of quasi-minimizers of (REF ), we have $ \\lim _{\\varepsilon \\rightarrow 0}\\overline{A}^\\flat _\\varepsilon =A_\\star .$ The proof of these results, which is postponed until Appendix , relies on two facts: The homogenized matrix $A_\\star \\in {\\cal S}_{\\alpha ,\\beta }\\subset {\\cal S}$ can be used as a test-matrix in (REF ).",
"In view of Lemma REF below, it satisfies $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0}\\Phi _\\varepsilon (A_\\star )=0$ , which directly implies (REF ); We show in Lemma REF below that there exist $d(d+1)/2$ right-hand sides $f_{\\star ,k}\\in {L^2_{\\rm n}(\\cal D)}$ such that the knowledge of $f_{\\star ,k}$ and of $u_{\\star ,k}$ solution to (REF ) with right-hand side $f_{\\star ,k}$ , $1 \\le k \\le d \\, (d+1)/2$ , is sufficient to uniquely reconstruct the constant symmetric matrix $A_\\star $ .",
"The proof of (REF ) relies on this argument and on (REF ).",
"We denote $ {\\cal F} = \\Big \\lbrace f_{\\star ,k}, \\quad 1 \\le k \\le d(d+1)/2 \\Big \\rbrace $ this set.",
"We do not know whether, for $\\varepsilon $ fixed, the infimum in (REF ) is attained, unless $\\varepsilon $ is sufficiently small (see Remark REF in Appendix REF below).",
"We will proceed throughout the article manipulating quasi-minimizers in the sense of (REF ).",
"Remark 6 The analysis of the approach (REF ) introduced in [16] relies on the same arguments as the approach introduced here: Lemma REF , and the equivalent of Lemma REF for the functional considered in [16], that is $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0} \\Psi _\\varepsilon (A^\\star ) = 0$ , where, for any $\\overline{A} \\in {\\cal S}$ , $\\Psi _\\varepsilon (\\overline{A}) = \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} {\\left\\Vert u_\\varepsilon (f) - \\overline{u}(f)\\right\\Vert }_{{L^2(\\cal D)}}^2.$ Remark 7 Note that the assumptions (REF ) and (REF ) are not necessary to prove the results (REF ) and (REF ).",
"All that needs to be assumed is that the sequence of matrices $\\lbrace A_\\varepsilon \\rbrace _{\\varepsilon >0}$ converges, in the sense of homogenization, to a constant and symmetric homogenized matrix $A_\\star $ .",
"In that vein, we will see in Section REF below that the conclusions of Proposition REF carry over to the specific stochastic case we consider there.",
"Remark 8 Consider the set ${\\cal F}$ defined by (REF ), and let $ I_\\varepsilon ^{\\rm max} = \\inf _{\\overline{A}\\in {\\cal S}} \\, \\max _{f\\in {\\cal F}} \\, \\Phi _\\varepsilon (\\overline{A},f).$ This problem is, in principle, easier to solve than (REF ), as we replaced the supremum over $f\\in {L^2_{\\rm n}(\\cal D)}$ by a maximization over the finite set ${\\cal F}$ .",
"Let $\\displaystyle \\Phi ^{\\rm max}_\\varepsilon (\\overline{A}) = \\max _{f\\in {\\cal F}}\\Phi _\\varepsilon (\\overline{A},f)$ .",
"For any quasi-minimizer $\\overline{A}^{\\rm max,\\flat }_\\varepsilon \\in {\\cal S}$ of (REF ), we have $I_\\varepsilon ^{\\rm max}\\le \\Phi ^{\\rm max}_\\varepsilon (\\overline{A}^{\\rm max,\\flat }_\\varepsilon )\\le I_\\varepsilon ^{\\rm max}+\\varepsilon \\le \\Phi ^{\\rm max}_\\varepsilon (A_\\star )+\\varepsilon \\le \\Phi _\\varepsilon (A_\\star )+\\varepsilon .$ Since $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0}\\Phi _\\varepsilon (A_\\star )=0$ , we get that $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0}I_\\varepsilon ^{\\rm max}=0$ .",
"In addition, one can show that $\\overline{A}^{\\rm max,\\flat }_\\varepsilon \\rightarrow A_\\star $ as $\\varepsilon \\rightarrow 0$ (we refer to Remark REF below for details).",
"Similarly to (REF ), the approach (REF ) is therefore asymptotically consistent.",
"Note however that, in practice, the functions of the set ${\\cal F}$ defined by (REF ) are unknown.",
"We note that Proposition REF provides, in the setting described in Section REF , a characterization of the homogenized matrix which is an alternative to the standard characterization of homogenization theory.",
"To the best of our knowledge, this characterization has never been made explicit in the literature."
],
[
"Approximation of $u_\\varepsilon $ in the {{formula:9828fa48-b487-4927-a6ad-a58b5eae9aea}} norm",
"As a consequence of Proposition REF , we note that $\\overline{u}_\\varepsilon $ , solution to (REF ) with matrix $\\overline{A}_\\varepsilon $ , is an accurate approximation of $u_\\varepsilon $ in the $L^2$ norm, but not in the $H^1$ norm.",
"Indeed, when $\\varepsilon $ goes to zero, $\\overline{A}_\\varepsilon $ converges to $A_\\star $ .",
"Hence, for $\\varepsilon $ sufficiently small, $\\overline{u}_\\varepsilon $ is an accurate $H^1$ -approximation of $u_\\star $ solution to (REF ).",
"In addition, from homogenization theory, we know that $u_\\star $ is an accurate $L^2$ -approximation of $u_\\varepsilon $ .",
"This implies that $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0} \\Vert \\overline{u}_\\varepsilon - u_\\varepsilon \\Vert _{L^2({\\cal D})} = 0$ .",
"Note also that $u_\\star $ and $u_\\varepsilon $ are not close to each other in the $H^1$ norm, and hence $\\overline{u}_\\varepsilon $ is not an accurate approximation of $u_\\varepsilon $ in the $H^1$ norm.",
"We present here an approach to reconstruct such an approximation.",
"In many settings of homogenization theory (and in particular in the periodic setting we consider here), once the corrector problems are solved to compute the homogenized matrix, one can consider the two-scale expansion (truncated at the first-order) $ u_\\varepsilon ^{1,\\theta }(x)=u_\\star (x) + \\varepsilon \\sum _{i=1}^d w^{\\theta _i}_{e_i}(x/\\varepsilon ) \\, \\partial _i u_\\star (x),$ where $w^{\\theta _i}_{e_i}$ is the unique solution with mean value $\\theta _i \\in \\mathbb {R}$ to the periodic corrector equation (REF ) for $p=e_i$ .",
"It is well-known that this two-scale expansion approximates $u_\\varepsilon $ in the $H^1$ norm, in the sense that, under some regularity assumptions (see e.g.",
"[1]), we have $ {\\left\\Vert u_\\varepsilon -u_\\varepsilon ^{1,\\theta }\\right\\Vert }_{H^1({\\cal D})}\\le C\\,\\sqrt{\\varepsilon }$ for a constant $C$ independent of $\\varepsilon $ .",
"Remark 9 From the theoretical perspective, the mean value $\\theta $ of the correctors is irrelevant, and the estimate (REF ) holds for any fixed $\\theta $ .",
"From the numerical perspective, the error ${\\left\\Vert u_\\varepsilon -u_\\varepsilon ^{1,\\theta }\\right\\Vert }_{H^1({\\cal D})}$ slightly depends on $\\theta $ , in particular when $\\varepsilon $ is not asymptotically small.",
"In view of the numerical tests described in Section below (see e.g.",
"(REF )), we keep track of this parameter.",
"Computing the gradient of (REF ), we deduce from (REF ) that $ \\nabla u_\\varepsilon = C_\\varepsilon \\, \\nabla u_\\star + \\text{h.o.t.",
"},$ where the $d \\times d$ matrix $C_\\varepsilon $ is given by $ \\left[C_\\varepsilon \\right]_{i,i} = 1 + \\partial _i w_{e_i}(\\cdot /\\varepsilon ),\\qquad \\qquad \\left[C_\\varepsilon \\right]_{i,j} = \\partial _i w_{e_j}(\\cdot /\\varepsilon ) \\quad \\text{if $j \\ne i$.",
"}$ Our idea for constructing an approximation of $\\nabla u_\\varepsilon $ is to mimick formula (REF ) and seek an approximation under the form $\\overline{C}_\\varepsilon \\nabla \\overline{u}_\\varepsilon $ .",
"Once the best matrix $\\overline{A}_\\varepsilon $ has been computed, we compute a surrogate $\\overline{C}_\\varepsilon $ of $C_\\varepsilon $ by solving the least-squares problem $ \\inf _{\\overline{C}\\in ({L^2(\\cal D)})^{d \\times d}} \\ \\ \\sum _{r=1}^R{\\left\\Vert \\nabla u_\\varepsilon (f_r)-\\overline{C} \\ \\nabla \\overline{u}_\\varepsilon (f_r)\\right\\Vert }_{{L^2(\\cal D)}^d}^2$ for a given number $R$ of right-hand sides.",
"In practice, the right-hand sides $f_r$ selected for (REF ) are the first $R$ basis functions of the space $V^P_{\\rm n}({\\cal D})$ defined by (REF ), with $R$ such that $R \\le P.$ This choice makes the $H^1$ -reconstruction an inexpensive post-processing procedure once the best matrix is computed, as we already have at our disposal $u_\\varepsilon (f_r)$ for $1\\le r\\le R$ .",
"Remark 10 In our numerical experiments, we have observed that the surrogate $\\overline{C}_\\varepsilon $ that we construct is indeed oscillatory, and essentially periodic when $A_\\varepsilon $ is periodic.",
"This is expected since $\\overline{C}_\\varepsilon $ is meant to be an approximation of $C_\\varepsilon $ .",
"In practice, we independently identify each row of $\\overline{C}_\\varepsilon $ , by considering (for any $1 \\le i \\le d$ ) the least-squares problem $\\inf _{\\overline{c}^i \\in ({L^2(\\cal D)})^d} \\ \\ \\sum _{r=1}^R {\\left\\Vert \\partial _i u_\\varepsilon (f_r) - \\overline{c}^i \\cdot \\nabla \\overline{u}_\\varepsilon (f_r)\\right\\Vert }_{{L^2(\\cal D)}}^2.$ We next define the matrix $\\overline{C}_\\varepsilon $ by $\\left[ \\overline{C}_\\varepsilon \\right]_{i,j} = \\left[ \\overline{c}^i_\\varepsilon \\right]_j$ .",
"In our numerical experiments, the functions $u_\\varepsilon $ and $\\overline{u}_\\varepsilon $ are approximated by $u_{\\varepsilon ,h}$ and $\\overline{u}_{\\varepsilon ,h}$ using a $\\mathbb {P}^1$ Finite Element Method, and $\\overline{c}^i_\\varepsilon $ is searched as a piecewise constant function.",
"The value of $\\overline{c}^i_\\varepsilon $ on an element $T$ is defined by the problem $ \\inf _{\\overline{c}^i_T \\in \\mathbb {R}^d} \\ \\ \\sum _{r=1}^R \\left| \\left[\\partial _i u_{\\varepsilon ,h}(f_r)\\right]_{\\mid T} - \\overline{c}^i_T \\cdot \\left[\\nabla \\overline{u}_{\\varepsilon ,h}(f_r)\\right]_{\\mid T} \\right|^2,$ where the restrictions of $\\partial _i u_{\\varepsilon ,h}$ and $\\nabla \\overline{u}_{\\varepsilon ,h}$ to any element $T$ are constant.",
"This problem is ill-posed if $R < d$ , since, in this case, there exists vectors in $\\mathbb {R}^d$ orthogonal to all $\\left[\\nabla \\overline{u}_{\\varepsilon ,h}(f_r)\\right]_{\\mid T}$ , $1 \\le r \\le R$ .",
"We thus always take $R \\ge d$ .",
"To avoid technicalities related to the $\\mathbb {P}^1$ discretization of $\\overline{u}_\\varepsilon $ , only mesh elements not contiguous to the boundary of ${\\cal D}$ are considered in the minimization (REF )."
],
[
"The stationary ergodic setting",
"We have focused in Sections REF , REF and REF on the periodic setting.",
"We now briefly turn to the stochastic ergodic setting.",
"We introduce the modified cost function $\\Phi ^{\\rm sto}_\\varepsilon $ defined, for any $\\overline{A} \\in \\mathbb {R}^{d \\times d}_{\\rm sym}$ and $f\\in {L^2(\\cal D)}$ , by $ \\Phi ^{\\rm sto}_\\varepsilon (\\overline{A},f) = {\\left\\Vert (-\\Delta )^{-1}\\left[{\\rm div}\\left(\\overline{A}\\nabla \\mathbb {E}(u_\\varepsilon (f))\\right)+f\\right]\\right\\Vert }_{{L^2(\\cal D)}}^2.$ Note that $\\Phi ^{\\rm sto}_\\varepsilon $ is a deterministic quantity.",
"The difference with the cost function $\\Phi _\\varepsilon $ defined by (REF ) in a deterministic context is that $\\Phi ^{\\rm sto}_\\varepsilon $ involves $\\mathbb {E}(u_\\varepsilon (f))$ rather than $u_\\varepsilon (f)$ .",
"We next amend the $\\inf \\sup $ problem (REF ) in the following way.",
"For a given value of $\\varepsilon $ , we look for a best deterministic matrix $\\overline{A}_\\varepsilon \\in {\\cal S}$ that solves the problem $ I^{\\rm sto}_\\varepsilon = \\inf _{\\overline{A}\\in {\\cal S}}\\,\\sup _{f\\in {L^2_{\\rm n}(\\cal D)}}\\Phi ^{\\rm sto}_\\varepsilon (\\overline{A},f).$ All the considerations of Sections REF , REF and REF carry over, up to minor adjustments, to the present stochastic setting.",
"Under assumptions (REF ) and (REF ), asymptotic consistency can be proved for any sequence $\\lbrace \\overline{A}^\\flat _\\varepsilon \\in {\\cal S} \\rbrace _{\\varepsilon >0}$ of quasi-minimizers of (REF ).",
"The adaptation of the proof of Proposition REF to the stochastic setting is straightforward.",
"It relies on the fact that, for any $f\\in {L^2(\\cal D)}$ , $\\mathbb {E}(u_\\varepsilon (f))$ is bounded in $H^1({\\cal D})$ .",
"Indeed, using that $\\alpha \\le A_\\varepsilon (\\cdot ,\\omega ) \\le \\beta $ almost surely, we have $\\displaystyle {\\left\\Vert u_\\varepsilon (\\cdot ,\\omega )\\right\\Vert }_{H^1({\\cal D})} \\le \\frac{C}{\\alpha } {\\left\\Vert f\\right\\Vert }_{L^1({\\cal D})}$ almost surely (where $C$ is a deterministic constant only depending on ${\\cal D}$ ), hence $\\mathbb {E} \\left[ {\\left\\Vert u_\\varepsilon \\right\\Vert }_{H^1({\\cal D})}^2 \\right]$ is bounded.",
"Using the Cauchy-Schwarz inequality, we infer that $\\mathbb {E}(u_\\varepsilon (f))$ is indeed bounded in $H^1({\\cal D})$ .",
"We eventually get that $\\nabla \\mathbb {E}(u_\\varepsilon (f))$ weakly converges, and $\\mathbb {E}(u_\\varepsilon (f))$ strongly converges, in ${L^2(\\cal D)}$ and when $\\varepsilon $ goes to zero, to $\\nabla u_\\star (f)$ and $u_\\star (f)$ , respectively, where $u_\\star (f)$ is the solution to (REF ).",
"The $H^1$ -reconstruction procedure presented in Section REF is adapted to the stationary ergodic setting as follows.",
"It is known that, almost surely, $u_\\varepsilon (\\cdot ,\\omega )$ weakly converges in $H^1({\\cal D})$ towards $u_\\star $ when $\\varepsilon $ goes to zero.",
"As in the periodic setting, the correctors allow to obtain a strong convergence in $H^1({\\cal D})$ , in the sense that (see [18]) $\\lim _{\\varepsilon \\rightarrow 0} \\mathbb {E} \\left[ {\\left\\Vert u_\\varepsilon (\\cdot ,\\omega ) - u_\\varepsilon ^1(\\cdot ,\\omega )\\right\\Vert }_{H^1({\\cal D})}^2 \\right] = 0,$ with $u_\\varepsilon ^1(x,\\omega )=u_\\star (x) + \\varepsilon \\sum _{i=1}^d w_{e_i}(x/\\varepsilon ,\\omega ) \\, \\partial _i u_\\star (x),$ where $w_{e_i}$ is the unique solution with vanishing mean value to the stochastic corrector equation (REF ) for $p=e_i$ (in contrast to the periodic case, see Remark REF , we only consider here correctors with vanishing mean, for the sake of simplicity).",
"The equations (REF )–(REF ) imply that $\\mathbb {E} \\left[ \\nabla u_\\varepsilon (\\cdot ,\\omega ) \\right] = C_\\varepsilon \\, \\nabla u_\\star + \\text{h.o.t.",
"},$ where the $d \\times d$ matrix $C_\\varepsilon $ is given by $ \\left[C_\\varepsilon \\right]_{i,i} = 1 + \\mathbb {E} \\left[ \\partial _i w_{e_i}(\\cdot /\\varepsilon ,\\omega ) \\right],\\qquad \\qquad \\left[C_\\varepsilon \\right]_{i,j} = \\mathbb {E} \\left[ \\partial _i w_{e_j}(\\cdot /\\varepsilon ,\\omega ) \\right] \\quad \\text{if $j \\ne i$.",
"}$ We have chosen to look for an approximation of $\\mathbb {E}(\\nabla u_\\varepsilon )$ as follows.",
"Once the best matrix $\\overline{A}_\\varepsilon $ has been computed, we compute a surrogate $\\overline{C}_\\varepsilon $ of $C_\\varepsilon $ by solving the least-squares problem $ \\inf _{\\overline{C}\\in ({L^2(\\cal D)})^{d \\times d}} \\ \\ \\sum _{r=1}^R {\\left\\Vert \\nabla \\mathbb {E} \\left[ u_\\varepsilon (f_r) \\right] -\\overline{C} \\ \\nabla \\overline{u}_\\varepsilon (f_r)\\right\\Vert }_{{L^2(\\cal D)}^d}^2$ for a given number $R$ of right-hand sides, which are selected as in the periodic setting (see Section REF ).",
"Eventually, $\\mathbb {E} \\left[ \\nabla u_\\varepsilon (\\cdot ,\\omega ) \\right]$ is approximated by $\\overline{C}_\\varepsilon \\, \\nabla \\overline{u}_\\varepsilon $ .",
"Remark 11 Criteria (REF ) and (REF ) are arbitrary and selected upon practical considerations.",
"Among the possible alternatives, we could have considered $\\Phi ^{\\rm sto}_\\varepsilon (\\overline{A},f) = \\mathbb {E} \\left[ {\\left\\Vert (-\\Delta )^{-1}\\left[{\\rm div}\\left(\\overline{A}\\nabla u_\\varepsilon (f) \\right)+f\\right]\\right\\Vert }_{{L^2(\\cal D)}}^2 \\right]$ instead of (REF ), and a similar alternative for the reconstruction (REF ).",
"We have not proceeded in any of these directions.",
"Note also that, in [16], we defined the minimization problems $\\omega $ by $\\omega $ and next took the expectation of the results.",
"Of course, considering expectations in the cost functions results in significant computational savings, besides actually improving accuracy and robustness."
],
[
"Implementation details to solve (",
"We detail here how problem (REF ), in the stationary ergodic setting, can be efficiently solved in practice.",
"Problem (REF ), in the periodic setting, is actually simpler to solve, and we skip the easy adaptation to that case.",
"The minimizer of (REF ) is denoted by $\\overline{A}_{\\varepsilon ,h}^{P,M}$ , where $h\\ll \\varepsilon $ denotes the size of a mesh ${\\cal T}_h = \\lbrace T\\rbrace $ of the domain ${\\cal D}$ , $P$ denotes the dimension of the subspace $V_{\\rm n}^P({\\cal D})$ of ${L^2_{\\rm n}(\\cal D)}$ used to approximate the $\\sup $ problem (see (REF )), and $M\\in \\mathbb {N}^\\star $ denotes the number of Monte Carlo realizations used to approximate $\\mathbb {E}(u_\\varepsilon )$ in (REF ).",
"The algorithm consists of three steps: Compute an approximation of $\\Big \\lbrace \\mathbb {E}[u_\\varepsilon (f_p)] \\Big \\rbrace _{1\\le p\\le P}$ (see Section REF ).",
"This is the most expensive step, as $M \\times P$ oscillatory problems of the type (REF ) are to be solved.",
"Compute an approximation of $(-\\Delta )^{-1} f_p$ and of $\\displaystyle \\left\\lbrace (-\\Delta )^{-1} \\left( \\partial _{ij} \\mathbb {E}[u_\\varepsilon (f_p)]\\right) \\right\\rbrace _{1\\le i,j\\le d}$ , for any $1\\le p\\le P$ (see Section REF ).",
"This amounts to solving $P \\left( 1+ d(d+1)/2 \\right)$ problems with constant coefficients.",
"Solve problem (REF ) iteratively (see Section REF ).",
"Each iteration involves diagonalizing a $P \\times P$ matrix and solving a linear system with $d(d+1)/2$ unknowns.",
"The cost of this third step is negligible.",
"We now successively detail these three steps."
],
[
"Approximation of $\\Big \\lbrace \\mathbb {E}[u_\\varepsilon (f_p)] \\Big \\rbrace _{1\\le p\\le P}$",
"For any basis function $f_p$ of $V^P_{\\rm n}({\\cal D})$ , $1\\le p\\le P$ , we approximate $\\mathbb {E}[u_\\varepsilon (f_p)]$ by the empirical mean $ u_{\\varepsilon ,h}^M(f_p) = \\frac{1}{M}\\sum _{m=1}^M u_{\\varepsilon ,h}(f_p;\\omega _m),$ where, for $1 \\le m \\le M$ , $u_{\\varepsilon ,h}(f_p;\\omega _m)$ is the $\\mathbb {P}^1$ approximation on ${\\cal T}_h$ of $u_\\varepsilon (f_p;\\omega _m)$ , unique solution to (REF ) with the oscillatory matrix-valued coefficient $A_\\varepsilon (\\cdot ,\\omega _m)$ and the right-hand side $f_p$ .",
"To compute (REF ) for all $1\\le p\\le P$ , one has to (i) assemble $M$ random stiffness matrices, (ii) assemble $P$ deterministic right-hand sides, and (iii) solve $M\\times P$ linear systems.",
"This step is the only one involving Monte Carlo computations, and is therefore the most expensive part of the whole procedure."
],
[
"Precomputation of tensorial quantities",
"Once the computations of Section REF have been performed, we assemble some tensors that are needed to efficiently solve the $\\sup $ and $\\inf $ problems involved in (REF ).",
"We first compute, for any $1\\le p\\le P$ , the approximations $z_h(f_p)$ and $\\left\\lbrace z^{M,ij}_{\\varepsilon ,h}(f_p) \\right\\rbrace _{1\\le i,j\\le d}$ on ${\\cal T}_h$ of $(-\\Delta )^{-1} f_p$ and $\\displaystyle \\left\\lbrace (-\\Delta )^{-1} \\left( \\partial _{ij} \\mathbb {E}[u_\\varepsilon (f_p)]\\right) \\right\\rbrace _{1\\le i,j\\le d}$ .",
"In particular, $z^{M,ij}_{\\varepsilon ,h}(f_p)$ is such that, for any $\\mathbb {P}^1$ function $w_h$ on ${\\cal T}_h$ that vanishes on $\\partial {\\cal D}$ , $ \\int _{\\cal D} \\nabla z^{M,ij}_{\\varepsilon ,h}(f_p) \\cdot \\nabla w_h = -\\int _{\\cal D} \\partial _j \\left[u_{\\varepsilon ,h}^M(f_p)\\right]\\;\\partial _i w_h.$ Note that the following symmetry identity holds: $z^{M,ij}_{\\varepsilon ,h}(f_p)=z^{M,ji}_{\\varepsilon ,h}(f_p)$ .",
"We next assemble, for all integers $1\\le i,j,k,l\\le d$ and $1\\le p,q\\le P$ , the quantities $\\left[{\\cal K}_{\\varepsilon ,h}^M\\right]_{i,j,k,l,p,q} & = & 2 \\int _{\\cal D} z_{\\varepsilon ,h}^{M,ij}(f_p) \\, z_{\\varepsilon ,h}^{M,kl}(f_q),\\\\\\left[\\mathbb {K}_{\\varepsilon ,h}^M\\right]_{i,j,p,q} & = & -\\int _{\\cal D} z_{\\varepsilon ,h}^{M,ij}(f_p) \\, z_h(f_q),\\\\\\left[K_h\\right]_{p,q}& = & \\int _{\\cal D} z_h(f_p) \\, z_h(f_q).$ We emphasize that the cost of this step depends on $P$ but is independent of the number $M$ of Monte Carlo realizations, and thus small in comparison to the cost of the operations described in Section REF for typical values of $M$ and $P$ (in the numerical results reported on in Section , we have worked with $M=100$ and $P \\le 9$ )."
],
[
"Formulation",
"At this stage, the original problem (REF ) has been approximated by its fully discrete version $ I^{P,M}_{\\varepsilon ,h} = \\inf _{\\overline{A}\\in {\\cal S}}\\,\\sup _{c \\in \\mathbb {R}^P, \\, |c|^2=1}\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c),$ where, for any $\\overline{A} \\in \\mathbb {R}^{d \\times d}_{\\rm sym}$ and $c= \\lbrace c_p \\rbrace _{1\\le p\\le P} \\in \\mathbb {R}^P$ , $\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c) = {\\left\\Vert \\sum _{p=1}^Pc_p\\left(\\sum _{1\\le i,j\\le d}\\overline{A}_{i,j}\\,z^{M,ij}_{\\varepsilon ,h}(f_p)+z_h(f_p)\\right)\\right\\Vert }_{{L^2(\\cal D)}}^2.$ Problem (REF ) is solved by iteratively considering the problem $\\sup _{c \\in \\mathbb {R}^P, \\, |c|^2=1} \\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)$ with $\\overline{A}\\in {\\cal S}$ fixed, and the problem $\\inf _{\\overline{A}\\in {\\cal S}} \\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)$ with $c \\in \\mathbb {R}^P$ fixed.",
"We successively explain how we solve the $\\sup $ problem (REF ) (for $\\overline{A}\\in {\\cal S}$ fixed), the $\\inf $ problem (REF ) (for $c\\in \\mathbb {R}^P$ fixed), and next describe the iterative algorithm that we have implemented to solve (REF )."
],
[
"The $\\sup $ problem (",
"Let $\\overline{A}\\in {\\cal S}$ be fixed.",
"One can easily observe that $ \\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)=c^T\\,G_{\\varepsilon ,h}^M(\\overline{A})\\,c,$ where $G_{\\varepsilon ,h}^M(\\overline{A})$ is a symmetric, positive semi-definite, $P\\times P$ matrix which can be assembled at no additional cost using the precomputed quantities defined in (REF )–()–() (see Appendix for its exact expression).",
"Solving the $\\sup $ problem (REF ) (with fixed matrix $\\overline{A}$ ) hence amounts to finding a normalized eigenvector in $\\mathbb {R}^P$ associated with the largest eigenvalue of the matrix $G_{\\varepsilon ,h}^M(\\overline{A})$ .",
"This is reminiscent of the eigenvalue problem discussed in Section REF .",
"Practically, this eigenvector is computed using the power method.",
"The cost of such a computation is negligible, owing to the small size of the matrix $G_{\\varepsilon ,h}^M(\\overline{A})$ (recall that $P$ is typically small in comparison to $M$ ).",
"We denote by $c(\\overline{A})$ its solution and hence have $ \\sup _{c\\in \\mathbb {R}^P,\\,|c|^2=1}\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)=c(\\overline{A})^T\\,G_{\\varepsilon ,h}^M(\\overline{A})\\,c(\\overline{A}).$"
],
[
"The $\\inf $ problem (",
"Let $c\\in \\mathbb {R}^P$ , $|c|^2=1$ , be fixed.",
"We observe that $\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c) = \\frac{1}{2} \\sum _{1\\le i,j,k,l\\le d} \\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j,k,l} \\ \\overline{A}_{i,j} \\ \\overline{A}_{k,l}-\\sum _{1\\le i,j\\le d} \\left[B_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j} \\ \\overline{A}_{i,j} + b_h^P(c),$ where $\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)$ is a $d\\times d\\times d\\times d$ fourth-order tensor, $B_{\\varepsilon ,h}^{P,M}(c)$ is a $d\\times d$ matrix and $b_h^P(c)$ is a scalar that can all be assembled at no additional cost using the precomputed quantities defined in (REF )–()–() (see Appendix for their exact expressions).",
"We recognize in $\\Phi ^{P,M}_{\\varepsilon ,h}$ the discrete equivalent of (REF ).",
"The $\\inf $ problem (REF ) (with fixed eigenvector $c$ ) is in practice solved as explained in Section REF , by considering the linear system (see (REF )) $\\forall \\,1\\le i, j\\le d, \\quad \\sum _{1\\le k,l\\le d} \\left[ \\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j,k,l} \\ \\overline{A}_{k,l} = \\left[B_{\\varepsilon ,h}^{P,M}(c) \\right]_{i,j}.$"
],
[
"Iterative algorithm",
"In the above description, we have considered either the $\\sup $ problem (on $c$ , with fixed $\\overline{A}$ ) or the $\\inf $ problem (on $\\overline{A}$ , for fixed $c$ ) involved in (REF ).",
"We now assemble these two building blocks to build an algorithm to solve (REF ).",
"Introducing $ \\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A}) = \\sup _{c\\in \\mathbb {R}^P,\\,|c|^2=1}\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A},c),$ we recast (REF ) as $ I^{P,M}_{\\varepsilon ,h} = \\inf _{\\overline{A}\\in {\\cal S}}\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A}).$ We have seen (see (REF )) that $\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A})=c(\\overline{A})^T\\,G_{\\varepsilon ,h}^M(\\overline{A})\\,c(\\overline{A})$ , where $c(\\overline{A})$ is an eigenvector of the matrix $G_{\\varepsilon ,h}^M(\\overline{A})$ .",
"One can easily prove that, for any $1\\le i,j\\le d$ , $\\left[\\nabla _{\\overline{A}}\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A})\\right]_{i,j} = c(\\overline{A})^T \\ \\partial _{\\overline{A}_{i,j}} G_{\\varepsilon ,h}^M(\\overline{A}) \\ c(\\overline{A}),$ which reads, using the expressions (REF ), (REF ) and (REF ) of $G_{\\varepsilon ,h}^M$ , $\\mathbb {B}_{\\varepsilon ,h}^{P,M}$ and $B_{\\varepsilon ,h}^{P,M}$ given in Appendix , as $ \\left[\\nabla _{\\overline{A}}\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A})\\right]_{i,j} = \\sum _{1\\le k,l\\le d} \\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c(\\overline{A}))\\right]_{i,j,k,l} \\ \\overline{A}_{k,l} - \\left[B_{\\varepsilon ,h}^{P,M}(c(\\overline{A}))\\right]_{i,j}.$ Let $0<\\mu <1$ .",
"In practice, we iterate as follows to solve problem (REF ).",
"Let $n\\in \\mathbb {N}$ and $\\overline{A}^n\\in {\\cal S}$ .",
"We compute $c^n = c(\\overline{A}^n)$ solution to the $\\sup $ problem (REF ) with fixed matrix $\\overline{A}^n$ .",
"We compute $\\overline{A}^{n+1}_\\flat \\in \\mathbb {R}^{d \\times d}_{\\rm sym}$ solution to the linear system (REF ) with fixed eigenvector $c^n$ .",
"As pointed out above, we assume that $\\overline{A}_\\flat ^{n+1}$ belongs to the convex subset ${\\cal S}$ of $\\mathbb {R}^{d \\times d}_{\\rm sym}$ .",
"It has always been the case in our numerical experiments.",
"We define the next iterate as $ \\overline{A}^{n+1} = (1-\\mu )\\,\\overline{A}^n+\\mu \\,\\overline{A}^{n+1}_\\flat .$ For the numerical results reported on in Section , we have worked with $\\mu \\le 0.1$ .",
"Since $\\overline{A}^{n+1}$ is a convex combination of $\\overline{A}^n \\in {\\cal S}$ and $\\overline{A}^{n+1}_\\flat \\in {\\cal S}$ , we have $\\overline{A}^{n+1} \\in {\\cal S}$ .",
"The iterations are initialized using, say, $\\overline{A}^0 = \\mathbb {E}\\left(\\frac{1}{|{\\cal D}|}\\int _{\\cal D} A_\\varepsilon (x,\\cdot )\\,{\\rm d}x\\right).$ Let us briefly explain, at least formally, why the algorithm defined above enables to find a minimizer of (REF ).",
"We assume the linear system (REF ) to be invertible, and we denote by $\\left[\\mathbb {B}^{P,M}_{\\varepsilon ,h}(c)\\right]^{-1}$ its formal inverse.",
"Since $\\overline{A}^{n+1}_\\flat $ is defined as the solution to (REF ) with eigenvector $c^n$ , we infer from (REF ) and (REF ) that $\\mathbb {B}^{P,M}_{\\varepsilon ,h}(c^n)\\overline{A}^{n+1}_\\flat =B_{\\varepsilon ,h}^{P,M}(c^n)=\\mathbb {B}^{P,M}_{\\varepsilon ,h}(c^n)\\overline{A}^n-\\nabla _{\\overline{A}}\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A}^n),$ and thus $\\overline{A}^{n+1}_\\flat =\\overline{A}^n - \\left[\\mathbb {B}^{P,M}_{\\varepsilon ,h}(c^n)\\right]^{-1}\\nabla _{\\overline{A}}\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A}^n).$ The iteration (REF ) can be recast under the form $\\overline{A}^{n+1}=\\overline{A}^n-\\mu \\, \\left[\\mathbb {B}^{P,M}_{\\varepsilon ,h}(c^n)\\right]^{-1} \\nabla _{\\overline{A}}\\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A}^n).$ This is a quasi-Newton algorithm for the minimization of the function $\\overline{A}\\mapsto \\Phi _{\\varepsilon ,h}^{P,M}(\\overline{A})$ , with a fixed step size $\\mu $ and where the Hessian of $\\Phi _{\\varepsilon ,h}^{P,M}$ with respect to $\\overline{A}$ is approximated by $\\mathbb {B}^{P,M}_{\\varepsilon ,h}$ .",
"Note that each iteration of the algorithm is inexpensive in comparison with the cost of the operations described in Sections REF and REF .",
"Consequently, there is no real advantage in improving the optimization algorithm (REF ) (e.g.",
"by optimizing the value of $\\mu $ by a line search)."
],
[
"Numerical results",
"As pointed out in Section , our approach targets practical situations where the information on the oscillatory coefficients in the equation may be incomplete, and thus the other available approaches cannot be applied.",
"It is nevertheless a legitimate question to investigate how our approach performs on standard test-cases in the periodic and stationary ergodic settings, and how it compares with the classical homogenization approach for small values of $\\varepsilon $ .",
"As already pointed out in Section REF , and as detailed below (see Section REF ), the aim of the numerical tests is different in the periodic setting and in the stochastic setting.",
"It is also different if $\\varepsilon $ is asymptotically small or if $\\varepsilon $ takes larger values.",
"This section is organized as follows.",
"In Section REF , we introduce the periodic and the stationary ergodic test cases considered.",
"In Section REF , we present the numerical results obtained in the case of small values of $\\varepsilon $ .",
"In Section REF , we address the case of larger values of $\\varepsilon $ ."
],
[
"Test-cases",
"We let $d=2$ and the domain ${\\cal D}$ be the unit square $(0,1)^2$ .",
"We fix the value of the parameter $\\overline{\\varepsilon }$ to ${\\rm size}({\\cal D})/10 = 10^{-1}$ ."
],
[
"Periodic setting",
"We consider the test-case introduced in [16], namely $ A_\\varepsilon (x,y)=A^{\\rm per}(x/\\varepsilon ,y/\\varepsilon ),$ with $A^{\\rm per}$ a $\\mathbb {Z}^2$ -periodic symmetric matrix field given by $ \\begin{alignedat}{1}\\left[A^{\\rm per}(x,y)\\right]_{1,1}&=2+\\frac{1}{2\\pi }(\\sin (2\\pi x)+\\sin (2\\pi y)),\\\\\\left[A^{\\rm per}(x,y)\\right]_{1,2}&=\\frac{1}{2\\pi }(\\sin (2\\pi x)+\\sin (2\\pi y)),\\\\\\left[A^{\\rm per}(x,y)\\right]_{2,2}&=1+\\frac{1}{2\\pi }(\\sin (2\\pi x)+\\sin (2\\pi y)).\\end{alignedat}$ The coefficients of the corresponding homogenized matrix (obtained by solving the periodic corrector problem (REF ) on a very fine mesh) are $ [A_\\star ]_{1,1}\\approx 1.9806,\\qquad [A_\\star ]_{1,2} = [A_\\star ]_{2,1} \\approx -0.019345,\\qquad [A_\\star ]_{2,2} \\approx 0.98065.$"
],
[
"Stationary ergodic setting",
"We consider the random checkerboard test-case (studied e.g.",
"in [16]), namely $ A_\\varepsilon (x,y,\\omega )=a^{\\rm sto}(x/\\varepsilon ,y/\\varepsilon ,\\omega )\\,{\\rm Id}_2,$ with $a^{\\rm sto}$ a discrete stationary field given by (recall that $Q=(0,1)^2$ ) $ a^{\\rm sto}(x,y,\\omega )=\\sum _{k\\in \\mathbb {Z}^2}\\mathbb {1}_{Q+k}(x,y)X_{k}(\\omega ),$ where the random variables $X_{k}$ are i.i.d.",
"and such that $\\mathbb {P}(X_{k}=4)=\\mathbb {P}(X_{k}=16)=1/2$ .",
"An explicit expression for the homogenized matrix is known in that case: $ A_\\star =8\\,{\\rm Id}_2.$"
],
[
"Objectives in the periodic case and in the stochastic case",
"In the regime $\\varepsilon <\\overline{\\varepsilon }$ , we know from Proposition REF that our method can be seen as a practical variational approach for computing the homogenized matrix $A_\\star $ .",
"The remaining question is whether this approach is efficient or not, and particularly, compared with the classical approach in homogenization.",
"Our approach (based on (REF )–(REF )) requires solving the highly oscillatory equations (REF ) set on the domain ${\\cal D}$ , for $P=d(d+1)/2$ right-hand sides.",
"In the periodic setting, the classical homogenization approach requires solving $d$ non-oscillatory equations set on the unit cell $Q$ .",
"There is thus no hope to outperform the latter approach in terms of computational time.",
"This setting is nonetheless considered as a validation and we investigate how our approach performs in terms of accuracy, for the approximation of the homogenized matrix, and for the approximation of $u_\\varepsilon $ in the $L^2$ and $H^1$ norms.",
"The real, discriminating, test-case for our approach is the stationary ergodic setting.",
"Indeed, classical homogenization then requires solving equations that are set on a truncated approximation $Q^N=(-N,N)^d$ of an asymptotically infinitely large domain (see (REF ) in Section REF ).",
"The coefficients of these equations vary at scale 1.",
"In that case, to hope for an accurate approximation of the homogenized matrix, one has to consider a meshsize $H\\ll 1$ .",
"On the other hand, we consider a meshsize $h \\ll \\varepsilon $ to solve the highly oscillatory equations (set on the domain ${\\cal D}$ ) involved in our approach.",
"We see that, up to an appropriate choice of the parameter $H$ such that $ \\frac{2N}{H} = \\frac{\\text{size}({\\cal D})}{h},$ where $\\text{size}({\\cal D})$ is typically the diameter of ${\\cal D}$ , the classical homogenization approach and ours involve solving linear systems of the same size.",
"The computational workload for the two approaches is thus of the same order of magnitude, although not identical.",
"We have decided to enforce (REF ) and to relate $N$ in (REF ) and $\\varepsilon $ in (REF ) by $ N={\\rm size}({\\cal D})/2\\varepsilon .$ Note that imposing (REF ) is equivalent to enforcing $\\varepsilon /h=1/H$ .",
"We then compare the two methods in terms of solution time and accuracy.",
"Obviously, for the two methods, the same number $M$ of Monte Carlo realizations is used, and the same $M$ realizations are considered.",
"Remark 12 Another possibility would have been to impose $\\varepsilon /h=1/H$ and to adjust the size $N$ of $Q^N$ in (REF ) so that both approaches exactly share the same workload.",
"We did not pursue in that direction.",
"The numerical experiments reported in Section REF show that, in the stochastic case, and for all the values of $\\varepsilon <\\overline{\\varepsilon }$ that have been considered, the approximation of $A_\\star $ obtained by the classical homogenization approach is slightly more accurate than that obtained with our approach.",
"In contrast, our approach provides a better $L^2$ -approximation and a better $H^1$ -approximation of $\\mathbb {E}(u_\\varepsilon )$ .",
"This is somewhat intuitive, as our approach is targeted toward the approximation of $u_\\varepsilon $ rather than $A_\\star $ .",
"In terms of computational cost, our approach is slightly less expensive for moderately small values of $\\varepsilon $ , and slightly more expensive for asymptotically small values of $\\varepsilon $ (in any cases, the ratio of costs remains close to 1, see Figure REF below)."
],
[
"Choice of the numerical parameters",
"We recall that the integer $M$ denotes the number of i.i.d.",
"realizations used to approximate the expectation in the cost function (REF ) (see (REF )).",
"We also recall that the integer $P$ denotes the dimension of the set $V^P_{\\rm n}({\\cal D})$ (defined in (REF )) that is used to approximate the space $L^2_{\\rm n}({\\cal D})$ in the $\\sup $ problem.",
"As explained in Section REF , we consider as basis functions of the set $V^P_{\\rm n}({\\cal D})$ the first $P$ normalized eigenvectors of the laplacian operator in the domain ${\\cal D}$ .",
"Because of the simple geometry of ${\\cal D}$ , they are here analytically known.",
"We take here $P=d \\, (d+1)/2$ , that is $P=3$ , which is the minimum dimension of the search space $V^P_{\\rm n}({\\cal D})$ ."
],
[
"Results in the periodic setting",
"We consider the parameters $\\lbrace \\varepsilon _k \\rbrace _{0\\le k\\le 6}$ such that $\\varepsilon _0=0.4$ and $\\varepsilon _k=\\varepsilon _{k-1}/2$ for $1\\le k\\le 6$ .",
"The associated meshsizes are $\\lbrace h_k \\rbrace _{0\\le k\\le 6}$ such that $h_k=\\varepsilon _k/r$ for $r\\approx 43$ , unless otherwise mentioned.",
"We focus on the values $\\lbrace \\varepsilon _k \\rbrace _{3\\le k\\le 6}$ , for which we have $\\varepsilon _k<\\overline{\\varepsilon }$ .",
"The error in the approximation of the homogenized matrix is defined by $ {\\tt err_per_mat} = \\left(\\frac{\\sum _{1\\le i,j\\le d}\\left| \\left[\\overline{A}_{\\varepsilon ,h}^P\\right]_{i,j}- [A_\\star ]_{i,j} \\right|^2}{\\sum _{1\\le i,j\\le d} \\left| [A_\\star ]_{i,j} \\right|^2 } \\right)^{1/2},$ where $A_\\star $ is taken equal to its reference value (REF ) and $\\overline{A}^P_{\\varepsilon ,h}$ is the best matrix computed by our approach.",
"The numerical results are collected in Table REF .",
"We observe that our approach provides an accurate approximation of the homogenized matrix.",
"The accuracy of the approximation improves (in the limit of spatial resolution) as $\\varepsilon $ decreases.",
"errpermat Table: Approximation of A ☆ A_\\star () in function of ε\\varepsilon (each line corresponds to a different value of the ratio ε/h\\varepsilon /h).",
"The test cases with ε\\varepsilon too small and ε/h\\varepsilon /h too large are prohibitively expensive to perform.",
"They are marked with an X.We now examine the approximation of $u_\\varepsilon $ in the $L^2$ norm.",
"We denote by $\\bullet $ $u_{\\varepsilon ,h}(f)$ the discrete solution to (REF ) with the periodic oscillatory coefficient given by (REF )–(REF ) and the right-hand side $f$ ; $\\bullet $ $u_{\\star ,h}(f)$ the discrete solution to (REF ) with the homogenized matrix (REF ) and the right-hand side $f$ ; $\\bullet $ $u_{\\varepsilon ,h}^{1,\\theta }(f)$ the two-scale expansion (truncated at first-order) built from $u_{\\star ,h}(f)$ (see (REF )), where we use the periodic correctors solution to (REF ); $\\bullet $ $\\overline{u}^P_{\\varepsilon ,h}(f)$ the discrete solution to (REF ) with the matrix $\\overline{A}_{\\varepsilon ,h}^P$ and the right-hand side $f$ (we recall that the matrix $\\overline{A}_{\\varepsilon ,h}^P$ has been computed using a small number $P$ of right-hand sides).",
"To assess the quality of the approximation of $u_{\\varepsilon ,h}$ by $\\widehat{u}^{\\theta }_h \\in \\left\\lbrace u_{\\star ,h}, \\ u_{\\varepsilon ,h}^{1,\\theta }, \\ \\overline{u}^P_{\\varepsilon ,h} \\right\\rbrace $ in the $L^2$ norm, we define the criterion $ {\\tt err_per_L2} = \\left( \\frac{\\displaystyle \\inf _{\\theta \\in \\mathbb {R}^2} \\left[ \\sup _{f\\in V^{\\cal Q}_{\\rm n}({\\cal D})} {\\left\\Vert u_{\\varepsilon ,h}(f) - \\widehat{u}^{\\theta }_h(f)\\right\\Vert }_{{L^2(\\cal D)}}^2 \\right]}{{\\left\\Vert u_{\\varepsilon ,h} \\left( \\widehat{f}_\\varepsilon \\right)\\right\\Vert }_{{L^2(\\cal D)}}^2} \\right)^{1/2}.$ Note that the supremum is taken over $f\\in V^{\\cal Q}_{\\rm n}({\\cal D})$ , where ${\\cal Q} \\gg P$ .",
"We take ${\\cal Q}=16$ , and we have checked, in all the cases considered below, that our results do not significantly change for a larger value of ${\\cal Q}$ .",
"The function $\\widehat{f}_\\varepsilon \\in V^{\\cal Q}_{\\rm n}({\\cal D})$ denotes the argument of the $\\inf \\sup $ problem in the numerator of (REF ).",
"We hence compare $u_\\varepsilon $ with its homogenized limit $u_\\star $ , its first-order two-scale expansion $u_\\varepsilon ^{1,\\theta }$ (recall in this case that the correctors are defined up to an additive constant $\\theta $ , over which we minimize the error in (REF )), and the approximation $\\overline{u}^P_\\varepsilon $ provided by our approach.",
"The numerical results are collected in Figure REF .",
"We observe that the solution associated with the best matrix we compute indeed converges towards the exact solution, in the $L^2$ norm.",
"We however recall that, in the present periodic setting, computing $\\overline{u}_{\\varepsilon ,h}^P$ is much more expensive than computing $u_{\\star ,h}$ or $u_{\\varepsilon ,h}^{1,\\theta }$ .",
"errperL2 Figure: Approximation of u ε u_\\varepsilon in the L 2 L^2 norm () by u ☆,h u_{\\star ,h} (red), u ε,h 1,θ u_{\\varepsilon ,h}^{1,\\theta } (brown) and u ¯ ε,h P \\overline{u}^P_{\\varepsilon ,h} (black) in function of ε\\varepsilon , for hh such that ε/h≈43\\varepsilon /h \\approx 43.We next examine the $H^1$ error.",
"For $f\\in {L^2(\\cal D)}$ , we denote by $C_{\\varepsilon ,h}\\nabla u_{\\star ,h}(f)$ the discrete equivalent of $C_\\varepsilon \\nabla u_\\star (f)$ , the homogenization-based approximation of $\\nabla u_\\varepsilon (f)$ , see (REF )–(REF ) in Section REF .",
"We recall that, in our approach, we seek an approximation of $\\nabla u_\\varepsilon (f)$ under the form $\\overline{C}_{\\varepsilon }\\nabla \\overline{u}_\\varepsilon (f)$ (see (REF )), the discrete equivalent of which is computed as $\\overline{C}^R_{\\varepsilon ,h}\\nabla \\overline{u}^P_{\\varepsilon ,h}(f)$ .",
"Recall that the integer $R$ is the number of right-hand sides used to define the least-squares minimization problem (REF ) giving $\\overline{C}^R_{\\varepsilon ,h}$ .",
"Here, we take $R=P=3$ .",
"To assess the quality of the approximation of $\\nabla u_{\\varepsilon ,h}$ , we define, for $\\widehat{C}_{\\varepsilon ,h}\\nabla \\widehat{u}_h \\in \\left\\lbrace C_{\\varepsilon ,h}\\nabla u_{\\star ,h}, \\ \\overline{C}^R_{\\varepsilon ,h}\\nabla \\overline{u}^P_{\\varepsilon ,h}\\right\\rbrace $ , the criterion $ {\\tt err_per_H1} = \\left( \\frac{ \\displaystyle \\sup _{f\\in V^{\\cal Q}_{\\rm n}({\\cal D})} {\\left\\Vert \\nabla u_{\\varepsilon ,h}(f) - \\widehat{C}_{\\varepsilon ,h} \\nabla \\widehat{u}_h(f)\\right\\Vert }_{L^2({\\cal D} \\setminus {\\cal B})}^2}{{\\left\\Vert \\nabla u_{\\varepsilon ,h} \\left( \\widehat{f}_\\varepsilon \\right)\\right\\Vert }_{L^2({\\cal D} \\setminus {\\cal B})}^2} \\right)^{1/2},$ where, here again, the supremum is taken over a space $V^{\\cal Q}_{\\rm n}({\\cal D})$ much larger than $V^P_{\\rm n}({\\cal D})$ (we take ${\\cal Q}=16$ ), and where $\\widehat{f}_\\varepsilon \\in V^{\\cal Q}_{\\rm n}({\\cal D})$ denotes the argument of the $\\sup $ problem.",
"In (REF ), ${\\cal B}$ represents the subset of ${\\cal D}$ formed by the boundary elements of the discretization ${\\cal T}_h$ .",
"We remove them in view of the discussion below (REF ).",
"We thus compare $\\nabla u_\\varepsilon $ with its approximation $C_\\varepsilon \\nabla u_\\star $ provided by the two-scale expansion and with the approximation $\\overline{C}^R_\\varepsilon \\ \\nabla \\overline{u}^P_\\varepsilon $ provided by our approach.",
"The numerical results are collected in Table REF .",
"We observe that our approach provides an accurate $H^1$ -approximation of $u_\\varepsilon $ .",
"As $\\varepsilon $ goes to zero, the surrogate we compute is (roughly) a first-order convergent approximation of $\\nabla u_\\varepsilon $ in the $L^2$ norm.",
"As far as the homogenization-based approximation is concerned, we expect it to converge with order at least one half (see (REF )).",
"This is what we observe in practice, as long as $\\varepsilon $ is not too small.",
"Otherwise, the error due to the meshsize dominates, and the error (REF ) does not decrease anymore when $\\varepsilon $ decreases.",
"errperH1 Table: Approximation of ∇u ε \\nabla u_\\varepsilon in the L 2 L^2 norm () by C ε,h ∇u ☆,h C_{\\varepsilon ,h}\\nabla u_{\\star ,h} and C ¯ ε,h R ∇u ¯ ε,h P \\overline{C}^R_{\\varepsilon ,h}\\nabla \\overline{u}^P_{\\varepsilon ,h} in function of ε\\varepsilon , for hh such that ε/h≈86\\varepsilon /h \\approx 86.",
"The test cases with ε\\varepsilon too small are prohibitively expensive to perform.",
"They are marked with an X."
],
[
"Results in the stationary ergodic setting",
"We consider the parameters $\\lbrace \\varepsilon _k \\rbrace _{0\\le k\\le 5}$ such that $\\varepsilon _k=2^{-(k+1)}$ for $0\\le k\\le 5$ .",
"In agreement with formula (REF ), we couple these parameters to the parameters $\\lbrace N_k \\rbrace _{0\\le k\\le 5}$ (defining the domain on which we solve the corrector problems (REF )) such that $N_k=2^k$ .",
"The associated meshsizes $\\lbrace h_k \\rbrace _{0\\le k\\le 5}$ and $\\lbrace H_k \\rbrace _{0\\le k\\le 5}$ are computed respectively letting $h_k=\\varepsilon _k/r$ for $r\\approx 27$ (unless otherwise stated) and using (REF ).",
"We focus on the values $\\lbrace \\varepsilon _k \\rbrace _{3\\le k\\le 5}$ and $\\lbrace N_k \\rbrace _{3\\le k\\le 5}$ , for which we have $\\varepsilon _k<\\overline{\\varepsilon }$ .",
"We consider $M=100$ Monte Carlo realizations.",
"Before discussing the accuracy of our approach, we first compare its cost with that of the classical approach.",
"We show on Figure REF the ratio of the time needed to compute $\\overline{A}^{P,M}_{\\varepsilon ,h}$ using our approach divided by the time needed to compute $A_{\\star ,H}^{N,M}$ by the classical homogenization approach.",
"To compare the computational times, we make use of an implementation that does not exploit parallelism, and we solve the linear systems by means of an iterative solver.",
"In view of Figure REF , for the choice of parameters discussed in Section REF , our method is slightly faster than the standard homogenization approach for values of $N$ up to approximately 14.",
"This observation can be explained as follows.",
"For the number $M=100$ of Monte Carlo realizations that we consider, we can neglect, in our procedure, the cost of the precomputation and final optimization stages, in comparison to the Monte Carlo step (see Section ).",
"Hence, to compute $\\overline{A}_{\\varepsilon ,h}^{P,M}$ , we have to (i) assemble $M=100$ stiffness matrices, (ii) assemble $P=3$ right-hand sides, and (iii) solve $P\\times M=300$ linear systems.",
"In contrast, to compute $A_{\\star ,H}^{N,M}$ , one has to solve $d\\times M=200$ approximate corrector equations (REF ), that is to say (i) assemble $M=100$ stiffness matrices, (ii) assemble $d\\times M=200$ right-hand sides, and (iii) solve $d\\times M=200$ linear systems.",
"Consequently, our approach necessitates solving 100 more linear systems, but assembling 200 less right-hand sides, than the classical homogenization approach.",
"This explains what we observe.",
"When the value of $N$ is not too large, the assembly cost is higher than the inversion cost, and our approach is faster.",
"Figure: Ratio of the computational times between our approach and the classical homogenization approach, in function of NN (here M=100M=100 and ε/h≈27\\varepsilon /h \\approx 27).We adapt to the stationary ergodic setting the accuracy criteria (REF ), (REF ) and (REF ) introduced in the periodic setting.",
"The error in the approximation of the homogenized matrix is defined, for $\\widehat{A}^M \\in \\left\\lbrace A_{\\star ,H}^{N,M}, \\overline{A}_{\\varepsilon ,h}^{P,M}\\right\\rbrace $ , by $ {\\tt err_sto_mat} = \\left(\\frac{\\sum _{1\\le i,j\\le d} \\left| \\left[\\widehat{A}^M\\right]_{i,j} - [A_\\star ]_{i,j} \\right|^2}{\\sum _{1\\le i,j\\le d} \\left| [A_\\star ]_{i,j} \\right|^2 }\\right)^{1/2},$ where $A_\\star $ is taken equal to the exact value (REF ).",
"We recall that $A_{\\star ,H}^{N,M}$ is the practical approximation of $A_\\star ^{N,M}$ defined in (REF ), and that our approach consists in computing the best matrix $\\overline{A}^{P,M}_{\\varepsilon ,h}$ following the procedure described in Section REF .",
"The numerical results are collected in Figure REF , for several choices of the meshsizes.",
"We observe that the matrix we compute converges to the homogenized matrix as $N$ increases.",
"However, for any value of $N$ in the range we consider, the approximation of $A_\\star $ obtained by the classical homogenization approach is slightly more accurate than the one obtained with our approach.",
"As shown on Figure REF , the former approach is as expensive as our approach for $N \\approx 14$ , and slightly less expensive for larger values of $N$ .",
"errstomat Figure: Approximation of A ☆ A_\\star by the classical homogenization approach (blue) and by our approach (black) in function of NN, for M=100M=100 realizations.",
"Since MM is finite, the error is actually random.",
"We compute it 100 times.",
"The thick line corresponds to the mean value over the 100 computations of the error.",
"The dashed lines show the 95%95\\% confidence interval.",
"Results obtained with hh such that ε/h≈27\\varepsilon /h \\approx 27 (resp.",
"ε/h≈108\\varepsilon /h \\approx 108) are denoted with x (resp.",
"o).Turning to the approximation of $\\mathbb {E}(u_\\varepsilon )$ in the $L^2$ norm, we denote by $\\bullet $ $u_{\\varepsilon ,h}^M(f)$ the expectation, as defined in (REF ), of the discrete solutions to (REF ) with the oscillatory coefficients given by (REF )–(REF ) and the right-hand side $f$ ; $\\bullet $ $u_{\\star ,h}(f)$ the discrete solution to (REF ) with the exact homogenized matrix (REF ) and the right-hand side $f$ (note that the exact matrix is usually unknown); $\\bullet $ $u_{\\star ,h}^{N,M}(f)$ the discrete solution to (REF ) with the matrix $A_{\\star ,H}^{N,M}$ and the right-hand side $f$ ; $\\bullet $ $\\overline{u}^{P,M}_{\\varepsilon ,h}(f)$ the discrete solution to (REF ) with the matrix $\\overline{A}_{\\varepsilon ,h}^{P,M}$ and the right-hand side $f$ .",
"The $M$ realizations of the field $A(\\cdot ,\\omega )$ we consider to compute $u^M_{\\varepsilon ,h}(f)$ , $u_{\\star ,h}^{N,M}(f)$ and $\\overline{u}^{P,M}_{\\varepsilon ,h}(f)$ are identical.",
"To assess the quality of the approximation of $u^M_{\\varepsilon ,h}$ by $\\widehat{u}_h \\in \\left\\lbrace u_{\\star ,h}, \\ u_{\\star ,h}^{N,M}, \\ \\overline{u}^{P,M}_{\\varepsilon ,h}\\right\\rbrace $ in the $L^2$ norm, we define the criterion $ {\\tt err_sto_L2} = \\left( \\frac{ \\displaystyle \\sup _{f\\in V^{\\cal Q}_{\\rm n}({\\cal D})} {\\left\\Vert u^M_{\\varepsilon ,h}(f) - \\widehat{u}_h(f)\\right\\Vert }_{{L^2(\\cal D)}}^2}{{\\left\\Vert u_{\\varepsilon ,h}^M\\left(\\widehat{f}_\\varepsilon \\right)\\right\\Vert }_{{L^2(\\cal D)}}^2}\\right)^{1/2}.$ As in the periodic case, the supremum is taken over $f\\in V^{\\cal Q}_{\\rm n}({\\cal D})$ with ${\\cal Q}=16 \\gg P$ , and $\\widehat{f}_\\varepsilon \\in V^{\\cal Q}_{\\rm n}({\\cal D})$ denotes the argument of the $\\sup $ problem.",
"The numerical results are collected in Figure REF , for several choices of the meshsizes and of the total number $M$ of realizations.",
"We observe that the solution associated with the best matrix we compute is a better $L^2$ -approximation (for the range of parameters considered here) of $\\mathbb {E}(u_\\varepsilon )$ than the solutions associated with the exact or approximate homogenized matrices.",
"Again, due to the small number $P$ of right-hand sides we consider to compute $\\overline{A}_{\\varepsilon ,h}^{P,M}$ , this good accuracy is not an immediate consequence of our practical procedure (it would have been if we had taken $P$ extremely large).",
"We also observe that the accuracy of the three approximations $u_{\\star ,h}$ , $u_{\\star ,h}^{N,M}$ and $\\overline{u}_{\\varepsilon ,h}^{P,M}$ improves when $h$ decreases or when $M$ increases, in somewhat a complex manner.",
"In terms of cost, our approach is again less expensive than the classical approach for $N \\le 14$ .",
"errstoL2 Figure: Approximation of 𝔼(u ε )\\mathbb {E}(u_\\varepsilon ) in the L 2 L^2 norm () by u ☆,h u_{\\star ,h} (red), u ☆,h N,M u_{\\star ,h}^{N,M} (blue) and u ¯ ε,h P,M \\overline{u}_{\\varepsilon ,h}^{P,M} (black) in function of NN (curves with x: ε/h≈27\\varepsilon /h \\approx 27 and M=100M=100; curves with o: ε/h≈108\\varepsilon /h \\approx 108 and M=100M=100; curves with +: ε/h≈27\\varepsilon /h \\approx 27 and M=400M=400; curves with □\\square : ε/h≈54\\varepsilon /h \\approx 54 and M=400M=400).We next turn to the $H^1$ -error.",
"We denote by $C^{N,M}_{\\varepsilon ,h}$ the approximation of the deterministic matrix $C_\\varepsilon $ defined by (REF ) by an empirical mean over $M$ realizations of the corrector functions, solution to (REF ): $\\left[ C^{N,M}_{\\varepsilon ,h} \\right]_{i,j}=\\delta _{ij} + \\frac{1}{M} \\sum _{m=1}^M \\partial _i w^N_{e_j}(\\cdot /\\varepsilon ,\\omega _m).$ For $f\\in {L^2(\\cal D)}$ , we denote by $C^{N,M}_{\\varepsilon ,h} \\nabla u_{\\star ,h}(f)$ and $C^{N,M}_{\\varepsilon ,h} \\nabla u^{N,M}_{\\star ,h}(f)$ the two discrete equivalents of $C_\\varepsilon \\, \\nabla u_\\star (f)$ , the homogenization-based approximation of $\\mathbb {E}\\left(\\nabla u_\\varepsilon (f)\\right)$ , obtained by using the exact homogenized matrix (REF ) and the matrix $A_{\\star ,H}^{N,M}$ , respectively, to compute an approximation of $u_\\star (f)$ .",
"In our approach, we seek a discrete approximation of $\\mathbb {E}\\left(\\nabla u_\\varepsilon \\right)$ under the form $\\overline{C}^{R,M}_{\\varepsilon ,h} \\nabla \\overline{u}^{P,M}_{\\varepsilon ,h}$ , with $R=P=3$ .",
"For $\\widehat{C}_{\\varepsilon ,h}^M \\, \\nabla \\widehat{u}_h \\in \\left\\lbrace C^{N,M}_{\\varepsilon ,h} \\, \\nabla u_{\\star ,h}, \\ C^{N,M}_{\\varepsilon ,h} \\, \\nabla u_{\\star ,h}^{N,M}, \\ \\overline{C}^{R,M}_{\\varepsilon ,h} \\, \\nabla \\overline{u}^{P,M}_{\\varepsilon ,h} \\right\\rbrace ,$ we define the criterion $ {\\tt err_sto_H1} = \\left( \\frac{ \\displaystyle \\sup _{f\\in V^{\\cal Q}_{\\rm n}({\\cal D})} {\\left\\Vert \\nabla u_{\\varepsilon ,h}^M(f) - \\widehat{C}_{\\varepsilon ,h}^M \\ \\nabla \\widehat{u}_h(f)\\right\\Vert }_{L^2({\\cal D} \\setminus {\\cal B})}^2}{{\\left\\Vert \\nabla u_{\\varepsilon ,h}^M \\left( \\widehat{f}_\\varepsilon \\right)\\right\\Vert }_{L^2({\\cal D} \\setminus {\\cal B})}^2}\\right)^{1/2},$ where, here again, the supremum is taken over the space $V^{\\cal Q}_{\\rm n}({\\cal D})$ for ${\\cal Q}=16 \\gg P$ , $\\widehat{f}_\\varepsilon \\in V^{\\cal Q}_{\\rm n}({\\cal D})$ denotes the argument of the $\\sup $ problem, and boundary elements ${\\cal B}$ are removed from the evaluation criterion, as in the periodic case (REF ).",
"We recall that, in (REF ), $u_{\\varepsilon ,h}^M(f)$ is the empirical mean (REF ) over $M$ realizations of $u_{\\varepsilon ,h}(f;\\omega )$ .",
"It is thus an approximation to $\\mathbb {E} \\left[ u_\\varepsilon (f) \\right]$ .",
"The numerical results are collected in Table REF .",
"We see that our surrogate defines an approximation of $\\mathbb {E}(\\nabla u_\\varepsilon )$ which is systematically better than that provided by the classical homogenization approach, for any choice of $h$ and $M$ .",
"errstoH1 Table: Approximation of 𝔼(∇u ε )\\mathbb {E}(\\nabla u_\\varepsilon ) in the L 2 L^2 norm () by C ε,h N,M ∇u ☆,h C^{N,M}_{\\varepsilon ,h} \\nabla u_{\\star ,h}, C ε,h N,M ∇u ☆,h N,M C^{N,M}_{\\varepsilon ,h} \\nabla u^{N,M}_{\\star ,h} and C ¯ ε,h R,M ∇u ¯ ε,h P,M \\overline{C}^{R,M}_{\\varepsilon ,h} \\nabla \\overline{u}^{P,M}_{\\varepsilon ,h} in function of NN (the various lines correspond to various values of hh and MM)."
],
[
"Results in the case $\\varepsilon \\ge \\overline{\\varepsilon }$",
"In the regime $\\varepsilon \\ge \\overline{\\varepsilon }$ , we quantitatively investigate whether the best constant matrix provided by our approach allows for an accurate approximation of the exact solution, in the $L^2$ norm in the sense of the criteria (REF ) or (REF ), and in the $H^1$ norm in the sense of the criteria (REF ) or (REF ).",
"We also consider below the criterion (REF ), only in the periodic setting.",
"It is indeed interesting to quantify the threshold value of $\\varepsilon $ above which $\\overline{A}_\\varepsilon $ is significantly different from $A_\\star $ (let alone to understand the practical limitation of homogenization theory).",
"When considering large values of the parameter $\\varepsilon $ , it is necessary to consider $P$ right-hand sides with $P$ larger than $d(d+1)/2=3$ , as pointed out in Section REF .",
"This value depends on $\\varepsilon $ and is denoted $P(\\varepsilon )$ ."
],
[
"Results in the periodic setting",
"We consider the set $\\lbrace \\varepsilon _k \\rbrace _{0\\le k\\le 2}$ of parameters introduced in Section REF .",
"For $0\\le k\\le 2$ , we have $\\varepsilon _k\\ge \\overline{\\varepsilon }$ .",
"We choose the number of right-hand sides as $P(\\varepsilon _0)=9$ and $P(\\varepsilon _1)=P(\\varepsilon _2)=5$ (we recall that $P(\\varepsilon _k)=3$ for $3\\le k\\le 6$ ).",
"Considering less right-hand sides significantly alters the approximation results, while considering more right-hand sides does not significantly improve these results.",
"We consider the evaluation criteria (REF ), (REF ) and (REF ).",
"We keep ${\\cal Q}=16$ functions in the test-space $V_{\\rm n}^{\\cal Q}({\\cal D})$ .",
"For the $H^1$ -reconstruction, we choose the number of right-hand sides $R(\\varepsilon )$ such that $R(\\varepsilon _0)=R(\\varepsilon _1)=5$ and $R(\\varepsilon _2)=3$ (which satisfies $R(\\varepsilon ) \\le P(\\varepsilon )$ ).",
"The numerical results for the approximation of the homogenized matrix, the $L^2$ -approximation and the $H^1$ -approximation, are respectively collected in Table REF , Figure REF and Table REF .",
"We observe on Table REF that the approximation of the homogenized matrix provided by our approach highly improves when decreasing $\\varepsilon $ from $\\varepsilon =0.4$ to $\\varepsilon =0.2$ .",
"For $\\varepsilon \\ge 0.4$ , the homogenized matrix does not correctly describe the medium.",
"errpermat Table: Approximation of A ☆ A_\\star () in function of ε\\varepsilon (here ε/h≈43\\varepsilon /h \\approx 43).Figure REF confirms this observation when it comes to the solution itself.",
"We have seen that, for $\\varepsilon =0.4$ , $A_\\star $ and $\\overline{A}_\\varepsilon $ are significantly different.",
"The solutions $u_\\star $ and $\\overline{u}_\\varepsilon = \\overline{u}(\\overline{A}_\\varepsilon )$ are also significantly different, the latter being a much better $L^2$ -approximation of $u_\\varepsilon $ than the former or the first-order two-scale expansion.",
"For smaller values of $\\varepsilon $ , we already observe the behavior we have described in Section REF .",
"Similar comments apply to the approximation of $\\nabla u_\\varepsilon $ (see Table REF ).",
"errperL2 Figure: Approximation of u ε u_\\varepsilon in the L 2 L^2 norm () by u ☆,h u_{\\star ,h} (red), u ε,h 1,θ u_{\\varepsilon ,h}^{1,\\theta } (brown) and u ¯ ε,h P \\overline{u}^P_{\\varepsilon ,h} (black) in function of ε\\varepsilon (here ε/h≈43\\varepsilon /h \\approx 43).",
"These quantities are defined in Section .errperH1 Table: Approximation of ∇u ε \\nabla u_\\varepsilon in the L 2 L^2 norm () by C ε,h ∇u ☆,h C_{\\varepsilon ,h}\\nabla u_{\\star ,h} and C ¯ ε,h R ∇u ¯ ε,h P \\overline{C}^R_{\\varepsilon ,h}\\nabla \\overline{u}^P_{\\varepsilon ,h} in function of ε\\varepsilon (here ε/h≈43\\varepsilon /h \\approx 43).",
"See Section for a definition of these quantities."
],
[
"Results in the stationary ergodic setting",
"We consider the sets $\\lbrace \\varepsilon _k \\rbrace _{0\\le k\\le 2}$ and $\\lbrace N_k \\rbrace _{0\\le k\\le 2}$ of parameters introduced in Section REF , for which we have $\\varepsilon _k>\\overline{\\varepsilon }$ .",
"We choose the number of right-hand sides as $P(\\varepsilon _0)=9$ and $P(\\varepsilon _1)=P(\\varepsilon _2)=5$ , and fix the number of Monte Carlo realizations to $M=100$ .",
"We consider the evaluation criteria (REF ) and (REF ), with ${\\cal Q}=16$ functions in the test-space $V_{\\rm n}^{\\cal Q}({\\cal D})$ .",
"For the $H^1$ -reconstruction, the number of right-hand sides is chosen to be $R(\\varepsilon _0)=R(\\varepsilon _1)=5$ and $R(\\varepsilon _2)=3$ .",
"Note that again $R(\\varepsilon ) \\le P(\\varepsilon )$ .",
"The numerical results for the $L^2$ - and $H^1$ -approximation are respectively collected in Figure REF and Table REF .",
"Remark 13 We note that, when working with $\\varepsilon = \\varepsilon _0 = 1/2$ , we have, in view of (REF ), $N=N_0=1$ .",
"In view of (REF )–(REF ), it turns out that, in this case, there are only 16 different realizations of the field $a^{\\rm sto}$ .",
"For this value of $\\varepsilon $ , the expectation is computed by a simple enumeration of all the possible realizations.",
"For $\\varepsilon = \\varepsilon _1 = 1/4$ , there are already 65,536 realizations, and expectations are computed by empirical means over $M$ realizations.",
"On Figure REF , we observe that the solution associated with the best matrix we compute is an approximation of $\\mathbb {E}(u_\\varepsilon )$ (in the $L^2$ norm) generally more accurate than the solution associated with the exact homogenized matrix (since here $N$ is small, the approximate matrix $A_\\star ^{N,M}$ is not expected to be an accurate approximation of $A_\\star $ ).",
"Table REF shows that our surrogate defines an approximation of $\\mathbb {E}(\\nabla u_\\varepsilon )$ , the accuracy of which is comparable, and often much better, to that provided by the homogenization approach.",
"For the small values of $N$ considered here, our approach is less expensive than the classical homogenization approach.",
"errstoL2 Figure: Approximation of 𝔼(u ε )\\mathbb {E}(u_\\varepsilon ) in the L 2 L^2 norm () by u ☆,h u_{\\star ,h} (red) and u ¯ ε,h P,M \\overline{u}_{\\varepsilon ,h}^{P,M} (black) in function of NN.",
"For N≥2N \\ge 2, all expectations are approximated by an empirical mean over M=100M=100 realizations.",
"Since MM is finite, results are random.",
"We have performed the overall computation 10 times and show the corresponding 95%95\\% confidence interval (here ε/h≈27\\varepsilon /h \\approx 27).errstoH1 Table: Approximation of 𝔼(∇u ε )\\mathbb {E}(\\nabla u_\\varepsilon ) in the L 2 L^2 norm () by C ε,h N,M ∇u ☆,h C^{N,M}_{\\varepsilon ,h} \\nabla u_{\\star ,h} and C ¯ ε,h R,M ∇u ¯ ε,h P,M \\overline{C}^{R,M}_{\\varepsilon ,h} \\nabla \\overline{u}^{P,M}_{\\varepsilon ,h} in function of NN, for M=100M=100 and ε/h≈27\\varepsilon /h \\approx 27 (see Section for a definition of these quantities)."
],
[
"Acknowledgments",
"The authors would like to thank Albert Cohen (Université Pierre et Marie Curie) for stimulating and enlightning discussions about the work reported in this article, and in particular for suggesting the cost function in (REF ) in replacement of that in (REF ), for providing the perspective of an optimization upon the class of matrices $\\overline{A}$ that are considered, as detailed in Section REF , and for carefully reading a preliminary version of this manuscript.",
"The authors also acknowledge several constructive comments by the two anonymous referees, which have allowed to improve (in particular with Remarks REF and REF ) the original version of this manuscript.",
"The work of CLB, FL and SL is partially supported by EOARD under Grant FA8655-13-1-3061.",
"The work of CLB and FL is also partially supported by ONR under Grants N00014-12-1-0383 and N00014-15-1-2777."
],
[
"Preliminary results",
"Before we are in position to show Proposition REF , we first need to prove the following two preliminary lemmas, namely Lemma REF and Lemma REF .",
"Lemma 14 Under the assumptions (REF ) and (REF ), the following convergence holds: $ \\lim _{\\varepsilon \\rightarrow 0} \\Phi _\\varepsilon (A_\\star ) = 0.$ We recall that $\\Phi _\\varepsilon $ is defined by (REF ): for any $\\overline{A}$ , $\\Phi _\\varepsilon (\\overline{A}) = \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} \\Phi _\\varepsilon (\\overline{A},f) = \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} {\\left\\Vert (-\\Delta )^{-1} \\left( {\\rm div}(\\overline{A}\\nabla u_\\varepsilon (f))+f\\right)\\right\\Vert }_{{L^2(\\cal D)}}^2.$ [Proof of Lemma REF ] We use the notations and results of Section REF .",
"Let $f_\\star ^\\varepsilon \\in L^2_{\\rm n}({\\cal D})$ such that $ \\Phi _\\varepsilon (A_\\star ) = {\\left\\Vert (-\\Delta )^{-1}\\left({\\rm div}(A_\\star \\nabla u_\\varepsilon (f_\\star ^\\varepsilon ))+f_\\star ^\\varepsilon \\right)\\right\\Vert }_{{L^2(\\cal D)}}^2,$ and let $C_{\\rm P}>0$ be a Poincaré constant for ${\\cal D}$ , namely a constant such that, for any $v \\in H^1_0({\\cal D})$ , we have ${\\left\\Vert v\\right\\Vert }_{{L^2(\\cal D)}} \\le C_{\\rm P} {\\left\\Vert \\nabla v\\right\\Vert }_{{L^2(\\cal D)}}$ .",
"Using standard a priori estimates, we have, for any $f \\in {L^2(\\cal D)}$ , that ${\\left\\Vert (-\\Delta )^{-1} f \\right\\Vert }_{{L^2(\\cal D)}} \\le C_{\\rm P}^2 \\, {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}.$ Using that $\\alpha \\le A_\\varepsilon \\le \\beta $ (see (REF )), we likewise get that, for any $f \\in {L^2(\\cal D)}$ , ${\\left\\Vert \\nabla u_\\varepsilon (f)\\right\\Vert }_{{L^2(\\cal D)}} \\le \\frac{C_{\\rm P}}{\\alpha } {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}.$ We now estimate $z_\\varepsilon = (-\\Delta )^{-1}\\left( {\\rm div}(A_\\star \\nabla u_\\varepsilon (f)) \\right)$ .",
"We recall that (REF ) implies that $\\alpha \\le A_\\star \\le \\beta .$ From the variational formulation satisfied by $z_\\varepsilon $ , we obtain ${\\left\\Vert \\nabla z_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}} \\le | A_\\star | \\, {\\left\\Vert \\nabla u_\\varepsilon (f)\\right\\Vert }_{{L^2(\\cal D)}}$ , which implies, using (REF ) and (REF ), that ${\\left\\Vert \\nabla z_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}} \\le C_{\\rm P} \\, \\beta /\\alpha \\, {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}$ , hence ${\\left\\Vert (-\\Delta )^{-1}\\left( {\\rm div}(A_\\star \\nabla u_\\varepsilon (f)) \\right)\\right\\Vert }_{{L^2(\\cal D)}} \\le C_{\\rm P}^2 \\, \\frac{\\beta }{\\alpha } \\, {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}.$ Using (REF ), (REF ), (REF ) and the fact that ${\\left\\Vert f_\\star ^\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}}=1$ for all $\\varepsilon >0$ , we deduce that the sequence $\\left\\lbrace \\Phi _\\varepsilon (A_\\star ) \\right\\rbrace _{\\varepsilon >0}$ is uniformly bounded.",
"There thus exists a subsequence, that we still denote by $\\left\\lbrace \\Phi _\\varepsilon (A_\\star ) \\right\\rbrace _{\\varepsilon >0}$ , that converges in $\\mathbb {R}$ .",
"Let us denote by $\\overline{\\Phi }$ its limit.",
"We prove in the sequel that $\\overline{\\Phi } = 0$ , which implies (REF ).",
"Since $\\left\\lbrace f_\\star ^\\varepsilon \\right\\rbrace _{\\varepsilon >0}$ is uniformly bounded in ${L^2(\\cal D)}$ , there exists a subsequence, again denoted $\\left\\lbrace f_\\star ^\\varepsilon \\right\\rbrace _{\\varepsilon >0}$ , that weakly converges in ${L^2(\\cal D)}$ when $\\varepsilon \\rightarrow 0$ to some function $f_\\star ^0\\in L^2({\\cal D})$ which satisfies ${\\left\\Vert f_\\star ^0\\right\\Vert }_{{L^2(\\cal D)}} \\le 1$ .",
"From (REF ), we infer, by the triangle inequality, $ \\left( \\Phi _\\varepsilon (A_\\star ) \\right)^{1/2} \\le I^\\varepsilon _1 + I^\\varepsilon _2 + I^\\varepsilon _3,$ with $I^\\varepsilon _1&=&{\\left\\Vert (-\\Delta )^{-1} \\left( {\\rm div}( A_\\star \\nabla u_\\varepsilon (f_\\star ^\\varepsilon - f_\\star ^0))\\right)\\right\\Vert }_{{L^2(\\cal D)}},\\\\I^\\varepsilon _2&=&{\\left\\Vert (-\\Delta )^{-1}\\left({\\rm div}(A_\\star \\nabla u_\\varepsilon (f_\\star ^0))+f_\\star ^0\\right)\\right\\Vert }_{{L^2(\\cal D)}},\\\\I^\\varepsilon _3&=&{\\left\\Vert (-\\Delta )^{-1}(f_\\star ^\\varepsilon -f_\\star ^0)\\right\\Vert }_{{L^2(\\cal D)}}.$ We successively show that $I^\\varepsilon _1$ , $I^\\varepsilon _2$ and $I^\\varepsilon _3$ vanish with $\\varepsilon $ .",
"Step 1: estimation of $I_1^\\varepsilon $ .",
"Let $z_\\varepsilon = (-\\Delta )^{-1}\\left({\\rm div}\\left[ A_\\star \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0)) \\right] \\right)\\in H^1_0({\\cal D})$ .",
"We have ${\\left\\Vert \\nabla z_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}}^2=-\\int _{\\cal D}A_\\star \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0))\\cdot \\nabla z_\\varepsilon \\le \\beta {\\left\\Vert \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0))\\right\\Vert }_{{L^2(\\cal D)}}{\\left\\Vert \\nabla z_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}},$ where we have used (REF ).",
"Using the Poincaré inequality, we deduce $I_1^\\varepsilon ={\\left\\Vert z_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}}\\le C_{\\rm P} \\, \\beta \\, {\\left\\Vert \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0))\\right\\Vert }_{{L^2(\\cal D)}},$ thus, using (REF ), we get that $ \\left( I_1^\\varepsilon \\right)^2\\le C_{\\rm P}^2 \\, \\frac{\\beta ^2}{\\alpha } \\, \\int _{\\cal D} A_\\varepsilon \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0))\\cdot \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0))=C_{\\rm P}^2 \\, \\frac{\\beta ^2}{\\alpha } \\, \\int _{\\cal D}(f_\\star ^\\varepsilon -f_\\star ^0)\\;u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0).$ From (REF ), we also deduce ${\\left\\Vert \\nabla (u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0))\\right\\Vert }_{{L^2(\\cal D)}}\\le \\frac{C_{\\rm P}}{\\alpha } \\, {\\left\\Vert f_\\star ^\\varepsilon -f_\\star ^0\\right\\Vert }_{{L^2(\\cal D)}}\\le 2 \\, \\frac{C_{\\rm P}}{\\alpha }.$ Using the Poincaré inequality, we obtain that the sequence $\\left\\lbrace u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0) \\right\\rbrace _{\\varepsilon >0}$ is uniformly bounded in $H^1({\\cal D})$ .",
"There thus exists a subsequence, that we again denote $\\left\\lbrace u_\\varepsilon (f_\\star ^\\varepsilon -f_\\star ^0) \\right\\rbrace _{\\varepsilon >0}$ , which is strongly convergent in ${L^2(\\cal D)}$ .",
"The right-hand side of (REF ) is therefore the $L^2$ product of a sequence that weakly converges to 0 times a sequence that strongly converges.",
"We hence deduce from (REF ) that $\\lim _{\\varepsilon \\rightarrow 0} I^\\varepsilon _1 = 0.$ Step 2: estimation of $I_2^\\varepsilon $ .",
"Let $w_\\varepsilon = {\\rm div}(A_\\star \\nabla u_\\varepsilon (f_\\star ^0))+f_\\star ^0$ , $r_\\varepsilon = (-\\Delta )^{-1} w_\\varepsilon \\in H^1_0({\\cal D})$ and $p_\\varepsilon = (-\\Delta )^{-1} r_\\varepsilon \\in H^1_0({\\cal D})$ .",
"Using the definition of $p_\\varepsilon $ , we have $\\left( I_2^\\varepsilon \\right)^2=\\int _{\\cal D} r^2_\\varepsilon =\\int _{\\cal D} \\nabla r_\\varepsilon \\cdot \\nabla p_\\varepsilon .$ Using the definition of $r_\\varepsilon $ , we have, for any $\\phi \\in H^1_0({\\cal D})$ , $\\int _{\\cal D} \\nabla r_\\varepsilon \\cdot \\nabla \\phi =- \\int _{\\cal D} A_\\star \\nabla u_\\varepsilon (f_\\star ^0) \\cdot \\nabla \\phi + \\int _{\\cal D} f_\\star ^0 \\ \\phi .$ Using (REF ) for $\\phi \\equiv p_\\varepsilon $ , (REF ) reads as $\\left( I_2^\\varepsilon \\right)^2=- \\int _{\\cal D} A_\\star \\nabla u_\\varepsilon (f_\\star ^0) \\cdot \\nabla p_\\varepsilon + \\int _{\\cal D} f_\\star ^0 \\ p_\\varepsilon .$ In order to pass to the limit $\\varepsilon \\rightarrow 0$ in (REF ), we establish some bounds.",
"Using (REF ) with $\\phi \\equiv r_\\varepsilon $ and the bounds (REF ), we deduce ${\\left\\Vert \\nabla r_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}} \\le \\beta {\\left\\Vert \\nabla u_\\varepsilon (f_\\star ^0)\\right\\Vert }_{{L^2(\\cal D)}} + C_{\\rm P} {\\left\\Vert f_\\star ^0\\right\\Vert }_{{L^2(\\cal D)}},$ which (together with the Poincaré inequality and (REF )) implies that $r_\\varepsilon $ is uniformly bounded in $H^1({\\cal D})$ .",
"There thus exists $r_0 \\in H^1_0({\\cal D})$ such that, up to some extraction, $r_\\varepsilon $ converges to $r_0$ , weakly in $H^1({\\cal D})$ and strongly in $L^2({\\cal D})$ .",
"Passing to the limit $\\varepsilon \\rightarrow 0$ in (REF ), and using that $\\nabla u_\\varepsilon (f)$ weakly converges to $\\nabla u_\\star (f)$ , we deduce that, for any $\\phi \\in H^1_0({\\cal D})$ , $\\int _{\\cal D} \\nabla r_0 \\cdot \\nabla \\phi =- \\int _{\\cal D} A_\\star \\nabla u_\\star (f_\\star ^0) \\cdot \\nabla \\phi + \\int _{\\cal D} f_\\star ^0 \\ \\phi = 0,$ in view of the variational formulation of (REF ).",
"We hence get that $r_0 \\equiv 0$ .",
"We now turn to $p_\\varepsilon $ .",
"We have $p_\\varepsilon = (-\\Delta )^{-1} r_\\varepsilon \\in H^1_0({\\cal D})$ and $r_\\varepsilon $ converges to $r_0 = 0$ , weakly in $H^1({\\cal D})$ and strongly in $L^2({\\cal D})$ .",
"Hence $p_\\varepsilon $ converges to 0 strongly in $H^1_0({\\cal D})$ .",
"We now pass to the limit $\\varepsilon \\rightarrow 0$ in (REF ), and obtain $\\lim _{\\varepsilon \\rightarrow 0} I_2^\\varepsilon = 0.$ Step 3: estimation of $I_3^\\varepsilon $ .",
"Let $k_\\varepsilon = (-\\Delta )^{-1}(f_\\star ^\\varepsilon -f_\\star ^0)$ .",
"We have ${\\left\\Vert \\nabla k_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}}^2 = \\int _{\\cal D} (f_\\star ^\\varepsilon -f_\\star ^0) k_\\varepsilon ,$ hence, using the Poincaré inequality, ${\\left\\Vert \\nabla k_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}} \\le C_{\\rm P} {\\left\\Vert f_\\star ^\\varepsilon -f_\\star ^0\\right\\Vert }_{{L^2(\\cal D)}} \\le 2 \\, C_{\\rm P}.$ The sequence $\\lbrace k_\\varepsilon \\rbrace _{\\varepsilon >0}$ is thus uniformly bounded in $H^1({\\cal D})$ and there exists a subsequence, that we again denote $\\lbrace k_\\varepsilon \\rbrace _{\\varepsilon >0}$ , which is strongly convergent in ${L^2(\\cal D)}$ .",
"Using that $f_\\star ^\\varepsilon -f_\\star ^0$ weakly converges to 0 in ${L^2(\\cal D)}$ , we deduce from (REF ) that $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0} {\\left\\Vert \\nabla k_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}}^2 = 0$ , thus, again using the Poincaré inequality, $\\lim _{\\varepsilon \\rightarrow 0} I_3^\\varepsilon = \\lim _{\\varepsilon \\rightarrow 0} {\\left\\Vert k_\\varepsilon \\right\\Vert }_{{L^2(\\cal D)}} = 0.$ Conclusion.",
"Collecting (REF ), (REF ), (REF ) and (REF ), we obtain that $\\Phi _\\varepsilon (A_\\star )$ converges to zero as $\\varepsilon \\rightarrow 0$ .",
"We thus have shown that $\\overline{\\Phi }=0$ .",
"The limit being independent of the subsequence that we have considered, we eventually deduce that the whole sequence $\\lbrace \\Phi _\\varepsilon (A_\\star ) \\rbrace _{\\varepsilon >0}$ converges to zero.",
"This completes the proof of Lemma REF .",
"In what follows, we identify the set of indices $\\left\\lbrace (i,j), \\ \\ 1 \\le i \\le j \\le d \\right\\rbrace $ with the set of indices $\\displaystyle \\left\\lbrace m, \\ \\ 1 \\le m \\le \\frac{d(d+1)}{2} \\right\\rbrace $ .",
"Lemma 15 There exist $\\displaystyle \\frac{d \\, (d+1)}{2}$ functions $f_{\\star ,k}\\in {L^2_{\\rm n}(\\cal D)}$ and $\\displaystyle \\frac{d \\, (d+1)}{2}$ functions $\\varphi _{\\star ,k}\\in C^\\infty _0({\\cal D})$ such that the matrix $Z_\\star \\in \\mathbb {R}^{\\frac{d(d+1)}{2} \\times \\frac{d(d+1)}{2}}$ defined by $ \\forall \\, 1\\le k\\le \\frac{d \\, (d+1)}{2},\\quad \\forall \\, 1\\le i<j\\le d, \\quad {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\left[Z_\\star \\right]_{k,(i,i)} = \\int _{\\cal D} u_{\\star ,k}\\;\\partial _{ii}\\varphi _{\\star ,k},\\\\ {\\vspace{4.0pt}}\\displaystyle \\left[Z_\\star \\right]_{k,(i,j)} = 2\\int _{\\cal D} u_{\\star ,k}\\;\\partial _{ij}\\varphi _{\\star ,k},\\end{array}\\right.",
"}$ where $u_{\\star ,k} = u_\\star (f_{\\star ,k})$ is the solution to (REF ) with right-hand side $f_{\\star ,k}$ , is invertible.",
"[Proof of Lemma REF ] In the Steps 1 and 2 below, we construct $f_{\\star ,k}\\in L^2_{\\rm n}({\\cal D})$ and $\\varphi _{\\star ,k}\\in C^\\infty _0({\\cal D})$ inductively for $\\displaystyle 1 \\le k \\le d(d+1)/2$ , such that the vector $E_\\star ^k\\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ defined by $ \\forall \\, 1\\le i<j\\le d,\\qquad {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\left[E_\\star ^k\\right]_{(i,i)} = \\int _{\\cal D}u_{\\star ,k}\\;\\partial _{ii}\\varphi _{\\star ,k},\\\\ {\\vspace{3.0pt}}\\displaystyle \\left[E_\\star ^k\\right]_{(i,j)} = 2\\int _{\\cal D}u_{\\star ,k}\\;\\partial _{ij}\\varphi _{\\star ,k},\\end{array}\\right.",
"}$ does not belong to $\\text{Span}(E_\\star ^1,\\dots ,E_\\star ^{k-1})$ .",
"The vectors $E_\\star ^1$ , ..., $E_\\star ^{d(d+1)/2}$ being the rows of the matrix $Z_\\star $ , we deduce that $Z_\\star $ is invertible.",
"Step 1: Construction of $E_\\star ^1$ .",
"Choose $f_{\\star ,1} \\in L^2_{\\rm n}({\\cal D})$ and $\\varphi _{\\star ,1} \\in C^\\infty _0({\\cal D})$ such that $\\displaystyle \\int _{\\cal D} f_{\\star ,1} \\, \\varphi _{\\star ,1}\\ne 0$ , and consider $E^1_\\star \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ defined by (REF ) (where we recall that $u_{\\star ,1}$ is the solution to (REF ) with right-hand side $f_{\\star ,1}$ ).",
"Recalling that $A_\\star $ is symmetric and constant, we have $\\sum _{1\\le i\\le j\\le d} \\left[A_\\star \\right]_{i,j} \\ \\left[E_\\star ^1\\right]_{(i,j)}=-\\int _{\\cal D} A_\\star \\nabla u_{\\star ,1} \\cdot \\nabla \\varphi _{\\star ,1}=-\\int _{\\cal D} f_{\\star ,1} \\ \\varphi _{\\star ,1} \\ne 0,$ hence $E_\\star ^1 \\ne 0$ .",
"Step 2: Induction.",
"We assume that we have constructed $f_{\\star ,1}$ , ..., $f_{\\star ,k-1}$ and $\\varphi _{\\star ,1}$ , ..., $\\varphi _{\\star ,k-1}$ such that the family $E_\\star ^1$ , ..., $E_\\star ^{k-1}$ is free, for $\\displaystyle k \\le d(d+1)/2$ .",
"We now construct $f_{\\star ,k}\\in L^2_{\\rm n}({\\cal D})$ and $\\varphi _{\\star ,k}\\in C^\\infty _0({\\cal D})$ such that the vector $E_\\star ^k\\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ defined in (REF ) does not belong to $\\text{Span}(E_\\star ^1,\\dots ,E_\\star ^{k-1})$ .",
"We proceed by contradiction and assume that, for any such $f_{\\star ,k}$ and $\\varphi _{\\star ,k}$ , there exist $\\lambda _\\ell (f_{\\star ,k},\\varphi _{\\star ,k}) \\in \\mathbb {R}$ , $1\\le \\ell \\le k-1$ , such that $E_\\star ^k = \\sum _{\\ell =1}^{k-1} \\lambda _\\ell (f_{\\star ,k},\\varphi _{\\star ,k}) \\, E_\\star ^\\ell .$ For any vector $S_\\star \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ , we have $\\sum _{1\\le i\\le j\\le d} \\int _{\\cal D} \\left[ \\widehat{S}_\\star \\right]_{(i,j)} \\; \\partial _{ij} u_{\\star ,k} \\; \\varphi _{\\star ,k}=\\sum _{1\\le i\\le j\\le d} [S_\\star ]_{(i,j)} \\left[E_\\star ^k\\right]_{(i,j)}=\\sum _{\\ell =1}^{k-1} \\lambda _\\ell (f_{\\star ,k},\\varphi _{\\star ,k}) \\, S_\\star \\cdot E_\\star ^\\ell ,$ where, for any $S \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ and $E \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ , we denote $\\displaystyle S \\cdot E = \\sum _{m=1}^{d(d+1)/2} [S]_m \\, [E]_m$ , and where $\\widehat{S_\\star } \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ is defined, for any $1 \\le i < j \\le d$ , by $\\left[ \\widehat{S}_\\star \\right]_{(i,i)} = \\left[ S_\\star \\right]_{(i,i)},\\qquad \\left[ \\widehat{S}_\\star \\right]_{(i,j)} = 2 \\left[ S_\\star \\right]_{(i,j)}.$ Since $\\displaystyle k-1< d(d+1)/2$ , there exists $S_\\star \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ , $S_\\star \\ne 0$ , such that $S_\\star \\cdot E_\\star ^\\ell =0$ for all $1\\le \\ell \\le k-1$ , and thus $\\forall \\varphi _{\\star ,k} \\in C^\\infty _0({\\cal D}), \\qquad \\sum _{1\\le i\\le j\\le d} \\int _{\\cal D} \\left[ \\widehat{S}_\\star \\right]_{(i,j)} \\; \\partial _{ij} u_{\\star ,k} \\; \\varphi _{\\star ,k}=0.$ Since $S_\\star $ (and thus $\\widehat{S}_\\star $ ) only depends on $E_\\star ^1$ , ..., $E_\\star ^{k-1}$ and not on $\\varphi _{\\star ,k}$ , this implies $\\sum _{1\\le i\\le j\\le d} \\left[ \\widehat{S}_\\star \\right]_{(i,j)} \\; \\partial _{ij} u_{\\star ,k}=0 \\ \\ \\text{in the sense of distributions,}$ thus $0=-\\sum _{1\\le i\\le j\\le d} \\left[ \\widehat{S}_\\star \\right]_{(i,j)} \\; \\partial _{ij} {\\rm div}\\left[ A_\\star \\nabla u_{\\star ,k} \\right]=\\sum _{1\\le i\\le j\\le d} \\left[ \\widehat{S}_\\star \\right]_{(i,j)} \\; \\partial _{ij}f_{\\star ,k},$ for any $f_{\\star ,k}\\in L^2_{\\rm n}({\\cal D})$ .",
"Since $\\displaystyle \\left[ \\widehat{S}_\\star \\right]_{(i,j)}$ does not depend on $f_{\\star ,k}$ , this shows that $\\widehat{S}_\\star $ , and thus $S_\\star $ , vanishes.",
"We reach a contradiction.",
"We thus obtain the existence of $f_{\\star ,k}\\in L^2_{\\rm n}({\\cal D})$ and $\\varphi _{\\star ,k}\\in C^\\infty _0({\\cal D})$ such that the vectors $E_\\star ^1$ , ..., $E_\\star ^{k-1}$ , $E_\\star ^k$ form a free family."
],
[
"Proof of Proposition ",
"We can now perform the proof of Proposition REF .",
"The convergence (REF ) proved in Lemma REF readily shows (REF ).",
"We are left with showing (REF ).",
"Using the functions $f_{\\star ,k} \\in {L^2_{\\rm n}(\\cal D)}$ and $\\varphi _{\\star ,k} \\in C^\\infty _0({\\cal D})$ defined by Lemma REF , we introduce the matrix $Z_\\varepsilon \\in \\mathbb {R}^{\\frac{d(d+1)}{2}\\times \\frac{d(d+1)}{2}}$ defined by $ \\forall \\, 1\\le k\\le \\frac{d(d+1)}{2},\\quad \\forall \\, 1\\le i<j\\le d,\\quad {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\left[Z_\\varepsilon \\right]_{k,(i,i)} = \\int _{\\cal D} u_{\\varepsilon ,k}\\;\\partial _{ii}\\varphi _{\\star ,k},\\\\ {\\vspace{4.0pt}}\\displaystyle \\left[Z_\\varepsilon \\right]_{k,(i,j)} = 2\\int _{\\cal D} u_{\\varepsilon ,k}\\;\\partial _{ij}\\varphi _{\\star ,k},\\end{array}\\right.",
"}$ where $u_{\\varepsilon ,k} = u_\\varepsilon (f_{\\star ,k})$ is the solution to (REF ) with right-hand side $f_{\\star ,k}$ .",
"Note that, for the second index of $Z_\\varepsilon $ , we have again identified the sets $\\left\\lbrace (i,j), \\ 1 \\le i \\le j \\le d \\right\\rbrace $ and $\\displaystyle \\left\\lbrace m, \\ 1 \\le m \\le \\frac{d(d+1)}{2} \\right\\rbrace $ .",
"Since $u_{\\varepsilon ,k}$ converges to $u_{\\star ,k}$ in $L^2({\\cal D})$ , the matrix $Z_\\varepsilon $ converges to the matrix $Z_\\star $ defined by (REF ) when $\\varepsilon $ goes to zero.",
"We have proved in Lemma REF that the matrix $Z_\\star $ is invertible.",
"This implies that the matrix $Z_\\varepsilon $ is invertible for $\\varepsilon $ sufficiently small, and that $Z_\\varepsilon ^{-1}$ is bounded independently of $\\varepsilon $ .",
"We now introduce the vectors $\\overline{V}^\\flat _\\varepsilon $ and $V_\\star $ in $\\mathbb {R}^{\\frac{d(d+1)}{2}}$ such that $\\forall \\, 1\\le i\\le j\\le d,\\qquad \\left[\\overline{V}^\\flat _\\varepsilon \\right]_{(i,j)} = \\left[\\overline{A}^\\flat _\\varepsilon \\right]_{i,j},\\qquad \\left[V_\\star \\right]_{(i,j)} = \\left[A_\\star \\right]_{i,j},$ where we recall that $\\overline{A}^\\flat _\\varepsilon $ is a quasi-minimizing sequence of the functional (REF ) (see (REF )).",
"It can easily be seen that, for any $\\overline{A} \\in {\\cal S}$ , denoting $\\overline{V} \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ the vector such that $[\\overline{V}]_{(i,j)} = \\overline{A}_{i,j}$ for any $1\\le i\\le j\\le d$ , the following holds: for any $1\\le k \\le d(d+1)/2$ , $ \\left[Z_\\varepsilon \\ \\overline{V}\\right]_k=\\int _{\\cal D}u_{\\varepsilon ,k}\\,{\\rm div}(\\overline{A}\\nabla \\varphi _{\\star ,k})=\\int _{\\cal D} {\\rm div}(\\overline{A}\\nabla u_{\\varepsilon ,k}) \\ \\varphi _{\\star ,k}=- \\int _{\\cal D} (-\\Delta )^{-1}\\left[{\\rm div}(\\overline{A}\\nabla u_{\\varepsilon ,k})\\right] \\, \\Delta \\varphi _{\\star ,k},$ where $Z_\\varepsilon \\ \\overline{V} \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ is the product of the matrix $Z_\\varepsilon \\in \\mathbb {R}^{\\frac{d(d+1)}{2}\\times \\frac{d(d+1)}{2}}$ by the vector $\\overline{V} \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ : for any $1\\le k \\le d(d+1)/2$ , $\\displaystyle \\left[Z_\\varepsilon \\ \\overline{V}\\right]_k = \\sum _{1 \\le i \\le j \\le d} \\left[Z_\\varepsilon \\right]_{k,(i,j)} [\\overline{V}]_{(i,j)}$ .",
"Now, for any $f\\in L^2_{\\rm n}({\\cal D})$ , we observe that $ \\begin{alignedat}{1}{\\left\\Vert (-\\Delta )^{-1} \\left[ {\\rm div}\\left(\\overline{A}^\\flat _\\varepsilon \\nabla u_\\varepsilon (f) \\right)-{\\rm div}\\Big (A_\\star \\nabla u_\\varepsilon (f) \\Big ) \\right]\\right\\Vert }_{{L^2(\\cal D)}}^2&\\le 2\\left(\\Phi _\\varepsilon (\\overline{A}^\\flat _\\varepsilon )+\\Phi _\\varepsilon (A_\\star )\\right) \\\\&\\le 2\\left(I_\\varepsilon +\\varepsilon +\\Phi _\\varepsilon (A_\\star )\\right) \\\\&\\le 2\\left(2\\Phi _\\varepsilon (A_\\star )+\\varepsilon \\right).\\end{alignedat}$ Hence, applying this to $f \\equiv f_{\\star ,k}$ , and owing to Lemma REF , ${\\left\\Vert (-\\Delta )^{-1}\\left[ {\\rm div}\\left( \\overline{A}^\\flat _\\varepsilon \\nabla u_{\\varepsilon ,k} \\right) \\right] - (-\\Delta )^{-1} \\Big [ {\\rm div}\\Big ( A_\\star \\nabla u_{\\varepsilon ,k} \\Big ) \\Big ]\\right\\Vert }_{{L^2(\\cal D)}}$ vanishes with $\\varepsilon $ , for any $1\\le k \\le d(d+1)/2$ .",
"We next deduce from (REF ) that $Z_\\varepsilon (\\overline{V}^\\flat _\\varepsilon -V_\\star )$ vanishes as $\\varepsilon \\rightarrow 0$ .",
"Since $Z_\\varepsilon $ is invertible when $\\varepsilon $ is sufficiently small (with $Z_\\varepsilon ^{-1}$ bounded independently of $\\varepsilon $ ), we obtain that $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0} \\overline{V}^\\flat _\\varepsilon = V_\\star $ , which is exactly the claimed convergence (REF ).",
"This concludes the proof of Proposition REF .",
"Remark 16 Since the above proof uses (REF ) precisely for the functions $f_{\\star ,k}$ , $1\\le k\\le d(d+1)/2$ (and not for all functions $f\\in L^2_{\\rm n}({\\cal D})$ ), we observe that, in the $\\inf \\max $ formulation introduced in Remark REF , we have $\\overline{A}^{{\\rm max},\\flat }_\\varepsilon \\rightarrow A_\\star $ when $\\varepsilon \\rightarrow 0$ .",
"Remark 17 We recall that our approach consists in considering the problem (REF ), that is $I_\\varepsilon = \\inf _{\\overline{A}\\in {\\cal S}} \\ \\Phi _\\varepsilon (\\overline{A}),$ where $\\Phi _\\varepsilon $ is defined by (REF ): for any $\\overline{A}$ , $\\Phi _\\varepsilon (\\overline{A}) = \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} \\Phi _\\varepsilon (\\overline{A},f) = \\sup _{f\\in {L^2_{\\rm n}(\\cal D)}} {\\left\\Vert (-\\Delta )^{-1} \\left( {\\rm div}(\\overline{A}\\nabla u_\\varepsilon (f))+f\\right)\\right\\Vert }_{{L^2(\\cal D)}}^2.$ We show here that, when $\\varepsilon $ is sufficiently small, the minimum $I_\\varepsilon $ is attained.",
"Consider indeed a minimizing sequence $\\overline{A}_\\varepsilon ^\\eta $ , satisfying, for any $\\eta >0$ , $I_\\varepsilon \\le \\Phi _\\varepsilon (\\overline{A}_\\varepsilon ^\\eta ) \\le I_\\varepsilon + \\eta .$ Similarly to (REF ), we observe that, for any $f\\in L^2_{\\rm n}({\\cal D})$ , $\\begin{alignedat}{1}{\\left\\Vert (-\\Delta )^{-1} \\left[ {\\rm div}\\left(\\overline{A}^\\eta _\\varepsilon \\nabla u_\\varepsilon (f) \\right)-{\\rm div}\\Big (A_\\star \\nabla u_\\varepsilon (f) \\Big ) \\right]\\right\\Vert }_{{L^2(\\cal D)}}^2&\\le 2\\left(\\Phi _\\varepsilon (\\overline{A}^\\eta _\\varepsilon )+\\Phi _\\varepsilon (A_\\star )\\right) \\\\&\\le 2\\left(I_\\varepsilon +\\eta +\\Phi _\\varepsilon (A_\\star )\\right) \\\\&\\le 2\\left(2\\Phi _\\varepsilon (A_\\star )+\\eta \\right).\\end{alignedat}$ Using (REF ), we have $\\Big | Z_\\varepsilon \\, \\left(\\overline{V}^\\eta _\\varepsilon - V_\\star \\right) \\Big |\\le C \\sup _{f \\in {L^2_{\\rm n}(\\cal D)}} {\\left\\Vert (-\\Delta )^{-1} \\left[ {\\rm div}\\left(\\overline{A}^\\eta _\\varepsilon \\nabla u_\\varepsilon (f) \\right)-{\\rm div}\\Big (A_\\star \\nabla u_\\varepsilon (f) \\Big ) \\right]\\right\\Vert }_{{L^2(\\cal D)}}$ where $C$ is a constant independent of $\\varepsilon $ and $\\eta $ and where the vector $\\overline{V}^\\eta _\\varepsilon \\in \\mathbb {R}^{\\frac{d(d+1)}{2}}$ is defined by $\\displaystyle \\left[\\overline{V}^\\eta _\\varepsilon \\right]_{(i,j)} = \\left[\\overline{A}^\\eta _\\varepsilon \\right]_{i,j}$ for any $1\\le i\\le j\\le d$ .",
"When $\\varepsilon $ is sufficiently small, the matrix $Z_\\varepsilon $ is invertible with $Z_\\varepsilon ^{-1}$ bounded independently of $\\varepsilon $ .",
"We thus deduce from the two above estimates that $\\Big | \\overline{V}^\\eta _\\varepsilon - V_\\star \\Big |^2 \\le C \\left(\\Phi _\\varepsilon (A_\\star )+\\eta \\right)$ for some $C$ independent of $\\varepsilon $ and $\\eta $ .",
"The vector $\\overline{V}^\\eta _\\varepsilon $ (resp.",
"$V_\\star $ ) is the representation (as a vector in $\\mathbb {R}^{\\frac{d(d+1)}{2}}$ ) of the symmetric matrix $\\overline{A}_\\varepsilon ^\\eta \\in \\mathbb {R}^{d \\times d}$ (resp.",
"$A_\\star $ ).",
"We hence equivalently write that $\\Big | \\overline{A}^\\eta _\\varepsilon - A_\\star \\Big |^2 \\le C \\left(\\Phi _\\varepsilon (A_\\star )+\\eta \\right).$ This shows that the sequence $\\overline{A}^\\eta _\\varepsilon $ is bounded independently of $\\eta $ .",
"Up to the extraction of a subsequence (that we still denote $\\eta $ for the sake of simplicity), it thus converges to some symmetric matrix $\\overline{A}^0_\\varepsilon $ when $\\eta \\rightarrow 0$ .",
"Since $A_\\star $ is positive definite and since $\\displaystyle \\lim _{\\varepsilon \\rightarrow 0} \\Phi _\\varepsilon (A_\\star ) = 0$ , we get that $\\overline{A}_\\varepsilon ^0$ is also positive-definite.",
"Passing to the limit $\\eta \\rightarrow 0$ in (REF ), and temporarily assuming that $\\Phi _\\varepsilon $ is continuous, we get that $I_\\varepsilon = \\Phi _\\varepsilon (\\overline{A}_\\varepsilon ^0)$ .",
"This concludes the proof that the minimum $I_\\varepsilon $ is indeed attained when $\\varepsilon $ is sufficiently small.",
"We are left with showing the continuity of $\\overline{A} \\mapsto \\Phi _\\varepsilon (\\overline{A})$ .",
"For any two matrices $\\overline{A}_1$ and $\\overline{A}_2$ and any $f \\in {L^2(\\cal D)}$ , we compute that $\\Phi _\\varepsilon (\\overline{A}_1,f) - \\Phi _\\varepsilon (\\overline{A}_2,f)={\\left\\Vert (-\\Delta )^{-1} \\left[ {\\rm div}\\left( (\\overline{A}_1-\\overline{A}_2) \\nabla u_\\varepsilon (f) \\right) \\right]\\right\\Vert }_{{L^2(\\cal D)}}^2\\\\+ 2 \\left\\langle (-\\Delta )^{-1} \\left[ {\\rm div}\\left( (\\overline{A}_1-\\overline{A}_2) \\nabla u_\\varepsilon (f) \\right) \\right], (-\\Delta )^{-1} \\left[ {\\rm div}\\left( \\overline{A}_2 \\nabla u_\\varepsilon (f) \\right) + f \\right] \\right\\rangle _{L^2({\\cal D})},$ hence $\\left| \\Phi _\\varepsilon (\\overline{A}_1,f) - \\Phi _\\varepsilon (\\overline{A}_2,f) \\right|\\le C \\, \\left| \\overline{A}_1-\\overline{A}_2 \\right|^2 \\ {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}^2+C \\, \\left| \\overline{A}_1-\\overline{A}_2 \\right| \\ {\\left\\Vert f\\right\\Vert }_{{L^2(\\cal D)}}^2,$ where $C$ is independent of $f$ and $\\overline{A}_1$ .",
"Taking the supremum over $f \\in {L^2_{\\rm n}(\\cal D)}$ , we thus deduce that $\\left| \\Phi _\\varepsilon (\\overline{A}_1) - \\Phi _\\varepsilon (\\overline{A}_2) \\right|\\le C \\, \\left| \\overline{A}_1-\\overline{A}_2 \\right|^2+C \\, \\left| \\overline{A}_1-\\overline{A}_2 \\right|,$ which implies that $\\displaystyle \\lim _{\\overline{A}_1 \\rightarrow \\overline{A}_2} \\Phi _\\varepsilon (\\overline{A}_1) = \\Phi _\\varepsilon (\\overline{A}_2)$ , and thus the continuity of $\\Phi _\\varepsilon $ ."
],
[
"Details on the algorithm to solve the discrete problem (",
"Let $\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)$ be given by (REF ).",
"Using the fact that $\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)$ is quadratic with respect to $c \\in \\mathbb {R}^P$ , one can easily observe that $ \\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)=c^T\\,G_{\\varepsilon ,h}^M(\\overline{A})\\,c,$ where $G_{\\varepsilon ,h}^M(\\overline{A})$ is the $P\\times P$ matrix defined, for any $1\\le p,q\\le P$ , by $ \\left[G_{\\varepsilon ,h}^M(\\overline{A})\\right]_{p,q} = \\frac{1}{2} \\sum _{1\\le i,j,k,l\\le d} \\left[{\\cal K}_{\\varepsilon ,h}^M\\right]_{i,j,k,l,p,q} \\ \\overline{A}_{i,j} \\ \\overline{A}_{k,l}\\\\-\\sum _{1\\le i,j\\le d} \\left( \\left[\\mathbb {K}_{\\varepsilon ,h}^M \\right]_{i,j,p,q} + \\left[\\mathbb {K}_{\\varepsilon ,h}^M\\right]_{i,j,q,p} \\right) \\overline{A}_{i,j} + \\left[K_h\\right]_{p,q},$ where ${\\cal K}_{\\varepsilon ,h}^M$ , $\\mathbb {K}_{\\varepsilon ,h}^M$ and $K_h$ are defined by (REF ), () and (), respectively.",
"Using the fact that $\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c)$ is also quadratic with respect to $\\overline{A}$ , we have that $\\Phi ^{P,M}_{\\varepsilon ,h}(\\overline{A},c) = \\frac{1}{2} \\sum _{1\\le i,j,k,l\\le d} \\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j,k,l} \\ \\overline{A}_{i,j} \\ \\overline{A}_{k,l}-\\sum _{1\\le i,j\\le d} \\left[B_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j} \\ \\overline{A}_{i,j} + b_h^P(c),$ where $\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)$ is the $d\\times d\\times d\\times d$ fourth-order tensor defined by $\\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j,k,l}=\\sum _{1\\le p,q\\le P} \\left[{\\cal K}_{\\varepsilon ,h}^M\\right]_{i,j,k,l,p,q} \\ c_p \\ c_q,$ $B_{\\varepsilon ,h}^{P,M}(c)$ is the $d\\times d$ matrix defined by $\\left[B_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j}=\\sum _{1\\le p,q\\le P} \\left(\\left[\\mathbb {K}_{\\varepsilon ,h}^M\\right]_{i,j,p,q} + \\left[\\mathbb {K}_{\\varepsilon ,h}^M \\right]_{i,j,q,p} \\right) c_p \\ c_q,$ and $b_h^P(c) = \\sum _{1\\le p,q\\le P} \\left[K_h\\right]_{p,q} \\ c_p \\ c_q,$ where ${\\cal K}_{\\varepsilon ,h}^M$ , $\\mathbb {K}_{\\varepsilon ,h}^M$ and $K_h$ are defined by (REF ), () and (), respectively.",
"We remark, in light of the expressions (REF ) and (), that $\\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j,k,l} = \\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{k,l,i,j},\\qquad \\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j,k,l} = \\left[\\mathbb {B}_{\\varepsilon ,h}^{P,M}(c)\\right]_{j,i,k,l},$ and $\\left[B_{\\varepsilon ,h}^{P,M}(c)\\right]_{i,j} = \\left[B_{\\varepsilon ,h}^{P,M}(c)\\right]_{j,i}$ ."
]
] | 1612.05807 |
[
[
"A monoidal representation for linearized gravity"
],
[
"Abstract We propose an alternative representation for linear quantum gravity.",
"It is based on the use of a structure that bears some resemblance to the Abelian loop representation used in electromagnetism but with the difference that space of extended object on which waves functions take values have a structure of commutative monoid instead of Abelian group.",
"The generator of duality of the theory is realized in this representation and a geometrical interpretation is discussed."
],
[
"Introduction",
"It is well known that the introduction of the loop representation has opened a new avenue for the quantization of gauge theories such as electromagnetism [1], [2], Yang-Mills [3] and General Relativity [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].",
"It allows to indroduce an elegant way of solving the constraints of the theories under study as well as, to obtain certain geometrical information about the space of extended objects on which the wave functionals take values.",
"An example this is given by the Abelian loop representation of the free Maxwell theory (MT).",
"It is well known that, in this case, the loop representation allows to solve the Gauss constraint immediately [1], [2].",
"Furthermore, despite as electromagnetism is not a topological theory, it has associated a topological invariant, namely the generator of duality, which realization in terms of loop-variables leads to a knot invariant, more precisely, the Gauss linking number [2].",
"Within the same spirit, in the context of Abelian field theories, the loop representation has been implemented to quantize the massless Fierz-Pauli theory (FPT) [17], [18] with the purpose of explore some geometrical aspects of it.",
"For example, in reference [18] a representation in terms of skein of loops is obtained.",
"There it was shown that, although the canonical algebra was fulfilled, a geometrical interpretation of the physical quantities in term of loops was unclear.",
"Further, the Abelian loop representation is not naturally adapted to the theory in the sense that tensor indexes of the fields play mixed roles labeling both space coordinates and loops.",
"In this paper, we consider a monoidal representation as an alternative to deal with the quantization of symmetric $(0,2)$ Fierz-Pauli tensor fields.",
"As we shall see, it allows to avoid the problem of the mixed indexes because it seems to be well-suited to linearized gravity.",
"As a consecuence of that, the wave functionals take values on a space generated by just one kind of extended object instead of on a list of three tangled Abelian loops as shown in reference [18].",
"The organization of the paper is as follows.",
"Section is devoted to the study of the Abelian loop representation and the subsequent quantization of both Maxwell and Fierz-Pauli models.",
"In Sec.",
", the monoidal representation is introduced and its application in quantizing the FPT is studied.",
"Finally, we end the paper with some conclusions in Sec.",
"."
],
[
"Abelian loop representation for Maxwell and Fierz-Pauli models",
"The goal of this section is to state the key features for the application of the Abelian loop representation in the quantization of the MT and the FPT.",
"We begin by recalling that the Abelian path-space can be described as the set of certain equivalence classes of curves $\\gamma $ in a manifold, which we take as $R^{3}$ .",
"The equivalence relation is given in terms of the so-called form factor $T^{a}(\\vec{x},\\gamma )$ of the curves $T^{a}(\\vec{x},\\gamma )=\\int \\limits _{\\gamma }dz^{a}\\delta ^{3}(\\vec{z}-\\vec{x}),$ as follows: $\\gamma $ and $\\gamma ^{\\prime }$ are said to be equivalent (i.e.",
"they represent the same path), if their form factor coincide.",
"Closed curves give raise to a subspace of path-space: the loop-space.",
"It can be seen that the standard composition of curves translates into a composition of paths that endows path-space with an Abelian group structure.",
"Now, we can define the path derivatives $\\delta _{a}(\\vec{x})$ by $u^{a}\\delta _{a}(\\vec{x})\\Psi [\\gamma ]&:=&\\Psi [\\gamma \\circ u_{\\vec{x}}]-\\Psi [\\gamma ]$ where $\\circ $ denote the path-space product.",
"The derivative $\\delta _{a}(\\vec{x})$ measures the change in the path-dependent wave functional when an infinitesimal path $\\delta u$ is attached to its argument $\\gamma $ at the point $\\vec{x}$ .",
"It is understood that these changes are considered up to first-order in the infinitesimal vector $u$ associated with the small path generated by it.",
"As an example of how these operators work, we calculate the path-derivative of the form factor.",
"One has $T^{a}[\\vec{x},\\gamma \\circ u_{\\vec{y}}]&=&\\int \\limits _{\\gamma \\circ u_{\\vec{y}}}dz^{a}\\delta ^{3}(\\vec{x}-\\vec{z})\\nonumber \\\\&=&T^{a}[\\vec{x},\\gamma ]+u^{b}\\delta _{b}^{a}\\delta ^{3}(\\vec{x}-\\vec{y})$ Hence, $\\delta _{a}T^{b}[\\vec{x},\\gamma ]=\\delta ^{b}_{a}\\delta ^{3}(\\vec{x}-\\vec{y})$ Now, a geometric representation of a vector field theory arises when the canonical fields are realized as operators acting onto path-dependent wave functionals $\\Psi [\\gamma ]$ .",
"In the MT the electric and vector potential field operators can be represented in the path-space such as they fulfill the canonical algebra $[\\hat{A}_{a}(\\vec{x},\\hat{E}^{b}(\\vec{y}))]=i\\delta ^{b}_{a}\\delta ^{3}(\\vec{x}-\\vec{y})$ In fact, the realization $\\hat{E}^{a}|\\Psi >&\\rightarrow & T^{a}[x,\\gamma ]\\Psi [\\gamma ]\\nonumber \\\\\\hat{A}_{a}|\\Psi >&\\rightarrow & i\\delta _{a}(\\vec{x})\\Psi [\\gamma ]$ together with equation (REF ), fulfills the algebra as can be easily verified.",
"In the other hand, the Gauss constraint $\\partial _{a}\\hat{E}^{a}=0$ is identically satisfied if we deal with closed paths, i.e, if we consider that wave functionals take values on the loop-space.",
"The program depicted above can be implemented in the Fierz-Pauli model as in reference [18].",
"In fact, it is immediate to realize that the canonical algebra $[\\hat{h}_{ab}(\\vec{x}),\\hat{p}^{cd}(\\vec{y})]=\\frac{i}{2}(\\delta ^{c}_{a}\\delta ^{d}_{b}+\\delta ^{c}_{b}\\delta ^{d}_{a})\\delta ^{3}(\\vec{x}-\\vec{y}).$ can be fulfilled with the follow realization $\\hat{h}_{ab}(\\vec{x})|&\\rightarrow &\\frac{i}{\\sqrt{2}}\\delta _{a}(\\vec{x},\\gamma _{b})-\\delta _{b}(\\vec{x},\\gamma _{a})\\nonumber \\\\\\hat{p}^{ab}(\\vec{x},\\gamma )&\\rightarrow &\\frac{1}{2\\sqrt{2}}(T^{a}(\\vec{x},\\gamma _{b})+T^{b}(\\vec{x},\\gamma _{a})),$ over wave functionals depending on lists of three closed paths, labeled with the same indexes used to denote spatial components.",
"Note that, this mixing of space indexes with “color” indexes is crucial for the realization of the algebra.",
"Unfortanely, this representation for the FPT does not satisfy automatically the constraints of the theory given by $\\partial _{a}\\hat{p}^{ab}|\\Psi >&=0&\\nonumber \\\\(\\partial _{a}\\partial _{b}\\hat{h}_{ab}-\\nabla ^{2}\\hat{h})|\\Psi >&=&0.$ However, as it is well-known, the linearized constraints state that only the transverse and traceless (TT) components of the Fierz-Pauli variables are observables in the sense of Dirac.",
"In other words, the TT components of the Fierz-Pauli fields fulfills the constraint automatically.",
"For this reason, we must compute the TT part for the canonical pair $(h_{ab},p^{ab})$ making use, for example, of the TT projector $P_{ab/cd}=P_{ac}P_{db}-\\frac{1}{2}P_{ab}P_{cd}$ defined in [18], [19], [20] (here $P_{ab}$ stands for the usual transverse projector of electromagentism ), in order to obtain the quantum observables of the theory, i.e, $\\hat{h}^{TT}_{ab}&=&P_{ab/cd}\\hat{h}_{cd}\\nonumber \\\\\\hat{p}^{abTT}&=&P_{ab/cd}\\hat{p}^{cd}.$ It is worth mentioning that considering the quantities defined in equation (REF ) as our basics quantum operators, the constraints (REF ) are “strong” equalities (quantization in the reduced phase space).",
"Before concluding this section, a comparison between Abelian loop representation for Maxwell and Fierz-Pauli models is mandatory.",
"On one hand, it is plausible to interpret Abelian loops in the MT as quantum Faraday's lines which are closed in absent of sources.",
"Note that this fact allows to solve automatically the Gauss constraint.",
"On the other hand, the quantum operator associated to the electric field can be interpreted as a measure of the density of electric flux and the loop-dependent wave functional depends only on those features of the loop that are captured by the form-factor, i.e.",
"$\\Psi [\\gamma ] = \\Psi [T^{a}[\\vec{x},\\gamma ]$ .",
"However, in the Fierz-Pauli case such an interpretations are unclear.",
"In the first place, we are not able to solve immediately the first class constraints by the very choice of closed paths.",
"As was explained above, the constraints are solved as long as we consider the quantization on the reduced phase space.",
"Secondly, it is not clear if $p^{abTT}$ can be interpreted as the flux density of gravitational Faraday's lines because it is a function of non-local terms involving the form factor of the loops.",
"In fact, a straighforward calculation reveals that $\\hat{p}^{abTT}\\Psi [\\gamma ]&=&\\frac{1}{\\sqrt{2}}(T^{a}[\\vec{x},\\gamma _{b}]+T^{b}[\\vec{x},\\gamma _{a}]-\\delta _{ab}T^{c}[\\vec{x},\\gamma _{c}]\\nonumber \\\\&&-\\nabla ^{-2}\\partial _{c}\\partial _{b}T^{a}[\\vec{x},\\gamma _{c}]-\\nabla ^{-2}\\partial _{c}\\partial _{a}T^{b}[\\vec{x},\\gamma _{c}]+\\nabla ^{-2}\\partial _{a}\\partial _{b}T^{c}[\\vec{x},\\gamma _{c}])\\equiv \\mathcal {T}^{ab}[\\gamma ]\\Psi [\\gamma ],$ where $\\mathcal {T}^{ab}[\\gamma ]$ is a function of the form factors.",
"For the reasons listed above, the only we can say about the wave functional is that it take values on a certain space of skein of colored loops, i.e, $\\Psi [\\gamma ]=\\Psi [\\mathcal {T}^{ab}[\\gamma ]]$ In the next section we shall develope an alternative for the loop representation that avoid the problem of the space formed by skein of different loops: the monoidal representation."
],
[
"Monoidal representation",
"In this section we proceed to construct the monoidal representation for the FPT.",
"As a starting point, let us then consider a set of parametrized curves $\\gamma $ on $R^{3}$ given by $\\vec{z}_{\\gamma }=\\vec{z}(l)$ , with $l$ the arc length parameter, and define $I^{ab}[\\vec{x},\\gamma ]:=\\int \\limits _{\\gamma }dlu^{a}_{T_{\\gamma }}u^{b}_{T_{\\gamma }}\\delta ^{3}(\\vec{z}_{\\gamma }-\\vec{x}),$ with $\\hat{u}_{T_{\\gamma }}:=d\\vec{z}_{\\gamma }/dl$ the tangent vector to $\\gamma $ and $a,b=1,2,3$ .",
"It is easy to check that (REF ) is symmetric and independent of the curve orientation.",
"It is worth mentioning that for any arbitrary parameter $\\tau $ we have $I^{ab}[\\vec{x},\\Gamma ]=\\int \\limits _{\\tau _{1}}^{\\tau _{2}}d\\tau \\frac{\\dot{z}^{a}\\dot{z}^{b}}{|\\ \\dot{\\vec{z}}\\ |}\\delta ^{3}(\\vec{x}-\\vec{z}(\\tau ))$ where $\\dot{z}^{a}:=dz^{a}/d\\tau $ and $\\tau _{1}<\\tau _{2}$ independent of the orientation of $\\gamma $ .",
"Let $\\mathcal {M}$ be the space which elements $\\Gamma $ are the union of disjoint curves, i.e, $\\Gamma =\\gamma _{1}\\cup ....\\cup \\gamma _{n}$ with arbitrary $n$ .",
"From the definition (REF ), it follows that $I^{ab}[\\vec{x},\\Gamma ]&=&I^{ab}[\\vec{x},\\gamma _{1}\\cup ...\\cup \\gamma _{n}]\\nonumber \\\\&=&\\sum \\limits _{i=1}^{n}I^{ab}[\\vec{x},\\gamma _{i}].$ We shall say that two curves $\\Gamma $ and $\\Gamma ^{\\prime }$ represent the same element or path if $I^{ab}[\\vec{x},\\Gamma ]=I^{ab}[\\vec{x},\\Gamma ^{\\prime }]$ .",
"The space formed by such a paths will be denoted by $\\mathfrak {M}$ .",
"For elements in $\\mathfrak {M}$ we shall define the product $\\circ $ between two paths $\\Gamma _{1}$ and $\\Gamma _{2}$ as follows $I^{ab}[\\vec{x},\\Gamma _{1}\\circ \\Gamma _{2}]&=&I^{ab}[\\vec{x},\\Gamma _{2}\\circ \\Gamma _{1}]\\nonumber \\\\&:=&I^{ab}[\\vec{x},\\Gamma _{1}\\cup \\Gamma _{2}]\\nonumber \\\\&=&I^{ab}[\\vec{x},\\Gamma _{1}]+I^{ab}[\\vec{x},\\Gamma _{1}].$ Note, from the above definition, that this product is commutative.",
"Furtheremore, for an element $\\Gamma _{e}\\in \\mathfrak {M}$ such as $I^{ab}[\\vec{x},\\Gamma _{e}]=0$ (the null path) occurs that $I^{ab}[\\vec{x},\\Gamma \\circ \\Gamma _{e}]=I^{ab}[\\vec{x},\\Gamma ].$ In this manner, $\\Gamma _{e}$ is the neutral element for the multiplication defined in (REF ).",
"In summary, the elements of $\\mathfrak {M}$ along with relation (REF ) form a commutative monoid, i.e, a semigroup with an identity.",
"Let us now consider square integrable functionals (at least formally) $\\Psi :\\mathfrak {M}\\rightarrow C$ .",
"We define the path derivative operator $\\delta _{cd}(\\vec{y},\\hat{v})$ as $v^{c}v^{d}\\delta _{cd}(\\vec{y},\\hat{v})\\Psi :=\\lim \\limits _{L\\rightarrow 0}\\frac{\\Psi [\\Gamma \\circ \\Gamma [\\vec{y},\\hat{v}]]-\\Psi [\\Gamma ]}{L}.$ As can be seen, this derivative computes the change of $\\Psi $ when an infinitesimal path $\\Gamma [\\vec{y},\\hat{v}]$ starting at $\\vec{y}$ in the $\\hat{v}$ direction, is appended to the path $\\Gamma $ .",
"Now, with the above definition, is straightforward to prove that the path derivative for $I^{ab}[\\vec{x},\\Gamma ]$ is given by $\\delta _{cd}(\\vec{y},\\hat{v})I^{ab}[\\vec{x},\\Gamma ]=\\frac{1}{2}(\\delta ^{a}_{c}\\delta ^{b}_{d}+\\delta ^{a}_{d}\\delta ^{b}_{c})\\delta ^{3}(\\vec{y}-\\vec{x})$ With these tools at hand we are ready to obtain a monoidal representation for linearized gravity.",
"In order to carry this out, let us consider the realization $\\hat{h}_{ab}\\Psi [\\Gamma ]&\\rightarrow & i\\delta _{ab}(\\vec{x},\\hat{v})\\Psi [\\Gamma ]\\nonumber \\\\\\hat{p}^{ab}\\Psi [\\Gamma ]&\\rightarrow & I^{ab}[\\vec{x},\\Gamma ]\\Psi [\\Gamma ],$ which fulfills the canonical algebra (see Eq.",
"(REF )).",
"Note that in the above realization, we have chosen the following “polarization” for the quantum operators: the momenta are diagonal, while the fields act by taking (path) derivatives.",
"This choice, inspired by the Maxwell case, is not mandatory, and the roles of momenta and fields could be interchanged (with an appropriate change of sign).",
"It is also noticeable that this prescription is naturally adapted to linearized gravity in the same way that the Abelian loop representation in the electromagnetic case, in the sense that no further intrepretation for the tensor indexes $a,b$ is needed.",
"In the same way, the wave functional depends on a non-trivial tangle of the same closed path $\\Gamma $ instead of skein of colored loops (as it was shown in the previous section).",
"Unfortunately, the first class constraints cannot be realized in this case either and therefore we must carry out the quantization in the reduced phase space as in the FPT case.",
"In spite of that, the monoidal representation can be used to obtain an interesting geometric interpretation of the generator of duality.",
"With this purpose, let us first recall how the generator arises in the theory and subsequently we shall proceed to quantize it in terms of monoid variables.",
"It is well known that, analogously to what occurs in the FMT, the Fierz-Pauli model shows invariance of the equations of motion under the transformations $p^{abTT}&\\rightarrow &\\frac{1}{2}(\\mathcal {O}h^{TT})_{ab}\\nonumber \\\\\\frac{1}{2}(\\mathcal {O}h^{TT})_{ab}&\\rightarrow &-p^{abTT},$ where $(\\mathcal {O}h)_{ab}=\\frac{1}{2}\\varepsilon ^{acd}\\partial _{c}h_{db} + \\varepsilon ^{bcd}\\partial _{c}h_{da}$ is the symmetrized curl.",
"As a guide of the comparison with the Maxwell case, it is useful to keep in mind that $p^{abTT}$ is the analogous to the transverse part of the electric field, while $h^{TT}_{ab}$ would relate with the transverse part of the vector potential and its symmetrized curl $(\\mathcal {O}h^{T T} )_{ab}$ is analogous to the magnetic field.",
"Infinitesimally, the duality transformations can be written down as $\\delta p^{abTT}&=&\\frac{\\theta }{2}(\\mathcal {O}h^{TT})_{ab}\\nonumber \\\\\\frac{1}{2}\\delta h_{ab}^{TT}&=&\\theta \\nabla ^{-2}(\\mathcal {O}p^{TT})^{ab}$ which, as in the Maxwell theory, correspond to a SO(2) rotation.",
"A straightforward application of Noether's theorem leads to the conserved quantity which generates the infinitesimal duality rotations (see reference [18] for details).",
"Now, in the monoidal representation this generator of duality can be shown to be given by $\\hat{G}&=&\\int d^{3}x\\delta _{ab}(\\vec{x},\\hat{v})\\varepsilon ^{acd}\\partial _{c}\\delta _{db}(\\vec{x},\\hat{v})+\\int d^{3}x\\partial _{b}\\delta _{ab}(\\vec{x},\\hat{v})\\varepsilon ^{acd}\\partial _{c}\\nabla ^{-2}\\partial _{e}\\delta _{de}(\\vec{x},\\hat{v})\\nonumber \\\\&+&\\int d^{3}xI^{ab}[\\vec{x},\\Gamma ]\\varepsilon ^{acd}\\nabla ^{-2}\\partial _{c}I^{db}[\\vec{x},\\Gamma ]+\\int d^{3}x\\partial _{b}I^{ab}[\\vec{x},\\Gamma ]\\varepsilon ^{acd}\\partial _{c}\\nabla ^{-4}\\partial _{e}I^{de}[\\vec{x},\\Gamma ].$ It is worth to mentioning that the expression (REF ) have formal similarities with generator of duality of the MT reported in reference [2].",
"One of these features is that both are written in term of products which involves either path derivatives or “form factors”.",
"Another one is that non-local terms appear as a consecuence of the inverse Laplacian operator $\\nabla ^{-2}$ which allowed to define the infinitesimal transformation in equation (REF ).",
"Finally, only the “form factor\" part of the generator can be interpreted in terms of features of the space of extended objectsThe path derivative only has a geometric meaning in the dual representation where $\\hat{h}^{TT}_{ab}$ is diagonal.",
"In order to show this, let us concentrate in the third term of the generator of dualityIt can be shown that the fourth term has not a simple interpretation which after some manipulation can be written as $\\hat{G}_{3}\\propto -\\frac{1}{4\\pi }\\oint \\limits _{\\Gamma _{1}}dl_{\\Gamma _{2}}\\oint \\limits _{\\Gamma _{2}}dl_{\\Gamma _{2}}(\\hat{u}_{T_{\\Gamma _{1}}}\\cdot \\hat{u}_{T_{\\Gamma _{2}}})\\int \\limits _{\\gamma }^{\\vec{z}_{\\Gamma _{1}}}dl_{\\gamma }(\\hat{u}_{T_{\\Gamma _{1}}}\\times \\hat{u}_{T_{\\Gamma _{2}}})\\cdot \\hat{u}_{T_{\\gamma }}\\delta ^{3}(\\vec{z}_{\\Gamma _{2}}-\\vec{w}_{\\gamma }).$ In the former expression, it is assumed that each line integrals have different supports, namely $\\Gamma _{1}$ and $\\Gamma _{2}$ (actually both integrals are defined on the same path $\\Gamma $ but, as we shall see, our choice of different paths will simplify the geometrical interpretation).",
"Furthermore, $\\gamma _{1}$ is an auxiliar infinte straight curve that does not belong to the monoide space $\\mathfrak {M}$ but it has been introduced for convenience (see for example reference [2] for details).",
"In the other hand, unitary tangent vectors $\\hat{u}_{T_{\\Gamma _{1}}}$ and $\\hat{u}_{T_{\\Gamma _{2}}}$ belong to the closed paths $\\Gamma _{1}$ and $\\Gamma _{2}$ respectively.",
"In contrast, $\\hat{u}_{T_{\\gamma }}$ is tangent to the open straight curve which starts in the spatial infinity and ends on $\\Gamma _{2}$ .",
"Now, the triple product $(\\hat{u}_{T_{\\Gamma _{1}}}\\times \\hat{u}_{T_{\\Gamma _{2}}})\\cdot \\hat{u}_{T_{\\gamma }}$ force the vectors $\\hat{u}_{T_{\\Gamma _{1}}}$ , $\\hat{u}_{T_{\\Gamma _{2}}}$ and $\\hat{u}_{T_{\\Gamma _{2}}}$ to construct a non-degenerated volume if we want to detect non-vanishing contributions.",
"We can see that these curves $\\Gamma _{1}$ and $\\Gamma _{2}$ are linked together by the fact that $\\Gamma _{1}$ is obligated to intersect once the open curve, and if in such intersection the three vectors involved form a non-degenerate figure-volume, then this will “count” out one contribution.",
"In base on the former discussion, we can realize the intriguing similarity between $\\hat{G}_{3}$ and the Gauss Linking Number (GLN).",
"The only difference with the GLN is that $\\hat{G}_{3}$ contain also the product $\\hat{u}_{T_{\\Gamma _{1}}}\\cdot \\hat{u}_{T_{\\Gamma _{2}}}$ which is a measure of the angle between the tangent vectors of the closed paths.",
"In this sense we can say that this term will contribute to the generator as long as tangent vectors $\\hat{u}_{T_{\\Gamma _{1}}},\\hat{u}_{T_{\\Gamma _{2}}},\\hat{u}_{T_{\\gamma }}$ form a trihedron but also the dihedral angle between planes intersecting on the line generated by $\\hat{u}_{T_{\\gamma }}$ must be neceseraly different from 0 and $\\pi /2$ .",
"However, instead of obtain $\\pm 1$ every time this occurs (as for the Gauss Linking Number) our result is angle dependent, or in other words, is metric dependent.",
"This fact was expected because the term from where it was derived, namely, $I^{ab}[\\vec{x},\\Gamma ]\\varepsilon ^{acd}\\partial _{c}I^{db}[\\vec{x},\\Gamma ]$ is metric dependent (although at the first glance, looks like a Chern-Simon term).",
"Note that a similar result was obtained for the generator of duality for the MT (see reference [2]).",
"In fact, the electric part of the generator can be written as $\\oint \\limits _{\\Gamma _{1}}dl_{\\Gamma _{2}}\\oint \\limits _{\\Gamma _{2}}dl_{\\Gamma _{2}}\\int \\limits _{\\gamma }^{\\vec{z}_{\\Gamma _{1}}}dl_{\\gamma }(\\hat{u}_{T_{\\Gamma _{1}}}\\times \\hat{u}_{T_{\\Gamma _{2}}})\\cdot \\hat{u}_{T_{\\gamma }}\\delta ^{3}(\\vec{z}_{\\Gamma _{2}}-\\vec{w}_{\\gamma }),\\nonumber \\\\$ which, in absence of the scalar product $\\hat{u}_{T_{\\Gamma _{1}}}\\cdot \\hat{u}_{T_{\\Gamma _{2}}}$ , it is an analytical expression of the Gauss Linking number between Abelian loops."
],
[
"Concluding remarks",
"In this work, a well-suited geometrical representation for the massless Fierz-Pauli theory has been obtained.",
"It presents advantages when is compared with previous attempts.",
"For instance it allows to represent symmetric quantum operators in a way that a misleading interpretation of tensor indexes of the theory is avoided.",
"As a consecuence, the space on which wave funtionals take values is expanded by monoidal closed paths instead of on skein of colored Abelian loops.",
"Moreover, the monoidal representation developed here, allows to carry out a kind of geometric interpretation of interesting objects like the generator of duality of the theory.",
"Although a knot invariant is not obtained, the resemblance with the Gauss Linking Number is intriguing and further research along this line is mandatory.",
"If the monoidal representation allows to obtain some hints that can be further applied or generalized to full gravity or if it can provide more illumination respect to the standard loop formulation is under investigation."
],
[
"Acknowledgments",
"E.C acknowledges Pedro Bargueño for his constructive and valuable suggestions."
]
] | 1612.05563 |
[
[
"Aeronomical constraints to the minimum mass and maximum radius of hot\n low-mass planets"
],
[
"Abstract Stimulated by the discovery of a number of close-in low-density planets, we generalise the Jeans escape parameter taking hydrodynamic and Roche lobe effects into account.",
"We furthermore define $\\Lambda$ as the value of the Jeans escape parameter calculated at the observed planetary radius and mass for the planet's equilibrium temperature and considering atomic hydrogen, independently of the atmospheric temperature profile.",
"We consider 5 and 10 $M_{\\oplus}$ planets with an equilibrium temperature of 500 and 1000 K, orbiting early G-, K-, and M-type stars.",
"Assuming a clear atmosphere and by comparing escape rates obtained from the energy-limited formula, which only accounts for the heating induced by the absorption of the high-energy stellar radiation, and from a hydrodynamic atmosphere code, which also accounts for the bolometric heating, we find that planets whose $\\Lambda$ is smaller than 15-35 lie in the \"boil-off\" regime, where the escape is driven by the atmospheric thermal energy and low planetary gravity.",
"We find that the atmosphere of hot (i.e.",
"$T_{\\rm eq}\\gtrapprox$ 1000 K) low-mass ($M_{\\rm pl}\\lessapprox$ 5 $M_{\\oplus}$) planets with $\\Lambda$ < 15-35 shrinks to smaller radii so that their $\\Lambda$ evolves to values higher than 15-35, hence out of the boil-off regime, in less than $\\approx$500 Myr.",
"Because of their small Roche lobe radius, we find the same result also for hot (i.e.",
"$T_{\\rm eq}\\gtrapprox$ 1000 K) higher mass ($M_{\\rm pl}\\lessapprox$ 10 $M_{\\oplus}$) planets with $\\Lambda$ < 15-35, when they orbit M-dwarfs.",
"For old, hydrogen-dominated planets in this range of parameters, $\\Lambda$ should therefore be $\\geq$15-35, which provides a strong constraint on the planetary minimum mass and maximum radius and can be used to predict the presence of aerosols and/or constrain planetary masses, for example."
],
[
"Introduction",
"Thanks to the large number of extra-solar planets (exoplanets) discovered to date by ground- and space-based facilities, such as SuperWASP [39], HATNet [3], CoRoT [2], Kepler [8], and K2 [23], we are beginning to classify the large variety of detected exoplanets on the basis of their properties.",
"One of the greatest recent surprises in planetary sciences was the discovery of a large population of planets with mass and radius in between that of terrestrial and giant planets of the solar system [33].",
"These planets, hereafter sub-Neptunes, typically have masses and radii in the 1.5–17 $M_{\\oplus }$ and 1.5–5 $R_{\\oplus }$ range.",
"Sub-Neptunes fill a gap of physical parameters that are absent from the solar system.",
"Accurately deriving their masses and radii is therefore crucial to our overall understanding of planets.",
"The high quality of the Kepler light curves allowed us to obtain precise transit radii, even for small planets, but for most of them, the low mass and faint apparent magnitude of their host stars hampers a precise enough determination of the planetary mass through radial velocity.",
"For several multi-planet systems, planetary masses have been inferred from transit-timing variations (TTVs), but some of the resulting values are at odds with those derived from radial velocity [48].",
"Sub-Neptunes for which both mass and radius have been measured present a large spread in bulk density ($\\approx $ 0.03–80 g cm$^{-3}$ ; low average densities imply the presence of hydrogen-dominated atmospheres), which finding is currently greatly debated [32], [21], [22], [30], [31], [37], [20].",
"It is therefore important to find external independent constraints to planetary masses and radii that could be applied to a large number of planets, for example to independently test the masses derived from TTVs, identify the possible presence of high-altitude aerosols, and estimate a realistic range of planetary radii/masses given a certain mass/radius.",
"We show here how basic aeronomical considerations, supported by hydrodynamic modelling and previous results [38], can constrain the mass/radius of old sub-Neptunes given their radius/mass and equilibrium temperature ($T_{\\rm eq}$ )."
],
[
"Generalisation of the Jeans escape parameter",
"The Jeans escape parameter $\\lambda $ is classically defined at the exobase and for a hydrostatic atmosphere.",
"It is the ratio between the escape velocity $\\upsilon _{\\infty }$ and the most probable velocity $\\upsilon _{\\rm 0}$ of a Maxwellian distribution at temperature $T$ , squared [24], [13], [35], [4].",
"We generalise the Jeans escape parameter at each atmospheric layer $r$ and corresponding temperature $T$ for a hydrodynamic atmosphere composed of atomic and molecular hydrogen as $\\begin{split}\\lambda ^*(r) \\equiv \\frac{\\upsilon _{\\infty }^2}{\\upsilon _{\\rm 0}^2} = \\frac{\\upsilon _{\\infty }^2}{\\left(\\upsilon _{\\rm hy}/2+\\sqrt{\\upsilon _{\\rm hy}^2/4+\\upsilon _{\\rm th}^2}\\right)^2} = \\\\ \\frac{2GM_{\\rm pl}}{r\\left(\\upsilon _{\\rm hy}/2+\\sqrt{\\upsilon _{\\rm hy}^2/4+2k_{\\rm B}T/m}\\right)^2}\\,,\\end{split}$ where $G$ is Newton's gravitational constant, $k_{\\rm B}$ is Boltzmann's constant, $M_{\\rm pl}$ is the planetary mass, $\\upsilon _{\\rm th}$ is the thermal velocity $\\sqrt{2k_{\\rm B}T/m}$ , and $\\upsilon _{\\rm hy}$ is the bulk velocity of the particles at each atmospheric layer.",
"In Eq.",
"REF , $m$ is the mean molecular weight $m = \\frac{\\sum n_{\\rm X}m_{\\rm X}}{\\sum n_{\\rm X}}$ where $n_{\\rm X}$ and $m_{\\rm X}$ are the density and mass of each atom/molecule (X) in the atmosphere.",
"In this work, we consider atomic and molecular hydrogen.",
"The value of $\\upsilon _{\\rm 0}$ in the hydrodynamic case is that of a shifted Maxwellian distribution, where $\\upsilon _{\\rm hy}$ is the shift.",
"The Maxwellian velocity distribution gives the number of particles between $\\upsilon $ and $\\upsilon +d\\upsilon $ and can be written as $F(\\upsilon )d\\upsilon = 4 \\pi n \\left(\\frac{m}{2 \\pi k_{\\rm B}T}\\right)^{3/2}\\upsilon ^2 \\exp \\left(-\\frac{m(\\upsilon -\\upsilon _{\\rm hy})^2}{2k_{\\rm B}T}\\right) d\\upsilon ,$ where $n$ is the number density and $m$ the particle mass.",
"The most probable velocity $\\upsilon _0$ is found where Eq.",
"(REF ) has its maximum and can therefore be derived by setting $dF/d\\upsilon $ = 0.",
"This condition results in a quadratic equation for $\\upsilon $ , $\\frac{\\upsilon ^2}{\\upsilon _{\\rm th}^2} - \\frac{\\upsilon \\,\\upsilon _{\\rm hy}}{\\upsilon _{\\rm th}^2} - 1 = 0.$ The solution of this equation is $\\upsilon _0 = \\frac{\\upsilon _{\\rm hy}}{2} + \\sqrt{\\frac{\\upsilon _{\\rm hy}^2}{4} + \\upsilon _{\\rm th}^2},$ where only this positive solution is physical (the negative solution yields a negative $\\upsilon _0$ ).",
"Note that a direct derivation of $\\upsilon _0$ by setting $\\upsilon _{\\rm hy}=0$ in Eq.",
"(REF ) or in Eq.",
"(REF ) yields $\\upsilon _0 = \\upsilon _{\\rm th}$ .",
"From Eq.",
"(REF ) it also follows that if $\\upsilon _{\\rm th}\\rightarrow 0$ , then $\\upsilon _0\\rightarrow \\upsilon _{\\rm hy}$ , as expected.",
"The formulation of the Jeans escape parameter given in Eq.",
"REF is reminiscent of the “solar breeze” used before Parker's solar wind model was accepted [11], [12].",
"If $\\upsilon _{\\rm hy}$ is negligible compared to $\\upsilon _{\\rm th}$ (i.e.",
"hydrostatic atmosphere), the Jeans escape parameter returns to the classical form of $\\lambda ^* = \\lambda = \\frac{GM_{\\rm pl}m}{k_{\\rm B}Tr}\\,.$ We recall that for the classical Jeans escape parameter (hydrostatic atmosphere), a layer is completely bound to a planet for $\\lambda $ $\\gtrsim $ 30 and escape is important for $\\lambda $ $<$ 15, while for $\\lambda $ $\\lesssim $ 1.5 the atmosphere is in hydrodynamic “blow-off” [24], [13], [35], [4].",
"This last condition occurs when the thermal energy of the gas is very close to, or even exceeds, the gravitational energy.",
"The vast majority of the exoplanets known to date orbits at close distance to their host stars.",
"We therefore consider Roche-lobe effects.",
"Following the procedure described in Sect.",
"2 of [17], in Eq.",
"(REF ) we substitute the gravitational potential difference between the planetocentric distance $r$ and infinity ($GM_{\\rm pl}/r$ ) by the gravitational potential difference between $r$ and the Roche-lobe radius ($\\Delta \\phi $ ).",
"We therefore obtain $\\tilde{\\lambda }^*(r) = \\frac{2\\Delta \\phi }{\\left(\\upsilon _{\\rm hy}/2+\\sqrt{\\upsilon _{\\rm hy}^2/4+2k_{\\rm B}T/m}\\right)^2}\\,,$ where $\\Delta \\phi = \\phi _{\\rm 0}\\frac{\\xi -1}{\\xi }\\left[1-\\frac{1}{\\delta }\\frac{\\xi }{\\gamma ^2}\\frac{\\gamma (1+\\xi )-\\xi }{(\\gamma -1)(\\gamma -\\xi )}-\\frac{\\xi (1+\\delta )(1+\\xi )}{2\\delta \\gamma ^3}\\right]$ [17] and $\\phi _{\\rm 0} = G\\frac{M_{\\rm pl}}{r}\\,,\\,\\,\\,\\delta = \\frac{M_{\\rm pl}}{M_{\\star }}\\,,\\,\\,\\,\\gamma = \\frac{d}{r}\\,,\\,{\\rm and}\\,\\,\\,\\xi = \\frac{R_{\\rm RL}}{r}\\,.$ In Eq.",
"(REF ), $M_{\\star }$ is the stellar mass, $d$ is the semi-major axis, and $R_{\\rm RL}$ is the Roche lobe radius.",
"Therefore, Eq.",
"REF gives the generalised form of the Jeans escape parameter."
],
[
"Planet atmosphere modelling",
"To draw profiles of $\\lambda ^*$ and $\\tilde{\\lambda }^*$ we derive the temperature, pressure, velocity, and density structure of planetary atmospheres employing a stellar high-energy (XUV; 1–920 Å) absorption and 1D hydrodynamic upper-atmosphere model that solves the system of hydrodynamic equations for mass, momentum, and energy conservation, and also accounts for ionisation, dissociation, recombination, and Ly$\\alpha $ cooling.",
"The full description of the hydrodynamic code adopted for the simulations is presented in [19].",
"Hydrodynamic modelling is valid in presence of enough collisions, which occurs for Knudsen number $Kn=l/H<0.1$ [46], where $l$ is the mean free path and $H$ is the local scale height; in the domain of our models, from $R_{\\rm pl}$ to $R_{\\rm RL}$ , this criterion is always fulfilled.",
"Throughout our calculations, we adopt a net heating efficiency ($\\eta $ ) of 15% [45] and use stellar XUV fluxes ($I_{\\rm XUV}$ ) estimated from the average solar XUV flux [40], scaled to the appropriate distance and stellar radius.",
"We note that X-ray heating is not relevant in our case, because we do not consider active young stars [36].",
"We also assumed that at $R_{\\rm pl}$ hydrogen is completely in molecular form (i.e.",
"H$_2$ ), which is true for planets with $T_{\\rm eq}$ $<$ 2000 K [26].",
"For all calculations, and throughout the paper, we consider that $R_{\\rm pl}$ lies at a fiducial atmospheric pressure ($p_0$ ) of 100 mbar.",
"To justify this assumption, we calculated the photospheric deposition level using an updated version of the radiative transfer code described in [14] and [5].",
"The model considers opacities from line-by-line transitions from HITEMP for H$_2$ O, CO, and CO$_2$ [42] and HITRAN for CH$_4$ [43].",
"In addition, it includes opacities for H$_2$ –H$_2$ and H$_2$ –He collision-induced absorption from [7], [6], and [25], H$_2$ Rayleigh scattering from [28], and sodium and potassium doublets from [9].",
"In Fig.",
"REF we present transmission spectra for a fiducial sub-Neptune with $M_{\\rm pl}$ = 5 $M_{\\oplus }$ , $R_{\\rm pl}$ = 4 $R_{\\oplus }$ , and an isothermal atmosphere at 1000 K, in hydrostatic and thermochemical equilibrium.",
"We explored three different cases varying the atmospheric elemental metallicities, considering 0.01, 1.0, and 100 times solar abundances (Figs.",
"REF and REF ).",
"We adjusted the pressure-radius reference level such that the resulting transmission radius (integrated over the optical band) matches the fiducial planetary radius, adopting the CoRoT spectral response curve, as an example.",
"We find that the planetary transmission radii correspond to pressure levels of 130, 50, and 10 mbar for the 0.01, 1.0, and 100.0$\\times $ solar-metallicity models, respectively (Fig.",
"REF , left panel).",
"After we obtained the pressure-radius relationship, we computed the contribution functions (in the optical band) for the vertical optical depth.",
"The barycenter (i.e., average) of the contribution functions indicate where the atmosphere becomes optically thick.",
"This is the position of the planetary photosphere, where the lower boundary for the hydrodynamic calculation would need to be set.",
"We find that for the planet considered here the photospheric deposition level is approximately located at 551, 159, and 33 mbar for the 0.01, 1.0, and 100.0$\\times $ solar-metallicity models, respectively (Fig.",
"REF , right panel).",
"We performed the same procedure for all planets analysed in this work and list the pressure corresponding to the barycenter of the contribution function in the fifth column of Table REF .",
"The pressure values range between about 100 and 700 mbar, where the lower pressure values are obtained for the cooler, lower density planets.",
"Figure REF shows that a higher metallicity, as expected for low-mass planets, would lead to a slight decrease in pressure values, hence justifying our assumption of placing $R_{\\rm pl}$ at an average 100 mbar pressure level.",
"Figure: Mole-mixing fractions of the atmospheric species (see legend in the bottom panel) in thermochemical equilibrium for isothermal (1000 K) models calculated for 0.01 (top), 1.0 (middle), and 100 (bottom) times solar metallicity.Our hydrodynamic model implicitly considers the stellar continuum absorption by setting the temperature at the lower boundary, hence at $R_{\\rm pl}$ (i.e.",
"where most of the stellar radiation is absorbed), equal to $T_{\\rm eq}$ .",
"We return to the validity of this approximation in Sect. .",
"The planets considered here are old, hence heating from the planet interior can be neglected."
],
[
"$\\lambda ^*$ and {{formula:a643cab1-1475-483e-9079-cb4496858524}} profiles",
"As an example to show the differences between $\\lambda ^*$ and $\\tilde{\\lambda }^*$ , we modelled a close-in low-density 5 $M_{\\oplus }$ and 4 $R_{\\oplus }$ (average density $\\rho $ of 0.4 g cm$^{-3}$ ) planet with $T_{\\rm eq}$ of 1000 K, orbiting an early K-type star (see Table REF ).",
"The parameters adopted for this idealised planet are similar to those of Kepler-87c [34].",
"We derived the mean molecular mass at each atmospheric layer from the modelled H and H$_2$ mixing ratios.",
"Figure REF shows the obtained profiles.",
"In the 1–2 $R_{\\rm pl}$ range, $\\lambda ^*$ decreases with increasing $r$ because the gravitational potential decreases and the H$_2$ molecules dissociate under the action of the stellar XUV flux.",
"All H$_2$ molecules are dissociated at $\\sim $ 2 $R_{\\rm pl}$ .",
"Then, at larger radii, as the temperature continues to decrease due to adiabatic cooling, $\\lambda ^*$ increases and remains above 30 for radii grater than 6.5 $R_{\\rm pl}$ .",
"This implies that no particles could escape, regardless of their proximity to $R_{\\rm RL}$ , which is non-physical.",
"Instead, $\\tilde{\\lambda }^*$ monotonically decreases with increasing $r$ .",
"The bottom panel of Fig.",
"REF shows that for such a close-in planet, despite the hydrodynamic nature of the atmosphere, in most layers $\\upsilon _{\\rm hy}$ is negligible compared to $\\upsilon _{\\rm th}$ , therefore $\\lambda ^*$ $\\approx $ $\\lambda $ and $\\tilde{\\lambda }^*$ $\\approx $ $\\tilde{\\lambda }$ .",
"Figure: Top: temperature (black solid line) and pressure (red dashed line) profiles as a function of radius rr in units of R pl R_{\\rm pl} for a 5 M ⊕ M_{\\oplus } and 4 R ⊕ R_{\\oplus } planet with T eq T_{\\rm eq} =1000=1000 K, orbiting an early K-type star (see Table ).",
"The right axis indicates the pressure scale.",
"Middle: λ * \\lambda ^* (black solid line) and λ ˜ * \\tilde{\\lambda }^* (red dashed line) profiles as a function of radius rr in units of R pl R_{\\rm pl}.",
"The horizontal lines mark the critical values of the Jeans escape parameter in the hydrostatic case: 1.5, 15, and 30.",
"The blue dotted lines show the λ * \\lambda ^* and λ ˜ * \\tilde{\\lambda }^* profiles calculated assuming that the whole atmosphere is made of atomic hydrogen.",
"The filled circle indicates the Λ\\Lambda value (see Sect. ).",
"Bottom: υ th \\upsilon _{\\rm th} (black solid line) and υ hy \\upsilon _{\\rm hy} (red dashed line) profiles in km s -1 \\mathrm {km\\,s}^{-1}.Figure REF shows that the value of $\\tilde{\\lambda }^*$ approaches unity at atmospheric layers where the pressure, hence density, is high enough to power high escape rates (see Table REF ).",
"These upper layers are in a blow-off regime where the escaping gas is continuously replenished by the hydrodynamically expanding atmosphere, with the expansion being driven by the high thermal energy and low planet gravity.",
"This escape regime, here presented from an aeronomical point of view, has been discovered and thoroughly described by [38], who called it “boil-off”, in relation to the study of the evolution of young planets that are just released from the protoplanetary nebula [20]."
],
[
"Using escape rates to identify planets in the boil-off regime",
"We define $\\Lambda $ as the Jeans escape parameter $\\lambda $ (without accounting for Roche-lobe effects and hydrodynamic velocities) at $R_{\\rm pl}$ , evaluated at the $T_{\\rm eq}$ of the planet and for an atomic-hydrogen gas (see the full dot and the blue dotted lines in Fig.",
"REF ) $\\Lambda = \\frac{GM_{\\rm pl}m_{\\rm H}}{k_{\\rm B}T_{\\rm eq}R_{\\rm pl}}\\,.$ This quantity, which we call the restricted Jeans escape parameter, is useful because it can be derived for any planet for which mass, transit radius, and $T_{\\rm eq}$ are measured, and without the need of any atmospheric modelling or calculation of $R_{\\rm RL}$ .",
"We aim here at roughly finding the threshold $\\Lambda $ values ($\\Lambda _{\\rm T}$ ), as a function of $M_{\\rm pl}$ , $R_{\\rm pl}$ , and $T_{\\rm eq}$ , below which the atmosphere transitions towards the boil-off regime.",
"For this we use escape rates, as described below.",
"In addition to the escape rates derived from the hydrodynamic model ($L_{\\rm hy}$ ), we consider the maximum possible XUV-driven escape rates, which can be analytically estimated using the energy-limited formula [47], [17], $L_{\\rm en}=\\frac{\\pi \\eta R_{\\rm pl}R_{\\rm XUV_{eff}}^2I_{\\rm XUV}}{GM_{\\rm pl}m_{\\rm H}K(\\xi )}\\,,$ where $R_{\\rm XUV_{eff}}$ is the effective radius at which the XUV energy is absorbed in the upper atmosphere [17], [18] and $\\eta $ is the heating efficiency (see Sect.",
"REF ).",
"The factor $K(\\xi )=1-\\frac{3}{2\\xi }+\\frac{1}{2\\xi ^3}$ accounts for Roche-lobe effects [17].",
"We note that Roche-lobe effects are also considered in the hydrodynamic model.",
"By construction, XUV heating and the intrinsic thermal energy of the atmosphere are considered in the computation of $L_{\\rm hy}$ , while only XUV heating is taken into account when deriving $L_{\\rm en}$ .",
"It follows that the boil-off regime, that is, when the intrinsic thermal energy of the atmosphere becomes the efficient main driver of the escape, occurs for $L_{\\rm hy}$ greater than $L_{\\rm en}$ .",
"For this situation, $L_{\\rm hy}$ /$L_{\\rm en}$ $>1$ cannot be achieved purely from XUV heating, implying that the outflow must be driven by the heat present at the lower boundary of the atmosphere.",
"We can therefore use the $L_{\\rm hy}$ /$L_{\\rm en}$ $\\approx 1$ as an empirical condition to estimate $\\Lambda _{\\rm T}$ .",
"To identify the $\\Lambda _{\\rm T}$ value, which is the $\\Lambda $ value satisfying the $L_{\\rm hy}$ /$L_{\\rm en}$ $\\approx 1$ condition, we ran a set of hydrodynamic simulations for two idealised old planets of 5 and 10 $M_{\\oplus }$ orbiting an early G-, K-, and M-type star at distances such that $T_{\\rm eq}$ is equal to 500 and 1000 K, assuming a Bond albedo of 0.3.",
"Table REF lists the complete set of input parameters and results, which are visually displayed in Fig.",
"REF .",
"Figure: Ratio between the hydrodynamic (L hy L_{\\rm hy}) and energy-limited (L en L_{\\rm en}) escape rates as a function of Λ\\Lambda for the modelled planets orbiting the G2 (top), K2 (middle), and M2 (bottom) star.",
"Within each panel, the legend indicates the mass (in M ⊕ M_{\\oplus }) and temperature (in K) of the modelled planets.",
"The dashed line indicates the equality between L hy L_{\\rm hy} and L en L_{\\rm en}, while the dotted line indicates where the L hy L_{\\rm hy}/L en L_{\\rm en} ratio is equal to 2.0.",
"The value of Λ T \\Lambda _{\\rm T} lies between 15 and 35.Table: Input parameters and results of the simulations performed with η\\eta = 15%.Figure REF shows that the $L_{\\rm hy}$ /$L_{\\rm en}$ $\\approx 1$ condition is reached for $\\Lambda $ values between 15 and 35, with a slight dependence on stellar type and $T_{\\rm eq}$ .",
"In particular, for the planets orbiting the G- and K-type stars, the $\\Lambda _{\\rm T}$ values appear to be lower at higher temperature, hence $\\Lambda _{\\rm T}$ decreases with increasing tidal gravity.",
"This does not seem to be the case for the planets orbiting the M-type star, particularly for the 10 $M_{\\oplus }$ planet.",
"We discuss here the uncertainties related to the computation of the $L_{\\rm hy}$ /$L_{\\rm en}$ ratio.",
"Since we do not consider real planets, there are no observational uncertainties connected to the system parameters.",
"The $R_{\\rm XUV_{eff}}$ value present in Eq.",
"(REF ) is an output of the hydrodynamic code, and it is used to calculate $L_{\\rm hy}$ as well.",
"For these reasons, there are no uncertainties on the $R_{\\rm XUV_{eff}}$ value.",
"The heating efficiency $\\eta $ is therefore the only input parameter for which its uncertainties may affect the $L_{\\rm hy}$ /$L_{\\rm en}$ ratio.",
"Generally, the heating efficiency varies with altitude, and [45] concluded that for hot Jupiters the value of $\\eta $ in the thermosphere varies between $\\approx $ 10% and 20%.",
"Because our model does not self-consistently calculate $\\eta $ with height, we assume an average value of 15% (Sect.",
"REF ).",
"This agrees well with calculations by [36], who also estimated that $\\eta $ values higher than 40% are unrealistically high.",
"More recently, [44] calculated the average heating efficiency for a set of planets with different masses and radii.",
"They concluded that for planets with $\\log (GM_{\\rm pl}/R_{\\rm pl})$ smaller than 13.11 (the case of the planets considered here), $\\eta $ is about 23%, independent of the planet parameters.",
"As discussed by [27], the heating efficiency enters in the calculation of both $L_{\\rm hy}$ and $L_{\\rm en}$ , although with a slightly different dependence.",
"To quantitatively estimate the effects of the uncertainty on the heating efficiency on the $L_{\\rm hy}$ /$L_{\\rm en}$ ratio, we ran a set of simulations for two planets orbiting the K2 star with two different $\\Lambda $ values ($\\Lambda $ = 21, $M_{\\rm pl}$ = 5 $M_{\\oplus }$ , $R_{\\rm pl}$ = 1.8 $R_{\\oplus }$ and $\\Lambda $ = 8, $M_{\\rm pl}$ = 5 $M_{\\oplus }$ , $R_{\\rm pl}$ = 4.5 $R_{\\oplus }$ ) and $T_{\\rm eq}$ = 1000 K, varying $\\eta $ between 10 and 40%, leaving all other parameters fixed.",
"The results, displayed in Fig.",
"REF , indicate that variations of $\\eta $ by a factor of two from the adopted value of 15% (e.g.",
"between 10 and 30%) modify the $L_{\\rm hy}$ /$L_{\\rm en}$ ratio by a factor of about 1.5 in the case of low $\\Lambda $ and of about 1.05 in the case of high $\\Lambda $ .",
"The sensitivity of the $L_{\\rm hy}$ /$L_{\\rm en}$ ratio on variations of $\\eta $ therefore decreases with increasing $\\Lambda $ .",
"On the basis of these results, to be conservative, we consider the $L_{\\rm hy}$ /$L_{\\rm en}$ $\\approx 1$ condition to be fulfilled when $L_{\\rm hy}$ /$L_{\\rm en}$$\\le 2.0$ .",
"Figure: Variation of the L hy L_{\\rm hy}/L en L_{\\rm en} ratio, normalised to the value of the L hy L_{\\rm hy}/L en L_{\\rm en} ratio obtained with η\\eta = 15% (adopted for our calculations), as a function of heating efficiency η\\eta for two planets orbiting the K2 star with two different Λ\\Lambda values (dashed line: Λ\\Lambda = 21, M pl M_{\\rm pl} = 5 M ⊕ M_{\\oplus }, R pl R_{\\rm pl} = 1.8 R ⊕ R_{\\oplus }; solid line: Λ\\Lambda = 8, M pl M_{\\rm pl} = 5 M ⊕ M_{\\oplus }, R pl R_{\\rm pl} = 4.5 R ⊕ R_{\\oplus }) and T eq T_{\\rm eq} = 1000 K.Figure REF shows the atmospheric structure of the 5 $M_{\\oplus }$ planet considered in Sect.",
"REF , but with a radius of 1.8 $R_{\\oplus }$ (i.e.",
"out of the boil-off regime).",
"Close to $R_{\\rm pl}$ the atmosphere is hydrostatic, as indicated by the temperature increase (i.e.",
"no adiabatic cooling), with the high-energy stellar flux providing a considerable amount of heating.",
"The rise in temperature close to the lower boundary in Fig.",
"REF is caused by XUV heating, which is the driver of the outflow.",
"In contrast, the monotonic temperature decrease (caused by adiabatic cooling) shown in Fig.",
"REF indicates that XUV heating is not important, implying that the outflow is driven by the high thermal energy of the planet.",
"In our modelling we do not consider cooling from H$_3^+$ .",
"However, H$_3^+$ cooling is not relevant in our case, because it does not affect the thermally driven escape rates in the boil-off regime [10].",
"Figure: Same as Fig.",
", but for a 5 M ⊕ M_{\\oplus } planet with a radius of 1.8 R ⊕ R_{\\oplus }.On the basis of detailed evolution modelling of young planets immediately after the disk dispersal, [38] concluded that planets exit the boil-off regime when their radius becomes smaller than 0.1 Bondi radii ($R_{\\rm B}$ ).",
"The Bondi radius is defined as $R_{\\rm B}=GM_{\\rm pl}/2c_{\\rm s}^2$ , where $c_{\\rm s}$ is the isothermal sound speed.",
"The $R_{\\rm pl}/R_{\\rm B}=0.1$ condition for the occurrence of boil-off given by [38] is therefore mathematically identical to the $\\Lambda _{\\rm T}$ $=20$ condition, when an adiabatic gas index $\\gamma $ equal to 1 is considered, or in other words, isothermal gas.",
"We arrived at a result similar to that of [38], who properly took into account the various heating and cooling sources, which indicates that the assumptions and simplifications we made for our modelling are robust.",
"In particular, it shows the validity of (i) simplifying the processes leading to the planet's thermal balance by setting the temperature of the atmosphere equal to $T_{\\rm eq}$ at the lower boundary, and (ii) setting the lower boundary at the pressure level where the optical depth is roughly unity, which is where most of the stellar radiation is absorbedThis is also how [38] set their upper boundary.. We note that modifications to these two assumptions affect the shape of the atmospheric profiles, but not the escape rates, if $L_{\\rm hy}$ is equal to or smaller than $L_{\\rm en}$ .",
"For example, Fig.",
"2 of [27] shows that by varying the pressure at the lower boundary from 100 mbar to 1 bar only affects the $L_{\\rm hy}$ /$L_{\\rm en}$ ratio in the boil-off regime (when $L_{\\rm hy}$ $>$ $L_{\\rm en}$ ), while the radius at which the $L_{\\rm hy}$ /$L_{\\rm en}$ $\\approx 1$ condition is reached (namely the value of $\\Lambda _{\\rm T}$ ) is not affected.",
"This implies that, within our scheme, the $\\Lambda _{\\rm T}$ values are independent of the two assumptions described above.",
"It should also be noted that our results apply to any planet, independent of the internal structure, for which the 100 mbar pressure level lies above the solid core, if any is present."
],
[
"Constraints on $M_{\\rm pl}$ and {{formula:8a148ff4-517e-48aa-846f-f811d32a037a}}",
"To explore whether the knowledge of the value of $\\Lambda _{\\rm T}$ , or equivalently of the $R_{\\rm pl}/R_{\\rm B}=0.1$ condition, can help to constrain the parameters of old planets, it is necessary to consider the atmospheric evolution of planets in the boil-off regime.",
"To roughly estimate how much time the modelled planets need to evolve out of the boil-off regime, we follow the same procedure as adopted by [27] to study the case of CoRoT-24b.",
"As an example, we take the simulations we carried out for the $M_{\\rm pl}$ = 5 $M_{\\oplus }$ planet with $T_{\\rm eq}$ = 1000 K orbiting the K-type star.",
"We assumed a core mass of 5 $M_{\\oplus }$ and used formation and structure models by [41] to estimate for each modelled radius the atmospheric mass fraction $f$ .",
"We then used the $L_{\\rm hy}$ values to roughly estimate the evolution of the atmospheric mass over time.",
"Figure REF shows that the atmospheric mass for a radius above 1.8 $R_{\\oplus }$ (where $L_{\\rm hy}$ /$L_{\\rm en}$ $\\approx 1$ ) would be lost within $\\approx $ 500 Myr.",
"This is therefore the timescale needed for this planet to evolve out of the boil-off regime.",
"Figure: Atmospheric mass M AT M_{\\rm AT} evolution normalised to the atmospheric mass corresponding to R pl R_{\\rm pl} =1.8 R ⊕ R_{\\oplus } (where L hy L_{\\rm hy}/L en L_{\\rm en} ≈1\\approx 1) estimated from the L hy L_{\\rm hy} escape rates obtained for the M pl M_{\\rm pl} = 5 M ⊕ M_{\\oplus } planet with T eq T_{\\rm eq} = 1000 K orbiting the K-type star.",
"The dashed line indicates M AT M_{\\rm AT} = M AT M_{\\rm AT}(1.8 R ⊕ R_{\\oplus }).",
"The initial time is arbitrarily set at 0.1 Myr.",
"The legend lists the atmospheric mass fraction corresponding to each radius.Table REF lists the timescales for each modelled planet from Table REF .",
"We find the shorter time scales for the less massive and hotter planets.",
"In particular, for planets with $M_{\\rm pl}$ = 5 $M_{\\oplus }$ and $T_{\\rm eq}$ = 1000 K, the timescale to evolve out of boil-off is shorter than 500 Myr.",
"The same also occurs for the hot (i.e.",
"$T_{\\rm eq}$ = 1000 K) 10 $M_{\\oplus }$ planet orbiting the M-type star, likely because of the effect of the smaller Roche-lobe radius compared to the case of the same planet orbiting the G- and K-type stars.",
"In general, we therefore find that hot (i.e.",
"$T_{\\rm eq}$ $\\gtrapprox $ 1000 K) low-mass ($M_{\\rm pl}$ $\\lessapprox $ 5 $M_{\\oplus }$ ) planets with hydrogen-dominated atmospheres, unless very young, should not have $\\Lambda $ $<$ $\\Lambda _{\\rm T}$ .",
"Because of their small Roche lobe, this conclusion also extends to hot (i.e.",
"$T_{\\rm eq}$ $\\gtrapprox $ 1000 K) higher mass ($M_{\\rm pl}$ $\\lessapprox $ 10 $M_{\\oplus }$ ) planets if they are orbiting M-dwarfs.",
"Table: Approximate time scales (in Myr) needed for the modelled planets to evolve out of the boil-off regime, following the analysis described in Sect. .",
"Timescales larger than 1 Gyr have been rounded to the nearest 100 Myr.",
"The fourth and fifth columns indicate the initial and final Λ\\Lambda values of the planets used to calculate the timescales.From the above considerations, it follows that for hot low-mass planets with hydrogen-dominated atmospheres with observed values leading to $\\Lambda $ $<$ 15–35 there must be problems with the estimation/interpretation of the measured mass (i.e.",
"too low), or radius (i.e.",
"too large), or both.",
"Large transit radii may be caused by the presence of aerosols lying far above $R_{\\rm pl}$ or by an incorrect estimation of the stellar radius.",
"We note, however, that the atmosphere of planets with a large enough atmospheric mass may stably lie in the boil-off regime, as described above.",
"Figure: Colour-scaled value of Λ\\Lambda as a function of planetary mass and radius for T eq T_{\\rm eq} = 1000 K. The white straight lines indicate equal Λ\\Lambda values given in the plot.",
"The red solid lines indicate lines of equal average densities of 0.6, 1.6, 3.2, and 5.5 g cm -3 ^{-3}.",
"The symbols correspond to the observed (blue bar and arrow) and possible mass-radius combination (black points) for CoRoT-24b.The presence of aerosols may indeed lead to a misinterpretation of the observed transit radius.",
"[29], for example, calculated from first principles the formation of aerosols in the atmosphere of the hot Jupiter HD 189733 b ($T_{\\rm eq}$ $\\approx $ 1000 K, similar to that of the hottest planets considered in this work), obtaining that clouds start forming in the 10–100 $\\mu $ bar pressure range.",
"For the planet considered in Fig.",
"REF ($M_{\\rm pl}$ = 5 $M_{\\oplus }$ ; $R_{\\rm pl}$ = 1.8 $R_{\\oplus }$ ; $T_{\\rm eq}$ = 1000 K), this pressure level corresponds to about 1.2–1.4 $R_{\\rm pl}$ , that is, a radius of 2.2–2.5 $R_{\\oplus }$ or 5.3–9.3 pressure scale heights above $R_{\\rm pl}$ .",
"The presence of high-altitude clouds/hazes in the atmosphere of such a planet would therefore lead to an overestimation of $R_{\\rm pl}$ measured through broad-band optical transit observations of about 20–40%.",
"[29] investigated a hot Jupiter, which has physical characteristics different from those of the planets considered here, but this is what is currently available, showing that similar cloud formation calculations, tuned for lower-mass planets, are clearly needed for a more appropriate interpretation of the results.",
"For hot low-mass planets it is therefore possible to use the $\\Lambda $ $\\ge $ $\\Lambda _{\\rm T}$ condition to constrain the minimum mass, given a certain radius, or maximum radius, given a certain mass.",
"The only assumption is the presence of a hydrogen-dominated atmosphere, which is likely for low-density planets, and an old age (i.e.",
"$>$ 1 Gyr).",
"Most of the extremely low-density planets discovered by Kepler fall into this regime.",
"Figure REF shows the $\\Lambda $ value as a function of planetary mass and radius (at the 100 mbar level) for $T_{\\rm eq}$ = 1000 K. We use the sub-Neptune CoRoT-24b as an example of the constraining power of this plot.",
"CoRoT-24b has a mass lower than 5.7 $M_{\\oplus }$ , a transit radius of 3.7$\\pm $ 0.4 $R_{\\oplus }$ , and an equilibrium temperature of 1070 K [1].",
"CoRoT-24b therefore has a $\\Lambda $ value lower than 10.9, well below $\\Lambda _{\\rm T}$ .",
"For a value of $\\Lambda _{\\rm T}$ of 25 and when we assume that $M_{\\rm pl}$ is equal to 5.7 $M_{\\oplus }$ [27], Fig.",
"REF (bottom black point) indicates that the 100 mbar pressure level, and hence where the transit radius would be if the planet were possessed of a clear atmosphere, lies around 2 $R_{\\oplus }$ ($\\approx $ 1.7 $R_{\\oplus }$ less than the transit radius), in agreement with the detailed analysis of [27].",
"When we instead assume a clear atmosphere, hence $R_{\\rm T}$ = $R_{\\rm 100\\,mbar}$ , $M_{\\rm pl}$ should be $\\gtrapprox 12$ $M_{\\oplus }$ (right black cross in Fig.",
"REF ), although this is unlikely given the non-detection of the planet in the radial-velocity measurements.",
"The atmospheric pressure profile of CoRoT-24b shown by [27] indicates that if we assume that the 100 mbar level lies at 2 $R_{\\oplus }$ , then the transit radius is at a pressure of 1–10 $\\mu $ bar, which is about 10 times smaller than the lowest pressure at which [29] predicts cloud formation.",
"For this particular planet, the most likely scenario is therefore a combined effect of the presence of aerosols and of a slight mass underestimation.",
"Table REF shows that for most of the more massive planets ($M_{\\rm pl}$ $\\gtrapprox $ 10 $M_{\\oplus }$ ) and all the cooler ($T_{\\rm eq}$ $\\lessapprox $ 500 K) ones, the timescale for the atmosphere to evolve out of the boil-off regime is longer than 10 Gyr and in some cases even longer than the main-sequence life time of the host stars.",
"This clearly shows that although the atmosphere of these planets may be in boil-off, the escape rates are not high enough to significantly affect the atmosphere in a short time, in agreement with the results of [20].",
"From the results of Table REF , it follows that in the 5–10 $M_{\\oplus }$ planetary mass and 500–1000 K equilibrium temperature range with increasing temperature and/or decreasing mass the escape rates start affecting the long-term evolution of the atmosphere.",
"This transition region depends not only on the planetary parameters, but also on the stellar properties and orbital separation, which affect the escape rates through the XUV flux and size of the Roche lobe.",
"We will explore this transition region in detail in a forthcoming work."
],
[
"Conclusions",
"We generalised the expression of the Jeans escape parameter to account for hydrodynamic and Roche-lobe effects, which is important for close-in exoplanets.",
"We use a planetary upper atmosphere hydrodynamic code to derive the atmospheric temperature, pressure, and velocity structure of sub-Neptunes with various masses and radii and draw the profiles of the Jeans escape parameter as a function of height.",
"We used our simulations and the generalised Jeans escape parameter to describe the boil-off regime [38], which is characterised by very high escape rates driven by the planet's high thermal energy and low gravity.",
"We introduce the restricted Jeans escape parameter ($\\Lambda $ ) as the value of the Jeans escape parameter calculated at the observed planetary radius and mass for the planet's equilibrium temperature, and considering atomic hydrogen.",
"We used the $L_{\\rm hy}$ /$L_{\\rm en}$ $\\le 1$ empirical condition, where $L_{\\rm en}$ is derived analytically from the energy-limited formula, to estimate $\\Lambda _{\\rm T}$ , the critical value of $\\Lambda $ below which efficient boil-off occurs.",
"We ran simulations with varying planetary mass, stellar mass, and equilibrium temperature, concluding that $\\Lambda _{\\rm T}$ lies between 15 and 35, depending on the system parameters.",
"This result, mostly based on aeronomical considerations, is in agreement with that obtained by [38], namely $R_{\\rm pl}/R_{\\rm B}>0.1$ .",
"From the analysis of our simulations, we find that the atmosphere of hot (i.e.",
"$T_{\\rm eq}$ $\\gtrapprox $ 1000 K) low-mass ($M_{\\rm pl}$ $\\lessapprox $ 5 $M_{\\oplus }$ ) planets with $\\Lambda $ $<$ $\\Lambda _{\\rm T}$ would be unstable against evaporation because they lie in an efficient boil-off regime that would shrink their radius within a few hundreds of Myr.",
"We find the same result also for hot (i.e.",
"$T_{\\rm eq}$ $\\gtrapprox $ 1000 K) higher mass ($M_{\\rm pl}$ $\\lessapprox $ 10 $M_{\\oplus }$ ) planets with $\\Lambda $ $<$ $\\Lambda _{\\rm T}$ , when they orbit M-dwarfs.",
"We conclude that for old hydrogen-dominated planets in this range of parameters, $\\Lambda $ should be $\\ge $ $\\Lambda _{\\rm T}$ , which therefore provides a strong constraint on the planetary minimum mass/maximum radius.",
"This information can be used to predict the presence of high-altitude aerosols on a certain planet without the need to obtain transmission spectra, or inform on the reliability of planetary masses.",
"Our results could also be used to indicate the possible presence of contaminants in the images used to derive the transit light curves, which would lead to the measurement of a planetary radius larger than what is in reality [16].",
"Our results are relevant because of the various present and future ground- and space-based planet-finding facilities (e.g.",
"K2, NGTS, CHEOPS, TESS, PLATO), which will detect sub-Neptunes orbiting bright stars, hence amenable to atmospheric characterisation.",
"Our results will help prioritisation processes: for instance, hot low-density, low-mass planets, with masses measured through radial velocity, are good targets for transmission spectroscopy, but their large radii may be caused by high-altitude clouds, which would therefore obscure the atmospheric atomic and molecular features.",
"An application of our results to the transiting sub-Neptune planets known to date is presented by [15].",
"The simulations presented in this work, only sparsely cover the typical parameter space of the discovered systems hosting sub-Neptunes, also in terms of high-energy stellar flux.",
"In the future, we will extend our work to a larger parameter space and aiming at its more homogeneous coverage.",
"In particular, we will better identify the dependence of the $\\Lambda _{\\rm T}$ value on the planetary (e.g.",
"mass, radius, and temperature/pressure at the lower boundary) and stellar (e.g.",
"mass and high-energy flux) parameters.",
"We acknowledge the Austrian Forschungsförderungsgesellschaft FFG projects “RASEN” P847963 and “TAPAS4CHEOPS” P853993, the Austrian Science Fund (FWF) NFN project S11607-N16, and the FWF project P27256-N27.",
"NVE acknowledges support by the RFBR grant No.",
"15-05-00879-a and 16-52-14006 ANF_a.",
"We thank the anonymous referee for the comments that led to a considerable improvement of the manuscript."
]
] | 1612.05624 |
[
[
"Effective bounds for the number of MMP-series of a smooth threefold"
],
[
"Abstract We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two."
],
[
"Introduction",
"Establishing the existence of minimal models is one of the first steps towards the birational classification of smooth projective varieties.",
"Moreover, starting from dimension three, minimal models are known to be non-unique, leading to some natural questions such as: does a variety admit a finite number of minimal models?",
"And if yes, can we fix some parameters to bound this number?",
"Thanks to the groundbreaking result [1], we know that varieties of general type admit a finite number of minimal models.",
"For varieties of non-general type this number can be infinite, see [13].",
"However, it is conjecture that the number of minimal models up to isomorphism is always finite.",
"This is known for threefolds of positive Kodaira dimension [7].",
"In [12] it is proved that it is possible to bound the number of minimal models of a smooth variety of general type and bounded volume.",
"Moreover, in dimension three Cascini and Tasin [3] bounded the volume using the Betti numbers.",
"This result can be used to show that the number of minimal models of a threefold of general type can be bounded using topological data, solving a conjecture of Cascini and Lazić [2].",
"In this note we address a closely related, although different, question.",
"In the case of a smooth threefold of positive Kodaira dimension we bound in an effective way a subset, obtained under some technical assumptions, of the minimal model programs of $X$ .",
"Specifically, we bound how many are the possible series of $K_X$ -negative birational contractions starting from $X$ .",
"Our main theorem is the following.",
"Theorem 1.1 Let $X$ be a smooth projective threefold of positive Kodaira dimension.",
"Then the number of minimal model programs of $X$ that can be obtained performing first a series of divisorial contractions followed by a series of flips is at most $\\max \\lbrace 1, 3^{\\,(\\rho (X) - 2)}[(\\rho (X) - 2)!",
"]^2\\rbrace $ , where $\\rho (X)$ is the Picard number of $X$ .",
"Theorem REF reduces quickly to finding a bound for the number of possible flipping curves passing through a terminal singularity, see [8].",
"Note that in the case of a smooth surface $S$ of positive Kodaira dimension, the minimal model is unique [8] and, therefore, we cannot have two (-1)-curves $E_1$ and $E_2$ passing through the same point.",
"Indeed, they both should be contracted before reaching the minimal model of $S$ , but the contraction of $C_1$ transforms $C_2$ in a curve with non-negative selfintersection, and so impossible to contract.",
"The assumption on the order of flips and divisorial contractions comes from the fact that we cannot control the number of flipping curves contained in a divisor that is later contracted in the MMP.",
"In the non-general type case there exists an example of a threefold $X$ where this number of curves is infinite, hence producing a new example of a variety with non-negative Kodaira dimension and an infinite number of $K_X$ -negative extremal rays, see [11].",
"In the general-type case this cannot happen because of the finiteness statement proved in [1], and there is still hope to bound the total number of minimal model programs using the topology of the variety.",
"Theorem REF is a technical result, in the sense that in general we cannot impose geometric conditions on $X$ so that the hypothesis is satisfied.",
"However, using Theorem REF we can obtain some effective bounds in the case of Picard number equal to three.",
"Theorem 1.2 Let $X$ be a smooth projective threefold of positive Kodaira dimension of Picard number $\\rho (X) = 3$ , then the number of minimal model programs of $X$ is at most 3.",
"The paper is organized as follows: in Section we collect some preliminary notions, mainly about the MMP in dimension three.",
"The reader in need of more details should refer to [8].",
"In Section we prove Theorem REF and Theorem REF .",
"We conclude with a possible strategy to bound the number of minimal model programs in the case of Picard number equal to four, see Section REF ."
],
[
"Acknowledgements",
"This work is part of my Ph.D. thesis.",
"I would like to thank my supervisor Paolo Cascini for proposing me to work on this problem and for constantly supporting me with his invaluable help and comments.",
"I also would like to thank Ivan Cheltsov, Alessio Corti, Claudio Fontanari, Stefan Schreieder and Luca Tasin for the many fruitful conversations we had about this subject.",
"The results of this paper were conceived when I was supported by a Roth Scholarship."
],
[
"Preliminary Results",
"We will always refer to an MMP for $X$ as a series of $K_X$ -negative birational contractions; in this context, a minimal model for $X$ is just an outcome of an MMP for $X$ .",
"Definition 3.1 Let $f \\colon X \\dashrightarrow Y$ be a birational map, we recall that the exceptional locus of $f$ , that we denote with $\\operatorname{Exc}(f)$ , is the locus of $X$ where $f$ is not an isomorphism."
],
[
"The Picard number",
"Let $X$ be a normal variety.",
"Two Cartier divisors $D_1$ and $D_2$ on $X$ are numerically equivalent, $D_1 \\equiv D_2$ , if they have the same degree on every curve on $X$ , i.e.",
"if $D_1 \\cdot C = D_2 \\cdot C$ for each curve $C$ in $X$ .",
"The quotient of the group of Cartier divisors modulo this equivalence relation is denoted by $N^1(X)$ .",
"We can also define $\\operatorname{N}^1(X)$ as the subspace of cohomology $H^2(X, \\mathbb {Z})$ spanned by divisors.",
"We write $\\operatorname{N}^1(X)_{\\mathbb {Q}} \\mathrel {\\mathop :}=\\operatorname{N}^1(X) \\otimes _{\\mathbb {Z}} \\mathbb {Q}$ .",
"$\\operatorname{N}^1(X)_{\\mathbb {Q}}$ is a finite dimensional vector space.",
"Definition 3.2 We define $\\rho (X) \\mathrel {\\mathop :}=\\dim _{\\mathbb {Q}} \\operatorname{N}^1(X)_{\\mathbb {Q}}$ and we call it the Picard number of $X$ .",
"We remark that $\\rho (X) \\le b_2$ , the second Betti number of $X$ , that depends only on topological information of $X$ .",
"Similarly, two 1-cycles $C_1$ and $C_2$ are numerically equivalent if they have the same intersection number with any Cartier divisor.",
"We call $N_1(X)$ the quotient group and we write $\\operatorname{N}_1(X)_{\\mathbb {Q}} \\mathrel {\\mathop :}=\\operatorname{N}_1(X) \\otimes _{\\mathbb {Z}} \\mathbb {Q}$ .",
"We can also see $N_1(X)$ as the subspace of homology $H_2(X, \\mathbb {Z})$ spanned by algebraic curves.",
"See for details [5].",
"We will also use the following defintion.",
"Definition 3.3 Let $E$ be an irreducible divisor contained in a projective variety $X$ .",
"We consider the following map $\\psi : \\operatorname{N}_1(E)_{\\mathbb {Q}} \\longrightarrow \\operatorname{N}_1(X)_{\\mathbb {Q}}$ and we denote with $N_1(E|X)_{\\mathbb {Q}} \\mathrel {\\mathop :}=\\psi (\\operatorname{N}_1(E)_{\\mathbb {Q}}) \\subseteq \\operatorname{N}_1(X)_{\\mathbb {Q}}$ , and $\\rho (E|X) \\mathrel {\\mathop :}=\\dim N_1(E|X)_{\\mathbb {Q}}$ .",
"Note that $\\ker {\\psi }$ might be not empty."
],
[
"The difficulty",
"In dimension three, the existence and termination of flips was proved by Mori and Shokurov.",
"A key ingredient for the proof of termination is the so called difficulty of $X$ , introduced by Shokurov.",
"Definition 3.4 [14], [8] Let $X$ be a projective threefold, then the difficulty of $X$ is defined as follows $d(X) \\mathrel {\\mathop :}=\\#\\lbrace E \\text{ prime divisor } | \\\\\\ a(E, X) < 1, E \\text{ is exceptional over } X\\rbrace ,$ where $a(E, X)$ is the discrepancy of $E$ with respect of $X$ .",
"Remark 3.5 Note that the difficulty always goes down under a flip, and if $X$ is smooth, then $d(X) = 0$ and we cannot have any flips.",
"See [8].",
"We also recall that minimal models are connected by a sequence of flops, i.e.",
"by an isomorphism in codimension one, [9].",
"For the definitions of divisorial contraction, flip and flop we refer to [8] ,[8].",
"We just recall that under a divisorial contraction $f \\colon X \\dashrightarrow X^{\\prime }$ the Picard number drops by one, i.e $\\rho (X^{\\prime }) = \\rho (X) - 1$ ; if $f$ is a flip instead, the Picard number is stable: $\\rho (X^{\\prime }) = \\rho (X)$ ."
],
[
"Proof of Theorem ",
"In this section, we prove Theorem REF and Theorem REF .",
"The strategy of the proof is to first bound the number of steps of the MMP and then count how many are the possible divisorial contractions to a point, to a curve and flips at a certain step.",
"Our starting point is a smooth projective threefold $X$ of positive Kodaira dimension.",
"As we recalled in Remark REF , this means that the difficulty of $X$ has to be zero and no flips are possible.",
"Then, the first operation of the MMP for $X$ has to be a divisorial contraction.",
"Let us assume that $\\rho (X) = 2$ (if $\\rho (X) = 1$ no contractions are possible).",
"Lemma 4.1 Let $X$ be a smooth projective threefold of positive Kodaira dimension such that $\\rho (X) = 2$ .",
"Then there is only a unique MMP for $X$ .",
"After one divisorial contraction $\\phi ^{\\prime }$ we reach a variety $X^{\\prime }$ with Picard number equal to one and we stop.",
"If there were an other divisorial contraction $\\phi ^{\\prime \\prime }$ onto a different variety $X^{\\prime \\prime }$ , we would have a sequence of flops connecting the minimal varieties $X^{\\prime }$ and $X^{\\prime \\prime }$ , see [9], but that is impossible since $\\rho (X^{\\prime }) = 1$ and so no contractions are possible.",
"Remark 4.2 Therefore, we can always assume $\\rho (X) \\ge 3$ and that $\\rho (X^{\\prime }) \\ge 2$ for $X^{\\prime }$ a minimal model for $X$ , since otherwise we have only one possible MMP.",
"Let us proceed now with the bound for the number of steps.",
"It is a calculation that follows from the termination of flips in dimension three, see [4].",
"Lemma 4.3 Let $X$ be a smooth projective threefold of positive Kodaira dimension such that $\\rho (X) \\ge 3$ .",
"Let $X^{\\prime }$ be the outcome of a MMP for $X$ , we suppose in addition that $\\rho (X^{\\prime }) \\ge 2$ .",
"Let $\\mathcal {S}$ be the total number of steps of a minimal model program of $X$ .",
"Then $\\mathcal {S}$ is at most $2(\\rho (X) - 2)$ .",
"We denote with $\\mathcal {D}_C$ the total number of divisorial contractions and with $\\mathcal {F}$ the total number of flips.",
"Let $X^{\\prime }$ be the outcome of an MMP for $X$ .",
"Clearly $\\mathcal {S} = \\mathcal {D}_C + \\mathcal {F}$ .",
"Under a divisorial contraction the Picard number drops by one.",
"Hence, $\\mathcal {D}_C = \\rho (X) - \\rho (X^{\\prime }) \\le \\rho (X) - 2$ .",
"To conclude the proof, we claim that $\\mathcal {F} \\le \\mathcal {D}_C$ .",
"Under a flip, the Picard number is stable and we need to consider the difficulty $d(X)$ , see Definition REF .",
"If $X$ is smooth, $d(X) = 0$ and no flips are possible, see Remark REF .",
"Moreover, if $X_{i-1} \\rightarrow X_i$ is a divisorial contraction, then $d(X_i) \\le d(X_{i - 1}) + 1,$ since the contraction might have created some singularities.",
"Otherwise, if $X_{i-1} \\dashrightarrow X_i$ is a flip, then $d(X_i) \\le d(X_{i - 1}) - 1,$ because flips strictly improve the singularities (see [8]).",
"We conclude that in order to have a flip, we first need to have had a divisorial contraction.",
"Thus, $ \\mathcal {F} \\le \\mathcal {D}_C$ and $\\mathcal {S} \\le 2(\\rho (X) - 2)$ .",
"Let $X$ be a smooth threefold of positive Kodaira dimension, satisfying all the assumptions of Theorem REF and Lemma REF .",
"Let $X^{\\prime }$ be the outcome of an MMP $\\phi ^{\\prime }$ for $X$ composed by a series of divisorial contractions followed by a series of flip.",
"Then we can represent $\\phi ^{\\prime }$ in the following way.",
"$\\phi ^{\\prime } \\colon X = \\underbrace{X_0 \\rightarrow X_1 \\rightarrow \\dots \\rightarrow X_i}_\\text{divisorial contractions} = \\underbrace{X^0 \\dashrightarrow X^1 \\dashrightarrow \\dots \\dashrightarrow X^j}_\\text{flips} = X^{\\prime }.$ We will always indicate with $X_i$ , $0\\le i \\le \\rho (X) - 2$ , a step in the minimal model program for $X$ that can be reached from $X$ with a series of divisorial contractions; with $X^j$ , $0\\le j \\le \\rho (X) - 2$ , a step in the minimal model program for $X$ from which we can reach the minimal model $X^{\\prime }$ , with a series of flips.",
"We now proceed to bound the number of divisorial contractions to a point.",
"Lemma 4.4 Let $X = X_0$ be a smooth projective threefold of positive Kodaira dimension satisfying the assumptions of Lemma REF and Theorem REF .",
"Let $X_i$ be a step in the minimal model program for $X$ , $1 \\le i \\le \\rho (X) - 2$ .",
"The number of ways to go from $X_i$ to the following step with a divisorial contraction to a point is at most $\\rho (X) - 2 - i$ .",
"We divide the proof into steps.",
"Step 1.",
"We consider the last divisorial contraction.",
"$\\psi \\colon X_{i-1} \\rightarrow X_i = X^0$ and we claim that there is a unique choice of divisor $E$ that can be contracted to a point by $\\psi $ .",
"In particular, $\\psi $ is uniquely determined by $X_{i-1}$ .",
"Let us assume by contradiction that there are two distinct divisors $E_1$ and $E_2$ contained in $X_{i-1}$ and that they can be both contracted to a point.",
"${5pc}{X_{i-1} [dr]_{\\psi ^2} [r]^{\\psi ^1} & X_i^{\\prime } \\\\& X_i^{\\prime \\prime } \\\\}$ We denote with $\\psi ^1$ the contraction of $E_1$ and with $\\psi ^2$ the contraction of $E_2$ .",
"But then, $X_i^{\\prime }$ contains $\\psi ^1(E_2)$ and $X_i^{\\prime \\prime }$ contains $\\psi ^2(E_1)$ , as divisors.",
"Since $X_i^{\\prime }$ and $X_i^{\\prime \\prime }$ are followed only by flips, $\\psi ^1(E_2)$ and $\\psi ^2(E_1)$ are not contracted and survive until the minimal models.",
"${5pc}{X_{i-1} [dr]_{\\psi ^2} [r]^{\\psi ^1} & X_i^{\\prime } @{-->}[r] & X^{\\prime } @{-->}[d]^{\\eta _{12}} \\\\& X_i^{\\prime \\prime } @{-->}[r] & X^{\\prime \\prime } \\\\}$ We have that $X^{\\prime } \\supseteq \\psi _1(E_2)$ and $X^{\\prime \\prime } \\supseteq \\psi _2(E_1)$ , (where by abuse of notation with still indicate with $\\psi _1(E_2)$ and $\\psi _2(E_1)$ the images of the divisors through the series of flips).",
"However, $X^{\\prime }$ and $X^{\\prime \\prime }$ are minimal models and are, therefore, connected by a sequence of flops $\\eta _{12}$ , i.e.",
"an isomorphism in codimension one, [9].",
"We reach a contradiction.",
"Step 2.",
"Let us now consider the preceding step.",
"In this case we have at least two divisors $E_1$ and $E_2$ that can be contracted to a point.",
"${5pc}{X_{i-2} [dr]_{\\psi _{i-1}^{\\prime \\prime }} [r]^{\\psi _{i-1}^{\\prime }} & X_{i-1}^{\\prime } [r]^{\\psi _i^{\\prime }} & X_i^{\\prime } \\\\& X_{i-1}^{\\prime \\prime } [r]_{\\psi _i^{\\prime \\prime }} & X_i^{\\prime \\prime } \\\\}$ There are just two possibilities, we can either contract first $E_1$ with $\\psi _{i-1}^{\\prime }$ and then $E_2$ with $\\psi _i^{\\prime }$ or we can invert the order and contract first $E_2$ with $\\psi _{i-1}^{\\prime \\prime }$ and then $E_1$ with $\\psi _i^{\\prime \\prime }$ .",
"We claim that there are no more possible divisors that can be contracted into a point.",
"Let us assume by contradiction that there exists an other divisor $E_3$ , such that $E_3$ is distinct from $E_1$ and $E_2$ and that can be contracted into a point by a divisorial contraction that we call $\\psi _{i-1}^{\\prime \\prime \\prime } \\colon X_{i-2} \\rightarrow X_{i-1}^{\\prime \\prime \\prime }$ .",
"Since $E_3$ is not contracted by $\\psi _{i-1}^{\\prime }$ and $\\psi _i^{\\prime }$ , $\\psi _{i}^{\\prime }(\\psi _{i-1}^{\\prime }(E_3))$ is contained in $X_i^{\\prime }$ and so also in the minimal model $X^{\\prime }$ because $X_i^{\\prime }$ is followed just by flips.",
"If we consider instead the minimal model $X^{\\prime \\prime \\prime }$ that follows $X_{i-1}^{\\prime \\prime \\prime }$ , in particular this means that $X^{\\prime \\prime \\prime }$ is obtained contracting $K_{X_{i-1}^{\\prime \\prime \\prime }}$ -negative extremal rays, $X^{\\prime \\prime \\prime }$ is not going to contain the image of $E_3$ .",
"But since $X^{\\prime }$ and $X^{\\prime \\prime \\prime }$ are minimal models and so connected by an isomorphism in codimension one, we reach a contradiction.",
"Step 3.",
"At the step $X_i$ , for $0 \\le i \\le \\rho (X) - 2$ , the number of ways to go from $X_i$ to the following step with a divisorial contraction to a point is at most $\\rho (X) - 2 - i$ .",
"We can iterate the argument of Step 2 in the case of a series of divisorial contractions.",
"Let $X_i$ be a step of the minimal model for $X$ , there are at least $\\rho (X) - 2 - i$ choices for a divisor to be contracted to a point.",
"If there were at least $\\rho (X) - 1 - i$ choices, there would be a divisor $E$ that survives until we reach the minimal model $X^{\\prime }$ .",
"But since at the step $X_i$ the divisor $E$ can be contracted, there exists another MMP $\\phi ^{\\prime \\prime }$ such that $\\phi ^{\\prime \\prime }$ contracts $E$ and again we reach a contradiction since minimal models are isomorphic in codimension one.",
"We conclude that the number of possible contractions to a point at the step $X_i$ is at most $\\rho (X) - 2 - i$ .",
"This conclude the proof of the lemma.",
"We want now to count how many are the possible choices for a divisorial contraction to a curve.",
"Lemma 4.5 Let $X = X_0$ be a smooth projective threefold of positive Kodaira dimension satisfying the assumptions of Lemma REF and Theorem REF .",
"Let $X_i$ be one step in the minimal model program for $X$ , $0 \\le i \\le \\rho (X) - 2$ .",
"The number of ways to go from $X_i$ to the following step with a divisorial contraction to a curve is at most $2(\\rho (X) - 2 - i)$ .",
"Again we divide the proof into steps.",
"Step 1.",
"We consider the last divisorial contraction.",
"$\\psi \\colon X_{i-1} \\rightarrow X_i = X^0$ and we claim that there are at most two divisorial contractions to a curve.",
"Proceeding as we did in Step 1 of Lemma REF , we can prove that there is a unique choice for a divisor $E$ to be contracted.",
"Then we need to understand in how many ways the divisor $E$ can be contracted into a curve $C$ .",
"We know that $\\rho (C)=1$ because two divisors on a curve are numerically equivalent if they have the same degree, then we obtain that $\\rho (E|X)=2$ , see Definition $\\ref {def:rhoRestr}$ , because $\\rho (X_{i-1}) - \\rho (X_{i}) = 1$ , since $\\psi _i$ is a divisorial contraction.",
"We then obtain two possible contractions: $\\psi ^1$ that contracts $E$ into a curve $C_1$ and $\\psi ^2$ that contracts $E$ into a curve $C_2$ .",
"${5pc}{X_{i-1} [dr]_{\\psi ^2} [r]^{\\psi ^1} & X_i^{\\prime } @{-->}[d]^{\\eta _{12}} \\\\& X_i^{\\prime \\prime } \\\\}$ Step 2.",
"Let $X_i$ be one step in the minimal model program for $X$ , $0 \\le i \\le \\rho (X) - 2$ .",
"The number of ways to go from $X_i$ to the following step with a divisorial contraction to a curve is at most $2(\\rho (X) - 2 - i)$ .",
"Let $X_i$ be a step of the minimal model for $X$ , there are at least $\\rho (X) - 2 - i$ choices for a divisor to be contracted.",
"If there were at least $\\rho (X) - 1 - i$ choices, there would be a divisor $E$ that survives until we reach the minimal model $X^{\\prime }$ .",
"But since at the step $X_i$ the divisor $E$ can be contracted, there exists another MMP $\\phi ^{\\prime \\prime }$ such that $\\phi ^{\\prime \\prime }$ contracts $E$ and again we reach a contradiction since minimal models are isomorphic in codimension one.",
"Moreover, each of this divisor can be contracted in at most two different ways as we explained in Step 1.",
"We conclude that the number of possible contractions to a point at the step $X_i$ is at most $2(\\rho (X) - 2 - i)$ .",
"This conclude the proof of the Lemma.",
"Remark 4.6 The simplest example of the situation in Step 1 of Lemma REF is the case of Atiyah's flop, see for instance [6], where $E \\cong \\mathbb {P}^1 \\times \\mathbb {P}^1$ and the map $\\eta _{12}$ between $X_i^{\\prime }$ and $X_i^{\\prime \\prime }$ is the flop that sends $C_1 \\cong \\mathbb {P}^1$ into $C_2 \\cong \\mathbb {P}^1$ .",
"In conclusion, we have obtained the following lemma.",
"Lemma 4.7 Let $X = X_0$ be a smooth projective threefold of positive Kodaria dimension satisfying the assumptions of Lemma REF and Theorem REF .",
"Let $X_i$ be a step in the minimal model program for $X$ , $0 \\le i \\le \\rho (X) - 2$ .",
"The number of ways to go from $X_i$ to the following step with a divisorial contraction is at most $3(\\rho (X) - 2 - i)$ .",
"The difficult part is to bound the number of possible ways to go from one step to the following with a flip.",
"Proposition 4.8 Let $X = X_0$ be a smooth projective threefold of positive Kodaira dimension satisfying the assumptions of Lemma REF and Theorem REF .",
"Let $X^j$ be a step in the minimal model program for $X$ , $0 \\le j \\le \\rho (X) - 2$ .",
"Then the number of ways to go from $X^j$ to the following step with a flip is at most $\\rho (X) - 2 - j$ .",
"We divide the proof into steps.",
"Step 1.",
"We consider the last flip in the minimal model program for $X$ .",
"$\\psi : X^{j-1} \\dashrightarrow X^j = X^{\\prime }$ where $0 \\le j \\le \\rho (X)-2$ .",
"and we claim that there is a unique choice of curve $\\xi $ that can be flipped by $\\psi $ .",
"Assume by contradiction that there are two possible flips into two distinct minimal models $X^{\\prime }$ and $X^{\\prime \\prime }$ .",
"${5pc}{X^{j-1} @{-->}[dr]^{\\psi ^2} @{-->}[r]^{\\psi ^1} & X^{\\prime } @{-->}[d]^{\\eta _{12}} \\\\& X^{\\prime \\prime } \\\\}$ Where we denoted by $\\eta _{12}$ the sequence of flops connecting the two minimal models $X^{\\prime }$ and $X^{\\prime \\prime }$ (see [9]) and by $\\xi ^2$ the curve that is flipped by $\\psi ^2$ .",
"Now, thanks to the Abundance Theorem [10], there exists an integer $m$ such that $|mK_{X^{\\prime }}|$ is free.",
"Therefore, we can choose a general surface $ S \\in |mK_{X^{\\prime }}|$ such that it does not contain any irreducible components of $\\operatorname{Exc}(\\eta _{12})$ .",
"Then let $S_0 \\mathrel {\\mathop :}=(\\psi _{1}^{-1})_*(S)$ be the strict transform of $S$ , since flips are isomorphisms in codimension one, $S_0 \\in |mK_{X^{j-1}}|$ .",
"We are assuming that there exists another flipping curve $\\xi _2$ such that $\\xi _1 \\ne \\xi _2$ and so $\\xi _2 \\nsubseteq \\operatorname{Exc}(\\psi ^1)$ .",
"Since $\\xi _2$ is a flipping curve, $K_{X^{j-1}} \\cdot \\xi _2 < 0$ , and so $\\xi _2 \\subseteq S_0$ .",
"Now we consider the restriction of $\\psi ^1$ to $S_0$ $g \\mathrel {\\mathop :}=\\psi ^1_{|S_0} \\colon S_0 \\dashrightarrow S$ and since $\\operatorname{Exc}(g) \\subseteq \\operatorname{Exc}(\\psi ^1) \\cap S_0$ , we obtain that $\\xi _2 \\nsubseteq \\operatorname{Exc}(g)$ and so $\\psi ^1(\\xi _2) \\subseteq S$ .",
"This is a contradiction, because $\\psi ^1(\\xi _2)$ is flopped by $\\eta _{12}$ into $\\psi ^2(\\xi _1)$ but $S$ was chosen in such a way that it does not contain any irreducible components of $\\operatorname{Exc}(\\eta _{12})$ .",
"Step 2.",
"Let $X^j$ be a step in the minimal model program for $X$ , $0 \\le j \\le \\rho (X) - 2$ .",
"Then the number of ways to go from $X^j$ to the following step with a flip is at most $\\rho (X) - 2 - j$ .",
"We can iterate the argument of Step 1 in the case of a series of flips.",
"Let $X_i$ be a step of the minimal model for $X$ , there are at least $\\rho (X) - 2 - i$ choices for possible flipping curves.",
"If there were at least $\\rho (X) - 1 - i$ choices, there would be a curve $\\xi $ that survives until we reach the minimal model $X^{\\prime }$ and so is contained in the surface $S$ chosen in (REF ).",
"But since at the step $X_i$ the curve $\\xi $ is a flipping curve, there exists another MMP $\\phi ^{\\prime \\prime }$ that flips $\\xi $ and again we reach a contradiction because $\\xi $ would be contained in $\\operatorname{Exc}({\\eta _{12}})$ .",
"We conclude that the number of possible flips at the step $X_i$ is at most $\\rho (X) - 2 - i$ .",
"This conclude the proof of the lemma.",
"Now we have all the ingredients to prove Theorem REF .",
"Let $X^{\\prime }$ be the outcome of an MMP $\\phi ^{\\prime }$ for $X$ .",
"If $\\rho (X) = 2$ or $\\rho (X^{\\prime }) = 1$ , then $X^{\\prime }$ has to be unique, see Remark REF , Lemma REF .",
"Otherwise, we are in the condition of Lemma REF .",
"Then the proof is elementary combinatorics.",
"After the sequence of divisorial contractions, using Lemma REF we have $3^{\\,(\\rho (X) - 2)}(\\rho (X) - 2)!$ end points.",
"Then after the sequence of flips, thanks to Proposition REF , we have the final number of minimal model programs: $3^{\\,(\\rho (X) - 2)}[(\\rho (X) - 2)!",
"]^2$ ."
],
[
"Bounds for low Picard number",
"In this section we apply Theorem REF to obtain explicit bounds for the number of minimal model programs in the case of threefolds of low Picard rank.",
"We will use explanatory graphs for the MMP that can be read as follows: $X^{a,b}$ denotes a variety $X$ such that $\\rho (X) = a$ and $d(X) = b$ , see Definition REF and REF .",
"Divisorial contractions are going to be denoted by continue arrows, and flips by dash arrows.",
"Let $X$ be a smooth projective threefold of positive Kodaira dimension, such that $\\rho (X) = 3$ .",
"We are in the conditions to apply Theorem REF .",
"Indeed, let $X^{\\prime }$ be an outcome of an MMP $\\phi ^{\\prime }$ for $X$ .",
"We can assume that $\\rho (X^{\\prime }) \\ge 2$ , because otherwise $X^{\\prime }$ is unique, see Remark REF .",
"In this case the graph of the MMP for $X$ is extremely simple: the first operation is a divisorial contraction $X^{3,0} \\rightarrow X^{2,\\le 1}.$ Then we can only have a flip $X^{2,\\le 1} \\dashrightarrow X^{2,0}$ and we reach an end point.",
"Hence, the condition of Theorem REF are satisfied and we can conclude that the number of minimal model programs for $X$ are at most three.",
"The following is an explicit graph of the MMP for $X$ in this case, ${2.5pc}{& X^{3,0} [dl] [d] [dr] \\\\X^{2,\\le 1} @{-->}[d] & X^{2,\\le 1} @{-->}[d] & X^{2, \\le 1} @{-->}[d]\\\\X^{2,0} & X^{2,0} & X^{2,0} \\\\}$ where after the first divisorial contraction we reach varieties characterized by Picard number equal to two and difficulty less or equal one, and then after the flip we stopped having reached varieties with difficulty equal to zero and Picard number equal to two."
],
[
"Strategy for $\\rho (X) = 4$",
"In this last section we present a strategy to find an explicit bound in the case of Picard number equal to four and we highlight the main difficulties.",
"Let $X$ be a smooth projective threefold of positive Kodaira dimension such that $\\rho (X) = 4$ .",
"Again we assume that if $X^{\\prime }$ is the outcome of an MMP $\\phi ^{\\prime }$ for $X$ , then $\\rho (X^{\\prime }) \\ge 2$ .",
"The situation is more complicated.",
"The graph in this case can be described in the following way ${2.5pc}{& X^{4,0} [d]^6 \\\\& X^{3,\\le 1} [dl]^3 @{-->}[dr]^?\\\\X^{2,\\le 2} @{-->}[d]^2 & & X^{3,0} [d]^3 \\\\X^{2,\\le 1} @{-->}[d]^1 & & X^{2,\\le 1} @{-->}[d]^1 \\\\X^{2,0} & &X^{2,0}\\\\}$ The numbers at the right of the arrows represent the valence of the arrow in the graph, i.e.",
"in how many ways can be realized the operation corresponding to that arrow.",
"The valence is computed applying all the results of the previous section.",
"In order to count the number of end points, we need to compute the valence of the missing arrow that corresponds to a flip followed by a divisorial contraction, and so we can not conclude using Proposition REF .",
"Let us fix some notation $X^{3,\\le 1} \\stackrel{\\phi _1}{\\dashrightarrow } X^{3,0} \\stackrel{\\phi _2}{\\rightarrow } X^{2,\\le 1}$ where $\\phi _1$ is a flip and $\\phi _2$ is a divisorial contractions.",
"Let $\\xi \\subseteq X^{3,\\le 1}$ be the flipping curve.",
"Let $\\xi ^+ \\subseteq X^{3,0}$ be the flipped curve.",
"Let $E \\subseteq X^{3,\\le 1}$ be the divisor that is contracted by $\\phi _2$ .",
"Let $E^+ \\subseteq X^{3,0}$ be the image of the divisor through the flip $\\phi _1$ .",
"Lemma 4.9 If $\\xi \\nsubseteq E$ , then the flip can be realized in at most two ways.",
"We consider the surface $S \\in |mK_{X^{2,0}}|$ , for $m>0$ as defined in (REF ), and its strict transform $S^{\\prime } \\subseteq X^{2,\\le 1}$ , since flips do not change divisors $S^{\\prime } \\in |mK_{X^{2,\\le 1}}|$ .",
"We have that $mK_{X^{3,\\le 1}} = mK_{X^{2,\\le 1}} + \\beta E$ , where $\\beta > 0$ , because $X^{2,\\le 1}$ is terminal, if we choose $S_1$ a general element in $|mK_{X^{3,\\le 1}}|$ , then $S_1 = (\\phi _1^{-1})_*S^{\\prime } + \\beta E$ .",
"Since $\\xi \\cdot K_{X^{3, \\le 1}} < 0$ , because $\\xi $ is a flipping curve, $\\xi \\subseteq S_1$ .",
"But we are assuming that $\\xi \\nsubseteq E$ , so this forces $\\xi \\in (\\phi _1^{-1})_*S^{\\prime }$ , but then we can conclude thanks to the same argument of Proposition REF .",
"The major problem if $\\xi \\subseteq E$ is that $E$ is not normal.",
"Indeed, since the discrepancies are increasing under a flip, thanks to [8] the multiplicity of $E$ along $\\xi $ is greater than one.",
"Then $E$ is singular along $\\xi $ and so not normal, since $E$ is a surface."
]
] | 1612.05449 |
[
[
"Reinforcement Learning Using Quantum Boltzmann Machines"
],
[
"Abstract We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks.",
"We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system.",
"We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the first and last layers of the deep network.",
"In absence of a transverse field, our simulations show that DBMs are trained more effectively than restricted Boltzmann machines (RBM) with the same number of nodes.",
"We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field for reinforcement learning.",
"This method also outperforms the reinforcement learning method that uses RBMs."
],
[
"[ Reinforcement Learning Using Quantum Boltzmann Machines Daniel Crawford$^1$Anna Levit$^1$Navid Ghadermarzy$^{1,2}$Jaspreet S. Oberoi$^{1,3}$Pooya Ronagh$^{1,2,\\ast }$ ] $^1$ 1QB Information Technologies (1QBit), Vancouver, British Columbia, V6C 2B5, Canada $^2$ Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada $^3$ School of Engineering Science, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada $^\\ast $ corresponding author: [email protected]"
],
[
"Introduction",
"In view of recent advancements in the manufacturing of superconducting qubits [1], [2] and systems of qubits with second-order interactions, an important question for quantum computation is the existence of quantum supremacy (e.g., in [3]), that is, whether near-term quantum devices can outperform classical computers in some—in fact, any—computational task.",
"With this motivation, we consider reinforcement learning as the computational task of interest, and design a method of reinforcement learning using a layout of quantum bits similar to that of a deep Boltzmann machine (DBM) (see Fig.",
"REF for a graphical representation).",
"We use simulated quantum annealing (SQA) to demonstrate the advantage of reinforcement learning using quantum Boltzmann machines over its classical counterpart, in small problem instances.",
"Figure: NO_CAPTIONReinforcement learning ([4], known also as neuro-dynamic programming [5]) is an area of optimal control theory at the intersection of approximate dynamic programming and machine learning.",
"It has been used successfully for many applications, in fields such as engineering [6], [7], sociology [8], [9], and economics [10], [11].",
"It is important to differentiate between reinforcement learning and common streams of research in machine learning.",
"For instance, in supervised learning, the learning is facilitated by training samples provided by a source external to the agent and the computer.",
"In reinforcement learning, the training samples are provided only by the interaction of the agent itself with the environment.",
"For example, in a motion planning problem in an uncharted territory, it is desired that the agent learns in the fastest possible way to correctly navigate with the least number of blind decisions required to be taken.",
"This is known as the dilemma of exploration versus exploitation; that is, neither exploration nor exploitation can be pursued exclusively without facing a penalty or failing at the task.",
"The goal is hence not just to design an algorithm that eventually converges to an optimal policy, but for it to be able to generate good policies early in the learning process.",
"We refer the reader to [4] for a thorough introduction to the use cases and problem scenarios addressed by reinforcement learning.",
"The core idea in reinforcement learning is defining an operator on the Banach space of real-valued functions on the set of states of a system such that a fixed point of the operator carries information about an optimal policy of actions for a finite or infinite number of decision epochs.",
"A numerical method for computing this fixed point is to explore this function space by travelling in a direction that minimizes the distance between two consecutive applications of the contraction mapping operator [5].",
"This optimization task, called learning in the context of reinforcement learning, can be performed by locally parametrizing the above function space using a set of auxiliary variables, and applying a gradient method to these variables.",
"One approach for such a parametrization, due to [12], is to use the weights of a restricted Boltzmann machine (RBM) (see Fig.",
"REF ) as the parameters, and the free energy of the RBM as an approximator for the elements in the function space.",
"The descent direction is then calculated in terms of the expected values of the nodes of the RBM.",
"It follows from the universal approximation theorem [13] that RBMs can approximate any joint distribution over binary variables [14], [15].",
"However, in the context of reinforcement learning, RBMs are not necessarily the best choice for approximating Q-functions relating to Markov decision processes because [14] and [15] show that RBMs may require an exponential number of hidden variables with respect to the number of visible variables in order to approximate the desired joint distribution.",
"On the other hand, DBMs have the potential to model higher-order dependencies than RBMs, and are more robust than deep belief networks [16].",
"One may, therefore, consider replacing the RBM with other graphical models and investigating the performance of the models in the learning process.",
"Except in the case of RBMs, calculating statistical data from the nodes of a graphical model amounts to sampling from a Boltzmann distribution, creating a bottleneck in the learning procedure if performed classically [17].",
"Figure: NO_CAPTIONAs we explain in what follows, DBMs are good candidates for reinforcement learning tasks.",
"Moreover, an important advantage of a DBM layout for a quantum annealing system is that the proximity and couplings of the qubits in the layout are similar to those of a sequence of bipartite blocks in D-Wave devices [18], and it is therefore feasible that such layouts could be manufactured in the near future.",
"This is why, instead of attempting to embed a Boltzmann machine structure on an existing quantum annealing system as in [19], [20], [21], [22], we work under the assumption that the network itself is the native connectivity graph of a near-future quantum annealer, and, using numerical simulations, we attempt to understand its applicability to reinforcement learning.",
"Quantum Monte Carlo (QMC) numerical simulations have been found to be useful in simulating time-dependant quantum systems.",
"Simulated quantum annealing (SQA) [23], [24], one of the many flavours of QMC methods, is based on the Suzuki–Trotter expansion of the path integral representation of the Hamiltonian of Ising spin models in the presence of a transverse field driver Hamiltonian.",
"Even though the efficiency of SQA for finding the ground state of an Ising model is topologically obstructed [25], we consider the samples generated by SQA to be good approximations of the Boltzmann distribution of the quantum Hamiltonian [26].",
"Experimental studies have shown similarities in the behaviour of SQA and that of quantum annealing [27], [28] and its physical realization [29], [30] by D-Wave Systems.",
"We expect that when SQA is set such that the final strength of the transverse field is negligible, the distribution of the samples approaches the classical limit one expects to observe in absence of the transverse field.",
"The classical counterpart of SQA is conventional simulated annealing (SA), which is based on thermal annealing.",
"This algorithm can be used to create Boltzmann distributions from the Ising spin model only in the absence of a transverse field.",
"It should, therefore, be possible to use SA or SQA to approximate the Boltzmann distribution of a classical Boltzmann machine.",
"However, unlike SA, it is possible to use SQA not only to approximate the Boltzmann distribution of a classical Boltzmann machine, but also that of a graphical model in which the energy operator is a quantum Hamiltonian in the presence of a transverse field.",
"These graphical models, called quantum Boltzmann machines (QBM), were first introduced in [31].",
"We use SQA simulations to demonstrate evidence that a quantum annealing device that approximates the distribution of a DBM or a QBM may improve the learning process compared to a reinforcement learning method that uses classical RBM techniques.",
"Other studies have shown that SQA is more efficient than thermal SA [23], [24].",
"Therefore, our method in conjunction with SQA can also be viewed as a quantum-inspired approach for reinforcement learning.",
"What distinguishes our work from current trends in quantum machine learning is that (i) we consider the use of quantum annealing in reinforcement learning applications rather than frequently studied classification or recognition problems; (ii) using SQA-based numerical simulations, we assume that the connectivity graph of a DBM directly maps to the native layout of a feasible quantum annealer; and (iii) the results of our experiments using SQA to simulate the sampling of an entangled system of spins suggest that using quantum annealers in reinforcement learning tasks has an advantage over thermal sampling."
],
[
"Results",
"Maze traversal is a problem typically used to develop and benchmark reinforcement learning algorithms [32].",
"A maze is structured as a two-dimensional grid of $r$ rows and $c$ columns in which a decision-making agent is free to move up, down, left, right, or stand still.",
"During this maze traversal, the agent encounters obstacles (e.g., walls), rewards (e.g., goals), and penalties (negative rewards, e.g., a pit).",
"Each cell of the maze can contain either a deterministic or stochastic reward, a wall, a pit, or a neutral value.",
"Fig.",
"REF and REF show examples of two mazes.",
"Fig.",
"REF shows the corresponding solutions to the maze in Fig.",
"REF .",
"The goal of the reinforcement learning algorithm in the maze traversal problem is for the agent to learn the optimal action to take in each cell of the maze by maximizing the total reward, that is, finding a route across the maze that avoids walls and pits while favouring rewards.",
"This problem can be modelled as a Markov decision process (MDP) determined by the following components (see Sec.",
"for an overview on MDP formulations).",
"[leftmargin=1pc, itemsep=-0pt] The state of the system is the agent's position within the maze.",
"The position state $s$ takes values in the set of states $S = \\lbrace 1,...,r\\rbrace \\times \\lbrace 1,...,c\\rbrace .$ In any state, the agent can decide to take one of the five actions $a \\in \\lbrace \\uparrow , \\downarrow , \\leftarrow , \\rightarrow , \\circlearrowleft \\rbrace .$ These actions will guide the agent through the maze.",
"An action that leads the agent into a wall ($W$ ) or outside of the maze boundary is treated as an inadmissible action.",
"Each action can be viewed as an endomorphism on the set of states $a: S \\rightarrow S.$ If $a=$ $\\circlearrowleft $ , then $a(s) = s$ ; otherwise, $a (s)$ is the state adjacent to $S$ in the direction shown by $a$ .",
"We do not consider training samples where $a$ is inadmissible.",
"The transition kernel determines the probability of the agent moving from one state to another given a particular choice of action.",
"In the simplest case, the probability of transition from $s$ to $a(s)$ is one: $\\mathbb {P}(a(s)|s,a)=1.$ We call the maze clear if the associated transition kernel is as above, as opposed to the windy maze, in which there is a nonzero probability that if the action $a$ is taken at state $s$ , the next state will differ from $a(s)$ .",
"The immediate reward $r(s, a)$ that the agent gains from taking an action $a$ in state $s$ is the value contained in the destination state.",
"Moving into a cell containing a reward returns the favourable value $R$ , moving into a cell containing a penalty returns the unfavourable value $P$ , and moving into a cell with no reward returns a neutral value in the interval $(P, R)$ .",
"A discount factor for future rewards is a non-negative constant $\\gamma <1$ .",
"In our experiments, this discount factor is set to $\\gamma = 0.8$.",
"The discount factor is a feature of the problem rather than a free parameter of an implementation.",
"For example, in a financial application scenario the discount factor might be a function of the risk-free interest rate.",
"The immediate reward for moving into a cell with a stochastic reward is given by a random variable $\\mathcal {R}$ .",
"If an agent has prior knowledge of this distribution, then it should be able to treat the cell as one with a deterministic reward of value $\\mathbb {E}[\\mathcal {R}]$ .",
"This allows us to find the set of all optimal policies in each maze instance.",
"This policy information is denoted by $\\alpha ^\\ast : S \\rightarrow 2^A$ associating with each state $s \\in S$ a set of optimal actions $\\alpha ^\\ast (s) \\subseteq A$ .",
"In our maze model, the neutral value is set to 100, the reward $R = 200$ , and the penalty $P= 0$ .",
"In our experiments, the stochastic reward $\\mathcal {R}$ is simulated by drawing a sample from the Bernoulli distribution $200\\, \\mathrm {Ber} (0.5)$ ; hence, it has the expected value $\\mathbb {E}[\\mathcal {R}]= 100$ , which is identical to the neutral value.",
"Therefore, the solutions depicted in Fig.",
"REF are solutions to the maze of Fig.",
"REF as well.",
"We study the performance of temporal-difference reinforcement learning algorithms (explained in detail in Sec. )",
"using Boltzmann machines.",
"We generalize the method introduced in [12], and compare the policies obtained from these algorithms to the optimal policy using a fidelity measure, which we define in (REF ).",
"Figure: NO_CAPTIONThe RBM reinforcement learning algorithm is due to Sallans and Hinton [12].",
"This algorithm uses the update rules (REF ) and (REF ) to update the weights of an RBM and (REF ) to calculate the expected values of random variables associated with the hidden nodes $\\langle h \\rangle $ (referred to as activations of the hidden nodes in machine learning terminology).",
"As explained in Sec.",
"REF , the main advantage of RBM is that it has explicit formulas for the hidden-node activations given the values of the visible nodes.",
"Moreover, only for RBMs can the entropy portion of the free energy (REF ) be written in terms of the activations of the hidden nodes.",
"More-complicated network architectures do not possess this property, so there is a need for a Boltzmann distribution sampler.",
"Since we are interested in the dependencies between states and actions, we consider a DBM architecture that has a layer of states connected to the first layer of hidden nodes, followed by multiple hidden layers, and a layer of actions connected to the final layer of hidden nodes (see Fig.",
"REF ).",
"We demonstrate the advantages of this deep architecture trained using SQA and the derivation in Sec.",
"REF of the temporal-difference gradient method for reinforcement learning using general Boltzmann machines (GBM).",
"For $T_r$ independent training runs of the same reinforcement learning algorithm, $T_s$ training samples are used for reinforcement learning.",
"The fidelity measure at the $i$ -th training sample is defined by $\\text{fid}(i) =(T_r \\times |S|)^{-1} \\sum _{l = 1}^{T_r} \\sum _{s\\in S} \\mathbb {1}_{A(s, i, l) \\in \\alpha ^*(s)},$ where $A(s, i, l)$ denotes the action assigned at the $l$ -th run and $i$ -th training sample to the state $s$ .",
"In our experiments, each algorithm is run 1440 times, and for each run of an algorithm, $T_s= 500$ training samples are generated.",
"Fig.",
"REF and REF show the fidelity of the generated policies obtained from various reinforcement learning experiments on two clear $3 \\times 5$ mazes.",
"In Fig.",
"REF , the maze includes one reward, one wall, and one pit, and in Fig.",
"REF , the maze includes two stochastic rewards in addition.",
"In these experiments, the training samples are generated by sweeping over the maze.",
"Each sweep iterates over the maze elements in the same order.",
"This explains the periodic behaviour of the fidelity curves (cf.",
"Fig.",
"REF ).",
"The curves labelled `QBM-RL' represent the fidelity of reinforcement learning using QBMs.",
"Sampling from the QBM is performed using SQA.",
"All other experiments use classical Boltzmann machines as their graphical model.",
"In the experiment labelled `RBM-RL', the graphical model is an RBM, trained classically using formula (REF ).",
"The remaining curve is labelled `DBM-RL' for classical reinforcement learning using a DBM.",
"In these experiments, sampling from configurations of the DBM is performed with SQA (with $\\Gamma _f = 0.01$ ).",
"The fidelity results of DBM-RL coincide closely with those of sampling configurations of the DBM using simulated annealing; therefore, we have not sketched them.",
"Fig.",
"REF regenerates the results of Fig.",
"REF using uniform random sampling (i.e., without sweeping through the maze).",
"Our next result, shown in Fig.",
"REF , compares RBM-RL, DBM-RL and QBM-RL for a windy maze of size $3 \\times 5$ .",
"The transition kernel for this experiment is chosen such that $\\mathbb {P}(a(s)|s,a)=0.8,$ and $\\mathbb {P}(s^{\\prime } | s, a)$ has a nonzero value for all $s^{\\prime } \\ne a(s)$ that are reachable from $s$ by taking some action, in which case all of the values are equal.",
"The transition probability is zero for all other states.",
"Fig.",
"REF shows examples of the transition probabilities in the windy problem.",
"Figure: NO_CAPTIONTo demonstrate the performance of RBM-RL and DBM-RL with respect to scaling, we define another measure called average fidelity, $av_\\ell $ , where we take the average fidelity over the last $\\ell $ training samples of the fidelity measure.",
"Given $T_s$ total training samples and $\\text{fid}(i)$ as defined above, we write $av_\\ell = \\frac{1}{\\ell } \\sum _{i=T_s - \\ell }^{T_s} \\text{fid}(i)\\,.$ In Fig.",
"REF , we report the effect of maze size on $av_\\ell $ for RBM-RL and DBM-RL for varying maze sizes.",
"We plot $av_\\ell $ for RBM-RL and DBM-RL with $\\ell = 500, 250,$ and 10 as a function of maze size.",
"We use nine $n \\times 5$ mazes in this experiment indexed by various values of $n$ .",
"In addition to the $av_\\ell $ plots, we include a dotted-line plot depicting the fidelity for a completely random policy."
],
[
"Discussion",
"The fidelity curves in Fig.",
"REF show that the DBM-RL algorithm outperforms the RBM-RL algorithm with respect to the number of training samples.",
"Therefore, we expect that in conjunction with a high-performance sampler of Boltzmann distributions (e.g., a quantum or a quantum-inspired oracle taken as such), the DBM-RL algorithm improves the performance of reinforcement learning.",
"The QBM-RL algorithm improves upon the DBM-RL results even further by taking advantage of sampling in presence of a significant transverse field.",
"In each experiment, the fidelity curves from DBM-RL produced using SQA with $\\Gamma _f = 0.01$ match the ones produced using SA.",
"This is consistent with our expectation that using SQA with $\\Gamma \\rightarrow 0$ produces samples from the same distribution as SA, namely, the Boltzmann distribution of the classical Ising Hamiltonian with no transverse field.",
"The best algorithm in our experiments is evidently QBM-RL using SQA with a significant final transverse field (in this case, $\\Gamma _f = 2.00$ ).",
"This is consistent with ideas found in [31] on sampling at freeze-out [33].",
"Fig.",
"REF shows that, whereas the maze can be solved with fewer training samples using ordered sweeps of the maze, the periodic behaviour of the fidelity curves is due to this periodic choice of training samples.",
"This effect disappears once the training samples are chosen uniformly randomly.",
"Fig.",
"REF shows that the improvement in the learning of DBM-RL and QBM-RL algorithms persists in the case of more-complicated transition kernels.",
"The same ordering of fidelity curves discussed earlier is observed: QBM-RL outperforms DBM-RL, and DBM-RL outperforms RBM-RL.",
"It is worth mentioning that, even though it may seem that more connectivity between the hidden nodes may allow a Boltzmann machine to capture more-complicated correlations between the visible nodes, the training process of the Boltzmann machine becomes more computationally involved.",
"In our reinforcement learning application, an RBM with $m$ hidden nodes, and $n= |S| + |A|$ visible nodes, has $mn$ weights to train.",
"A DBM with two hidden layers of equal size has $\\frac{1}{4} m( 2 n + m)$ weights to train.",
"Therefore, when $m < 2 n$ , the training of the DBM is in a domain of a lower dimension.",
"Further, a GBM with all of its hidden nodes forming a complete graph requires $mn + {m \\atopwithdelims ()2}$ weights to train, which is always larger than that of an RBM or a DBM with the same number of hidden nodes.",
"One can observe from Fig.",
"REF that, as the maze size increases and the complexity of the reinforcement learning task increases, $av_\\ell $ decreases for each algorithm.",
"The RBM algorithm, while always outperformed by DBM-RL, shows a much faster decay in average fidelity as a function of maze size compared to DBM-RL.",
"For larger mazes, the RBM algorithm fails to capture maze traversal knowledge, and approaches $av_\\ell $ of a random action allocation (the dotted line), whereas the DBM-RL algorithm continues to be trained reasonably well.",
"DBM-RL is capable of training the agent to traverse larger mazes, whereas the RBM algorithm, utilizing the same number of hidden nodes and a larger number of weights, fails to converge to an output better than a random policy.",
"Given the ordering of fidelity curves discussed above, we expect QBM-RL to perform even better than DBM-RL as problem sizes increase, especially in the presence of a non-zero final transverse field.",
"The runtime and computational resources needed for DBM-RL and QBM-RL in comparison with RBM-RL are not investigated here.",
"We expect that in view of [15] the size of the RBMs needed to solve larger maze problems will grow exponentially.",
"It is, therefore, an interesting research path to pursue the extrapolation of the asymptotic complexity and size of the DBM-RL and QBM-RL algorithms in the quest for quantum supremacy."
],
[
"Method",
"In this section, we present the details of classical reinforcement learning using RBM, a semi-classical approach base on a DBM (using SA and SQA), and quantum reinforcement learning (using SQA).",
"Pseudo-code for these methods is provided in Algorithms , , and below.",
"SQA methods are a class of quantum-inspired algorithms that perform discrete optimization by classically simulating the quantum tunnelling phenomena (see [34] for an introduction).",
"The algorithm used in this paper is a single spin-flip version of quantum Monte Carlo numerical simulation based on the Suzuki–Trotter formula, and uses the Metropolis acceptance probabilities.",
"The SQA algorithm simulates the quantum annealing phenomena of an Ising spin model with a transverse field, that is, $\\mathcal {H}(t) = - \\sum J_{ij} \\sigma ^z_i \\sigma ^z_j - \\sum h_i \\sigma ^z_i- \\Gamma (t) \\sum \\sigma ^x_i\\,,$ where $\\sigma ^z$ and $\\sigma ^x$ represent the Pauli $z$ - and $x$ -matrices, respectively, and time $t$ ranges from 0 to 1.",
"In this quantum evolution, the strength of the transverse field is slowly reduced to zero at finite temperature.",
"In our implementations, we have used a linear transverse field schedule for the SQA algorithm as in [35] and [36].",
"Based on the Suzuki–Trotter formula, the key idea of this algorithm is to approximate the partition function of the Ising model with a transverse field as a partition function of a classical Hamiltonian denoted by $\\mathcal {H}^{\\mathrm {eff}}$ , corresponding to a classical Ising model of one dimension higher.",
"More precisely, $\\mathcal {H}^{\\mathrm {eff}}({\\sigma })=&-\\sum _{\\lbrace i,j\\rbrace }\\sum _{k=1}^{r}\\frac{J_{ij}}{r}\\sigma _{ik}\\sigma _{jk}-J^+\\sum _{i}\\sum _{k=1}^{r}\\sigma _{ik}\\sigma _{i,k+1}\\\\&-\\sum _{i}\\sum _{k=1}^{r}\\frac{h_i}{r}\\sigma _{ik}\\,, \\nonumber $ where $r$ is the number of replicas, $J^+ = \\frac{1}{2\\beta } \\log \\coth \\left(\\frac{\\Gamma \\beta }{r}\\right)$ , and $\\sigma _{ik}$ represent spins of the classical system of one dimension higher.",
"In our experiments, the strength $\\Gamma $ of the transverse field is scheduled to linearly decrease from $20.00$ to one of $\\Gamma _f = 0.01$ or $2.00$ .",
"The inverse temperature $\\beta $ is set to the constant $2.00$ .",
"Each spin is replicated 25 times to represent the Trotter slices in the extra dimension.",
"The simulation is set to iterate over all replications of all spins one time per sweep, and the number of sweeps is set to 300.",
"For each instance of input, the SQA algorithm is run 150 times.",
"After termination, the configuration of each replica, as well as the configuration of the entire classical Ising model of one dimension higher, is returned.",
"[t] RBM-RL [1] initialize weights of RBM training samples $(s_1, a_1)$ $s_2 \\leftarrow a_1 (s_1)$ , $a_2 \\leftarrow \\operatornamewithlimits{argmax}_a Q(s_2, a)$ calculate $\\langle {h}_i \\rangle \\text{ for }(i=1, 2)$ using (REF ) calculate $F(\\mathbf {s}_i, \\mathbf {a}_i) \\text{ for }(i=1, 2)$ using (REF ) $Q(s_i, a_i) \\leftarrow -F (\\mathbf {s}_i, \\mathbf {a}_i)$ for $(i=1, 2)$ update RBM weights using (REF ) and (REF ) $\\pi (s_1) \\leftarrow \\operatornamewithlimits{argmax}_a Q (s_1, a)$ return $\\pi $ Although the SQA algorithm does not follow the dynamics of a physical quantum annealer explicitly, it is used to simulate this process since it captures major quantum phenomena such as tunnelling and entanglement [27].",
"In [27], for example, it is shown that quantum Monte Carlo simulations can be used to understand the tunnelling behaviour in quantum annealers.",
"As mentioned previously, it readily follows from the results of [26] that the limiting distribution of SQA is the Boltzmann distribution of $\\mathcal {H}^{{\\mathrm {eff}}}$ .",
"This makes SQA a candidate classical algorithm for sampling from Boltzmann distributions of classical and quantum Hamiltonians.",
"The former is achieved by setting $\\Gamma _f \\simeq 0$ , and the latter by constructing an effective Hamiltonian of the system of one dimension higher, representing the quantum Hamiltonian with non-negligible $\\Gamma _f$ .",
"Alternatively, a classical Monte Carlo simulation used to sample from the Boltzmann distribution of the classical Ising Hamiltonian is the SA algorithm, based on thermal fluctuations of classical spin systems.",
"In Algorithm , we recall the steps of the classical reinforcement learning algorithm using an RBM with a graphical model similar to that shown in Fig.",
"REF .",
"We set the initial Boltzmann machine weights using Gaussian zero-mean values with a standard deviation of $1.00$ , as is common practice for implementing Boltzmann machines [37].",
"Consequently, this initializes an approximation of a Q-function and a policy $\\pi $ given by $\\pi (s) = \\operatornamewithlimits{argmax}_a Q(s, a)\\,.$ We associate a classical spin variable $\\sigma _h$ to each hidden node $h$ .",
"Then the activations of the hidden nodes are calculated via $\\langle h\\rangle = \\mathbb {P}(\\sigma _h = 1 | {s,a}) = \\sigma \\left(\\sum _{s \\in S} w^{sh}s+ \\sum _{a \\in A} w^{ah}a\\right),$ where $\\sigma (\\cdot )$ is the sigmoid function.",
"Here $\\mathbf {s}$ and $\\mathbf {a}$ are encodings of state $s$ and action $a$ , respectively, using the state and action nodes of the Boltzmann machine.",
"In our experiments, all Boltzmann machines have as many state nodes as $|S|$ and as many action nodes as $|A|$ .",
"We associate one node for every state $s \\in S$ , and the corresponding binary encoding is $\\mathbf {s} = (0, 0, \\ldots , 1, \\ldots , 0)$ , with zeroes everywhere except at the index of the node corresponding to $s$ .",
"We used similar encoding for the actions, using the action nodes.",
"The free energy of the RBM is calculated using $- F({s},{a}) = \\sum _{\\begin{array}{c}s \\in S\\\\h \\in H\\end{array}}w^{sh}s \\langle h\\rangle + \\sum _{\\begin{array}{c}a \\in A\\\\h \\in H\\end{array}} w^{ah}a \\langle h\\rangle \\\\- \\sum _{h \\in H} \\left[ \\langle h\\rangle \\log \\langle h\\rangle + (1-\\langle h\\rangle )\\log (1-\\langle h\\rangle )\\right].$ We refer the reader to Remark REF or directly to [12] for more details.",
"This results in an approximation of the Q-function (see Sec.",
"REF ) defined on the state–action space $S \\times A$ [12]: $Q ({s}, {a}) \\approx -F (\\mathbf {s}, \\mathbf {a})\\,.$ We then use the update rules $\\Delta w^{sh} &= \\varepsilon (r_n({s}_n, {a}_n) \\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) s \\langle h \\rangle \\nonumber $ and $\\Delta w^{ah} &=\\varepsilon ( r_n({s}_n, {a}_n)\\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n)) a \\langle h \\rangle \\, , \\nonumber $ with a learning rate $\\varepsilon $ to update the weights of the RBM.",
"[t] DBM-RL [1] initialize weights of DBM training samples $(s_1, a_1)$ $s_2 \\leftarrow a_1 (s_1)$ , $a_2 \\leftarrow \\operatornamewithlimits{argmax}_a Q(s_2, a)$ approximate $\\langle {h}_i \\rangle , \\langle {h}_i {h^{\\prime }}_i \\rangle , \\mathbb {P}(\\mathbf {h}|\\mathbf {s}_i, \\mathbf {a}_i)$ using SA or SQA for $(i=1, 2)$ calculate $F(\\mathbf {s}_i, \\mathbf {a}_i)$ using (REF ) for ($i = 1,2$ ) $Q(s_i, a_i) \\leftarrow -F (\\mathbf {s}_i, \\mathbf {a}_i)$ for $(i=1, 2)$ update DBM weights using (REF ), (REF ), and (REF ) $\\pi (s_1) \\leftarrow \\operatornamewithlimits{argmax}_a Q (s_1, a)$ return $\\pi $ In Algorithm , we summarize the DBM-RL method.",
"Here, the graphical model of the Boltzmann machine is similar to that shown in Fig.",
"REF .",
"The initialization of the weights of the DBM is performed in a similar fashion to the previous algorithm.",
"According to lines and of Algorithm , the samples from the SA or SQA algorithm are used to approximate the free energy of the classical DBM at points $(s_1, a_1)$ and $(s_2, a_2)$ using $- F({s},{a}) = \\\\\\sum _{\\begin{array}{c}s \\in S\\\\h \\in H\\end{array}}w^{sh}s \\langle h\\rangle + \\sum _{\\begin{array}{c}a \\in A\\\\h \\in H\\end{array}} w^{ah}a \\langle h\\rangle +\\sum _{\\lbrace h, h^{\\prime }\\rbrace \\subseteq H} u^{hh^{\\prime }}\\langle hh^{\\prime }\\rangle \\\\- \\frac{1}{\\beta }\\sum _{{h}}\\mathbb {P}({h}|{s}, {a}) \\log \\mathbb {P}({h}|{s}, {a})\\,.$ If SQA is used, averages are taken over each replica of each run; hence, there are 3750 samples of configurations of the hidden nodes for each state–action pair.",
"The strength $\\Gamma $ of the transverse field is scheduled to linearly decrease from $20.00$ to $\\Gamma _f = 0.01$ .",
"The SA algorithm is used with a linear inverse temperature schedule that increases from $0.01$ to $2.00$ , in 50,000 sweeps, and is run 150 times.",
"So, if SA is used, there are only 150 sample points used in the above approximation.",
"The results of DBM-RL using SA or SQA match, with no significant difference.",
"The final difference between Algorithm and Algorithm is that the update rule now includes updates of weights between two hidden nodes given by $\\Delta w^{hh^{\\prime }} &=\\varepsilon ( r_n({s}_n, {a}_n)\\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) \\langle hh^{\\prime } \\rangle \\,, \\nonumber $ in addition to the previous rules REF and REF (see Sec.",
"REF ).",
"[b] QBM-RL [1] initialize weights of DBM training samples $(s_1, a_1)$ $s_2 \\leftarrow a_1 (s_1)$ , $a_2 \\leftarrow \\operatornamewithlimits{argmax}_a Q(s_2, a)$ approximate $\\langle {h}_i \\rangle , \\langle {h}_i {h^{\\prime }}_i \\rangle , \\langle \\mathcal {H}^{\\mathrm {eff}}_{{s_i},{a_i}} \\rangle ,$ and $\\mathbb {P}(c|{s_i},{a_i})$ using SQA for ($i = 1,2$ ) calculate $F(\\mathbf {s}_i, \\mathbf {a}_i)$ using (REF ) for ($i = 1,2$ ) $Q(s_i, a_i) \\leftarrow -F (\\mathbf {s}_i, \\mathbf {a}_i)$ for $(i=1, 2)$ update DBM weights using (REF ), (REF ), and (REF ) $\\pi (s_1) \\leftarrow \\operatornamewithlimits{argmax}_a Q (s_1, a)$ return $\\pi $ The last algorithm is QBM-RL, presented in Algorithm .",
"The initialization is performed as in Algorithms and .",
"However, according to lines and , the samples from the SQA algorithm are used to approximate the free energy of a QBM at points $(s_1, a_1)$ and $(s_2, a_2)$ by computing the free energy corresponding to an effective classical Ising spin model of one dimension higher representing the quantum Ising spin model of the QBM, $&F (\\mathbf {s},\\mathbf {a}) = \\langle \\mathcal {H}^{\\mathrm {eff}}_{{s},{a}} \\rangle +\\frac{1}{\\beta }\\sum _{c}\\mathbb {P}(c|{s},{a}) \\log \\mathbb {P}(c|{s},{a})\\,.$ In this case, $\\langle \\mathcal {H}^{\\mathrm {eff}}_{{s},{a}} \\rangle $ is approximated by the average energy of the entire system of one dimension higher and $\\mathbb {P}(c|{s},{a})$ is approximated by the normalized frequency of the configuration $c$ of the entire system of one dimension higher (hence, there are only 150 sample points for each input instance in this case).",
"The strength $\\Gamma $ of the transverse field in SQA is scheduled to linearly decrease from $20.00$ to $\\Gamma _f = 2.00$ .",
"In this algorithm, the weights are updated as in Algorithm .",
"However, $\\langle h \\rangle $ and $\\langle h h^{\\prime } \\rangle $ in this algorithm represent expectations of measurements in the $z$ -basis.",
"See Sec.",
"for more details.",
"In each training iteration, we select a state–action pair $(s_1, a_1) \\in S \\times A$ .",
"Every node corresponding to a state or an action is removed from this graph and the configurations of the spins corresponding to the hidden nodes are sampled using SQA on an Ising spin model constructed as follows: the state $s_1$ contributes to a bias of $w^{s_1h}$ to $\\sigma _h$ if $h$ is adjacent to $s_1$ ; and the action $a_1$ contributes to a bias of $w^{a_1h}$ to $\\sigma _h$ if $h$ is adjacent to $a_1$ .",
"The bias on any spin $\\sigma _h$ for which $h$ is a hidden node not adjacent to state $s_1$ or action $a_1$ is zero.",
"A subsequent state $s_2$ is obtained from a state–action pair $(s_1, a_1)$ using the transition kernel outlined in Sec.",
", and a corresponding action $a_2$ is chosen via policy $\\pi $ .",
"Another SQA sampling is performed in a similar fashion to the above for this pair.",
"In Fig.",
"REF , REF , the selection of $(s_1, a_1)$ is performed by sweeping over the set of state–action pairs.",
"In Fig.",
"REF , the selection of $(s_1, a_1)$ and $s_2$ is performed by sweeping over $S \\times A \\times S$ .",
"In Fig.",
"REF , the selection of $s_1$ , $a_1$ , and $s_2$ are all performed uniformly randomly.",
"We experiment with a variety of learning-rate schedules, including exponential, harmonic, and linear; however, we found that for the training of both RBMs and DBMs, an adaptive learning-rate schedule performed best (for information on adaptive subgradient methods, see [38]).",
"In our experiments, the initial learning rate is set to 0.01.",
"In all of our studied algorithms, training terminates when a desired number of training samples have been processed, after which the updated policy is returned."
],
[
"Supplementary Material",
"In this section, we derive the Q-learning method for Markov decision processes (MDP) with a function approximator represented by a general Boltzmann machine (GBM).",
"To the best of our knowledge, this derivation has not been previously given, although it can be readily derived from the ideas presented in [12] and [31]."
],
[
"Markov Decision Process",
"The stochastic control problem of interest to us is an MDP, defined as follows: finite sets of states $S$ and of actions $A$ ; When both $S$ and $A$ are finite, the MDP is said to be finite.",
"a controlled Markov chain [39], defined by a transition kernel $\\mathbb {P}( s^{\\prime } \\in S | s \\in S, a \\in A)$ ; The transition kernel does not need to be time-homogeneous; however, this definition suffices for the purposes of this work.",
"a real-valued function $r: S \\times A \\rightarrow \\mathbb {R}$ known as the immediate reward structure; and a constant $\\gamma \\in [0, 1)$ known as the discount factor.",
"A function $\\pi : S \\rightarrow A$ is called a stationary policy; that is, it is a choice of action $\\pi (s)$ for every state $s$ independent of the point in time that the controlled process reaches $s$ .",
"A stationary policy $\\pi $ reduces the MDP into a time-homogeneous Markov chain, $\\Pi $ , with a transition probability $\\mathbb {P} (s^{\\prime } | s , \\pi (s))$ .",
"For more-general statements, see [39].",
"The random process $\\Pi $ with initial condition $\\Pi _0 = s$ we denote by $\\Pi ^s$ .",
"Our Markov decision problem is to find $\\pi ^* (s) = \\operatornamewithlimits{argmax}_{\\pi \\in A} V (\\pi , s),$ where $V (\\pi , s) = \\mathbb {E} \\left[ \\sum \\limits _{i=0}^{\\infty } \\gamma ^{i}\\,r\\, (\\Pi ^s_i,\\, \\pi (\\Pi ^s_i)) \\right].$"
],
[
"Value Iteration",
"Bellman [40] writes $V(\\pi , s)$ recursively in the following manner using the monotone convergence theorem.",
"$V(\\pi , s)= \\mathbb {E} \\left[ \\sum \\limits _{i=0}^{\\infty } \\gamma ^{i}\\, r\\, (\\Pi ^s_i,\\, \\pi (\\Pi ^s_i)) \\right] \\\\= \\mathbb {E}[ r\\, (\\Pi ^s_0,\\, \\pi (\\Pi ^s_0))] + \\gamma \\, \\mathbb {E} \\left[ \\sum \\limits _{i=0}^{\\infty } \\gamma ^{i}\\, r\\, (\\Pi ^s_{i+1},\\, \\pi (\\Pi ^s_{i+1})) \\right] \\\\= \\mathbb {E}[r\\, (s,\\, \\pi (s)) ]+ \\gamma \\sum _{s^{\\prime } \\in S} \\mathbb {P}(s^{\\prime } |s, \\pi (s))\\, V(\\pi , s^{\\prime })\\,$ In particular, $ V^\\ast (s) = V(\\pi ^\\ast , s)\\\\= \\max _a \\left( \\mathbb {E}[r\\, (s,\\, a)] + \\gamma \\sum _{s^{\\prime } \\in S} \\mathbb {P}(s^{\\prime } |s, a)\\, V^\\ast (s^{\\prime })\\right)\\,.$ Hence, $V^\\ast $ is a fixed point for the operator $T (f): s \\mapsto \\max _a \\left(\\mathbb {E}[r (s,a)] + \\gamma \\int f\\right)$ on the space $L_\\infty (S)$ of bounded functions $S \\rightarrow \\mathbb {R}$ endowed with the max norm.",
"Here, the integral is with respect to the probability measure on $S$ , induced by the conditional probability distribution $\\mathbb {P}(s^{\\prime }|s, a)$ .",
"It is easy to check that $T$ is a contraction mapping and thus $V^\\ast $ is the unique fixed point of $T$ and the uniform limit of any sequence of functions $\\lbrace T^k f\\rbrace _{k}$ .",
"Numerical computation of this limit using (REF ), called value iteration, is a common method of solving the Markov decision problem (REF )."
],
[
"Q-functions",
"For a stationary policy $\\pi $ , the Q-function (also known as the state–value function) is defined by mapping a pair $(s, a)$ to the expected value of the reward of the Markov chain that begins with taking action $a$ at initial state $s$ and continuing according to $\\pi $ [4]: $Q (\\pi , s, a) &= \\mathbb {E}[ r\\, (s,\\, a)] + \\mathbb {E} \\left[ \\sum \\limits _{i=1}^{\\infty } \\gamma ^{i}\\,r\\, (\\Pi ^s_{i},\\, \\pi (\\Pi ^s_{i})) \\right].$ In particular, it is straightforward to check that $V(\\pi , s)= \\max _a Q(\\pi , s, a),$ and for $Q^\\ast (s, a)= \\max _\\pi Q(\\pi , s, a)$ , the optimal policy for the MDP can be retrieved via the following: $\\pi ^\\ast (s) = \\operatornamewithlimits{argmax}_a Q^\\ast (s, a).$ This reduces the Markov decision problem to computing $Q^\\ast (s, a)$ .",
"Through a Bellman-like recursion, we get $\\!\\!\\!Q(\\pi , s, a) = \\mathbb {E}[r\\,(s, a)] + \\mathbb {E} \\left[ \\sum \\limits _{i=1}^{\\infty } \\gamma ^{i}\\, r\\, (\\Pi ^s_i,\\, \\pi (\\Pi ^s_i)) \\right] \\\\= \\mathbb {E}[r\\, (s, a) ] + \\gamma \\sum _{s^{\\prime }} \\mathbb {P}(s^{\\prime } | s, a) \\max _{a^{\\prime }} Q (\\pi , s^{\\prime }, a^{\\prime }),$ which makes $Q^\\ast $ the fixed point of a different operator $T(f): (s, a) \\mapsto \\mathbb {E}[ r\\, (s, a)] + \\gamma \\int \\max _{a^{\\prime }} f\\,$ defined on $L_\\infty (S \\times A)$ .",
"One can adapt a value iteration approach for $Q$ similar to that for $V$ .",
"Even then, ${\\varepsilon }$ -optimal algorithms for this approach depend heavily on the cardinality of ${S}$ and ${A}$ , and suffer from the curse of dimensionality [40], [41]."
],
[
"Temporal-Difference Gradient Descent",
"From the previous section, we know that starting from an initial $Q_0: S \\times A \\rightarrow \\mathbb {R}$ , the sequence $\\lbrace Q_n = T^n Q\\rbrace $ converges to $Q^\\ast $ .",
"The difference $Q_{n+1}(s, a) -Q_n(s,a) = \\mathbb {E}[r(s, a)] \\\\+ \\gamma \\sum _{s^{\\prime }} \\mathbb {P}(s^{\\prime } | s, a)\\max \\limits _a \\underbrace{Q_n(s^{\\prime }, a)}_{(\\ast )} - Q_n (s, a)$ is called the temporal difference of Q-functions, and is denoted by $E_\\mathrm {TD}$ .",
"Employing a gradient approach to find the fixed point of $T$ on $L_\\infty (S \\times A)$ involves locally parametrizing the functions in this space by a vector of parameters $\\theta $ , that is, $Q(s, a) = Q(s, a; \\theta ),$ and travelling in the direction that minimizes $\\Vert E_\\mathrm {TD}\\Vert ^2$ : $\\Delta \\theta \\propto - E_\\mathrm {TD} \\nabla _{\\theta } E_\\mathrm {TD}\\,.$ The method TD(0) consists of treating the $(\\ast )$ in (REF ) as constant with respect to the parameterization $\\theta $ , in which case we may write $\\Delta \\theta \\mathchoice{\\mathrel {\\displaystyle \\sim }\\unknown.",
"{\\hbox{$\\displaystyle \\propto $}}}{0.27\\box 2}{}{}$ $\\textstyle \\propto $$\\textstyle \\sim $$\\scriptstyle \\propto $$\\scriptstyle \\sim $$\\scriptscriptstyle \\propto $$\\scriptscriptstyle \\sim $ ETD(s,a) Q(s,a;).",
"$$ For an agent agnostic of the transition kernel or the distribution of the reward $r(s, a)$ , this update rule for $\\theta $ is not possible.",
"The alternative is to substitute, at each iteration, the expected value $\\sum _{s^{\\prime }} \\mathbb {P}(s^{\\prime } | s, a)\\max \\limits _a Q_n(s_{n+1}, a)$ by $\\max _a Q_n (s_{n+1}, a)$ , where $s_{n+1}$ is drawn from the probability distribution $\\mathbb {P}(s^{\\prime }|s, a)$ , and substitute $\\mathbb {E}[r(s_n, a_n)]$ by a sample of $r(s_n, a_n)$ .",
"This leads to a successful Monte Carlo training method called Q-learning.",
"In the following section, we explain the case where $\\theta $ comprises the weights of a Boltzmann machine."
],
[
"Clamped Boltzmann Machines",
"A classical Boltzmann machine is a type of stochastic neural network with two sets $V$ and $H$ of visible and hidden nodes, respectively.",
"Both visible and hidden nodes represent binary random variables.",
"We use the same notation for a node and the binary random variable it represents.",
"The interactions between the variables represented by their respective nodes are specified by real-valued weighted edges of the underlying undirected graph.",
"A GBM, as opposed to models such as RBMs and DBMs, allows weights between any two nodes.",
"The energy of the classical Boltzmann machine is ${E}(v, h)= -\\sum _{v \\in V,\\, h \\in H} w^{vh}v h\\\\ -\\sum _{\\lbrace v, v^{\\prime }\\rbrace \\subseteq V } w^{vv^{\\prime }} vv^{\\prime } -\\sum _{\\lbrace h, h^{\\prime }\\rbrace \\subseteq H} w^{hh^{\\prime }} hh^{\\prime },$ with $w^{vh}$ , $w^{vv^{\\prime }}$ , and $w^{hh^{\\prime }}$ denoting the weights between visible and hidden, visible and visible, and hidden and hidden nodes of the Boltzmann machine, respectively, defined as a function of binary vectors $\\bf v$ and $\\bf h$ corresponding to the visible and hidden variables, respectively.",
"A clamped GBM is a neural network whose underlying graph is the subgraph obtained by removing the visible nodes for which the effect of a fixed assignment $v$ of the visible binary variables contributes as constant coefficients to the associated energy ${E}_{v}(h)= -\\sum _{v \\in V,\\, h \\in H} w^{vh}v h\\\\ -\\sum _{\\lbrace v, v^{\\prime }\\rbrace \\subseteq V } w^{vv^{\\prime }} vv^{\\prime } -\\sum _{\\lbrace h, h^{\\prime }\\rbrace \\subseteq H} w^{hh^{\\prime }} hh^{\\prime }\\,.",
"$ A clamped quantum Boltzmann machine (QBM) has the same underlying graph as a clamped GBM, but instead of a binary random variable, a qubit is associated to each node of the network.",
"The energy function is substituted by the quantum Hamiltonian $\\mathcal {H}_{v}= -\\sum _{v \\in V,\\, h \\in H} w^{vh}v \\sigma ^z_h -\\sum _{\\lbrace v, v^{\\prime }\\rbrace \\subseteq V } w^{vv^{\\prime }} vv^{\\prime } \\\\ -\\sum _{\\lbrace h, h^{\\prime }\\rbrace \\subseteq H} w^{hh^{\\prime }} \\sigma ^z_h\\sigma ^z_{h^{\\prime }} - \\Gamma \\sum _{h \\in H} \\sigma _h^x\\,,$ where $\\sigma _h^z$ represent the Pauli $z$ -matrices and $\\sigma ^x_h$ represent the Pauli $x$ -matrices.",
"Thus, a clamped QBM with $\\Gamma = 0$ is equivalent to a clamped classical Boltzmann machine.",
"This is because $\\mathcal {H}_{v}$ is a diagonal matrix in the $\\sigma ^z$ -basis, the spectrum of which is identical to the range of $E_{v}$ .",
"The remainder of this section is formulated for the clamped QBMs, acknowledging that it can easily be specialized to clamped classical Boltzmann machines.",
"Let $\\beta = \\frac{1}{k_B T}$ be a fixed thermodynamic beta.",
"For an assignment of visible variables $v$ , $F({v})$ denotes the equilibrium free energy, and is defined as $F({v}) := - \\frac{1}{\\beta }\\ln Z_{v} = \\langle \\mathcal {H}_{v} \\rangle + \\frac{1}{\\beta }\\operatornamewithlimits{tr}(\\rho _{v} \\ln \\rho _{v})\\,.$ Here, $Z_{v} = \\operatornamewithlimits{tr}(e^{-\\beta \\mathcal {H}_{v}})$ is the partition function of the clamped QBM and $\\rho _{v}$ is the density matrix $\\rho _{v} = \\frac{1}{Z_{v}}e^{-\\beta \\mathcal {H}_{v}}$ .",
"The term $- \\operatornamewithlimits{tr}(\\rho _{v} \\ln \\rho _{v})$ is the entropy of the system.",
"The notation $\\langle \\cdots \\rangle $ is used for the expected value of any observable with respect to the Gibbs measure, in particular, $\\langle \\mathcal {H}_{v} \\rangle = \\frac{1}{Z_{\\mathbf {v}}} \\operatornamewithlimits{tr}( \\mathcal {H}_{v} e^{-\\beta \\mathcal {H}_{v}}).$"
],
[
"Reinforcement Learning with Clamped Boltzmann Machines",
"Following the ideas in [12], the goal is to use the negative free energy of a Boltzmann machine to approximate the Q-function through the relationship $Q (s, a) \\approx -F (\\mathbf {s}, \\mathbf {a}) = - F (\\mathbf {s}, \\mathbf {a}; \\theta )$ for each admissible state–action pair $(s, a) \\in S\\times A$.",
"Here, $\\mathbf {s}$ and $\\mathbf {a}$ are binary vectors encoding the state $s$ and action $a$ on the state nodes and action nodes, respectively, of the Boltzmann machine.",
"Recall that in reinforcement learning, the visible nodes of the GBM are partitioned into two subsets of state nodes $S$ and action nodes $A$ .",
"The parameters $\\theta $ , to be trained according to a TD(0) update rule (see Sec.",
"REF ), are the weights in a Boltzmann machine.",
"For every weight $w$ , the update rule is $\\Delta w =-\\varepsilon (r_n({s}_n, {a}_n)+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) \\frac{\\partial F}{\\partial w}\\,.$ From (REF ), we obtain $\\frac{\\partial F(\\mathbf {s}, \\mathbf {a})}{\\partial w} &= \\frac{1}{Z_{\\mathbf {s}, \\mathbf {a}}} \\frac{\\partial }{\\partial w} \\operatornamewithlimits{tr}(e^{-\\beta \\mathcal {H}_{\\mathbf {s},\\mathbf {a}}})\\\\&= - \\frac{1}{Z_{\\mathbf {s}, \\mathbf {a}}} \\operatornamewithlimits{tr}(\\beta e^{-\\beta \\mathcal {H}_{\\mathbf {s}, \\mathbf {a}}} \\frac{\\partial }{\\partial w} \\mathcal {H}_{\\mathbf {s}, \\mathbf {a}} ) \\\\&= - \\beta \\left\\langle \\frac{\\partial }{\\partial w} \\mathcal {H}_{\\mathbf {s}, \\mathbf {a}} \\right\\rangle \\,.$ Therefore, the update rule for TD(0) for the clamped QBM can be rewritten as $\\Delta w^{vh} =\\,&\\varepsilon (r_n({s}_n, {a}_n) \\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) v \\langle \\sigma _h^z \\rangle \\nonumber $ and $\\Delta w^{hh^{\\prime }} =\\,&\\varepsilon (r_n({s}_n, {a}_n) \\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) \\langle \\sigma _h^z\\sigma _{h^{\\prime }}^z \\rangle .",
"\\nonumber $ Here, $h$ and $h^{\\prime }$ denote two distinct hidden nodes and (by a slight abuse of notation) the letter $v$ stands for a visible (state or action) node, and also the value of the variable associated to that node.",
"To approximate the right-hand side of each of (REF ) and (REF ), we use SQA experiments.",
"By [42], we may find the expected values of the observables $\\langle \\sigma _h^z\\rangle $ and $\\langle \\sigma _h^z \\sigma _{h^{\\prime }}^z \\rangle $ by averaging the corresponding spins in the classical Ising model of one dimension higher constructed in SQA.",
"To approximate the Q-function, we take advantage of [42] and use (REF ) applied to this classical Ising model.",
"More precisely, let $\\mathcal {H}^{\\mathrm {eff}}_{v}$ represent the Hamiltonian of the classical Ising model of one dimension higher and the associated energy function $E^{\\mathrm {eff}}_{v}$ .",
"The free energy of this model can be written $&F ({v}) = \\langle \\mathcal {H}^{\\mathrm {eff}}_{v} \\rangle +\\frac{1}{\\beta }\\sum _{c}\\mathbb {P}(c|{v}) \\log \\mathbb {P}(c|{v})\\,,$ where $c$ ranges over all spin configurations of the classical Ising model of one dimension higher.",
"The above argument holds in the absence of the transverse field, that is, for the classical Boltzmann machine.",
"In this case, the TD(0) update rule is given by $\\Delta w^{vh} =\\,&\\varepsilon (r_n({s}_n, {a}_n) \\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) v \\langle h \\rangle \\nonumber $ and $\\Delta w^{hh^{\\prime }} =\\,&\\varepsilon (r_n({s}_n, {a}_n)\\\\&+\\gamma Q ({s}_{n+1}, {a}_{n+1}) - Q({s}_n,{a}_n) ) \\langle hh^{\\prime } \\rangle \\,, \\nonumber $ where $\\langle h \\rangle $ and $\\langle h h^{\\prime } \\rangle $ are the expected values of the variables and the product of variables, respectively, in the binary encoding of the hidden nodes with respect to the Boltzmann distribution given by $\\mathbb {P}({h}|{v}) = {\\exp (-{E}_{{v}} ({h}))}/{\\sum _{{{h}}^{\\prime }}\\exp (-{E}_{{v}} ({{h}}^{\\prime }))}$ .",
"Therefore, they may be approximated using SA or SQA when $\\Gamma \\rightarrow 0$ .",
"The values of the Q-functions in (REF ) and (REF ) can also be approximated empirically since in a classical Boltzmann machine, $&F ({v}) = \\sum _{{h}}\\mathbb {P}({h}|{v}) {E}_{v} ({h}) + \\frac{1}{\\beta }\\sum _{{h}}\\mathbb {P}({h}|{v}) \\log \\mathbb {P}({h}|{v})\\\\&\\quad = - \\sum _{\\begin{array}{c}s \\in S\\\\h \\in H\\end{array}}w^{sh}s \\langle h\\rangle - \\sum _{\\begin{array}{c}a \\in A\\\\h \\in H\\end{array}} w^{ah}a \\langle h\\rangle - \\sum _{\\lbrace h, h^{\\prime }\\rbrace \\subseteq H} u^{hh^{\\prime }}\\langle hh^{\\prime }\\rangle \\nonumber \\\\&\\quad \\qquad \\qquad \\qquad \\qquad \\qquad + \\frac{1}{\\beta }\\sum _{{h}}\\mathbb {P}({h}|{s}, {a}) \\log \\mathbb {P}({h}|{s}, {a}).",
"\\nonumber $ Remark 6.1 In the case of an RBM, Sallans and Hinton [12] show that the free energy is given by $- F({s},{a}) = \\sum _{\\begin{array}{c}s \\in S\\\\h \\in H\\end{array}}w^{sh}s \\langle h\\rangle + \\sum _{\\begin{array}{c}a \\in A\\\\h \\in H\\end{array}} w^{ah}a \\langle h\\rangle \\\\- \\sum _{h \\in H} \\left[ \\langle h\\rangle \\log \\langle h\\rangle + (1-\\langle h\\rangle )\\log (1-\\langle h\\rangle )\\right].$ The update rule for the weights of the RBM is (REF ) alone.",
"Moreover, in the case of RBMs, the equilibrium free energy $F({s,a})$ and its derivatives with respect to the weights can be calculated without the need for Boltzmann distribution sampling, according to the closed formula $\\langle h\\rangle &= \\mathbb {P}(\\sigma _h = 1 | {s,a}) = \\sigma \\left(\\sum _{s \\in S} w^{sh}s+ \\sum _{a \\in A} w^{ah}a\\right) \\\\&= \\left\\lbrace 1 + \\text{exp}\\left(- \\sum _{s \\in S} w^{sh}s - \\sum _{a \\in A} w^{ah}a\\right)\\right\\rbrace ^{-1}\\,.\\nonumber $ Note that, in the general case, since the hidden nodes of a clamped Boltzmann machine are not independent, the calculation of the free energy is intractable."
],
[
"Acknowledgements",
"We would like to thank Hamed Karimi, Helmut Katzgraber, Murray Thom, Matthias Troyer, and Ehsan Zahedinejad for reviewing this work and providing many helpful suggestions.",
"The idea of using SQA to run experiments involving measurements with a non-zero transverse field was communicated in person by Mohammad Amin.",
"We would also like to thank Marko Bucyk for editing this manuscript."
]
] | 1612.05695 |
[
[
"Predicting Completeness in Knowledge Bases"
],
[
"Abstract Knowledge bases such as Wikidata, DBpedia, or YAGO contain millions of entities and facts.",
"In some knowledge bases, the correctness of these facts has been evaluated.",
"However, much less is known about their completeness, i.e., the proportion of real facts that the knowledge bases cover.",
"In this work, we investigate different signals to identify the areas where a knowledge base is complete.",
"We show that we can combine these signals in a rule mining approach, which allows us to predict where facts may be missing.",
"We also show that completeness predictions can help other applications such as fact prediction."
],
[
"Motivation",
"Knowledge Bases (KBs) such as DBpedia [8], NELL [1], Wikidata [18], the Google Knowledge Vault [3], or YAGO [17] contain billions of machine-readable facts about the world.",
"They know for instance that Paris is the capital of France and that Barack Obama won the Nobel Peace Prize.",
"KBs have applications in information retrieval, question answering, machine translation, and data maintenance, among others.",
"However, the data quality of KBs is not always perfect.",
"Problems include false data, missing information, or schema inconsistencies.",
"Hence, many approaches aim to clean up erroneous information [19].",
"In contrast, the completeness (recall) of the KBs has remained relatively unexplored.",
"While we often know what proportion of the facts in the KB are correct, we usually do not know what proportion of the facts in the real world they cover.",
"For example, as of 2016, Wikidata knows the father of only 2% of all people in the KB – even though in the real world everyone has a father.",
"DBpedia contains only 6 Dijkstra Prize winners – but in the real world there are 35.",
"Likewise, according to YAGO, the average number of children per person is 0.02.",
"In general, between 69% and 99% of instances in popular KBs lack at least one property that other entities in the same class have [16], [10].",
"Thus, we know that today's KBs are highly incomplete, but we do not know where the information is missing.",
"This unknown degree of completeness poses several problems [13].",
"First, users do not have any guarantee that a query run against the KB yields all the results that match the query in the real world.",
"Second, the data providers themselves may not know where the data is incomplete, and thus cannot determine where to focus their efforts.",
"If they knew, e.g., which people are missing their alma mater, they could focus on tracing these pieces of information and adding them.",
"Third, completeness information could help identify wrong facts.",
"If we knew, e.g., that people always have only 2 parents, then a KB that contains 3 parents for an individual has to be erroneous.",
"Finally, completeness information can be insightful on its own, to know which missing facts are known to be wrong.",
"This is useful, e.g., for machine learning algorithms that require counter-examples.",
"Thus, it would be of tremendous use for both data providers and data consumers if we could know where the information in the KB is complete.",
"In the ideal case, we would want to make what we call completeness assertions, which say, e.g., This KB contains all children of Barack Obama."
],
[
"Challenges",
"The main obstacle to establish such completeness assertions is the Open World Assumption (OWA), which nearly all KBs make.",
"The OWA says that if the KB does not contain a certain piece of information, then this information is not necessarily false – it may be true in the real world, but absent from the KB.",
"This means that every part of the KB could be potentially incomplete.",
"Furthermore, today's KBs mostly consist of subject-predicate-object triples.",
"These formalisms usually provide very limited means to store negative information (if at all).",
"For example, YAGO says that Barack Obama is married to Michelle Obama, but it does not say that Barack Obama is not (and was never) married to any other person.",
"In fact, there is not even a way that YAGO and similar KBs could express this idea.",
"The KBs are not just incomplete, but also, by design, unable to provide any indications of completeness."
],
[
"Contribution",
"In this paper, we make a first step towards generating completeness information automatically.",
"Our goal is to determine automatically whether certain properties of certain objects are complete: whether a person has more children in reality than in the KB, whether a person graduated from a university in real life even though the KB does not know about it, or whether a person has more spouses in reality than are known to the KB.",
"More precisely: We conduct a systematic study of signals that can indicate completeness of properties of objects in a KB.",
"We show how completeness assertions can be learned through a rule mining system, AMIE; we further show how the necessary training data for AMIE can be obtained easily through crowdsourcing.",
"We find that completeness can be predicted for some relations with up to 100% precision on real KBs (YAGO and Wikidata).",
"As a use case, we show that our completeness assertions can increase the precision of rule mining.",
"This paper is structured as follows.",
"We first discuss related work in Section , and introduce preliminaries in Section .",
"We then present in Section the different signals that we use to predict completeness, and leverage them in Section to mine completeness rules with the AMIE system.",
"Section presents detailed evaluations of the signals in isolation and in combination.",
"We showcase in Section an application of completeness assertions, before concluding in Section ."
],
[
"Knowledge Bases",
"Many of today's KBs provide estimations of their precision.",
"The YAGO KB [17] was manually evaluated and found to be 95% correct.",
"NELL [1] is regularly checked by humans for precision.",
"Facts in the Knowledge Vault [3] are annotated with an estimated precision.",
"However, little is known about the recall/completeness of these KBs.",
"Of course, larger sizes may indicate higher completeness, but size is only a very coarse proxy for completeness."
],
[
"Incompleteness Studies",
"Some studies have found that KBs are indeed quite incomplete.",
"For instance, a watermarking study [16] reports that 69%–99% of instances in popular KBs lack at least one property that other entities in the same class have.",
"Google found that 71% of people in Freebase have no known place of birth, and 75% have no known nationality [3].",
"This allows us to know that KBs are incomplete in general, but not which parts are complete."
],
[
"Manual Indicators",
"The Wikidata community maintains lists that explain where information is still missing – e.g., a list of people without birth dateshttps://www.wikidata.org/wiki/Wikidata:Database_reports/top_missing_properties_by_number_of_sitelinks/P569.",
"Also, Wikidata contains no-value statements, which say that an empty relation is complete for an entity [4].",
"An extension for Wikidata allows contributors to manually add recall information [2].",
"However, these annotations are mostly provided manually: our work aims at deducing such annotations automatically."
],
[
"Partial Completeness Assumption",
"Some approaches simply assume that KBs are complete in certain areas.",
"For instance, the AMIE project used the partial completeness assumption (PCA) [5] (re-used as the local closed world assumption in [3]).",
"We discuss the PCA in detail in Section ."
],
[
"Rule Mining",
"Inductive Logic Programming and Rule Mining approaches [7] find rules such as If a person lives in a city, then their spouse probably lives in the same city.",
"These rules can then be used to predict new information (here: where the spouse lives).",
"As a side effect, this procedure determines where the KB is incomplete.",
"However, such approaches can only ever mine new facts between instances that are already known to the KB.",
"They cannot tell us that a spouse is missing if that spouse is not in the KB.",
"We will show in our experiments how rule mining approaches can benefit from the techniques we develop in this paper."
],
[
"Completeness Reasoning",
"On the database side, some work has investigated how to combine completeness information about parts of databases to deduce completeness annotations on query results [11], [9], [12].",
"However, this work assumes that the KB has already been annotated with completeness assertions.",
"Our goal, in contrast, is to generate such assertions."
],
[
"Knowledge Bases",
"In this paper, we target KBs in RDFS format [14].",
"We assume that the reader is familiar with RDFS.",
"We write facts as $r(s, o)$ , where $r$ is a relation, $s$ is the subject, and $o$ is the object.",
"For instance, $\\mathit {president}(\\mathit {Obama}, \\mathit {USA})$ is a fact.",
"We assume a fixed KB $\\mathcal {K}$ , and thus write $r(s, o)$ to mean $r(s, o) \\in \\mathcal {K}$ ."
],
[
"Functionality",
"The functionality [15] of a relation $r$ is defined as: $\\mathit {fun}(r) \\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\frac{\\#x: \\exists y: r(x, y)}{\\#(x, y): r(x, y)}$ where $\\#\\alpha : \\mathcal {A}$ denotes the number of $\\alpha $ that fulfill the condition $\\mathcal {A}$ .",
"For relations such as placeOfBirth which are functions, we have $\\mathit {fun}(r)=1$ .",
"For “quasi-functions” such as isCitizenOf, the value $\\mathit {fun}(r)$ is close to 1.",
"If $r$ has many objects for a subject, then $\\mathit {fun}(r)$ is closer to 0."
],
[
"Completeness",
"In line with work in databases [11], [13], we define completeness via a hypothetical ideal KB $\\mathcal {K}^*$ , which contains all facts of the real world.",
"A KB $\\mathcal {K}$ is complete for a query $q$ , if $q$ returns the same results on $\\mathcal {K}$ as on $\\mathcal {K}^*$ .",
"In this paper, we focus on a particular type of queries, namely those that ask for the objects of a given subject and relation.",
"Thus, a pair of an entity $s$ and a relation $r$ is complete in a KB $\\mathcal {K}$ , if $\\lbrace o : r(s, o) \\in \\mathcal {K}\\rbrace \\supseteq \\lbrace o : r(s, o) \\in \\mathcal {K}^*\\rbrace $ .",
"For example, a KB is complete for the subject Barack Obama and the relation hasChild, if it contains both of Obama's children.",
"If the KB is complete for a subject $s$ and a relation $r$ , we make a completeness assertion of the form $\\mathit {complete}(s, r)$ .",
"Our goal is to establish such completeness assertions.",
"In general, completeness assertions make less sense for relations with low functionality.",
"For example, it does not make sense to ask a KB if it knows all citizens of France.",
"It is more sensible to ask whether the KB knows all nationalities of one person.",
"Therefore, we consider completeness primarily for relations with high functionality.",
"In particular, if a relation has low functionality (such as countryHasCitizen), and its inverse has high functionality (personHasNationality), then we consider the inverse.",
"When a relation is incomplete for a subject, we could also try to estimate how many objects are missing.",
"This would amount to a cardinality estimation.",
"In this paper, however, we focus on the simpler task of establishing completeness assertions, and leave cardinality estimations for future work."
],
[
"Completeness Considerations",
"The notion of completeness is not well-defined for all relations [13].",
"Take, e.g., the relation hasHobby.",
"It is not always clear whether an activity counts as a hobby or not.",
"Thus, it is difficult to establish whether a KB is complete on the hobbies of a certain person.",
"Even for seemingly well-defined relations such as hasOfficialLanguage, completeness is not easy to establish: a country may have de facto official languages that are not legally recognized (e.g., the US); languages that are official in some regions but not in the country (e.g., India); or an official language that is not a spoken language (e.g., New Zealand).",
"In this paper, we manually selected relations for which completeness is well-defined, and concentrate on these."
],
[
"Completeness Oracles",
"A completeness oracle tries to guess whether a given relation is complete for a given subject in the fixed KB $\\mathcal {K}$ .",
"Technically, a completeness oracle is a binary relation on entities and relations that holds whenever the oracle predicts that a given subject is complete for a given relation.",
"The Partial Completeness Assumption (PCA) is an example of a simple completeness oracle.",
"It predicts completeness for a subject $s$ and a relation $r$ if there exists an object $x$ with $r(s, x)$ .",
"For instance, if in a KB, Barack Obama has one child, the PCA oracle will (wrongly) state that Barack Obama is complete for the relation hasChild, i.e., pca(BarackObama, hasChild) will be true.",
"The precision and recall of an oracle $o$ are defined as follows, where complete denotes the completeness assertions on $\\mathcal {K}$ that are true relative to the ideal KB $\\mathcal {K}^*$ : $\\mathit {precision}(o) & \\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\frac{\\#(e, r): o(e, r) \\wedge \\mathit {complete}(e, r)}{\\#(e, r): o(e, r)}\\\\[.5em]\\mathit {recall}(o) & \\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\frac{\\#(e, r): o(e, r) \\wedge \\mathit {complete}(e, r)}{\\#(e, r): \\mathit {complete}(e, r)}$ The F1 measure is defined as usual from precision and recall."
],
[
"Completeness Oracles",
"We now present various completeness oracles, of which we study two kinds: simple oracles and parameterized oracles."
],
[
"Closed World Assumption",
"The Closed World Assumption (CWA) assumes that any fact that is not in the KB does not hold in the real world.",
"That is, the CWA assumes that the entire KB is complete.",
"In general, the CWA is incompatible with the philosophy of the Semantic Web.",
"Still, the CWA may be suitable under certain conditions.",
"For instance, if a person is not known to be the president of any country, then most likely the person is indeed not the president of any country.",
"Formally, the CWA completeness oracle is simply defined as: $\\mathit {cwa}(s, r)\\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\mathit {true}$"
],
[
"Partial Closed World Assumption (PCA)",
"The PCA [6] is an oracle that has proven useful for rule mining [3], [5].",
"Under the PCA, a subject-relation pair $s, r$ is considered complete if there is at least an object $o$ with $r(s, o)$ .",
"In other words, we assume that, if the KB knows some $r$ -values for $s$ , then it knows all its values.",
"The PCA is more cautious at predicting completeness than the CWA: it predicts completeness only if objects are already known.",
"This implies that the PCA makes predictions only for those entities that have an object for the relationship, and remains silent otherwise.",
"For instance, according to the CWA, a person that has no nationality in the KB has no nationality in reality, but the PCA will not make such claims.",
"Formally, the PCA oracle is: $\\mathit {pca}(s, r)\\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\exists o: r(s, o)$ The PCA is especially well suited for functional relations, where an entity can have at most one object.",
"Indeed, if an entity has some object for a functional relation, then it is necessarily complete."
],
[
"Cardinality",
"A more cautious oracle than the PCA is the cardinality oracle.",
"For an integer value $k$ , the cardinality oracle for value $k$ says that a subject $s$ is complete for a relation $r$ if $s$ has at least $k$ different objects for $r$ .",
"Formally: $\\mathit {card}_k(s, r)\\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\#(o: r(s, o)) \\ge k$ This oracle subsumes the CWA and PCA: $\\mathit {card}_0$ is $\\mathit {cwa}$ , and $\\mathit {card}_1$ is $\\mathit {pca}$ .",
"Other values for $k$ can be useful, e.g., $\\mathit {card}_2$ can be effectively used as a predictor for the $\\mathit {hasParent}$ relation.",
"In our experience, however, larger values of $k$ are rarely useful, and hence we categorize this oracle as a simple oracle."
],
[
"Popularity",
"The previous oracles look at properties of entities in isolation, but we can also look at entities in the context of the entire KB.",
"For example, we can hypothesize that entities which are popular (by some measure) are more likely to be complete.",
"For example, we expect that Wikipedia-based KBs are more complete for famous entities (e.g., Albert Einstein) than for entities that have only stub-articles.",
"From a Boolean measure $\\mathit {pop}$ indicating whether an entity is popular or not, we define the popularity oracle as: $\\mathit {popularity}_{\\mathit {pop}}(s, r) \\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\mathit {pop}(s)$"
],
[
"No Change",
"So far, we have only looked at a single snapshot of the KB, but we can also study how the KB changes over time.",
"If the objects of a particular subject do not change, then this may suggest that the subject is complete.",
"Given a Boolean measure of change $\\mathit {chg}$ , where $\\mathit {chg}(s, r)$ indicates whether the set of objects for entity $s$ and relation $r$ has changed over time, we define the no-change oracle by: $\\mathit {nochange}_{\\mathit {chg}}(s, r)\\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\lnot \\mathit {chg}(s, r)$"
],
[
"Parameterized Oracles",
"We now move on to the study of oracles that depend on parameters that are difficult to determine upfront, such as classes and relations."
],
[
"Star Patterns",
"Instead of estimating the completeness for a relation by looking only at that relation, we can look at facts involving other relations.",
"For example, if someone has won a Nobel Prize, then we probably know their alma mater.",
"Formally, we consider “star-shaped patterns” of certain relations around the subject, and define the star oracle, that predicts completeness if these patterns are all present: $\\mathit {star}_{r_1 \\ldots r_n}(s, r)\\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\forall i \\in \\lbrace 1, \\ldots , n\\rbrace : \\exists o_i : r_i(s, o_i)$"
],
[
"Class Information",
"In some cases, the class to which an entity belongs can indicate completeness with respect to some relations.",
"For example, the instances of the class LivingPeople should not have a death date.",
"If we assume that the KB is correct, this implies that any instance of that class is complete with respect to that relation.",
"Formally, the class oracle for a class expression $c$ on our KB $\\mathcal {K}$ is: $\\mathit {class}_c(s, r)\\mathrel {\\unknown.",
"{\\raisebox {0.3ex}{\\m@th \\cdot }}\\raisebox {-0.3ex}{\\m@th \\cdot }}=\\mathit {type}(s, c) \\in \\mathcal {K}$ We conduct our study with two types of class expressions: plain class names such as LivingPeople and negated class expressions of the form $\\hat{t} \\wedge \\lnot t$ where $t$ is a subclass of $\\hat{t}$ , like in $\\mathit {Person} \\wedge \\lnot \\mathit {Adult}$ ."
],
[
"Others",
"Many other completeness oracles can be envisaged.",
"For example, we could extract information from the Web to find out whether we can find more objects; we could ask a crowd of users for more objects; we could compare two KBs to see if one contains more information than the other; or we could check against external sources.",
"In this paper, however, we limit ourselves to a single source, and leave other such approaches to future work."
],
[
"Combining Oracles",
"Some completeness oracles cannot be used out-of-the-box.",
"For example, to use the star oracle and the class oracle, we must try out a huge number of possible parameters: YAGO, e.g., has 200,000 classes.",
"Furthermore, oracles may work best in combination: in some cases, the PCA may be the best oracle, while in others, the cardinality oracle may be better.",
"Our goal is thus to generalize and learn more complex completeness oracles from the simple ones that we presented.",
"Towards this goal, we assume that we already have a certain number of gold standard completeness assertions as training data.",
"We show in Section how to obtain such assertions from a crowd of users with good precision.",
"Based on these gold standard annotations, we can then learn combinations and parametrizations of the oracles.",
"To this end, we adapt the AMIE rule mining approach [6], [5]."
],
[
"AMIE",
"AMIE [6], [5] is an inductive logic programming system that is particularly geared towards KBs.",
"The source code of AMIE is available onlinehttps://www.mpi-inf.mpg.de/departments/databases-and-information-systems/research/yago-naga/amie/.",
"Given a KB, AMIE finds Horn rules such as $\\mathit {marriedTo}(X, Y) \\wedge \\mathit {livesIn}(X, Z) \\Rightarrow \\mathit {livesIn}(Y, Z)$ .",
"These rules do not hold in all cases, and therefore come with a confidence value.",
"In AMIE, an atom is a binary fact where at least one of the arguments is a variable – as in $\\mathit {livesIn}(\\mathit {Obama}, Y)$ .",
"We write the variables of atoms as capital letters.",
"A rule is an expression of the form $\\mathbf {B} \\Rightarrow H$ , where $\\mathbf {B}$ is the body (a conjunction of atoms $B_1 \\wedge \\dots \\wedge B_n$ ), and $H$ is the head (a single atom).",
"The support of a rule is the number of different instantiations of the head variables that satisfy all atoms of the rule in the KB.",
"If $H=r(x,y)$ , the support is defined by: $ supp(\\mathbf {B} \\Rightarrow r(x, y)) = \\#(x, y) : \\mathbf {B} \\wedge r(x, y) $"
],
[
"Rule Mining",
"AMIE starts with rules with an empty body (i.e., rules of the form “${\\Rightarrow r(X, Y)}$ ”), and refines them using a number of operators.",
"Each of the operators takes a rule as input, and produces a set of refined rules as output, by adding one particular type of atom to the body of the rule: Add Dangling Atom: A dangling atom joins the rule on an existing variable and introduces a new variable in the other position.",
"Add Closing Atom: A closing atom is an atom that joins on two existing variables in the rule.",
"Add Instantiated Atom: An instantiated atom has one instantiated argument (a constant/entity) and joins with the rule on the other argument.",
"The operators always produce rules with less support than the original rule.",
"AMIE applies them iteratively to find all rules above a given support threshold."
],
[
"Enhancing AMIE",
"Our goal is now to teach AMIE to learn rules such as $\\mathit {moreThan}_1(X, \\mathit {hasParent}) \\Rightarrow \\mathit {complete}(X, \\mathit {hasParent})$ This rule says that if $X$ has more than one object for the relation hasParent, then $X$ is probably complete on that relation.",
"For this purpose, we assume that we have training data, i.e.",
"known assertions of the form $\\mathit {complete}(x, r)$ and $\\mathit {incomplete}(x, r)$ .",
"We show in Section how to obtain such training data from the crowd.",
"Then, all of the completeness oracles (Section ) have to be translated into the AMIE framework.",
"For this purpose, we define the following new types of atoms : complete(x, r), incomplete(x, r): These assertions represent our training data.",
"We add them to the KB.",
"isPopular(x): The popularity oracle relies on an external measure $\\mathit {pop}$ of entity popularity.",
"We considered three such measures: (i) number of facts for that entity, (ii) length of the article in the English Wikipedia, and (iii) number of ingoing links to the Wikipedia page.",
"Manual inspection revealed that (i) correlated best with completeness.",
"Thus, we add $\\mathit {isPopular}(x)$ to the KB if $x$ is among the 5% entities with the most facts in the KB.",
"hasNotChanged(x, r): Given an older version of the KB, we add the fact $\\mathit {hasNotChanged}(x, r)$ to the new KB if $x$ has exactly the same $r$ -objects in the new KB as in the old KB.",
"In our experiments, we applied this to the YAGO KB, for which we used the oldest version (YAGO1) and the newest one (YAGO3).",
"notype(x, t): The $\\mathit {notype}(x, t)$ atom states that an entity is not an instance of class $t$ .",
"Such atoms are always used in conjunction with instantiated atoms of the form $type(x, \\hat{t})$ where $\\hat{t}$ is a super-class of $t$ .",
"These types of atoms allow us to integrate class expressions of the form $\\hat{t} \\wedge \\lnot t$ as defined for the class oracle.",
"lessThan$_n$ (x, r), moreThan$_n$ (x, r): An atom of the form $\\mathit {lessThan}_n(x, r)$ with $n > 0$ is satisfied if $x$ has less than $n$ objects for relation $r$ in the KB.",
"The moreThan$_n$ atom is defined analogously.",
"Such atoms allow AMIE to learn the cardinality oracles that we introduced.",
"To use AMIE for the task of learning completeness, we also made some changes to the system.",
"We let AMIE mine only rules with heads of the form $c(X, r)$ , where $c$ is either $\\mathit {complete}$ or $\\mathit {incomplete}$ , $r$ is a relation, and $X$ is a variable.",
"We represent unary atoms $p(x)$ as $p(x, \\mathit {true})$ since AMIE only supports binary atoms.",
"For performance reasons, we enable the “Add Instantiated Atom” operator only for $\\mathit {isPopular}(x)$ , $\\mathit {hasNotChanged}(x, r)$ , $\\mathit {type}(x, t)$ and $\\mathit {notype}(x, t)$ .",
"Another problem is that AMIE's rule language enforces closed Horn rules, which are rules where each variable is closed, i.e., it appears in at least two atoms in the rule.",
"We drop this constraint for variables in the body of rules, in order to allow for rules with star patterns such as: $\\mathit {wonPrize}(x, z) \\wedge \\mathit {politicianOf}(x, w) \\Rightarrow \\mathit {complete}(x, \\mathit {citizenOf})$ Still, we do not allow non-closed variables in the new kinds of atoms, e.g., $\\mathit {isPopular}$ and $\\mathit {hasNotChanged}$ .",
"We also forbid atoms with the relation $r$ in the body of the rules.",
"The last change is that we define five additional mining operators to capture the oracles that we defined: Add Type: Given a rule $\\mathbf {B} \\Rightarrow c(X, r)$ , this operator adds an atom of the form $\\mathit {type}(X, t)$ , where $t$ is the domain of $r$ .",
"The operator is applied only if the rule does not yet contain a type atom.",
"Specialize Type: Given a rule $\\mathit {type}(X, t) \\wedge \\mathbf {B} \\Rightarrow c(X, r)$ , this operator yields a new rule $\\mathit {type}(X, t^{\\prime }) \\wedge \\mathbf {B} \\Rightarrow c(X, r)$ where $t^{\\prime }$ is a subclass of $t$ .",
"Add Negated Type: Given a rule $\\mathit {type}(X, t) \\wedge \\mathbf {B} \\Rightarrow c(X, r)$ , this operator produces a new rule $\\mathit {notype}(X, t^{\\prime }) \\wedge \\mathit {type}(X, t) \\wedge \\mathbf {B} \\Rightarrow c(X, r)$ , where $t^{\\prime }$ is a subclass of $t$ .",
"Add Cardinality Constraint: Given a rule $\\mathbf {B} \\Rightarrow c(X, r)$ , this operator adds an atom of the form $\\mathit {moreThan}_0(X, r)$ or $\\mathit {lessThan}_k(X, r)$ , where $k$ is the highest number of objects seen for any subject in the relation $r$ .",
"Tighten Cardinality Constraint: Given a rule $\\mathit {lessThan}_k(X, r) \\wedge \\mathbf {B} \\Rightarrow c(X, r)$ , this operator replaces $k$ by the largest value $k^{\\prime }$ (with $k^{\\prime } < k$ ) that decreases the support of the original rule.",
"Likewise, given a rule $\\mathit {moreThan}_k(X, r) \\wedge \\mathbf {B} \\Rightarrow c(X, r)$ , we replace $k$ by the smallest value $k^{\\prime }$ ($> k$ ) that decreases the support.",
"For example, given the rule $\\mathit {moreThan}_0(X, \\mathit {hasParent}) \\Rightarrow \\mathit {complete}(X, \\mathit {hasParent})$ , the operator will replace 0 by 1."
],
[
"Learning",
"With our supplementary atoms and new mining operators, and up to the changes that we described, the actual learning of completeness rules works exactly as the mining of normal rules in [6], [5].",
"We exemplify this by showing how AMIE mines the following rules: $\\mathit {notype}(X, \\mathit {Adult}) \\wedge \\mathit {type}(X, \\mathit {Person})\\Rightarrow \\mathit {complete}(X, \\mathit {hasChild})$ $\\mathit {lessThan_1}(X, \\mathit {isCitizenOf}) \\Rightarrow \\mathit {incomplete}(X, \\mathit {isCitizenOf})$ The first rule says that if a person is not an adult, then the KB is complete for the children of that person (most likely zero).",
"To mine this rule, AMIE starts with the simple rule “$\\Rightarrow \\mathit {complete}(X, \\mathit {hasChild})$ ” and applies all the mining operators described in Sections REF and REF .",
"Among the different new rules generated by this step, the “Add Type” operator produces the rule $\\mathit {type}(X, \\mathit {Person}) \\Rightarrow \\mathit {complete}(X, \\mathit {hasChild})$ .",
"In the next step, the operator “Add Negated Type” produces new rules of the form $\\mathit {notype}(X, t) \\wedge \\mathit {type}(X, \\mathit {Person}) \\Rightarrow \\mathit {complete}(X, \\mathit {hasChild})$ , where $t$ is a subclass of $\\mathit {Person}$ .",
"In particular, for $t=\\mathit {Adult}$ , we obtain our example rule.",
"The second rule states that if a person has less than one citizenship, then the KB is incomplete in the citizenship relation for that person.",
"AMIE starts with the rule $\\Rightarrow \\mathit {incomplete}(X, \\mathit {isCitizenOf})$ , and applies the “Add Cardinality Constraint”.",
"Assuming that in the KB nobody has more than 3 nationalities, the operator produces the rule $\\mathit {lessThan_3}(X, \\mathit {isCitizenOf}) \\Rightarrow \\mathit {incomplete}(X, \\mathit {isCitizenOf})$ .",
"This rule has support $s$ .",
"In a later step, AMIE tries to apply the `Tighten Cardinality Constraint” operator.",
"The operator searches for the largest $k < 3$ such that the support of the new rule drops.",
"If the number of incomplete people with less than 2 nationalities is smaller than $s$ , the system will chose $k= 2$ and the rule becomes $\\mathit {lessThan_2}(X, \\mathit {isCitizenOf}) \\Rightarrow \\mathit {incomplete}(X, \\mathit {isCitizenOf})$ .",
"Using again the “Tighten Cardinality Constraint” operator on the new rule produces $\\mathit {lessThan_1}(X, \\mathit {isCitizenOf}) \\Rightarrow \\mathit {incomplete}(X, \\mathit {isCitizenOf})$ .",
"We remark that depending on the data distribution, AMIE may need a single call to the “Tighten Cardinality Constraint” to produce the target rule, i.e., we may skip the intermediate step where $k=2$ ."
],
[
"AMIE as completeness oracle",
"AMIE will learn rules that predict completeness as well as rules that predict incompleteness.",
"For the first type of rules, AMIE uses the $\\mathit {complete}(x, r)$ atoms of the training data as examples, and the $\\mathit {incomplete}(x, r)$ atoms as counter-examples.",
"For the second type of rules, the roles are reversed.",
"This implies that confidence for completeness and incompleteness rules follows the formula: $\\textit {conf}(\\mathbf {B} \\Rightarrow c(X, r)) = \\frac{supp(\\mathbf {B} \\Rightarrow c(X,r))}{supp(\\mathbf {B} \\Rightarrow c(X, r)) + supp(\\mathbf {B} \\Rightarrow \\lnot c(X, r)) }$ where $c \\in \\lbrace \\mathit {complete}, \\mathit {incomplete} \\rbrace $ .",
"Once the rules have been learned, we can define a new completeness oracle, the AMIE oracle.",
"For a given entity $e$ and a relation $r$ , the AMIE oracle checks whether any of the learnt rules predicts $\\mathit {complete}(e, r)$ .",
"If so, and if there is no rule with higher or equal confidence that predicts $\\mathit {incomplete}(e, r)$ , the oracle returns true.",
"If there is a rule with equal confidence that predicts $\\mathit {incomplete}(e, r)$ , the oracle returns true if the support of the completeness rule is higher.",
"In all other cases, the oracle returns false.",
"By restricting AMIE to only star atoms or only class atoms, we can obtain a Star oracle and a Class oracle, respectively, analogously to the AMIE oracle."
],
[
"Knowledge bases",
"Our goal is to measure the precision and recall of the completeness oracles on real data.",
"We conducted our study on two KBs: YAGO3, released in September 2015, and a dump of Wikidata from December 2015.",
"For both datasets, we used the facts between entities, the facts with literal object values (except for the relation rdfs:label) and the instance information.",
"These choices leave us with a KB of 89M facts (78M type statements) for YAGO, and a KB of 15M facts (3.6M type statements) for Wikidata.",
"We studied completeness on a set of relations covering a large variety of cases, and including people, movies, and locations: For one type of relations, basically every entity of the domain has to have exactly one object: hasGender, wasBornIn in YAGO; sex_or_gender (P21), mother (P25), father (P22), place_of_birth (P19) in Wikidata.",
"For some other relations, entities do not need to have an object, but can have at most one: diedIn in YAGO; place_of_death (P20) in Wikidata.",
"Some other relations usually have one object, but can have more: isCitizenOf and director(Movie, Person) in YAGO; country_of_citizenship (P27) and director (P57) in Wikidata.",
"In the most general case, a subject can have zero, one, or several objects: hasChild, graduatedFrom, isConnectedTo(Airport, Airport), and isMarriedToDespite the name, this relation captures also past spouses.",
"in YAGO; child (P40), alma_materWe use the same semantics as in YAGO: places a person graduated from.",
"(P69), brother, and spouse (P26) in Wikidata.",
"One relation has to have 2 objects: hasParentThis is how we call the inverse of hasChild in YAGO.",
"in YAGO."
],
[
"Ground Truth",
"In order to evaluate our completeness oracles, we need a set of completeness assertions and incompleteness assertions as a gold standard.",
"For some relations, we could generate this gold standard automatically.",
"Namely, for the relations where every subject has to have exactly one object, we have $\\mathit {complete}(s, r)$ iff $\\exists o: r(s, o)$ .",
"For the relations where every subject must have at least one object, we can directly label as incomplete all subjects without a value.",
"For the relations with at most one object, all subjects with one object are considered complete.",
"For the relation isConnectedTo, we used the OpenFlightshttp://openflights.org/data.html dataset as ground truth, which we assumed to be complete for all airports in this dataset (identified by their IATA code).",
"However, due to data inaccuracies, for some airports YAGO knew more flights than OpenFlights: we discarded these airports."
],
[
"Crowdsourcing",
"For the remaining relations, we used crowdsourcing to obtain ground truth data.",
"Given an entity, a relation, and the objects known in the KB, we asked crowd workers whether they could find any additional objects on the Web.",
"If they could, we labelled the entity-relation pair as incomplete, otherwise as complete.",
"To make the task well-defined and manageable, we asked workers to look only at a set of given Web pages.",
"We manually defined queries for each relation (e.g., “$x$ died” for $\\mathit {diedIn}(x, y)$ or “$x$ child” for $\\mathit {hasChild}(x, y)$ ), and then gave workers the first 4 URLs retrieved using the Bing search API.",
"We used the Crowdflower platformhttps://www.crowdflower.com for crowdsourcing, and paid 1 cent per answer.",
"For every relation, we annotated 200 random entities.",
"For each entity, we collected 3 opinions."
],
[
"Quality Control",
"To monitor quality, we manually generated 20–29 test questions for each relation.",
"Annotators had to pass a qualification test of 10 questions with at least 80% correct answers; further, the remaining test questions were mixed with the data, and annotators had to maintain 80% correctness while working.",
"About a quarter of annotators failed at the initial tests, and about 5% fell below the correctness threshold while working.",
"Their answers were discarded.",
"Furthermore, we used only the annotations where all 3 answers were unanimous.",
"These make up 55% of the annotations."
],
[
"Sampling",
"In our experiments with AMIE, we use 80% of our gold standard for training, and the rest for testing.",
"This gold standard was produced by randomly picking 200 entities in the domain of the studied relations.",
"We call this sample uniform.",
"The uniform sample is not always useful.",
"For example, only 1% of people have a citizenship in YAGO.",
"Thus, in a sample of 200 people, we may expect a citizenship for only 2 of them, which is not enough to learn a rule.",
"Therefore, for relations where less than 10% of the subjects have an object, we construct a biased sample instead.",
"Rather than choosing 200 entities randomly, we choose 100 entities randomly among those that have an object, and 100 among those that do not.",
"In our experiments, we mark the relations where we used the biased sample.",
"For the calculation of precision and recall, we carried out a de-biasing step.",
"This means that the values we report reflect the true population of entities in the KBs, and not the biased population."
],
[
"Experiment",
"Our completeness oracles from Section try to guess whether a pair of a subject and a relation is complete.",
"We considered the subject–relation pairs where we had a gold standard, and computed precision and recall values as described in Section .",
"Table REF shows the results for the oracles for YAGO3, and Table REF for Wikidata.",
"Table REF and Table REF show the corresponding F1 measures."
],
[
"Cardinality Oracles",
"The first column in the tables shows the CWA.",
"It trivially achieves a recall of 100%: for all pairs that are complete in reality, it makes a correct prediction.",
"However, its precision is lower.",
"This precision value corresponds to the actual completeness of the KB with respect to the real world.",
"We see, e.g., that YAGO is complete for the death place for 44% of the people.",
"This means that these people are either alive, or dead with a known death place in YAGO.",
"We also observe that Wikidata is generally more complete than YAGO.",
"The next oracle is the PCA.",
"It achieves 100% precision for all functional relations: if a subject has an object, the PCA rightly assumes that the subject is complete.",
"For quasi-functions, such as isCitizenOf, the PCA still performs decently, failing only for people with several nationalities.",
"The PCA has a recall of 100% for relations that are mandatory (such as hasGender): whenever this relation is complete in the gold standard, the PCA indeed predicts it.",
"For the other relations, the PCA has a much lower precision and recall.",
"The $card_2$ oracle has a much lower recall.",
"We could not compute it for relations where the sample did not contain any entity with sufficiently many objects.",
"This oracle basically makes sense only for the hasParent relation, where it performs perfectly.",
"As $\\mathit {card}_3$ behaves worse that $card_2$ on both datasets, we omitted it for space reasons."
],
[
"Popularity Oracle",
"The fourth column shows the popularity oracle.",
"The oracle was not computed for isConnectedTo due to noise in the data.",
"The popularity oracle generally has a low recall, because there are not many popular entities.",
"Its precision is generally good, indicating that popular entities (those that have many facts in general) are indeed more complete than unpopular ones.",
"However, even popular entities are incomplete for parents and citizenship in YAGO, and for parents in Wikidata."
],
[
"No-Change Oracle",
"The next column shows the no-change oracle on YAGO, for those relations that exist in both YAGO1 and YAGO3.",
"It has a very low recall, indicating that most entities did indeed change their objects over time (they most likely gained more objects).",
"The precision is decent, but not extraordinary.",
"Table: Precision and recall of all completeness oracles on YAGO3.",
"Relations with a biased sample are marked with *.Table: F1 measure of all completeness oracles on YAGO3.",
"Relations with a biased sample are marked with *.Table: Precision and recall of all completeness oracles on Wikidata.",
"Relations with a biased sample are marked with *.Table: F1 measure of all completeness oracles on Wikidata.",
"Relations with a biased sample are marked with *."
],
[
"Learning",
"We took 80% of our gold standard to train our modified AMIE approach (Section ) with 4-fold cross-validation.",
"The training phase measures the performance of AMIE at different configurations, i.e., different values for the support and confidence thresholds.",
"We tested values for support in the range from 10 to 100 entities (in steps of 10), while confidence was tested on values from 0.3 to 1.0 (in steps of 0.1).",
"We report the best configuration in terms of F1 measure for each relation, and use it to measure performance in the testing set (the remaining 20% of the gold standard).",
"Training took 44 hours on YAGO, and 4 hours in Wikidata.",
"This difference is mainly due to the much larger type hierarchy in YAGO (78M type assertions as opposed to 3.6M in Wikidata).",
"Table REF shows some of the rules that AMIE learned.",
"The first rule says that a person who has a date of death, but no place of death, is incomplete for the place of death.",
"In the second rule, the IMDb id acts as a substitute for the type movie, which is not always consistently used in Wikidata.",
"Thus, the rule basically says that if a movie has a producer, then it is most likely complete on the director.",
"Many of our rules are specific to our dataset.",
"Others (such as the first) may apply to different datasets.",
"We leave the study of cross-dataset rules for future work, and concentrate on each individual dataset here."
],
[
"Results",
"After the rules have been learned, making the actual oracle predictions on the gold standard takes only seconds.",
"We evaluated these predictions against the remaining 20% of our gold standard, and report the precision, recall, and F1 values in the three last columns of Tables REF and REF for YAGO, and in Tables REF and REF for Wikidata.",
"For the star oracle, we used a star size of $n=1$ for YAGO and $n=3$ for Wikidata.",
"We observe that this oracle can improve the F1 value for the isMarriedTo relation.",
"The class oracle, likewise, performs well for certain relations.",
"In particular, the oracle learned that the YAGO class LivingPeople means that the diedIn relation must be complete, boosting F1 from 60% to 99%.",
"This shows that parametrized oracles can be useful.",
"In general, the oracles complement each other.",
"Only the complete AMIE approach can nearly always perform best.",
"This is because AMIE learns the strengths of the individual oracles, and combines them as is most useful.",
"For functional relations, AMIE learned a rule that mimics the PCA, predicting completeness for a subject whenever one object is present: $\\mathit {moreThan}_0(X, r) \\Rightarrow \\mathit {complete}(X, r)$ .",
"For diedIn, AMIE learned a rule that mimics the Class oracle: $type(X, \\mathit {LivingPeople}) \\Rightarrow complete(X, \\mathit {diedIn})$ .",
"In this way, our oracle achieves an F1-measure of over 90% for more than half of the relations – on both YAGO and Wikidata.",
"When such relation-specific rules are not available, AMIE learns the CWA.",
"This is the case for difficult relations such as brother, graduatedFrom or isConnectedTo.",
"In particular, AMIE learns the CWA in rules of the form $\\mathit {type}(X, \\mathit {domain}(r)) \\Rightarrow \\mathit {complete}(X, r)$ All in all, our results show that it is indeed possible to predict completeness with very good precision and recall for a large number of relations.",
"We can predict whether people are alive, whether they graduated, or whether they have siblings – all by just looking at the incomplete KB.",
"The hasChild and marriedTo relations are the only ones where our oracles perform less well.",
"However, guessing whether someone is married, or whether someone has children, is close to impossible even for a human."
],
[
"Application",
"Having studied the experimental performance of our approach, we now show how the completeness assertions that we generate can prove useful in applications.",
"We focus on fact prediction, which we first define."
],
[
"Goal",
"Rule mining is generally used to find arbitrary rules in a KB, not just completeness rules.",
"We can use these rules to perform fact prediction, i.e., predict which person lives where, or which city is located in which country [6], [5].",
"We can compare the predictions to the real world and thus measure the precision of the approach.",
"We will show how the precision of fact prediction can be improved by completeness assertions.",
"For this purpose, we use the standard AMIE approach to make fact predictions, but we use the completeness assertions to filter out some of them: we filter out predicted facts $r(s, o)$ whenever $\\mathit {complete}(s, r)$ holds.",
"For example, if fact prediction says that a person has a parent, but the KB already knows two parents, then we discard the prediction."
],
[
"Setup",
"We followed the experimental setup from [5] and ran the standard AMIE system on YAGO3, using the obtained rules to predict new facts.",
"Each rule (and thus each prediction) comes with a confidence score.",
"We grouped the predictions in buckets by confidence score, as in [5].",
"For each bucket, we resorted to crowd workers to evaluate the precision of the predictions on a sample of 100 facts.",
"The lower line in Figure REF shows the number of predictions versus the cumulative precision estimated on the samples.",
"Each data point corresponds to a bucket of predictions, i.e., the first point on the left corresponds to the predictions with confidence score between 0.9 and 1, the second point to those with confidence between 0.8 and 0.9, etc.",
"In the second phase of the experiment, we used completeness assertions to filter out predictions.",
"We produced completeness assertions as in Section REF , by training AMIE with cross-validation on our entire set of gold standard completeness assertions.",
"The upper line in Figure REF shows the cumulative precision and number of predictions for each bucket after filtering.",
"Figure: Precision of fact prediction"
],
[
"Results",
"As we can observe, the filtering could successfully prune all wrong predictions.",
"The remaining predictions have a precision of 100%.",
"This high precision has to be taken with a grain of salt: the remaining predictions are mainly about citizenship, which is guessed from the place of residence or place of birth.",
"The completeness assertions filter out any predictions that try to assign a second citizenship to a person, and thus drastically increase the precision.",
"However, there are also a few other relations among the predictions.",
"These are, e.g., the death place, or the alma mater (guessed from the workplace of the academic advisor).",
"This precision comes at a price.",
"In total, AMIE made 1.05M predictions.",
"Of these, 400K were correct.",
"From these, the filtering incorrectly removed 110K.",
"Thus, the filtering removes roughly 25% of the correct predictions as a side-effect.",
"Still, we believe that our experiments make the case that completeness assertions can significantly improve the performance of fact prediction."
],
[
"Conclusion",
"To the best of our knowledge, our work is the first systematic study of the problem of completeness in knowledge bases.",
"Completeness is an important dimension of quality, which is orthogonal to the dimension of correctness, and which has so far received less attention.",
"In our paper, we have defined and analyzed a range of simple and parametrized completeness oracles.",
"We have also shown how to combine these oracles into more complex oracles by rule mining.",
"Our experiments on YAGO and Wikidata prove that completeness can indeed be predicted with high precision for many relations.",
"These completeness estimations can then be used to improve fact prediction to 100% precision in specific cases.",
"We hope that our work can lead to new research avenues, aiming to design knowledge bases that are not only highly accurate, but also highly complete.",
"The experimental results of this paper are available at http://luisgalarraga.de/completeness-in-kbs."
],
[
"Acknowledgment",
"This work has been partially supported by the project “MAGIC”, funded by the Province of Bozen-Bolzano, and “TQTK”, funded by the Free University of Bozen-Bolzano."
]
] | 1612.05786 |
[
[
"Neutrino Trident Production at the Intensity Frontier"
],
[
"Abstract We have calculated cross sections for the production of lepton pairs by a neutrino incident on a nucleus using both the equivalent photon approximation, and deep inelastic formalism.",
"We find that production of mixed flavour lepton pairs can have production cross sections as high as 35 times those of the traditional muon pair-production process.",
"Rates are estimated for the SHiP and DUNE intensity frontier experiments.",
"We find that multiple trident production modes, some of which have never been observed, represent observable signals over the lifetime of the detectors.",
"Our estimates indicate that the SHiP collaboration should be able to observe on the order of 300 trident events given $2\\cdot 10^{20}$ POT, and that the DUNE collaboration can expect approximately 250 trident events in their near detector given 3X10^{22} POT.",
"We also discuss possible applications of the neutrino trident data to be collected at SHiP and DUNE for SM and BSM physics."
],
[
"Introduction ",
"Neutrino physics has traditionally been dominated by the measurement of oscillation parameters and the study of neutrino nucleus scattering.",
"These experimental signals are largely dominated by charged current (CC), and neutral current (NC) interactions whose cross sections scale as $\\sigma \\sim sG_F^2$ .",
"Traditionally, limits on beam luminosity have resulted in event counts that leave sub-dominant processes with expected event rates less than unity in the lifetime of an experiment.",
"As a result these processes are often omitted in the discussions of neutrino physics.",
"One such neglected process is neutrino trident production which has been previously observed at CHARM II, CCFR, and NuTev , , .",
"These measurements provided evidence at the $3\\sigma $ level for the contribution of Z bosons in weak interactions , and more recently have been used to constrain BSM physics.",
"Specifically, measurements from CCFR currently provide the best constraints on the mass and coupling of a heavy $Z^{\\prime }$ force-mediator charged under $L_\\mu -L_\\tau $ .",
"Both of these applications are successful because the neutrino trident production of leptons is sensitive to both the vector and axial current couplings (see sec:Lepton-Matrix).",
"The aforementioned collaborations only measured one possible mode of trident production; specifically $\\nu A \\rightarrow \\nu \\mu ^+ \\mu ^- A$ .",
"The leading order contribution to this process involves the production of a muon-anti-muon pair, which can then interact with the target nucleus $A$ electromagnetically (see fig:schematic-trident).",
"For low momentum transfers ($Q\\ll R_A^{-1}$ ) the nucleus interacts coherently with the virtual photons ($\\sigma \\propto Z^2$ ), and there is a strong enhancement due to the infrared divergence in the photon propagator; it is this kinematic regime which dominates the cross section.",
"Other qualitatively similar processes, such as $e^+ e^-$ or $\\mu ^+e^-$ trident production, were kinematically accessible, however, due to technological limitations in the detector design, the required vertex resolution for trident identification was not achievable for electrons.",
"This would not be an issue with modern detectors.",
"Figure: Leading hadronic contribution to trident production.",
"Arrows denote direction of momentum.The cross section for $\\mu ^+\\mu ^-$ neutrino trident production is approximately five orders of magnitude smaller than the charged current cross section ($\\sigma \\approx 10^{-5} \\sigma _{CC}$ ) for a $50~\\text{GeV}$ neutrino scattering off an iron nucleus ; high $Z$ materials will have an even larger cross section relative to CC scattering.",
"This means that practically trident production can only be observed in experiments with very large neutrino fluxes.",
"Additionally the leading contribution to the cross section discussed in the preceding paragraph can be calculated using the equivalent photon approximation and scales as $\\sigma \\sim G_F^2 E_\\nu Q_\\text{max} \\log (E_\\nu Q_\\text{max}/m^2_\\ell )$ , where $m_\\ell $ is related to the lepton masses, and $Q_\\text{max}$ is a characteristic momentum transfer set by the radius of the nucleus .",
"These considerations imply that for trident to be a useful tool one needs to consider experiments with both a high energy neutrino beam ($\\langle E_\\nu \\rangle \\gtrsim 1~\\text{GeV}$ ), and high statistics.",
"This can be achieved via beam luminosity, or target-mass considerations.",
"Fixed target and beam dump experiments—where neutrino energies can be in excess of $100~\\text{GeV}$ , and charged current event counts can exceed $10^6$ —are an ideal setting to study neutrino trident production.",
"The Search for Hidden Particles (SHiP) experiment and the Deep Underground Neutrino Experiment (DUNE) both fall into these categories, and, as we show in this paper, represent the newest frontier in the study of trident production.",
"SHiP's program of study, as it relates to neutrino physics, is largely focused on tau neutrino, and anti-tau neutrino events, and is therefore optimized to observe tau leptons .",
"This represents a qualitatively new opportunity in the study of trident production, because the high mass of the tau leptons results in a threshold effect, wherein coherent production of a single tau lepton is not possible unless the following inequality holds $E_\\nu >(1/2)m_\\tau ^2 R_{A}$ ; the bound for tau lepton pair production is given by $E_\\nu >2 m_\\tau ^2 R_{A}$ .",
"As a result we also investigate the incoherent contribution to the cross section using both a diffractive and deep-inelastic approach.",
"The experiment will use beams with $\\langle E_\\nu \\rangle \\approx 30~\\text{GeV}-60~\\text{GeV}$ , and expects a lifetime collection of charged current events on the order of $N_{\\text{CC}}\\approx 2.7\\cdot 10^{6}$ .",
"It is therefore reasonable to assume that mixed flavour trident production, possibly including tau leptons, should be observable at the SHiP experiment.",
"Although the focus of its program of study is neutrino oscillations, the Deep Underground Neutrino Experiment (DUNE) will use sufficiently high luminosities, and neutrino energies to induce trident production.",
"DUNE consists of a near detector on site at FERMILAB and a far detector at Sanford Lab, both composed of liquid argon.",
"This technology allows for the observation of both electrons, and muons.",
"The far detector is exposed to a flux of neutrinos after a $1300~\\text{km}$ transit through earth.",
"The near detector will be used to account for systematic uncertainties in the neutrino beam and to record the initial neutrino flux.",
"It is designed to obtain ten times the statistics of the far detector .",
"The expected charged current event count in the far detector over the lifetime of the experiment is on the order of $1 \\cdot 10^5$ , and so it is reasonable to expect an observable signal of trident events for some of the processes; especially given the enhanced statistics of the planned near detector.",
"Trident production has proven itself a useful tool for constraining BSM physics by virtue of its sensitivity to modifications of $C_A$ and $C_V$ .",
"Additionally it represents an experimental signal that would provide an obvious background to searches of lepton flavour violation in the case of multi-flavour charged-lepton tridents.",
"If these new experiments (SHiP and DUNE) are to use trident production to probe BSM physics, then it is imperative to understand the relevant Standard Model backgrounds.",
"The rest of this article is organized as follows: In sec:Lepton-Matrix we discuss the basic structure of the trident amplitude in the Standard Model.",
"In sec:DIS-Vs-Coherent we describe how to obtain the cross sections for three distinct kinematic regimes; each receiving a separate theoretical treatment.",
"In sec:Prospects we calculate expected rates, and cross sections for both DUNE and SHiP.",
"We also present differential distributions with respect to the invariant mass of the charged lepton pair.",
"In sec:Discussion we review the qualitative features of our results and outline possible applications of trident for both SHiP and DUNE.",
"Finally in sec:Conclusion we discuss future directions for trident production for the upcoming generation of accelerator based neutrino experiments.",
"Our treatment of trident production varies over kinematic regimes, characterized by the four-momentum transfer to the nucleus $Q^2$ .",
"In every approach we treat the leptonic matrix element involving the EM current consistently.",
"Our treatment of the nucleus' interaction with the EM field, however, varies, and so will be treated separately in each section.",
"In the lower $Q^2$ regimes we relate the cross section to that of a neutrino-photon collision (photo-trident production), while for large $Q^2$ we employ the parton model.",
"The amplitudes for photo-trident production and parton-trident production can be written $\\begin{split}I\\mathcal {M}_{\\gamma \\nu }&= \\epsilon ^\\mu L_\\mu ~~~~~~~~~~~~~\\text{(EPA)} \\\\I\\mathcal {M}_{h\\nu }&= \\frac{-\\eta ^{\\mu \\nu }}{q^2} h_\\nu L_\\mu ~~~~~\\text{(DIS)}\\end{split}$ where $\\epsilon ^\\mu $ is an on-shell polarization tensor, and $h_\\nu $ is the hadronic matrix element in the parton model.",
"The leptonic matrix element $L_\\mu $ is calculated explicitly below.",
"We study both neutrino, and anti-neutrino induced trident production, and for the remainder of this section all reactions will contain an implicit hadronic initial and final state.",
"We use Latin flavour indices $i,j,k\\in \\lbrace e,\\mu ,\\tau \\rbrace $ and consider reactions of the form $\\lbrace \\nu _i \\rightarrow \\nu _{i~\\text{or}~k} + \\ell ^-_{j} +\\ell ^+_{k}~,~~ \\overline{\\nu }_i \\rightarrow \\overline{\\nu }_{i~\\text{or}~j} +\\ell ^-_{j} + \\ell ^+_k\\rbrace $ with the constraint that generational lepton number is conserved.",
"Both mono-flavour, and multi-flavour charged lepton pairs (i.e.",
"$\\mu ^+ \\mu ^-$ and $\\mu ^+\\tau ^-$ ) are included in our analysis.",
"Assigning the labels $\\lbrace 1,2,3,4,5\\rbrace \\rightarrow \\lbrace \\nu ,\\gamma ,\\nu ^{\\prime },\\ell ^+,\\ell ^-\\rbrace $ with $\\nu ^{\\prime }$ the outgoing neutrino (see fig:tridentpicture) and generalizing the analysis of , to multi-flavour lepton pairs we find $\\begin{split}L^\\mu _{ijk}=-&\\frac{Ie G_F}{\\sqrt{2}}\\left\\lbrace \\overline{u}_3\\gamma ^\\alpha (1-\\gamma ^5) u_1~~,~~\\overline{v}_1\\gamma ^\\alpha (1-\\gamma ^5) v_3\\right\\rbrace \\times \\\\&\\overline{u}_5\\bigg [\\gamma _\\alpha \\left(V_{ijk}-A_{ijk}\\gamma ^5\\right)\\frac{1}{{q}-{p}_4-m_4}\\gamma ^\\mu \\\\+&\\gamma ^\\mu \\frac{1}{{p}_5-{q}-m_5}\\gamma _\\alpha \\left(V_{ijk}-A_{ijk}\\gamma ^5\\right)\\bigg ]v(p_4),\\end{split}$ where the first line contains the appropriate spinor wavefunctions for an incident neutrino and anti-neutrino beam respectively.",
"$V_{ijk}$ and $A_{ijk}$ are the flavour dependent vector and axial coupling strengths, which are typically denoted $C_V$ and $C_A$ respectively.",
"We use non-standard notation to stress that these couplings carry flavour indices because some processes are mediated exclusively by $W$ bosons, others exclusively by $Z$ bosons, and some a mixture of the two.",
"As we see from fig:tridentpicture, these mediators modify the coupling to the vector and axial currents, as can be verified by use of Fierz identities.",
"As noted in the interference between the neutral and charged current channels in the Standard Model results in a $40\\%$ reduction in the cross section compared to the $V-A$ theory prediction.",
"Thus by considering different combinations of leptons in the final state the cross section can be enhanced, or suppressed, significantly.",
"The constants $A_{ijk}$ and $V_{ijk}$ are presented in tab:neutrinoME for $\\nu _\\mu \\rightarrow \\nu _\\mu \\tau ^+\\tau ^-$ and for all trident processes with lifetime event counts greater than $0.01$ at either SHiP or DUNE.",
"Table: Modified vector and axial coupling constants for different combinations of incident neutrino flavours and final states"
],
[
"Coherent, Diffractive and Deep Inelastic Regimes ",
"We will begin by reviewing conventional scattering of neutrinos off of nuclei to emphasize the qualitative differences in trident production.",
"Neutrino-nucleus scattering is dominated by charged current events, which can be loosely partitioned into three classes for $E_\\nu $ >$$ 100 MeV$: quasi-elastic scattering, hadronic resonance production, and deep inelastic scattering \\cite {Formaggio2012}.",
"It is only at low centre of mass energies $ E$\\;\\stackrel{\\textstyle <}{\\sim }\\;$ 50 MeV$ that coherent scattering via the neutral current is possible such that the reaction^{\\prime }s cross section scales as $ (A-Z)2E2$ with $ A-Z$ the number of neutrons.",
"In this energy regime coherent scattering cross sections can be as much as three orders of magnitude larger than that predicted by a na{ï}ve sum of the nucleon cross sections \\cite {Drukier1984}.$ Figure: An example of a process which takes place exclusively through the neutral current channel.",
"The mismatch in flavour between the incident neutrino and outgoing leptons prohibits a charged current interaction.This limited kinematic window stands in sharp contrast to trident production where coherent contributions are possible at all energies, because the reaction is not $2\\rightarrow 2$ and the phase space is therefore less kinematically constrained.",
"This scattering is mediated electromagnetically, and, in addition to the coherent $Z^2$ amplification, the photon's propagator introduces an infrared divergence that further enhances the amplitude.",
"As is the case for coherent neutrino scattering this regime is characterized by small momentum transfers ($Q^2\\sim R^{-2}_A$ ) wherein the phases of the various amplitudes are nearly commensurate, and the amplitudes interfere constructively.",
"Kinematic considerations constrain the momentum transfer via $Q>s/(2E_\\nu )$ , with $s$ the invariant mass of the neutrino-photon pair .",
"When combined with the lepton pair's mass threshold, this regulates the infrared divergence mentioned above.",
"The three regimes typically considered in charged-current scattering for high energy neutrinos (mentioned in the first paragraph) also exist for trident production.",
"Quasi-elastic-like diffractive scattering can contribute significantly to trident production, especially when threshold effects related to lepton masses are important.",
"We expect the deep inelastic contribution to be suppressed, but for many of the neutrino energies at SHiP it is the only kinematically allowed production mechanism for tau leptons, and so we also include this regime in our analysis."
],
[
"Coherent Regime ",
"The coherent contribution to neutrino trident production can be accurately calculated using the equivalent photon approximation (EPA) , , , .",
"In the EPA the cross section for the full scattering process is decomposed into two pieces.",
"First the cross section corresponding to the scattering of a neutrino and photon creating a lepton trident, denoted by $\\sigma _{\\gamma \\nu }$ , is calculated.",
"Next, this cross section is weighted against a universal probability distribution $P(s,Q^2)$ that measures the likelihood of the nucleus producing a virtual photon with virtual-mass $Q^2$ , and neutrino-photon centre of mass energy $s$ .",
"The full cross section is given by $\\begin{split}\\sigma _{\\nu A}&= \\int \\mathrm {d}s~ \\sigma _{\\gamma \\nu }(s) \\int \\mathrm {d}Q^2 P(s,Q^2)\\\\&=\\frac{Z^2\\alpha }{\\pi }\\int _{m_{jk}^2}^{s_{\\text{max}}}\\frac{ \\mathrm {d}s }{s}\\sigma _{\\gamma \\nu }(s)\\int _{(s/2E_\\nu )^2}^\\infty \\frac{\\mathrm {d}Q^2}{Q^2}F^2(Q^2)\\end{split}$ with $m_{jk}=m_j+m_k$ the sum of the lepton pair's masses.",
"A fairly good, albeit crude, approximation is to treat the form-factor for the nucleus $F(Q^2)$ as a Heaviside function $\\Theta (Q^2_\\text{max}-Q^2)$ where the scale $Q_\\text{max}=\\Lambda _\\text{QCD}/A^{1/3}$ corresponds to characteristic momentum transfer at which one would expect the dissolution of the nucleus .",
"This sets a maximum centre-of-mass energy for the photon-neutrino interaction $s_\\text{max}=2E_\\nu Q_\\text{max}$ .",
"With these approximations, suppressing flavour indices and working in the leading log approximation, eq2:Coherent-Scattering simplifies to , $\\sigma _{\\nu A} \\approx \\frac{1}{2}(A^2+V^2)\\frac{2 ~Z^2\\alpha ^2 ~G_F^2 }{9\\pi ^3} s_\\text{max}\\log \\left(\\frac{s_\\text{max}}{4m^2} \\right)$ where $2m=m_{j}+m_{k}$ .",
"There are additional terms resulting from the interference between the vector and axial currents, but these are suppressed by two powers of the lepton mass, and are therefore small.",
"A more realistic implementation is to use the Woods-Saxon form-factor, which is what we used in all of our calculations (this changes the answer by order 10%, see sec:Three-Body for details).",
"We can write the coherent contribution to the neutrino-nucleus cross section as $\\mathrm {d}\\sigma _{\\gamma \\nu }=\\frac{1}{2s}\\frac{1}{2} \\sum _{\\text{pol} }\\left|\\epsilon _\\mu L^\\mu \\right|^2 \\mathrm {d}\\Phi _3$ where $\\Phi _3$ is the three-body phase space of final states, the factor of $1/2$ averages over photon polarizations, and $2s$ is the Lorentz invariant flux factor.",
"For details on the treatment of the three-body phase space see sec:Three-Body."
],
[
"Diffractive Regime",
"At intermediate $Q^2$ it is possible to interact with the individual protons of the nucleus, both without coherent interference of their individual amplitudes, and without probing their inner parton structure.",
"Our treatment of this regime follows the approach outlined in , and is identical to the coherent regime with the following changes: $\\begin{split}\\sigma _{\\nu A}&=Z\\int \\mathrm {d}s~ \\sigma _{\\gamma \\nu }(s) \\int \\mathrm {d}Q^2 P(s,Q^2)\\\\&=Z\\frac{\\alpha }{\\pi }\\int _{m_{jk}^2}^{s_{\\text{max}}}\\frac{ \\mathrm {d}s }{s}\\sigma _{\\gamma \\nu }(s)\\int _{Q^2_\\text{min}}^{1~\\text{GeV}^2} \\frac{\\mathrm {d}Q^2}{Q^2}F_\\text{dip}^2(Q^2).\\end{split}$ The charge of the nucleus now appears as an overall multiplicative factor as opposed to appearing in $P(s,Q^2)$ , we cut off our integral at $Q_\\text{min}=\\text{max}\\left(s/2E_\\nu ,R_A^{-1}\\right)$ to avoid double counting amplitudes included in the coherent calculation, and we use the standard dipole fit to the proton's electromagnetic form factor (see sec:Three-Body).",
"We introduce an explicit UV cut-off for the $Q^2$ integration to avoid double counting with the DIS amplitudes.",
"This was not necessary for the coherent regime due to the exponential, as opposed to power law, decay of the Wood-Saxon form factor at high $Q^2$ ."
],
[
"Deep Inelastic Regime",
"Our treatment of the deep inelastic case is fairly standard, with a few exceptions that are highlighted in sec:DIS-Hadron.",
"We treat this regime by convoluting the parton cross sections with nucleon parton distribution functions (PDFs) $f(\\xi ,Q)$ , taking into account the $u,d,c,s$ quarks.",
"The phase space integrals are sensitive to the lepton masses, and so although their effects on the matrix element are often sub-leading, we include their full dependence throughout our calculations.",
"All of the quarks are treated as massless in our analysis.",
"We take care to include a cut on momentum transfers so as not to double count contributions already accounted for by the EPA.",
"Additionally we place a cut on the momentum fraction $\\xi $ to ensure the parton carries enough four-momentum to both be able to produce the appropriate pair of charged leptons and to satisfy the double-counting-cut on momentum transfer.",
"The resulting cross sections for the various nucleons are then summed to obtain the scattering cross section with the nucleus.",
"We can write $\\sigma _{\\nu A}$ as a weighted sum of the cross sections with the constituent nucleons $\\sigma _{\\nu A}=Z\\sigma _{\\nu p}+(A-Z)\\sigma _{\\nu n}.$ These can in turn be written in terms of the parton-level cross sections $\\sigma _{h\\nu }$ via $\\sigma _{\\nu H}=\\sum _h\\int _{\\xi _\\text{min}}^1 \\mathrm {d}\\xi \\int ^{Q_\\text{max}}_{Q_\\text{min}} \\mathrm {d}Q~ \\frac{\\mathrm {d}\\sigma _{h\\nu }}{\\mathrm {d}Q}(\\xi ,Q) ~f^{(H)}_h(\\xi ,Q)$ where $f^{(H)}_h(\\xi ,Q)$ is the PDF for parton $h$ in the nucleon $H\\in \\lbrace n,p\\rbrace $ .",
"More details can be found in sec:Four-Body.",
"In the following, we calculate trident rates at SHiP, and at the DUNE far and near detectors.",
"We calculate the rates for momentum transfers $Q<0.217/(A)^\\frac{1}{3}~\\mathrm {GeV}\\approx R_A^{-1}$ regime using the coherent EPA method.",
"For intermediate momentum transfers $0.217/(A)^\\frac{1}{3}~\\mathrm {GeV}$ <$$ Q$\\;\\stackrel{\\textstyle <}{\\sim }\\;$ Mp$ transfers, we use the diffractive EPA treatment.",
"Finally for $ Q$\\;\\stackrel{\\textstyle >}{\\sim }\\;$ 1 GeVMp $ we employ the deep inelastic formalism.",
"We use PDFs from the MSTW collaboration (2008 NNLO best fit) \\cite {Martin2009}.",
"To calculate the rates, we estimate the number of SM neutrino trident events for each flavour of incident neutrino $ i$ producing a lepton pair composed of $ j-$ and $ k+$ with $ i,j,k{e,,}$.",
"We estimate the luminosity in terms of charged current events $ NCCi$ using\\begin{equation}N^{ijk}_\\text{Trident}=\\sum _E \\frac{N^i_{CC}(E)}{\\sigma _{CC}(E,A)}\\sigma ^{ijk}_{\\nu A}(E,Z,A)\\times \\epsilon ^j_-\\times \\epsilon ^k_+,\\end{equation}where $ CC$ is the neutrino charged current cross sections \\cite {Agashe2014} and $ i,j,k$ are flavours denoting the incident neutrino, outgoing $ -$ and outgoing $ +$ respectively.",
"Additionally $ +$ and $ -$ are the identification efficiencies for $ +$ and $ -$ respectively.",
"We do an analogous procedure for anti-neutrinos.$ There will be a background contribution to trident from resonant production of charged pions and charm production from $D$ mesons, whose leptonic modes are both dominated by muon flavoured final states.",
"In the different flavour opposite sign di-lepton final states, backgrounds can arise from $\\bar{\\nu }_\\mu $ CC scattering in combination with an elastic NC event releasing an electron, and also by muon final states in which one of the muons fake an electron.",
"As coherent-scattering is quasi-elastic, the backgrounds for the dominant contribution to the cross section (see sec:DIS-Vs-Coherent) can be greatly reduced by imposing hadronic vetoes in the analysis.",
"Further background suppression can be achieved by selecting oppositely charged leptons that fall within the vertex resolution of the detectors and selecting events with low $M_{\\ell ^+\\ell ^-}$ invariant masses.",
"We leave the background estimates to the collaborations' detailed and sophisticated simulations.",
"Our signal results are shown in tab:ratesship,tab:ratesdunefar,tab:ratesdunenear.",
"Table: Number of expected trident events for coherent (Coh) and diffractive (Diff) scattering, using the EPA, in the SHiP ν τ \\nu _\\tau detector, assuming 2×10 20 2\\times 10^{20} POT on molybdenum."
],
[
"Calibrations and Tests",
"The details of our calculations can be found in the Appendices.",
"We calibrated our EPA cross section calculations with previous theoretical and experimental work , , , and reproduced the analytic results of .",
"Our DIS work was calibrated with MadGraph5 for trident induced muon pair production.",
"MadGraph5 treats light leptons as massless, and due to infrared singularities in the propagators this necessitates a careful treatment; it also introduces questions of reliability.",
"We imposed the following cuts to replicate the effects of finite muon masses: $p_{T}>m_\\mu $ for the muons, $p_{T}>1.5~\\text{GeV}$ for the jets, and $\\Delta R=\\sqrt{\\Delta \\eta ^2+\\Delta \\phi ^2}>0.4$ for the lepton pairs.",
"With these cuts we found our calculations to agree with MadGraph5 to within a factor of $0.5-2.5$ for $E_\\nu =\\lbrace 20~\\text{GeV},200~\\text{GeV},1000~\\text{GeV}\\rbrace $ .",
"We believe our calculation to be more reliable than MadGraph5 in the low $Q^2$ regions of phase space which dominate the cross sections due to infrared divergences, which we treat carefully."
],
[
"Rates for SHiP",
"SHiP will be a lead based neutrino detector , .",
"It will utilize an emulsion cloud chamber for its electron detection and a muon magnetic spectrometer for muons.",
"It is estimated to have a 90% $e$ and $\\mu $ identification efficiency, and a micron vertex resolution.",
"Under nominal operating conditions, after 5 years of operation it will have collected data from $2\\times 10^{20}$ POT using a 400$~\\mathrm {GeV}$ SPS proton beam.",
"We quote all the rates assuming this normalization.",
"The energy spectrum at SHiP is very broad, and reaches sufficiently high energies such that trident production of tau leptons becomes kinematically allowed in the coherent, diffractive, and deep inelastic regimes.",
"The latter is allowed at almost all incident neutrino energies available at SHiP with the only requirement being the centre of mass energy exceed the lepton pair's mass-gap.",
"Despite being kinematically allowed, we find the large momentum transfer in the deep inelastic regime renders the contribution to the cross section negligible.",
"The diffractive and coherent regimes rely on the high energy tail of the quoted beam distribution .",
"For electrons and muons, coherent, and diffractive production are not only possible but extremely viable, while for tau leptons we find only diffractive production to be viable, but only marginally so.",
"In fig:smsigmavseshipEPA and fig:smsigmavseshipDIS, we show the cross section per nucleon as a function of the incoming neutrino energy for a variety of processes.",
"The coherent cross sections computed via the EPA are normalized by $Z^2$ while the deep inelastic contribution is normalized by $A$ .",
"There are small differences in these plots for various materials, as the EPA Woods-Saxon form factor and the relative number of protons to neutrons in DIS both introduce a sub-leading dependence on the ratio of protons to neutrons that is not removed by the per nucleon normalization.",
"Figure: σ/E ν \\sigma /E_\\nu trident cross sections normalized by Z 2 Z^2 for various SM flavours as a function of the incoming neutrino energy on a lead target (SHiP).In tab:ratesship we show the expected number of events in the various production modes for both low-$Q^2$ events calculated within the coherent EPA and intermediate-$Q^2$ events calculated using the diffractive EPA.",
"DIS rates are not included, because the cumulative lifetime event-count for all production modes in the deep inelastic regime is $N^{(\\text{tot})}_\\text{DIS}\\approx 0.1$ .",
"The basic features of our analysis can be understood by looking at tab:neutrinoME,eq3:Coherent-Scattering and remembering that the neutrino beam is dominated by $\\nu _\\mu $ and $\\overline{\\nu }_\\mu $ .",
"This is discussed in greater detail in sec:Discussion."
]
] | 1612.05642 |
[
[
"The VIMOS Public Extragalactic Redshift Survey (VIPERS): The growth of\n structures at $0.5<z<1.2$ from redshift-space distortions in the clustering\n of the PDR-2 final sample"
],
[
"Abstract We present measurements of the growth rate of cosmological structure from the modelling of the anisotropic galaxy clustering measured in the final data release of the VIPERS survey.",
"The analysis is carried out in configuration space and based on measurements of the first two even multipole moments of the anisotropic galaxy auto-correlation function, in two redshift bins spanning the range $0.5 < z < 1.2$.",
"We provide robust and cosmology-independent corrections for the VIPERS angular selection function, allowing recovery of the underlying clustering amplitude at the percent level down to the Mpc scale.",
"We discuss several improvements on the non-linear modelling of redshift-space distortions (RSD) and perform detailed tests of a variety of approaches against a set of realistic VIPERS-like mock realisations.",
"This includes using novel fitting functions to describe the velocity divergence and density power spectra $P_{\\theta\\theta}$ and $P_{\\delta\\theta}$ that appear in RSD models.",
"These tests show that we are able to measure the growth rate with negligible bias down to separations of $5h^{-1}Mpc$.",
"Interestingly, the application to real data shows a weaker sensitivity to the details of non-linear RSD corrections compared to mock results.",
"We obtain consistent values for the growth rate times the matter power spectrum normalisation parameter of $f\\sigma_8=0.55\\pm 0.12$ and $0.40\\pm0.11$ at effective redshifts of $z = 0.6$ and $z=0.86$ respectively.",
"These results are in agreement with standard cosmology predictions assuming Einstein gravity in a $\\Lambda \\rm{CDM}$ background."
],
[
"Introduction",
"The discovery of the accelerated expansion of the Universe in the late stages of the 20th Century has given us a self-consistent standard cosmological model, which is in close agreement with virtually all current cosmological observations.",
"Multiple lines of evidence, from cosmic microwave background anisotropies [34], [57], baryon acoustic oscillations in the galaxy distribution [4], [9], [1], to SNe Ia luminosity distances [60], [55], require most of the energy content of the Universe to be in form of a repulsive `dark energy' that is empirically close in behaviour to the classical cosmological constant (see e.g.",
"[73] [73] for some history and a review of current constraints).",
"The nature of dark energy is naturally a question of huge interest, with possibilities ranging from a fixed vacuum energy density with equation of state $w=P/\\rho c^2=-1$ , to dynamical models based on evolving scalar fields varying both in space and time.",
"Such models motivate an effort to measure $w$ and its evolution.",
"But independently of the outcome of this exercise, it remains the puzzle that a very large vacuum density seems to be necessary – so the much smaller observed value therefore requires a challenging degree of fine tuning [74].",
"A more radical explanation for the observed acceleration could be that the theory of gravity itself is modified on cosmological scales [13], [36], [14].",
"Commonly discussed alternatives include $f(R)$ gravity, where the gravitational Lagrangian is made more complicated than a simple Ricci scalar $R$ ; chameleon models that invoke a fifth fundamental force to drive the acceleration; and DGP (Dvali-Gabadadze-Porrati) models, which postulate a higher-dimensional Minkowski space-time, within which the ordinary 3+1 space-time is embedded.",
"For an appropriate choice of model parameters, dark energy and modified gravity can both reproduce the observed expansion history $H(z)$ .",
"In principle this degeneracy can be lifted by measuring the growth rate of cosmic structure.",
"Modifications of gravity involve a variation in the strength of the gravitational force with scale or environment, and thus a key question is whether density fluctuations are growing at the rate predicted by models involving General Relativity and a homogeneous dark energy.",
"Among observational methods to estimate the growth rate of structure, redshift-space distortions (RSD) in the clustering pattern of galaxies [38] have assumed a growing importance in the last decade [29].",
"RSD arise when the Doppler effect of galaxy peculiar velocities supplements the isotropic Hubble expansion.",
"Peculiar velocities are inevitably associated with gravitational growth of inhomogeneities, which can be described by the logarithmic growth rate of density perturbations: $f\\equiv \\frac{d\\ln \\delta }{d\\ln a}\\,\\,\\,\\,\\, ,$ where $\\delta $ is the fractional density fluctuation, and $a$ is the cosmic scale factor.",
"For many (but not all) theories of gravity, this growth rate can be well approximated by an empirical relation as $f(z)=[\\Omega (z)]^\\gamma $ [51], [40], provided the fluctuations are in the linear regime and in the growing mode.",
"For Einstein gravity, $\\gamma \\simeq 0.55$ ; but this parameter can vary by around 0.1 between different commonly-discussed models of late-time dark energy and modified gravity [23], [44].",
"Measurements of linear RSD from galaxy redshift surveys constrain the combination $\\beta =f/b$ , where $b$ is an unknown linear galaxy bias parameter.",
"But the real-space galaxy autocorrelation function, $b^2\\xi _{\\rm mass}$ , is observable – so this can be eliminated to yield an estimate of a quantity that purely concerns dark matter: $f\\sigma _8$ , with $\\sigma _8$ being the rms linear matter fluctuations within spheres of radius $8\\,h^{-1}{\\rm Mpc}$ .",
"Figure: Footprint of the VIPERS observations within the W1 (top) andW4 (bottom) fields, as reconstructed from the final galaxysample.",
"The VIMOS pointings and quadrants are marked by blackrectangles.",
"Galaxies are colour-coded according to their value ofthe Target Sampling Rate (TSR: see Sect.",
"), which can be considered as a proxy forthe inverse of the projected galaxy density field.",
"Empty rectanglescorrespond to failed quadrants, for which the spectroscopic maskinsertion failed or was incorrect, leading to no data beingcollected.Unfortunately, extracting the linear RSD signal from galaxy redshift surveys is non-trivial, because much of the RSD signal lies on quasi-linear and non-linear scales.",
"A simple and widely-used extension of the linear Kaiser model is the `dispersion model' [50], which accounts for radial convolution by a random velocity dispersion plus non-linear corrections to the real-space power spectrum.",
"This model was successfully applied to several galaxy surveys in the past [49], [29], but is insufficiently accurate to be trusted when the precision allowed by the data goes below 10% ([48] [48], [8] [8]; see also the companion paper by [75] [75]).",
"There have been a number of attempts to derive improved RSD models.",
"As shown by [66], the dispersion model is a simplification of the original streaming model [51], [24], in which the full redshift-space correlation function is obtained by convolution with a proper scale-dependent pairwise velocity distribution.",
"But predicting this distribution function is hard [7], [72], and typical applications simplify the problem by adopting a (scale-dependent) Gaussian pairwise distribution function [58].",
"[66] proposed an influential alternative, in which the linear Kaiser term is generalised by including the velocity and velocity-density power spectra.",
"This concept was extended by the TNS model [71], which takes better into account the non-linear coupling between the density and the velocity field.",
"This model is currently considered as one of the best descriptions of RSD down to the quasi-linear regime.",
"These theoretical developments have been stimulated by a growing number of new measurements from larger datasets.",
"These included in particular the 6dfGS [5], WiggleZ [9], [17] and BOSS [59], [6], [65], [64], [27].",
"The present paper is one in a series aimed at extending this RSD work to higher redshifts by analysing the final PDR-2 release of the VIMOS Public Extragalactic Redshift Survey , , .",
"This survey has collected redshifts for about $90\\,000$ galaxies in the range $0.4\\lesssim z \\lesssim 1.2$ with sampling and volume comparable to those of local surveys, such as the Two-degree Field Galaxy Redshift Survey (2dFGRS) at $z\\simeq 0.1$ [16].",
"The prime original goal of VIPERS was an accurate measurement of the growth rate of structure at redshift around unity.",
"An early measurement was performed using the Public Data Release 1 (PDR-1: [25] [25]), setting a reference measurement of $f\\sigma _8$ at $z = 0.8$ [19].",
"Having nearly doubled the sample, this analysis is now revisited, and expanded in a number of ways.",
"[21] performs a configuration space joint analysis involving RSD and galaxy-galaxy lensing, while [75] develops a direct Fourier-space approach coupled with the so-called `clipping' linearisation of the density field; with a similar aim, [45] identifies optimal sub-classes of RSD tracers, focusing on luminous blue galaxies; the analysis presented here uses the configuration-space information contained in the first two even multipole moments of the anisotropic correlation function, implementing the currently most advanced non-linear corrections and testing their performances on VIPERS-like mocks.",
"The paper is organised as follows.",
"In Sect.",
"2 we give a description of the final VIPERS dataset and of the corresponding mock catalogues used throughout the analysis, while in Sect.",
"3 we describe the estimation of the two-point correlation function of galaxies in redshift space.",
"Section 4 describes the target selection biases and how these are mitigated.",
"In Sect.",
"5 we present the VIPERS measurements.",
"The error estimates are described in Sect.",
"6 along with the fitting procedure.",
"Section 7 gives a description of the RSD models that are used in Sect.",
"8 to understand the level of systematics in the recovery of the growth rate of structure.",
"The results are presented in Sect.",
"9 and discussed in Sect.",
"10 with our conclusions.",
"Throughout this analysis, if not specified otherwise, we assume a fiducial flat $\\Lambda {\\rm CDM}$ cosmological model with $(\\Omega _m,\\Omega _b,n_s)=(0.30,0.045,0.96)$ and parametrise the Hubble constant as $H_0=100\\,h\\,\\rm {km\\,s^{-1}Mpc^{-1}}$ .",
"The VIPERS survey covers an overall area of $ 23.5$ deg$^2$ over the W1 and W4 fields of the Canada-France-Hawaii Telescope Legacy Survey Wide (CFHTLS-Wide).",
"The VIMOS multi-object spectrograph [43] was used to cover these two fields with a mosaic of 288 pointings, 192 in W1 and 96 in W4 (see Fig.",
"REF ).",
"Galaxies are selected from the CFHTLS catalogue to a faint limit of $i_{\\rm AB}=22.5$ , applying an additional $(r-i)$ vs $(u-g)$ colour pre-selection that efficiently and robustly removes galaxies at $z<0.5$ .",
"Coupled with a highly optimised observing strategy [68], this doubles the mean galaxy sampling efficiency in the redshift range of interest, compared to a purely magnitude-limited sample, bringing it to 47%.",
"Spectra are collected at moderate resolution ($R\\simeq 220$ ) using the LR Red grism, providing a wavelength coverage of 5500-9500$\\smash{\\rm {Å}}$ .",
"The typical redshift error for the sample of reliable redshifts is $\\sigma _z=0.00054(1+z)$ , which corresponds to an error on a galaxy peculiar velocity at any redshift of 163 $\\,{\\rm km\\, s^{-1}}$ .",
"These and other details are given in the full PDR-2 release accompanying paper [69].",
"A discussion of the data reduction and management infrastructure was presented in [25], while a complete description of the survey design and target selection was given in the survey description paper [30].",
"The dataset used in this paper is an early version of the PDR-2 data, from which it differs by a few hundred redshifts revised during the very last period before the release.",
"In total it includes $89\\,022$ objects with measured redshifts.",
"As in all statistical analyses of the VIPERS data, only measurements with quality flags 2 to 9 inclusive are used, corresponding to a sample with a redshift confirmation rate of $96.1\\%$ (for a description of the quality flag scheme, see [69] [69]).",
"In the analysis presented here we shall analyse two redshift sub-samples of the whole survey (W1 + W4) in the ranges $0.5<z<0.7$ and $0.7<z<1.2$ , including respectively 30 764 and 35 734 galaxies.",
"Figure: Redshift distribution of the final VIPERS galaxy sample.",
"Thedistributions of redshifts collected separately within the twoCFHTLS fields are plotted together with the combined distributionusing different colours.",
"The red and purple solid lines showrespectively the best fit using the analytic template inEq.",
"() and the predicted V max V_{\\rm max} profile ofthe combined redshift distribution.",
"The peculiar distribution of theVIPERS galaxy sample differs from the typical expectation from amagnitude-limited sample.",
"This deviation is the result of thecolour-colour pre-selection adopted to reject most galaxies locatedat z<0.5z<0.5."
],
[
"Redshift distribution",
"The redshift distribution of the galaxy sample is shown in Fig.",
"REF .",
"At $z>0.6$ , it follows the typical decay in the number of objects expected for a magnitude-limited survey, while the rapid fall of the counts at $z<0.5$ is the result of the colour-colour pre-selection.",
"In [19] it was shown that this histogram can be modelled analytically by the functional form $N(z)=A\\bigg (\\frac{z}{z_0}\\bigg )^\\alpha \\exp \\bigg (-\\bigg (\\frac{z}{z_0}\\bigg )^\\beta \\bigg )\\,{\\rm CSR}(z)\\,\\, ,$ where $A$ , $z_0$ , $\\alpha $ and $\\beta $ are fitting parameters.",
"The term ${\\rm CSR}(z)$ (Colour Sampling Rate) describes the colour-colour pre-selection in terms of an error function transitioning between 0 and 1 around redshift $z=0.5$ , i.e.",
"${\\rm CSR}(z)=\\left(1-\\mathrm {erf}\\big [b(z_t-z)\\big ]\\right)\\,/\\,2 $ where the transition redshift $z_t$ and the transition width $b$ are free parameters.",
"As shown in [69], ${\\rm CSR}(z)$ is unity for $z\\ge 0.6$ , corresponding to a purely magnitude-limited selection.",
"The best fit of Eq.",
"REF to the final VIPERS data is shown by the red curve in Fig.",
"REF .",
"Such modelling of the redshift distribution is an important and sensitive ingredient when estimating galaxy clustering, as we discuss in Sect.",
"and in [19].",
"We compare it with the $V_{\\rm max}$ technique [15], [19] shown in Fig.",
"REF with the purple curve.",
"Although we find no significant difference in the resulting clustering between the two methods, here we chose to use the $V_{\\rm max}$ method, as in the companion paper of [21].",
"A further method often used in the literature is that of smoothing the observed redshift distribution with a Gaussian kernel (as for instance in the parallel papers by [62] [62] and [75] [75])."
],
[
"Mock galaxy samples",
"In order to test the details of the analysis as well as the modelling of RSD, we make use of a suite of mock galaxy catalogues designed to be a realistic match to the VIPERS final dataset.",
"These have been constructed from the Big MultiDark N-body simulation [39], which assumes a flat $\\Lambda $ CDM cosmology with $(\\Omega _m,\\Omega _\\Lambda , \\Omega _b, h, n_s, \\sigma _8)=(0.307, 0.693, 0.0482,0.678, 0.960, 0.823)$ and covers a volume of $15.625\\,h^{-3}\\,\\mathrm {Gpc}^3$ .",
"The construction of the mock samples is described in [21] and is based on the method detailed in [19].",
"We refer the reader to these papers for details and only give a brief overview of the adopted method in the following.",
"153 independent lightcones have been extracted from the simulation volume, which follow the geometry of the VIPERS W1+W4 fields.",
"The dark matter haloes identified in the simulation have been populated with galaxies using the halo occupation distribution (HOD) technique.",
"Because of the halo mass resolution of the simulation which is too large to host the faintest galaxies observed in VIPERS, the method of [20] has been applied to reconstruct haloes below the resolution limit.",
"Each halo has then been populated with galaxies according to its mass as described by the HOD.",
"The latter has been calibrated directly on the VIPERS data as presented in [19].",
"To obtain fully realistic VIPERS mocks one needs to reproduce the VIPERS survey selection function.",
"This has been done following several steps.",
"First, the magnitude cut $i_{\\rm AB}<22.5$ and the effect of the colour selection on the radial distribution of the mocks have been applied.",
"The mock catalogues thus obtained are similar to the parent photometric sample used as target galaxy sample for spectroscopy in VIPERS.",
"The slit-positioning algorithm with the same setting as for the data has further been applied to parent mock catalogues.",
"This allows us to reproduce the VIPERS footprint on the sky, the small-scale angular pair incompleteness, and the variation of TSR across the fields.",
"Finally, random redshift errors has been added to mock galaxy redshifts, similar to that present in the data.",
"This procedure allows us to produce realistic mock galaxy catalogues that contain the detailed survey completeness function and observational biases of VIPERS.",
"We quantify galaxy clustering in redshift space by estimating the anisotropic two-point correlation function $\\xi (s,\\mu )$ , where $s$ is the redshift-space separation of galaxy pairs and $\\mu $ is the cosine of the angle between the separation vector and the line of sight.",
"We generate a catalogue of randomly distributed objects subject to the same angular and radial selection as the true data, and use the [41] estimator: $\\xi (s,\\mu )=\\frac{GG(s,\\mu )-2GR(s,\\mu )+RR(s,\\mu )}{RR(s,\\mu )},$ where $GG(s,\\mu )$ , $GR(s,\\mu )$ , and $RR(s,\\mu )$ are respectively the normalized galaxy-galaxy, galaxy-random, and random-random pair counts in bins of $s$ ($\\Delta (\\log _{10}s)=0.1$ ) and $\\mu $ ($\\Delta \\mu =0.01$ ).",
"This estimator has been shown to provide a nearly unbiased estimate of the two-point correlation function, while minimising its variance [41].",
"We typically use random samples with 30 times more objects than in the true data, to reduce their shot noise contribution to a negligible amount.",
"In this work we shall estimate the growth rate by fitting RSD models not to the full shape of $\\xi (s,\\mu )$ , but rather to its first two even multipole moments, $\\xi ^{(0)}(s)$ and $\\xi ^{(2)}(s)$ , defined as $\\xi ^{(\\ell )}(s)=\\frac{2\\ell +1}{2}\\int _{-1}^{+1}\\xi (s,\\mu )\\mathcal {L}_\\ell (\\mu )d\\mu ,$ where $\\mathcal {L}_\\ell $ is the $\\ell $ -th order Legendre polynomials.",
"Such an approach is normally preferred in order to prevent the size of data vectors and the resulting covariance matrix from becoming too large for practical computation (but see [46] [46] for discussion of some drawbacks of this choice)."
],
[
"Systematic selection effects",
"The VIPERS angular selection function is the result of combining several different angular completeness functions.",
"Two of these are binary masks (i.e.",
"describing areas that are fully used or fully lost).",
"The first mask is related to defects in the parent photometric sample (mostly areas masked by bright stars) and the other to the specific footprint of VIMOS and how the different pointings are tailored together to mosaic the VIPERS area.",
"These masks are easily accounted for when defining the area and the auxiliary random samples for clustering measurements.",
"A more complex selection is related to the incomplete target sampling of VIPERS: on average 47% of the targets satisfying the VIPERS selection criteria can be placed behind a slit and observed, defining what we call the average Target Sampling Rate (TSR).",
"In principle, we should also account for the colour-colour pre-selection of the target sample, which introduces a Colour Sampling Rate (CSR: see [69] [69]).",
"In practice, since the CSR can be safely assumed to be constant over the survey area (thanks to the particularly careful homogenization of the parent sample photometry – see [30] [30]), its effect is absorbed into the fit or model describing the smoothed redshift distribution, as in Eq.",
"(REF ).",
"In any case, the CSR is consistent with being unity for $z \\ge 0.6$ .",
"Finally, we have also to take into account how the probability of measuring the redshift of a targeted galaxy depends on observational conditions or technical issues (which can be location-dependent), which we call the Spectroscopic Success Rate (SSR).",
"The relative relevance, modelling and overall impact of all these effects is described in more detail the following sections."
],
[
"Slit collisions",
"A multi-object spectrograph survey must inevitably face the limitations imposed by the mechanics of how light from the targets is collected on the focal plane.",
"Either fibres or `slitlets' (as in the case of VIMOS) impose a minimum physical size below which the spectrum of two adjacent galaxies on the sky cannot be collected at the same time.",
"This suppresses completely the small-scale clustering amplitude, unless multiple telescope visits of the same field are performed (which is not the case with VIPERS).",
"Furthermore, the same limit on close pairs causes high-density regions on the sky to be more poorly sampled with respect to low-density regions; this introduces a mismatch that, as we shall show, affects the amplitude of clustering on all scales.",
"In VIMOS, this effect is further enhanced by the slit-positioning optimisation software [11], which attempts to maximise the number of slits observed in each quadrant and as such tends to homogenize the angular distribution of targets.",
"Furthermore, in a multi-slit spectrograph such as VIMOS the dispersed spectrum is imaged directly onto the detector.",
"As is evident from Fig.",
"REF , this creates another `forbidden zone' perpendicular to the slit, where no other target can be observed without causing two spectra to overlap (unlike in fibre spectrographs, where fibres are typically taken away from the telescope to a standing spectrograph and the spectra conveniently aligned and packed on the CCD).",
"Since the projected length of the spectrum on the detector is much larger than the corresponding size of the slit, this introduces another typical scale below which the number of measured angular pairs will be reduced, again limiting the sampling of overdensities on the sky.",
"In VIPERS, the spectral dispersion is always oriented along the North-South direction, so the depletion of galaxy pairs will be anisotropic on the sky and will be larger along the declination direction.",
"The impact of these effects on angular clustering is quantified in Fig.",
"REF , where in the top panel we have plotted for both the average of 153 mocks (solid lines) and the VIPERS data (filled points) the angular correlation function of the parent and spectroscopic samples ($w_p(\\theta )$ and $w_s(\\theta )$ , respectively).",
"The bottom panel shows instead the ratio of the corresponding numbers of pairs (bottom panel), defined as $C(\\theta )=\\frac{1+w_s(\\theta )}{1+w_p(\\theta )}\\,\\,\\,\\, .$ In this figure we find clear evidence of the two angular scales discussed earlier, which are related to the width and length of the spectra; these and identified in the figure by the vertical dashed lines.",
"The origin of this effect can be better identified if we split the separation angle $\\theta $ into its components along the right ascension and declination directions, $\\Delta _{\\rm RA}$ and $\\Delta _{\\rm DEC}$ .",
"The angular completeness map $C(\\Delta _{\\rm RA},\\Delta _{\\rm DEC})$ , corresponding to Eq.",
"(REF ) is shown in Fig.",
"REF .",
"Here the `shadow' of the target spectra is recognisable as the rectangular region with nearly zero counts at small separations.",
"The few residual counts in this area are produced by the small variations in the slit length, together with the effect of the few serendipitous targets observed by chance within the slit of a primary target.",
"Translated to spatial scales, this angular selection function results in a strong suppression of the clustering amplitude below $1\\,h^{-1}{\\rm Mpc}$ , as shown by the dotted line in Fig.",
"REF .",
"In [19], we corrected for this effect by up-weighting each galaxy-galaxy pair at a given angular separation $\\theta _{ij}$ by the inverse of the corresponding value of $C(\\theta _{ij})$ , i.e.",
"$w^A(\\theta )=\\frac{1}{C(\\theta _{ij})}.$ We shall discuss the effectiveness of this weight together with the correction of the large-scale effect of the TSR, at the end of the next section."
],
[
"Larger-scale effects",
"Along with the drastic suppression at small separations, the physical size of the slits is responsible for the inhomogeneous sampling between high- and low-density regions across a single VIMOS quadrant.",
"This translates in an almost constant suppression of the clustering amplitude on scales above $1\\,h^{-1}{\\rm Mpc}$ .",
"The correcting scheme we discuss here builds upon the original approach of [19], in which galaxies are assigned a further weight $w_i=\\frac{1}{\\mathrm {TSR}_i}.$ In that paper, however, the TSR used for each galaxy was simply the average value over the corresponding VIMOS quadrant; in this way, all target galaxies in a quadrant were up-weighted by the same factor.",
"As shown by the dot-dashed curve in Fig.",
"REF , when considering the real-space correlation function $\\xi (r)$ this procedure has limited effect (note however than when combined with the $w^A(\\theta )=1/{C(\\theta _{ij})}$ small-scale boost, it provides a better correction: see Fig.",
"8 of [19] [19]).",
"The improved correction adopted here uses instead a local estimate of the TSR$ _i$ , defined as the ratio of the local surface densities of target and parent galaxies (i.e.",
"before and after applying the target selection); these are estimated as detailed below and then averaged within an aperture of a given shape and size.",
"If we call these quantities $\\delta _i^p$ and $\\delta _i^s$ , the TSR$_i$ is defined as $\\mathrm {TSR}_i=\\frac{\\delta _i^s}{\\delta _i^p}.$ The continuous $\\delta $ fields are obtained, starting from the discrete distributions of parent and target galaxies, using a Delaunay tessellation [22] to estimate the density at the position of each galaxy, and then linearly interpolating.",
"These two continuous fields are then used to compute the values of $\\delta _i^p$ and $\\delta _i^s$ within an aperture of a given shape and size.",
"Figure: Optimising the correction for the TargetSampling Rate on large-scales; the tests are based on the mean of153 mock samples.",
"Top: systematic error on the real-spacetwo-point correlation function introduced by the TSR (dottedline), confronted to the results of different strategies toestimate its local value and the corresponding weight (see textfor details).",
"Circular apertures with varying radius (r=90r=90,70 and 50 arcsec 50\\,\\mathrm {arcsec}), and a rectangular aperture60×100 arcsec 2 60\\times 100\\,\\mathrm {arcsec^2} are compared.",
"The dot-dashedline also shows the result of using a weight based only on thequadrant-averaged TSR.",
"Note that here the small-scale furthercorrection based on Eq.",
"() has not been appliedyet.",
"Bottom: corresponding scatter of the differentcorrections.",
"To allow comparison with the systematic error, thisis also reported, for the rectangular aperture, as the shaded areain the top panel.We identified the best-performing geometry for this aperture through the tests shown in Fig.",
"REF .",
"The overall correction is remarkable, since we are able to accurately recover the parent $\\xi (r)$ at large separations, both with a circular and a rectangular aperture.",
"The rectangular aperture is the one providing the best correction to real-space clustering, which can be understood in terms of the anisotropy of the spectral `shadows' discussed earlier.",
"The optimal size of the rectangular aperture is found to be $60\\times 100\\,\\mathrm {arcsec^2}$ .",
"The resulting distribution of the TSR$_i$ values over the survey regions is shown in Fig.",
"REF .",
"Figure: Impact of the Target Sampling Rate and the SpectroscopicSuccess Rate on the radial profile of the VIPERS galaxy samples.",
"Inthe bottom panel we plot the relative difference of the V max V_{\\rm max} fits to the redshift distribution after applying thecorrection, to the same obtained from the observedhistogram.",
"Dashed, dotted and solid lines give the results for W1,W4 and the combined measurement, respectively.",
"The smoothed radialprofile is estimated using the V max V_{\\rm max} method.",
"While the TSRdoes not affect the redshift distribution, the SSR enhances thenumber counts at z>0.95z>0.95."
],
[
"Redshift dependence of angular corrections",
"Some of the corrections for angular selection biases do have an effect also on the redshift distribution.",
"Fig.",
"REF shows the effect of correcting for the TSR and SSR on the observed redshift distribution of the VIPERS data.",
"While the TSR does not introduce a significant redshift dependence, the application of the SSR boosts the expected number of galaxies in the distant ($z>1$ ) part of the sample.",
"This clearly reflects the increased inefficiency to measure redshifts for more and more distant objects.",
"To be fully consistent with the data, then, the random samples used for the clustering analyses will have to be weighted accordingly."
],
[
"Two-point correlations from the VIPERS data",
"We thus proceed to estimate the redshift space correlation function and its moments for the VIPERS survey, adopting the weighting scheme discussed in the previous sections, which we recap for convenience: each galaxy is upweighted by the inverse of its TSR defined by Eqs.",
"(REF ) and (REF ), $w^{{\\rm TSR}}_i$ , as well as by the inverse of its SSR, $w^{{\\rm SSR}}_i$ , each galaxy-galaxy pair with angular separation $\\theta $ is upweighted by the angular weight $w^A(\\theta )$ defined in Eqs.",
"(REF ) and (REF ).",
"Pair counts in the two-point correlation function estimator of Eq.",
"REF are then expressed as GG(s,)=i=1NGj=i+1NGwA(ij)wTSRiwTSRjwSSRiwSSRjij(s,), GR(s,)=i=1NGj=1NRwTSRiwSSRiij(s,), RR(s,)=i=1NRj=i+1NRij(s,) , where $\\Theta _{ij}(s,\\mu )$ is equal to unity for $\\log (s_{ij})$ in $[\\log (s)-\\Delta \\log (s)/2,\\log (s)+\\Delta \\log (s)/2]$ and $\\mu _{ij}$ in $[\\mu -\\Delta \\mu /2,\\mu +\\Delta \\mu /2]$ , and null otherwise.",
"The final performance of this weighting scheme on the recovered monopole and quadrupole of the redshift space correlation function are shown in Fig.",
"REF , for the two redshift ranges considered in the analysis.",
"The combined correction recovers the amplitude of the monopole at the $2\\%$ level, down to the $\\mathrm {Mpc}$ scale, yielding a quasi-unbiased estimate of $\\xi ^{(0)}(s)$ on all comoving scales that will be used for the RSD fitting.",
"As for the quadrupole, we are able to have a reliable measurement of $\\xi ^{(2)}(s)$ ($<5\\%$ deviation from the fiducial value) down to a few $\\mathrm {Mpc}$ .",
"This is an encouraging result: any uncorrected anisotropy from selection effects would be in danger of inducing a spurious contribution to the quadrupole, since this is our main measure of anisotropy.",
"Fig.",
"REF shows the measurement of the anisotropic correlation function $\\xi (r_p,\\pi )$ obtained from the full VIPERS data at $0.5<z<0.7$ and $0.7<z<1.2$ .",
"A bin size $\\Delta s=0.5$ $\\,h^{-1}$ Mpc has been used in both $r_p$ and $\\pi $ directions.",
"We combine the results coming from the two VIPERS fields W1 and W4 simply by summing up the pair counts in each bin of separation and normalising for the total number of objects."
],
[
"Covariance matrix and error estimation",
"Given the intrinsic correlation among different bins of the two-point correlation function (and consequently of its multipoles), it is essential to obtain a reliable estimate of the covariance matrix to be used during the fitting procedure.",
"The fit is carried out performing a maximum likelihood analysis of the data given the RSD model, that can be more easily described as the search throughout the parameter space of the position minimising the likelihood function $\\mathcal {L}$ defined as $-2\\ln \\mathcal {L}=\\sum _{i=0}^{N_b-1}\\sum _{j=0}^{N_b-1}(y^d_i-y^m_i){\\Psi _{ij}}(y^d_j-y^m_j).$ Here the observable $y=(\\xi ^0,\\xi ^2)$ is the monopole-quadrupole combined vector; $\\Psi \\equiv {C}^{-1}$ is the precision matrix (the inverse of the covariance matrix); $N_b$ is the total number of data points; and indices $d$ and $m$ stand respectively for data and model.",
"The covariance matrix ${C}$ is organised in four blocks corresponding to the monopole-monopole, quadrupole-quadrupole and monopole-quadrupole cross covariance (two identical blocks in the latter case).",
"The full monopole-quadrupole covariance matrix is estimated from the 153 mock realisations as $\\hat{C}_{ij}=\\frac{1}{N_s-1}\\sum _{k=1}^{N_s}\\left(y^k_i-\\bar{y}_i\\right)\\left(y^k_j-\\bar{y}_j\\right),$ where $N_s$ is the number of independent realisations used to estimate the covariance, $y$ is the monopole-quadrupole vector, indices $i,j$ run over the data points and index $k$ runs over different realisations.",
"The mean value $\\bar{y}$ is estimated by averaging the measured values from different realisations, namely $\\bar{y}=\\frac{1}{N_s}\\sum _{k=1}^{N_s}y^k.$ The corresponding correlation matrices obtained in this way for the two redshift sub-samples are shown in Fig.",
"REF .",
"Given the large number of mock samples, the estimate and the inversion of the covariance matrices can be achieved with good accuracy.",
"However, the use of a finite number of mocks has two implications.",
"Firstly, the estimated precision matrix obtained by taking the inverse of $\\hat{C}$ is biased with respect to the true one, $\\Psi $ , with the difference being well-represented by an inverse Wishart distribution.",
"Furthermore, the precision matrix $\\Psi $ contains statistical errors that propagate to the parameter space, affecting the derived errors on the cosmological parameters.",
"We follow [52] and correct for these effects by applying two correction factors.",
"In the first case, we can remove the systematic bias of the precision matrix by rescaling $\\hat{C}^{-1}$ as $\\Psi =\\left(1-\\frac{N_{b}+1}{N_{s}-1}\\right)\\hat{C}^{-1}.$ The latter correction factor involves the total number of data points $N_b$ and realisations $N_s$ .",
"It takes into account the typical skewness characterising an inverse Wishart distribution and is capable of providing an unbiased estimate of the precision matrix [31].",
"In the second case, the propagation of errors from the precision matrix to the derived parameters can be corrected by defining $A=\\frac{2}{(N_s-N_b-1)(N_s-N_b-4)},$ $B=\\frac{(N_s-N_b-2)}{(N_s-N_b-1)(N_s-N_b-4)},$ and applying the correction factor $m_1=\\frac{1+B(N_b-N_p)}{1+A+B(N_p+1)}$ to the estimated parameter covariance.",
"In the previous equation, $N_p$ is the total number of free parameters.",
"Figure: Correlation matrices for the combined monopole-quadrupoledata vector, in the low (top) and high (bottom) redshiftbin.",
"Correlation matrices are computed asR ij =C ij /C ii C jj R_{ij}=C_{ij}/\\sqrt{C_{ii}C_{jj}}, where CC is the covariancematrix estimated from a set of 153 independent mock samples.",
"Thebottom left and top right squares correspond respectively to theauto-covariance of the monopole s 2 ξ (0) s^2\\xi ^{(0)} and the quadrupoles 2 ξ (2) s^2\\xi ^{(2)}, while the remaining squares show thecross-covariance terms.",
"The scales under consideration range froms min =5h -1 Mpc s_{\\rm min}=5\\,h^{-1}{\\rm Mpc} to s max =50h -1 Mpc s_{\\rm max}=50\\,h^{-1}{\\rm Mpc} (from left toright)."
],
[
"Modelling redshift-space distortions",
"Redshift-space distortions arise because the apparent position of galaxies is modified by the Doppler effect of their peculiar velocity ${v}$ .",
"In this way, the redshift-space position ${s}$ of galaxies located at ${r}$ becomes ${s}={r}+\\frac{v_\\parallel }{aH(a)}{\\hat{e}}_\\parallel ,$ where $a$ is the scale factor, $H(a)$ is the expansion rate and $v_\\parallel ={v}\\cdot {\\hat{e}}_\\parallel $ is the component of the galaxy peculiar velocity along the line of sight.",
"Invoking mass conservation, the redshift-space density field $\\delta ^s({s})$ can be expressed as a function of its real-space counterpart $\\delta ({r})$ as $\\delta ^s({s})=[1+\\delta ({r})] \\bigg |\\frac{d^3{s}}{d^3{r}}\\bigg |^{-1}-1.$ The targeting of high-redshift galaxies in VIPERS means that the largest pair separations are much smaller than the distance from the observer, so we can use the small-angle plane-parallel approximation; the Jacobian of the real-to-redshift space transformation then reduces to $\\bigg |\\frac{d^3{s}}{d^3{r}}\\bigg |=1-f\\partial _\\parallel u_\\parallel ,$ where the normalized velocity field is defined as ${u}({r})=-{v}({r})/[faH(a)]$ .",
"Substituting this expression inside Equation (REF ) it follows that $\\delta ^s({s})=\\frac{\\delta ({r})+f\\partial _\\parallel u_\\parallel }{1-f\\delta _\\parallel u_\\parallel }.$ Taking the Fourier transform of this equation and making explicit the dependence on $\\mu ={\\bf \\hat{k}\\cdot \\hat{r}}$ , we obtain $\\delta ^{s}(k,\\mu ) &=& \\int \\frac{d^3{s}}{(2\\pi )^3} e^{-i{{k}}\\cdot {{s}}} \\delta ^s({s}) \\nonumber \\\\&=& \\int \\frac{d^3{r}}{(2\\pi )^3} e^{-i{{k}}\\cdot {{r}}} e^{ik\\mu fu_\\parallel }\\big [\\delta ({{r}})+f\\partial _\\parallel u_\\parallel \\big ].$ The redshift-space power spectrum can thus be written as [67] $P^s(k,\\mu ) &=& \\int \\frac{d^3{{r}}}{(2\\pi )^3} e^{-i{{k}}\\cdot {{r}}} \\bigg \\langle e^{-ik\\mu f \\Delta u_\\parallel }\\times \\nonumber \\\\&& \\times \\Big [\\delta ({{x}})+ f\\partial _\\parallel u_\\parallel \\Big ]\\,\\Big [\\delta ({{x^{\\prime }}})+ f\\partial _\\parallel u_\\parallel \\Big ]\\,\\bigg \\rangle ,$ with $\\Delta u_\\parallel =u_\\parallel ({x})-u_\\parallel ({x^{\\prime }})$ and ${r}={x}-{x^{\\prime }}$ .",
"This last equation completely describes the anisotropies produced by peculiar velocities on the clustering of matter particles at each separation.",
"Here, the only assumption is the plane-parallel approximation limit.",
"It is possible to identify two main regimes within which distortions manifest themselves.",
"At large separations, matter has a coherent flow towards overdense regions.",
"In this regime, the velocity field is mainly irrotational [3] and can thus be described by its divergence $\\theta ({x})={\\nabla }\\cdot {u}({x})$ .",
"These motions produce a systematic distortion of the large-scale distribution along the line of sight.",
"This `Kaiser effect' [38] is basically produced by the terms inside the square brackets in Eq.",
"(REF ).",
"In contrast, within the typical scale of haloes, galaxy orbits cross each other: there is a random dispersion in velocities at a given point, which convolves the redshift-space structure in the radial direction.",
"The clustering amplitude is thus suppressed on small scales, and structures appear stretched along the line of sight in the so called `Fingers of God' [35].",
"This effect is mainly generated by the exponential pre-factor involving the moment generating function of the velocity field.",
"Eq.",
"(REF ) is hard to use in its given form, because we lack an analytic formula for the ensemble average term inside the integral, particularly in the strongly non-linear regime.",
"But a number of simpler approximate forms have been suggested, which aim to provide a satisfactory representation of the redshift-space power spectrum measured from galaxy surveys: – Kaiser model [38]: within the linear theory approximation, the exponential pre-factor can be suppressed since its impact on the largest scales is negligible and $\\theta \\propto \\delta $ .",
"If the galaxy-matter bias relation is also assumed to be linear ($\\delta _g=b\\delta $ ), it follows that $P^s(k,\\mu )=\\Bigg (1+\\frac{f}{b}\\mu ^2\\Bigg )^2b^2P_{\\delta \\delta }(k),$ where $P_{\\delta \\delta }=P$ is the linear real-space matter power spectrum and $b$ is the linear galaxy bias.",
"– Dispersion model [50]: although the previous model can reproduce the apparent enhancement of clustering at large separations, it fails in the description of the non-linear regime.",
"The latter can be treated in a phenomenological way, by artificially suppressing the linear theory predictions to account for the effect of the Fingers of God.",
"Eq.",
"(REF ) can thus be written as $P^s(k,\\mu )=D\\big (k\\mu \\sigma _{12}\\big )\\,\\Bigg (1+\\frac{f}{b}\\mu ^2\\Bigg )^2b^2P_{\\delta \\delta }(k),$ where $D\\big (k\\mu \\sigma _{12}\\big )$ is an analytical damping factor.",
"This term depends on a nuisance parameter $\\sigma _{12}$ , which plays the role of a pairwise velocity dispersion.",
"The basic assumption of the dispersion model is that $\\sigma _{12}$ is not scale-dependent, but rather can be fitted as a free parameter.",
"An useful extension of this model is to replace the linear $P_{\\delta \\delta }$ by a non-linear version (using an analytic approximation such as HALOFIT).",
"This then allows the dispersion model to give the correct prediction for $\\mu =0$ : such modes run transverse to the line of sight and undergo no RSD effect.",
"Note that some of the alternatives discussed here fail to match the real-space power exactly at $\\mu =0$ : this is because they are attempting the harder task of predicting the non-linearities, rather than taking them from a fit to $N$ -body simulation data.",
"– Scoccimarro model [66]: as soon as the mildly non-linear regime is entered, the density and velocity divergence fields must be treated separately to account for the non-linear mode coupling between them.",
"The ansatz proposed by Scoccimarro is that the exponential pre-factor inside Eq.",
"(REF ) can be decoupled from the Kaiser term, so that its impact on the clustering is limited only to the smallest scales.",
"In this case, it can be replaced with a damping factor similar to the one already used in the dispersion model, leading to $P^s(k,\\mu )=D\\big (k\\mu \\sigma _{12}\\big )\\,\\Big (b^2P_{\\delta \\delta }(k)+2fb\\mu ^2P_{\\delta \\theta }(k)+f^2\\mu ^4P_{\\theta \\theta }(k)\\Big ),$ where $P_{\\delta \\theta }$ and $P_{\\theta \\theta }$ are respectively the density-velocity divergence cross-spectrum and the velocity divergence auto-spectrum.",
"When applying this (and the following) model to real data, these quantities cannot be obtained from the data under analysis.",
"As such, applications of this (and the following) model have used empirical fitting functions calibrated using numerical simulations [37].",
"In a parallel paper [2], new, more general formulas are proposed: $P_{\\delta \\theta }(k)=\\bigg (P_{\\delta \\delta }(k)P^{lin}(k)e^{-k/k^*}\\bigg )^{\\frac{1}{2}},$ $P_{\\theta \\theta }(k)=P^{lin}(k)e^{-k/k^*},$ where $P^{lin}(k)$ is the linear matter power spectrum and $k^*$ is a parameter representing the typical damping scale of the velocity power spectra.",
"This can be well described as $\\frac{1}{k^*}=p_1\\sigma _8^{p_2},$ where $p_1,\\,p_2$ are the only free parameters of the fit.",
"These forms for $P_{\\delta \\theta }$ and $P_{\\theta \\theta }$ have valuable, physically motivated properties: they naturally converge to $P_{\\delta \\delta }(k)$ in the linear regime, including a dependence on redshift through $\\sigma _8(z)$ .",
"Full details on the derivation and performances of these fitting formulas are presented in [2].",
"Their use in the analysis presented in the following sections is a significant improvement over previous applications of the Scoccimarro and TNS [71] models, as it allows us to extend our tests to smaller scales and apply the models to high redshifts as sampled by VIPERS.",
"– Taruya (or TNS) model [71]: the non-linear mode coupling between the density and velocity divergence fields is responsible for a systematic bias between measurements of the power spectrum and its prediction using the previous RSD model.",
"The origin of this deviation is the additional terms inside Eq.",
"(REF ), which are not accounted for within the previous ansatz.",
"The corrected model can be written as $\\begin{split}P^s(k,\\mu )=D\\big (k\\mu \\sigma _{12}\\big )\\,\\Big (b^2P_{\\delta \\delta }(k)&+2fb\\mu ^2P_{\\delta \\theta }(k)+f^2\\mu ^4P_{\\theta \\theta }(k)+\\\\&+C_A(k,\\mu ,f,b)+C_B(k\\,u,f,b)\\Big ),\\end{split}$ where $C_A$ and $C_B$ are terms derived using perturbation theory, which aim to account for the density and velocity divergence couplings with the exponential pre-factor in Eq.",
"(REF ).",
"See [18] for the details of its application to biased tracers.",
"Figure: Systematic errors on the measurement of the linear growthrate from the mean of 153 mock samples, using the three modelsdiscussed in the text.",
"Here we used the parent mocks, to focus onthe intrinsic performances of the models.",
"Relative systematicerrors are plotted as a function of the minimum fitting scales min s_{\\rm min}.",
"s max s_{\\rm max} is always fixed at 50h -1 Mpc 50\\,h^{-1}{\\rm Mpc}.",
"Thefilled symbols correspond to the use of a Lorentzian form for thenon-linear damping factor in the models, whereas dashed lines to aGaussian one.All the tested RSD models feature a phenomenological damping factor $D(k\\mu \\sigma _{12})$ .",
"The function $D(k\\mu \\sigma _{12})$ damps the power spectra in the Kaiser term but also partially mimics the effects of the pairwise velocity distribution in virialised systems.",
"The expected analytic form of the damping factor on large enough scales assuming the Scoccimarro ansatz is Gaussian [66]; but analyses of simulated galaxy samples [18] have shown that a Lorentzian template provides a better practical fit.",
"Models in equations REF , REF and REF are all tested in the next sections to understand their impact on the recovery of the growth rate.",
"In all cases, at each step of our Monte Carlo Markov chains we generate the full anisotropic redshift-space power spectrum.",
"For this we make use of CAMB with the latest HALOFIT prescription for the non-linear $P_{\\delta \\delta }$ [70], and Eqs.",
"REF and REF to generate the $P_{\\delta \\theta }$ and $P_{\\theta \\theta }$ power spectra.",
"The normalisation of the latter real-space power spectra, which can be set by $\\sigma _8$ , is degenerate with $f$ and $b$ .",
"This is why one generally parametrises RSD models in terms $f\\sigma _8$ and $b\\sigma _8$ parameters.",
"In the case of the TNS model, however, this is not possible directly since the $C_A$ term involves sub-terms that are not multiples of the $f\\sigma _8$ or $b\\sigma _8$ parameters [71], [18].",
"Therefore for the TNS model, and for the others for consistency, we decide to treat $f$ , $b$ , $\\sigma _8$ , $\\sigma _{12}$ as free distinct parameters in the fit, and provide derived constraints on $f\\sigma _8$ a posteriori from the MCMC chains.",
"It is important to emphasize that $\\sigma _8(z)$ not only plays a role in shaping the $C_A$ term, it also controls the level of non-linearity in $P_{\\delta \\delta }$ , $P_{\\delta \\theta }$ , and $P_{\\theta \\theta }$ .",
"In particular for $P_{\\delta \\delta }$ , the HALOFIT non-linear correction to the linear matter power spectrum is computed at each step of the MCMC according to the tested value of $\\sigma _8(z)$ .",
"This represents a significant improvement over what is usually done in RSD analyses, where $\\sigma _8(z)$ is fixed to its fiducial value for the description of $P_{\\delta \\delta }$ .",
"In the end, we measure the Fourier-space multipole moments as $P^{(\\ell )}(k)=\\frac{2\\ell +1}{2}\\int _{-1}^{+1}P^s(k,\\mu )\\mathcal {L}_\\ell (\\mu )d\\mu ,$ and convert them to their configuration space counterparts as $\\xi ^{(\\ell )}(s)=i^\\ell \\int \\frac{dk}{2\\pi ^2}k^2P^{(\\ell )}(k)j_\\ell (ks),$ where $j_\\ell $ denotes the spherical Bessel functions."
],
[
"Tests of RSD models",
"We test in this section the RSD models introduced previously on our set of $N_s=153$ mock catalogues.",
"In practice, analysing each mock and averaging the measurements would be computationally infeasible, considering the large number of configurations to be tested.",
"We thus chose to average the monopole and quadrupole measurements over the mocks, scale the covariance matrix properly, and fit the models to these average measurements.",
"The aim is to reach a statistical uncertainty that is a factor $\\smash{1/\\sqrt{N_s}}$ smaller than a single VIPERS survey, to be able to detect potential systematics as small as $1\\%$ .",
"This process is more revealing and can show how well a given model performs in recovering the detailed shapes of the quadrupole and monopole correlation function.",
"We perform likelihood analyses of the mock mean measurements in different configurations, starting the ideal case and moving on to that in fully realistic conditions.",
"All likelihood analyses are carried out using an MCMC code, whose output has been cross-checked with the independent MCMC code used in [21].",
"We select flat priors for the full set of free parameters, using boundaries that allow a large set of late-time evolution cosmological models to be considered as possible alternatives to standard $\\Lambda $ CDM.",
"The full list of priors is shown in Table REF , while the best-fit values for the parameters are listed in Table REF .",
"We vary the minimum scale $s_{\\rm min}$ of the fit to understand how to select the best fitting range for the VIPERS data – we expect all RSD models to fail at sufficiently small and non-linear scales.",
"The maximum scale of the fit is fixed at $s_{\\rm max}=50\\,h^{-1}{\\rm Mpc}$ , above which errors on the VIPERS measured monopole and quadrupole become too large.",
"Figure: Effect of redshift errors on the recovered monopole andquadrupole from the galaxy mocks, obtained by adding to the mockredshifts a random Gaussian deviate with dispersion equal to therms redshift error of the VIPERS.Figure: Same as Fig.",
", but now includingGaussian redshift errors with dispersion equal to the rms value measured for the VIPERS data, added to the mock galaxyredshifts.",
"Here the dashed lines correspond to the use of aLorentzian damping only, which in Fig.",
"wasfound to perform at best.",
"With redshift errors, this needs to besupplemented by a further Gaussian damping factor with dispersionfixed to the above rms error value, to yield the valuesdescribed by the filled symbols."
],
[
"Ideal case",
"We first study the ideal case that neglects the complex VIPERS angular selection by using the parent mocks.",
"Here we consider mock measurements from the full redshift range probed by VIPERS, i.e.",
"$0.5<z<1.2$ , to avoid tuning the procedure towards small systematic deviations on the two redshift bins.",
"The relative difference of the recovered $f\\sigma _8$ with respect to the fiducial one is shown in Fig.",
"REF .",
"Redshift errors are not considered here to understand how different RSD models behave in the absence of any observational bias.",
"Two types of small-scale damping factor $D(k\\mu \\sigma _{12})$ are tested: the Lorentzian (filled symbols) and Gaussian (dashed lines) forms.",
"The overall trend of models using Lorentzian damping favours the TNS model: it yields almost unbiased measurements of the growth rate down to $s_{\\rm min} =5 \\,h^{-1}{\\rm Mpc}$ .",
"Some overestimation is however seen for minimum scales close to $s_{\\rm min}=8\\,h^{-1}{\\rm Mpc}$ , which in fact corresponds to the zero-crossing scale of the quadrupole $\\xi ^{(2)}(s)$ .",
"In contrast, both dispersion and Scoccimarro models consistently underestimate the growth rate with an error that fluctuates between 5–10%.",
"Evidently, in all the cases the choice of a Lorentzian damping yields smaller systematic deviations than with a Gaussian damping.",
"This is reflected by the trend of the dashed lines, which are close to the corresponding markers only when the minimum fitting separation $s_{\\rm min}$ is larger than $\\sim 15\\,\\,h^{-1}{\\rm Mpc}$ , while rapidly deteriorating when smaller separations are included in the fit.",
"This is highlighted in Fig.",
"REF , where the different best-fitting models for the monopole and quadrupole using $s_{\\rm min}=5\\,h^{-1}{\\rm Mpc}$ are directly compared to the mock data.",
"Using a Gaussian damping, the model is no longer able to provide a good description of $\\xi ^{(0)}$ and $\\xi ^{(2)}$ .",
"Actually, the fit is mostly dominated by the small scales, whose data points have the smallest errors, and this explains why separations above $7\\,h^{-1}{\\rm Mpc}$ are apparently the ones giving the strongest deviation between model and data.",
"This result is in close agreement with previous work on the subject [8], [18]."
],
[
"Case with redshift errors",
"So far no redshift error was assumed in the mock samples.",
"However, real VIPERS redshifts have a significant uncertainty, which clearly impact observed redshift-space distortions.",
"We know from the multiple redshift measurements [69] that the redshift error probability distribution for the VIPERS sample of reliable redshifts used here, is well described by a Gaussian with standard deviation $\\sigma _z=0.00054(1+z)$ .",
"This corresponds to a dispersion in galaxy peculiar velocity of 163 km s$^{-1}$ .",
"Figure: The same as Fig.",
", but now usingthe fully realistic `observed' mocks, on which all observationaleffects (masks, SPOC selection and redshift errors) have beenincluded.",
"As before, error bars correspond to the error on theaverage of the 153 samples.Figure: Monopole and quadrupole of ξ(r p ,π)\\xi (r_p,\\pi ) for the tworedshift sub-sample of the final VIPERS dataset (solid points),together with the final best-fitting curves obtained using the TNSmodel, corresponding to the values reported inTable .",
"The likelihood computation has useddata down to s min =5h -1 Mpc s_{\\rm min}=5\\,h^{-1}{\\rm Mpc}, as indicated by the tests.Error bars are 1-σ1-\\sigma deviations, and correspond to thedispersion of the mock measurements.",
"Each of these is also shown asa faint background line.By applying random errors to mock galaxy redshifts following the VIPERS observed distribution, we can effectively see additional distortions.",
"These are shown in Fig.",
"REF , where one can see how the shapes of the monopole $\\xi ^{(0)}(s)$ and the quadrupole $\\xi ^{(2)}(s)$ are affected.",
"The imprint of redshift errors is similar to that of a small-scale damping of the power spectrum.",
"While the monopole is damped below $4\\,h^{-1}{\\rm Mpc}$ , the quadrupole is corrupted over a range extending out to $\\sim 20\\,h^{-1}{\\rm Mpc}$ .",
"Clearly, this effect needs to be carefully handled or modelled, if one hopes to recover an unbiased value for the growth rate.",
"The consequences of not correcting for this effect are shown by the dashed lines in Fig.",
"REF , where we are repeating the tests of Fig.",
"REF , but now including redshift errors.",
"As feared, there is a significant deviation from the values of $f\\sigma _8$ previously measured with the models in the best configuration, i.e.",
"with the Lorentzian damping.",
"Rather than correcting for redshift errors in the measurements, as performed for the angular selection selection, it is better to include it in the modelling.",
"It is intuitive to supplement the models with a convolution by an extra Gaussian distribution with standard deviation fixed to the VIPERS rms value of $\\sigma _z=163~\\rm {km}~s^{-1}$ , which corresponds to $\\sigma _\\pi =\\frac{c\\sigma _z}{H(z)},$ in terms of line-of-sight comoving separation.",
"The filled symbols in Fig.",
"REF show how with this extra damping term we recover a performance similar to the more idealised case of Fig.",
"REF .",
"We therefore adopt the TNS model with Lorentzian damping and Gaussian error damping, as our reference model for the final RSD analysis of the VIPERS data.",
"However, we will also test for consistency the behaviour of the other two models on the actual data, to verify whether the trends seen in the mocks are confirmed."
],
[
"Fully realistic case",
"We now turn to the case including fully realistic observing conditions.",
"This means considering the target selection (masks, TSR, SSR, etc.)",
"and limiting the samples to the same redshift ranges covered by the data and including redshift errors.",
"The results that we obtain are shown in Fig.",
"REF .",
"The trends of the systematic error as a function of $s_{\\rm min}$ are less stable than in the previous case, although the general behaviour and relative performances of the models are the same.",
"The variations gives us an idea of the impact of the selection function on samples this size.",
"Again, we see some instability in the TNS model (at least in the bin $0.7<z<1.2$ ) when the minimum scale of the fit is chosen around $s_{\\rm min}\\simeq 8\\,h^{-1}{\\rm Mpc}$ .",
"When we look into the MCMC results in more detail, we see that in this case the Markov chain tends to drift towards unrealistic values of $\\sigma _8$ , which are outside of the prior range defined in Table REF .",
"This seems to be related to the difficulty of TNS model to reach a stable fit in the region where the quadrupole crosses zero.",
"As soon as we include smaller scales (or we move away to larger ones), the regular trend is recovered.",
"Nevertheless, we confirm the TNS model as the best performing one, with systematics $\\lesssim 5\\%$ down to the smallest probed minimum scales.",
"Overall, the different tests performed on the mock catalogues indicate that we can safely use the TNS model with the appropriate damping functions as well as a minimum fitting scale of $s_{\\rm min}=5\\,h^{-1}{\\rm Mpc}$ .",
"This is the configuration that we adopt for the analysis of the data."
],
[
"VIPERS RSD results",
"We present in this section the results of the RSD analysis of the VIPERS final dataset.",
"We apply the methodology described in the previous sections to the VIPERS galaxy sample.",
"In the likelihood analysis we impose rather broad uniform priors on the sampling parameters.",
"These are reported in Table REF .",
"Since $f$ and $\\sigma _8$ are treated as separate parameters in the modelling, despite their intrinsic degeneracy, we need to impose sensible priors on them.",
"In fact the most sensitive prior is that on $\\sigma _8$ , as it is the main parameter entering the non-linear modelling of RSD.",
"To define a sensible and realistic prior, while allowing room for deviations from GR, we base our choice on the Effective Field Theory of dark energy formalism [28], [10], [26], which allows a description of various kinds of dark energy models and modifications of gravity to be expressed in a self-consistent framework that includes the growth rate of structure [56], [54].",
"The latter work shows that the range spanned by $\\sigma _8(z)$ for stable theories can vary significantly, suggesting a range [$0.2$ , $0.65$ ] as appropriate to account for early- and late-time dark energy models at the redshifts covered by VIPERS (for definitions, see [53] [53]).",
"This excludes some more extreme modified gravity models, but avoids non-physical degeneracies that arise in the likelihood for some particular values of $\\sigma _8$ outside of this range.",
"This choice is corroborated by our parallel complementary analysis using the same data by [21], in which the combination of RSD with galaxy-galaxy lensing constrains directly $\\sigma _8(z)$ , allowing a broader prior at the outset.",
"The $f\\sigma _8$ measurements that we finally obtain using our standard configuraton and previously discussed parameter priors are $f\\sigma _8(z=0.6)=0.55\\pm 0.12$ and $f\\sigma _8(z=0.86)=0.40\\pm 0.11$ .",
"We consider these as our reference measurements in this work and discuss their cosmological implications in the next section.",
"The measurements and best-fitting model monopole and quadrupole correlation functions obtained in the two considered redshift bins are shown in Fig.",
"REF .",
"The corresponding best-fit values for the derived parameters are reported in Table REF .",
"It is interesting to verify a posteriori whether the trends and relative RSD model performances as a function of $s_{\\rm min}$ established from the mock catalogues are similar to those seen in the real data.",
"It is of course clear that any trend will be less significant, as the data are statistically equivalent to considering just one of the 153 mock catalogues.",
"In the left panel of Fig.",
"REF , we show the result of this exercise, where the measured values of $f\\sigma _8$ as a function of $s_{\\rm min}$ are shown for the different tested models.",
"To ease comparison, we have reported in the right panel and using the same scale, the corresponding results from the mock test for the realistic case (i.e.",
"those of Fig.",
"REF ).",
"Apart from the different statistical errors, it is surprising to note how the three tested RSD models provide virtually identical results in the real data, as opposed to the behaviour seen in the mock catalogues.",
"Moreover, it seems that in the data the variation of the $f\\sigma _8$ measurements with minimum scale are not driven by the adequacy of the model down to those scales, but rather by statistical uncertainties in the measured galaxy correlation functions.",
"The similarity in the results obtained from the different models is confirmed directly by the values of the reduced $\\chi ^2$ , which turn out to be very similar.",
"By directly looking at the posterior likelihood distributions of the parameters obtained with the three models in Fig.",
"REF (for the high-redshift bin), we can see that each model provides slightly different parameters degeneracies, although after marginalization, $f\\sigma _8$ posterior likelihood distributions are almost identical for the three RSD models, with only a slightly larger statistical uncertainty with the TNS model.",
"However, some trends seen in the mock results are recognised in the data, as for example the preference of the TNS model in the high-redshift sample to deliver larger values of $f\\sigma _8$ when $s_{\\rm min}$ is close to the zero-crossing scale of the quadrupole.",
"Figure: The one- and two-dimensional posterior likelihooddistribution of the derived parameters fσ 8 f\\sigma _8, bσ 8 b\\sigma _8 andσ 12 \\sigma _{12} for the 0.5<z<0.70.5 <z < 0.7 redshift bin.",
"It corresponds tothe result of the analysis of VIPERS data using dispersion,Scoccimarro, and Taruya model and s min =5s_{\\rm min}=5 h -1 \\,h^{-1} Mpc.",
"The dark-and light-shaded areas correspond respectively to the 68%68\\% and95%95\\% joint two-parameter confidence levels.",
"The lower redshiftsample shows comparable contours and shapes.Finally, it is important to emphasize the global non-linear approach to RSD that has been used in this analysis.",
"We have used rather small non-linear scales in the fit, and by adopting a consistent modelling for the non-linearities in the real-space density and velocity divergence power spectra, we can obtain further cosmological insight.",
"The level of non-linearity in our analysis is controlled by one single parameter, $\\sigma _8(z)$ , and we find that by letting this parameter vary, we can partly break the standard degeneracy that dominates on linear scales between $f$ , $\\sigma _8$ , and $b$ parameters.",
"If we marginalise the posterior likelihood function over the $\\sigma _{12}$ , $\\sigma _8$ , $b$ parameters, we obtain the following direct growth rate and $\\sigma _8$ constraints: $[f(z=0.6),\\sigma _8(z=0.6)]=[1.048\\pm 0.199,0.528\\pm 0.076]$ and $[f(z=0.86),\\sigma _8(z=0.86)]=[0.742\\pm 0.179,0.539\\pm 0.068]$ .",
"A similar approach has been adopted in [21], where this is strengthened by additional constraints from galaxy-galaxy lensing.",
"In particular, the latter allows improving $\\sigma _8$ constraints while keeping similar uncertainties on $f$ .",
"A detailed discussion of these results is given in [21].",
"Overall, these findings demonstrate the additional constraining power encapsulated in quasi-linear scales, which can be used to break degeneracies and further improve the precision of measurement of the growth rate of structure."
],
[
"Discussion and conclusions",
"The measurements of the growth rate of structure times $\\sigma _8$ that we obtained are $f\\sigma _8(z=0.6)&=&0.55\\pm 0.12 \\\\f\\sigma _8(z=0.86)&=&0.40\\pm 0.11.$ These values are confronted in Fig.",
"REF with different measurements, including results from other surveys, the VIPERS earlier PDR-1 dataset, and parallel works analysing with complementary techniques analogous subsets of the VIPERS PDR-2 dataset.",
"It may look surprising that there is no appreciable improvement in the error bars between the former measurement from the PDR-1 [19] and the new PDR-2 estimate in a comparable redshift bin, despite a $\\sim 30\\%$ increase in the sample size.",
"As discussed in [18], this is essentially a price to pay for the more sophisticated treatment of nonlinear effects through the TNS model, which increases the degrees of freedom.",
"Figure: Plot of fσ 8 f\\sigma _8 versus redshift, showing the VIPERSresults together with a compilation of recent measurements.",
"Theprevious results from 2dFGRS , 2SLAQ, VVDS , SDSS LRG, , WiggleZ , BOSS, 6dFGS and FastSound surveys are shown withthe different symbols (see inset).",
"The solid curve and associatederror correspond to the prediction for General Relativity in aΛ CDM \\Lambda \\rm {CDM} model set to Planck 2015 cosmologicalparameters .The parallel PDR-2 results include measurements obtained from the combination of RSD with galaxy-galaxy lensing [21] or using the void-galaxy cross-correlation [32].",
"In forthcoming papers, we shall additionally present further pieces of this combined approach, using specific colour-selected subsamples [45] or the linearised density field in Fourier space [75], to minimise the need for non-linear corrections.",
"All these papers represent complementary approaches towards understanding the current limitations we face in our ability to extract in practice the value of these parameters from the modelling of RSD.",
"The values measured by these different techniques on the same VIPERS data as well as from other surveys at similar redshifts are virtually all compatible within 1-$\\sigma $ and agree with the predictions of a $\\Lambda $ CDM model governed by Einstein gravity.",
"But on a larger sample, with much smaller statistical errors, greater care would be needed to test for possible systematic biases that might still be hidden in one or more of the analyses.",
"The application of such a variety of approaches to VIPERS has been made possible by the specific properties of the survey, in particular its dense sampling and rich content of information.",
"With a sparse sampled survey, which is the approach of most of the cosmologically-oriented surveys, it would have been impossible to characterise accurately the density field and apply the clipping linearisation technique of [75], or reliably detect cosmic voids such as those used in [32].",
"At the same time, a survey with limited imaging information would not permit investigation of the selection of optimal sub-populations (or the combination of different ones), as we are pursuing in [45], or exploit the combination of RSD with lensing, as we have done in [21] and which should be exploited to the fullest by Euclid mission [42] in the next decade.",
"We therefore believe that the detailed investigation of the properties of RSD within VIPERS should serve as a valuable foundation for next-generation studies of greater statistical power.",
"We acknowledge the crucial contribution of the ESO staff for the management of service observations.",
"In particular, we are deeply grateful to M. Hilker for his constant help and support of this program.",
"Italian participation to VIPERS has been funded by INAF through PRIN 2008, 2010, 2014 and 2015 programs.",
"LG and BRG acknowledge support from the European Research Council through grant n. 291521.",
"OLF acknowledges support from the European Research Council through grant n. 268107.",
"JAP acknowledges support of the European Research Council through the COSFORM ERC Advanced Research Grant (# 670193).",
"GDL acknowledges financial support from the European Research Council through grant n. 202781.",
"RT acknowledges financial support from the European Research Council through grant n. 202686.",
"AP, KM, and JK have been supported by the National Science Centre (grants UMO-2012/07/B/ST9/04425 and UMO-2013/09/D/ST9/04030).",
"EB, FM and LM acknowledge the support from grants ASI-INAF I/023/12/0 and PRIN MIUR 2010-2011.",
"LM also acknowledges financial support from PRIN INAF 2012.",
"SDLT and MP acknowledge the support of the OCEVU Labex (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the \"Investissements d'Avenir\" French government program managed by the ANR.",
"Research conducted within the scope of the HECOLS International Associated Laboratory, supported in part by the Polish NCN grant DEC-2013/08/M/ST9/00664.",
"TM and SA acknowledge financial support from the ANR Spin(e) through the French grant ANR-13-BS05-0005."
]
] | 1612.05645 |
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